Integrand size = 24, antiderivative size = 1150 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{d+e x} \, dx =\text {Too large to display} \] Output:
-1/15*d*(6*b*e^2+5*c*d^2)*x*(c*x^4+b*x^2+a)^(1/2)/e^4-1/5*c*d*x^3*(c*x^4+b *x^2+a)^(1/2)/e^2-1/15*d*(21*a*c*e^4+3*b^2*e^4+20*b*c*d^2*e^2+15*c^2*d^4)* x*(c*x^4+b*x^2+a)^(1/2)/c^(1/2)/e^6/(a^(1/2)+c^(1/2)*x^2)+1/16*(8*c^2*d^4+ 10*b*c*d^2*e^2+b^2*e^4+8*a*c*e^4+2*c*e^2*(b*e^2+2*c*d^2)*x^2)*(c*x^4+b*x^2 +a)^(1/2)/c/e^5+1/6*(c*x^4+b*x^2+a)^(3/2)/e+1/2*(a*e^4+b*d^2*e^2+c*d^4)^(3 /2)*arctanh((a*e^4+b*d^2*e^2+c*d^4)^(1/2)*x/d/e/(c*x^4+b*x^2+a)^(1/2))/e^7 +1/32*(b*e^2+2*c*d^2)*(12*a*c*e^4-b^2*e^4+8*b*c*d^2*e^2+8*c^2*d^4)*arctanh (1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/c^(3/2)/e^7-1/2*(a*e^4+b*d ^2*e^2+c*d^4)^(3/2)*arctanh(1/2*(b*d^2+2*a*e^2+(b*e^2+2*c*d^2)*x^2)/(a*e^4 +b*d^2*e^2+c*d^4)^(1/2)/(c*x^4+b*x^2+a)^(1/2))/e^7+1/15*a^(1/4)*d*(21*a*c* e^4+3*b^2*e^4+20*b*c*d^2*e^2+15*c^2*d^4)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x ^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1 /4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(3/4)/e^6/(c*x^4+b*x^2+a)^(1/2)-1 /30*a^(1/4)*d*(30*c^(5/2)*d^6+10*a^(1/2)*c^2*d^4*e^2+3*a^(1/2)*b^2*e^6+2*a ^(1/2)*c*e^4*(3*a*e^2+7*b*d^2)+6*b*c^(1/2)*e^4*(4*a*e^2+3*b*d^2)+c^(3/2)*( 46*a*d^2*e^4+50*b*d^4*e^2))*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2 )+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2 -b/a^(1/2)/c^(1/2))^(1/2))/c^(3/4)/e^6/(c^(1/2)*d^2+a^(1/2)*e^2)/(c*x^4+b* x^2+a)^(1/2)-1/4*(c^(1/2)*d^2-a^(1/2)*e^2)*(a*e^4+b*d^2*e^2+c*d^4)^2*(a^(1 /2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*Ellipt...
Result contains complex when optimal does not.
Time = 23.86 (sec) , antiderivative size = 12989, normalized size of antiderivative = 11.29 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{d+e x} \, dx=\text {Result too large to show} \] Input:
Integrate[(a + b*x^2 + c*x^4)^(3/2)/(d + e*x),x]
Output:
Result too large to show
Time = 5.91 (sec) , antiderivative size = 1131, normalized size of antiderivative = 0.98, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {2266, 1529, 27, 1576, 1162, 1231, 27, 1269, 1092, 219, 1154, 219, 1786, 27, 414, 2207, 2207, 27, 1511, 27, 1416, 1509}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{d+e x} \, dx\) |
| \(\Big \downarrow \) 2266 |
| \(\displaystyle d \int \frac {\left (c x^4+b x^2+a\right )^{3/2}}{d^2-e^2 x^2}dx-e \int \frac {x \left (c x^4+b x^2+a\right )^{3/2}}{d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 1529 |
| \(\displaystyle d \left (\frac {\left (e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {2 c x^2+b-\sqrt {b^2-4 a c}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{4 c e^6}-\frac {\int \frac {2 \left (2 c^3 e^4 x^6+2 c^2 e^2 \left (c d^2+2 b e^2\right ) x^4+2 c \left (c^2 d^4+b^2 e^4+2 c \left (a e^4+b d^2 e^2\right )\right ) x^2+c \left (b c d^4+b^2 e^2 d^2+2 a c e^2 d^2+3 a b e^4-\sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{4 c e^6}\right )-e \int \frac {x \left (c x^4+b x^2+a\right )^{3/2}}{d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\left (e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {2 c x^2+b-\sqrt {b^2-4 a c}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{4 c e^6}-\frac {\int \frac {2 c^3 e^4 x^6+2 c^2 e^2 \left (c d^2+2 b e^2\right ) x^4+2 c \left (c^2 d^4+b^2 e^4+2 c \left (a e^4+b d^2 e^2\right )\right ) x^2+c \left (b c d^4+b^2 e^2 d^2+2 a c e^2 d^2+3 a b e^4-\sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{2 c e^6}\right )-e \int \frac {x \left (c x^4+b x^2+a\right )^{3/2}}{d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle d \left (\frac {\left (e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {2 c x^2+b-\sqrt {b^2-4 a c}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{4 c e^6}-\frac {\int \frac {2 c^3 e^4 x^6+2 c^2 e^2 \left (c d^2+2 b e^2\right ) x^4+2 c \left (c^2 d^4+b^2 e^4+2 c \left (a e^4+b d^2 e^2\right )\right ) x^2+c \left (b c d^4+b^2 e^2 d^2+2 a c e^2 d^2+3 a b e^4-\sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{2 c e^6}\right )-\frac {1}{2} e \int \frac {\left (c x^4+b x^2+a\right )^{3/2}}{d^2-e^2 x^2}dx^2\) |
| \(\Big \downarrow \) 1162 |
| \(\displaystyle d \left (\frac {\left (e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {2 c x^2+b-\sqrt {b^2-4 a c}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{4 c e^6}-\frac {\int \frac {2 c^3 e^4 x^6+2 c^2 e^2 \left (c d^2+2 b e^2\right ) x^4+2 c \left (c^2 d^4+b^2 e^4+2 c \left (a e^4+b d^2 e^2\right )\right ) x^2+c \left (b c d^4+b^2 e^2 d^2+2 a c e^2 d^2+3 a b e^4-\sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\int \frac {\left (b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2\right ) \sqrt {c x^4+b x^2+a}}{d^2-e^2 x^2}dx^2}{2 e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle d \left (\frac {\left (e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {2 c x^2+b-\sqrt {b^2-4 a c}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{4 c e^6}-\frac {\int \frac {2 c^3 e^4 x^6+2 c^2 e^2 \left (c d^2+2 b e^2\right ) x^4+2 c \left (c^2 d^4+b^2 e^4+2 c \left (a e^4+b d^2 e^2\right )\right ) x^2+c \left (b c d^4+b^2 e^2 d^2+2 a c e^2 d^2+3 a b e^4-\sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {-\frac {\int -\frac {\left (2 c d^2+b e^2\right ) \left (4 b c d^2+b^2 e^2+4 a c e^2\right ) d^2+4 c \left (2 a e^3+b d^2 e\right )^2+\left (2 c d^2+b e^2\right ) \left (8 c^2 d^4+8 b c e^2 d^2-b^2 e^4+12 a c e^4\right ) x^2}{2 \left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx^2}{4 c e^4}-\frac {\sqrt {a+b x^2+c x^4} \left (8 a c e^4+b^2 e^4+2 c e^2 x^2 \left (b e^2+2 c d^2\right )+10 b c d^2 e^2+8 c^2 d^4\right )}{4 c e^4}}{2 e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\left (e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {2 c x^2+b-\sqrt {b^2-4 a c}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{4 c e^6}-\frac {\int \frac {2 c^3 e^4 x^6+2 c^2 e^2 \left (c d^2+2 b e^2\right ) x^4+2 c \left (c^2 d^4+b^2 e^4+2 c \left (a e^4+b d^2 e^2\right )\right ) x^2+c \left (b c d^4+b^2 e^2 d^2+2 a c e^2 d^2+3 a b e^4-\sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {\int \frac {\left (2 c d^2+b e^2\right ) \left (4 b c d^2+b^2 e^2+4 a c e^2\right ) d^2+4 c \left (2 a e^3+b d^2 e\right )^2+\left (2 c d^2+b e^2\right ) \left (8 c^2 d^4+8 b c e^2 d^2-b^2 e^4+12 a c e^4\right ) x^2}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx^2}{8 c e^4}-\frac {\sqrt {a+b x^2+c x^4} \left (8 a c e^4+b^2 e^4+2 c e^2 x^2 \left (b e^2+2 c d^2\right )+10 b c d^2 e^2+8 c^2 d^4\right )}{4 c e^4}}{2 e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle d \left (\frac {\left (e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {2 c x^2+b-\sqrt {b^2-4 a c}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{4 c e^6}-\frac {\int \frac {2 c^3 e^4 x^6+2 c^2 e^2 \left (c d^2+2 b e^2\right ) x^4+2 c \left (c^2 d^4+b^2 e^4+2 c \left (a e^4+b d^2 e^2\right )\right ) x^2+c \left (b c d^4+b^2 e^2 d^2+2 a c e^2 d^2+3 a b e^4-\sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {16 c \left (a e^4+b d^2 e^2+c d^4\right )^2 \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx^2}{e^2}-\frac {\left (b e^2+2 c d^2\right ) \left (12 a c e^4-b^2 e^4+8 b c d^2 e^2+8 c^2 d^4\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{e^2}}{8 c e^4}-\frac {\sqrt {a+b x^2+c x^4} \left (8 a c e^4+b^2 e^4+2 c e^2 x^2 \left (b e^2+2 c d^2\right )+10 b c d^2 e^2+8 c^2 d^4\right )}{4 c e^4}}{2 e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle d \left (\frac {\left (e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {2 c x^2+b-\sqrt {b^2-4 a c}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{4 c e^6}-\frac {\int \frac {2 c^3 e^4 x^6+2 c^2 e^2 \left (c d^2+2 b e^2\right ) x^4+2 c \left (c^2 d^4+b^2 e^4+2 c \left (a e^4+b d^2 e^2\right )\right ) x^2+c \left (b c d^4+b^2 e^2 d^2+2 a c e^2 d^2+3 a b e^4-\sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {16 c \left (a e^4+b d^2 e^2+c d^4\right )^2 \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx^2}{e^2}-\frac {2 \left (b e^2+2 c d^2\right ) \left (12 a c e^4-b^2 e^4+8 b c d^2 e^2+8 c^2 d^4\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{e^2}}{8 c e^4}-\frac {\sqrt {a+b x^2+c x^4} \left (8 a c e^4+b^2 e^4+2 c e^2 x^2 \left (b e^2+2 c d^2\right )+10 b c d^2 e^2+8 c^2 d^4\right )}{4 c e^4}}{2 e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle d \left (\frac {\left (e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {2 c x^2+b-\sqrt {b^2-4 a c}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{4 c e^6}-\frac {\int \frac {2 c^3 e^4 x^6+2 c^2 e^2 \left (c d^2+2 b e^2\right ) x^4+2 c \left (c^2 d^4+b^2 e^4+2 c \left (a e^4+b d^2 e^2\right )\right ) x^2+c \left (b c d^4+b^2 e^2 d^2+2 a c e^2 d^2+3 a b e^4-\sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {16 c \left (a e^4+b d^2 e^2+c d^4\right )^2 \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx^2}{e^2}-\frac {\left (b e^2+2 c d^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \left (12 a c e^4-b^2 e^4+8 b c d^2 e^2+8 c^2 d^4\right )}{\sqrt {c} e^2}}{8 c e^4}-\frac {\sqrt {a+b x^2+c x^4} \left (8 a c e^4+b^2 e^4+2 c e^2 x^2 \left (b e^2+2 c d^2\right )+10 b c d^2 e^2+8 c^2 d^4\right )}{4 c e^4}}{2 e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle d \left (\frac {\left (e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {2 c x^2+b-\sqrt {b^2-4 a c}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{4 c e^6}-\frac {\int \frac {2 c^3 e^4 x^6+2 c^2 e^2 \left (c d^2+2 b e^2\right ) x^4+2 c \left (c^2 d^4+b^2 e^4+2 c \left (a e^4+b d^2 e^2\right )\right ) x^2+c \left (b c d^4+b^2 e^2 d^2+2 a c e^2 d^2+3 a b e^4-\sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {-\frac {32 c \left (a e^4+b d^2 e^2+c d^4\right )^2 \int \frac {1}{4 \left (c d^4+b e^2 d^2+a e^4\right )-x^4}d\left (-\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{\sqrt {c x^4+b x^2+a}}\right )}{e^2}-\frac {\left (b e^2+2 c d^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \left (12 a c e^4-b^2 e^4+8 b c d^2 e^2+8 c^2 d^4\right )}{\sqrt {c} e^2}}{8 c e^4}-\frac {\sqrt {a+b x^2+c x^4} \left (8 a c e^4+b^2 e^4+2 c e^2 x^2 \left (b e^2+2 c d^2\right )+10 b c d^2 e^2+8 c^2 d^4\right )}{4 c e^4}}{2 e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle d \left (\frac {\left (e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {2 c x^2+b-\sqrt {b^2-4 a c}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{4 c e^6}-\frac {\int \frac {2 c^3 e^4 x^6+2 c^2 e^2 \left (c d^2+2 b e^2\right ) x^4+2 c \left (c^2 d^4+b^2 e^4+2 c \left (a e^4+b d^2 e^2\right )\right ) x^2+c \left (b c d^4+b^2 e^2 d^2+2 a c e^2 d^2+3 a b e^4-\sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {16 c \left (a e^4+b d^2 e^2+c d^4\right )^{3/2} \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{e^2}-\frac {\left (b e^2+2 c d^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \left (12 a c e^4-b^2 e^4+8 b c d^2 e^2+8 c^2 d^4\right )}{\sqrt {c} e^2}}{8 c e^4}-\frac {\sqrt {a+b x^2+c x^4} \left (8 a c e^4+b^2 e^4+2 c e^2 x^2 \left (b e^2+2 c d^2\right )+10 b c d^2 e^2+8 c^2 d^4\right )}{4 c e^4}}{2 e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 1786 |
| \(\displaystyle d \left (\frac {\sqrt {\frac {2 a}{b-\sqrt {b^2-4 a c}}+x^2} \sqrt {-\sqrt {b^2-4 a c}+b+2 c x^2} \left (e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {\sqrt {2} \sqrt {2 c x^2+b-\sqrt {b^2-4 a c}}}{\sqrt {x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}} \left (d^2-e^2 x^2\right )}dx}{4 \sqrt {2} c e^6 \sqrt {a+b x^2+c x^4}}-\frac {\int \frac {2 c^3 e^4 x^6+2 c^2 e^2 \left (c d^2+2 b e^2\right ) x^4+2 c \left (c^2 d^4+b^2 e^4+2 c \left (a e^4+b d^2 e^2\right )\right ) x^2+c \left (b c d^4+b^2 e^2 d^2+2 a c e^2 d^2+3 a b e^4-\sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {16 c \left (a e^4+b d^2 e^2+c d^4\right )^{3/2} \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{e^2}-\frac {\left (b e^2+2 c d^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \left (12 a c e^4-b^2 e^4+8 b c d^2 e^2+8 c^2 d^4\right )}{\sqrt {c} e^2}}{8 c e^4}-\frac {\sqrt {a+b x^2+c x^4} \left (8 a c e^4+b^2 e^4+2 c e^2 x^2 \left (b e^2+2 c d^2\right )+10 b c d^2 e^2+8 c^2 d^4\right )}{4 c e^4}}{2 e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\sqrt {\frac {2 a}{b-\sqrt {b^2-4 a c}}+x^2} \sqrt {-\sqrt {b^2-4 a c}+b+2 c x^2} \left (e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {\sqrt {2 c x^2+b-\sqrt {b^2-4 a c}}}{\sqrt {x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}} \left (d^2-e^2 x^2\right )}dx}{4 c e^6 \sqrt {a+b x^2+c x^4}}-\frac {\int \frac {2 c^3 e^4 x^6+2 c^2 e^2 \left (c d^2+2 b e^2\right ) x^4+2 c \left (c^2 d^4+b^2 e^4+2 c \left (a e^4+b d^2 e^2\right )\right ) x^2+c \left (b c d^4+b^2 e^2 d^2+2 a c e^2 d^2+3 a b e^4-\sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {16 c \left (a e^4+b d^2 e^2+c d^4\right )^{3/2} \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{e^2}-\frac {\left (b e^2+2 c d^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \left (12 a c e^4-b^2 e^4+8 b c d^2 e^2+8 c^2 d^4\right )}{\sqrt {c} e^2}}{8 c e^4}-\frac {\sqrt {a+b x^2+c x^4} \left (8 a c e^4+b^2 e^4+2 c e^2 x^2 \left (b e^2+2 c d^2\right )+10 b c d^2 e^2+8 c^2 d^4\right )}{4 c e^4}}{2 e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle d \left (\frac {\left (b-\sqrt {b^2-4 a c}\right )^{5/2} \left (\frac {2 a}{b-\sqrt {b^2-4 a c}}+x^2\right ) \left (e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \operatorname {EllipticPi}\left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) e^2}{2 c d^2}+1,\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ),\frac {1}{2} \left (4-\frac {b \left (b-\sqrt {b^2-4 a c}\right )}{a c}\right )\right )}{8 a c^{3/2} d^2 e^6 \sqrt {\frac {\left (b-\sqrt {b^2-4 a c}\right )^2 \left (\frac {2 a}{b-\sqrt {b^2-4 a c}}+x^2\right )}{a \left (-\sqrt {b^2-4 a c}+b+2 c x^2\right )}} \sqrt {a+b x^2+c x^4}}-\frac {\int \frac {2 c^3 e^4 x^6+2 c^2 e^2 \left (c d^2+2 b e^2\right ) x^4+2 c \left (c^2 d^4+b^2 e^4+2 c \left (a e^4+b d^2 e^2\right )\right ) x^2+c \left (b c d^4+b^2 e^2 d^2+2 a c e^2 d^2+3 a b e^4-\sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {16 c \left (a e^4+b d^2 e^2+c d^4\right )^{3/2} \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{e^2}-\frac {\left (b e^2+2 c d^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \left (12 a c e^4-b^2 e^4+8 b c d^2 e^2+8 c^2 d^4\right )}{\sqrt {c} e^2}}{8 c e^4}-\frac {\sqrt {a+b x^2+c x^4} \left (8 a c e^4+b^2 e^4+2 c e^2 x^2 \left (b e^2+2 c d^2\right )+10 b c d^2 e^2+8 c^2 d^4\right )}{4 c e^4}}{2 e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle d \left (\frac {\left (b-\sqrt {b^2-4 a c}\right )^{5/2} \left (2 c d^2+\left (b+\sqrt {b^2-4 a c}\right ) e^2\right ) \left (c d^4+b e^2 d^2+a e^4\right ) \left (x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}\right ) \operatorname {EllipticPi}\left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) e^2}{2 c d^2}+1,\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ),\frac {1}{2} \left (4-\frac {b \left (b-\sqrt {b^2-4 a c}\right )}{a c}\right )\right )}{8 a c^{3/2} d^2 e^6 \sqrt {\frac {\left (b-\sqrt {b^2-4 a c}\right )^2 \left (x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}\right )}{a \left (2 c x^2+b-\sqrt {b^2-4 a c}\right )}} \sqrt {c x^4+b x^2+a}}-\frac {\frac {2}{5} c^2 x^3 \sqrt {c x^4+b x^2+a} e^4+\frac {\int \frac {2 c^3 e^2 \left (5 c d^2+6 b e^2\right ) x^4+2 c^2 \left (5 c^2 d^4+10 b c e^2 d^2+5 b^2 e^4+7 a c e^4\right ) x^2+5 c^2 \left (b^2 d^2 e^2+2 a c d^2 e^2-\sqrt {b^2-4 a c} \left (c d^4+a e^4\right )+b \left (c d^4-\sqrt {b^2-4 a c} e^2 d^2+3 a e^4\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{5 c}}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {16 c \left (c d^4+b e^2 d^2+a e^4\right )^{3/2} \text {arctanh}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b e^2 d^2+a e^4} \sqrt {c x^4+b x^2+a}}\right )}{e^2}-\frac {\left (2 c d^2+b e^2\right ) \left (8 c^2 d^4+8 b c e^2 d^2-b^2 e^4+12 a c e^4\right ) \text {arctanh}\left (\frac {2 c x^2+b}{2 \sqrt {c} \sqrt {c x^4+b x^2+a}}\right )}{\sqrt {c} e^2}}{8 c e^4}-\frac {\left (8 c^2 d^4+10 b c e^2 d^2+b^2 e^4+8 a c e^4+2 c e^2 \left (2 c d^2+b e^2\right ) x^2\right ) \sqrt {c x^4+b x^2+a}}{4 c e^4}}{2 e^2}-\frac {\left (c x^4+b x^2+a\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle d \left (\frac {\left (b-\sqrt {b^2-4 a c}\right )^{5/2} \left (2 c d^2+\left (b+\sqrt {b^2-4 a c}\right ) e^2\right ) \left (c d^4+b e^2 d^2+a e^4\right ) \left (x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}\right ) \operatorname {EllipticPi}\left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) e^2}{2 c d^2}+1,\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ),\frac {1}{2} \left (4-\frac {b \left (b-\sqrt {b^2-4 a c}\right )}{a c}\right )\right )}{8 a c^{3/2} d^2 e^6 \sqrt {\frac {\left (b-\sqrt {b^2-4 a c}\right )^2 \left (x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}\right )}{a \left (2 c x^2+b-\sqrt {b^2-4 a c}\right )}} \sqrt {c x^4+b x^2+a}}-\frac {\frac {2}{5} c^2 x^3 \sqrt {c x^4+b x^2+a} e^4+\frac {\frac {2}{3} c^2 \left (5 c d^2+6 b e^2\right ) x \sqrt {c x^4+b x^2+a} e^2+\frac {\int \frac {c^3 \left (15 b c d^4+15 b^2 e^2 d^2+20 a c e^2 d^2+33 a b e^4+2 \left (15 c^2 d^4+20 b c e^2 d^2+3 b^2 e^4+21 a c e^4\right ) x^2-15 \sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{3 c}}{5 c}}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {16 c \left (c d^4+b e^2 d^2+a e^4\right )^{3/2} \text {arctanh}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b e^2 d^2+a e^4} \sqrt {c x^4+b x^2+a}}\right )}{e^2}-\frac {\left (2 c d^2+b e^2\right ) \left (8 c^2 d^4+8 b c e^2 d^2-b^2 e^4+12 a c e^4\right ) \text {arctanh}\left (\frac {2 c x^2+b}{2 \sqrt {c} \sqrt {c x^4+b x^2+a}}\right )}{\sqrt {c} e^2}}{8 c e^4}-\frac {\left (8 c^2 d^4+10 b c e^2 d^2+b^2 e^4+8 a c e^4+2 c e^2 \left (2 c d^2+b e^2\right ) x^2\right ) \sqrt {c x^4+b x^2+a}}{4 c e^4}}{2 e^2}-\frac {\left (c x^4+b x^2+a\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\left (b-\sqrt {b^2-4 a c}\right )^{5/2} \left (2 c d^2+\left (b+\sqrt {b^2-4 a c}\right ) e^2\right ) \left (c d^4+b e^2 d^2+a e^4\right ) \left (x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}\right ) \operatorname {EllipticPi}\left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) e^2}{2 c d^2}+1,\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ),\frac {1}{2} \left (4-\frac {b \left (b-\sqrt {b^2-4 a c}\right )}{a c}\right )\right )}{8 a c^{3/2} d^2 e^6 \sqrt {\frac {\left (b-\sqrt {b^2-4 a c}\right )^2 \left (x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}\right )}{a \left (2 c x^2+b-\sqrt {b^2-4 a c}\right )}} \sqrt {c x^4+b x^2+a}}-\frac {\frac {2}{5} c^2 x^3 \sqrt {c x^4+b x^2+a} e^4+\frac {\frac {1}{3} \int \frac {15 b c d^4+15 b^2 e^2 d^2+20 a c e^2 d^2+33 a b e^4+2 \left (15 c^2 d^4+20 b c e^2 d^2+3 b^2 e^4+21 a c e^4\right ) x^2-15 \sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )}{\sqrt {c x^4+b x^2+a}}dx c^2+\frac {2}{3} e^2 \left (5 c d^2+6 b e^2\right ) x \sqrt {c x^4+b x^2+a} c^2}{5 c}}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {16 c \left (c d^4+b e^2 d^2+a e^4\right )^{3/2} \text {arctanh}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b e^2 d^2+a e^4} \sqrt {c x^4+b x^2+a}}\right )}{e^2}-\frac {\left (2 c d^2+b e^2\right ) \left (8 c^2 d^4+8 b c e^2 d^2-b^2 e^4+12 a c e^4\right ) \text {arctanh}\left (\frac {2 c x^2+b}{2 \sqrt {c} \sqrt {c x^4+b x^2+a}}\right )}{\sqrt {c} e^2}}{8 c e^4}-\frac {\left (8 c^2 d^4+10 b c e^2 d^2+b^2 e^4+8 a c e^4+2 c e^2 \left (2 c d^2+b e^2\right ) x^2\right ) \sqrt {c x^4+b x^2+a}}{4 c e^4}}{2 e^2}-\frac {\left (c x^4+b x^2+a\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle d \left (\frac {\left (b-\sqrt {b^2-4 a c}\right )^{5/2} \left (2 c d^2+\left (b+\sqrt {b^2-4 a c}\right ) e^2\right ) \left (c d^4+b e^2 d^2+a e^4\right ) \left (x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}\right ) \operatorname {EllipticPi}\left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) e^2}{2 c d^2}+1,\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ),\frac {1}{2} \left (4-\frac {b \left (b-\sqrt {b^2-4 a c}\right )}{a c}\right )\right )}{8 a c^{3/2} d^2 e^6 \sqrt {\frac {\left (b-\sqrt {b^2-4 a c}\right )^2 \left (x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}\right )}{a \left (2 c x^2+b-\sqrt {b^2-4 a c}\right )}} \sqrt {c x^4+b x^2+a}}-\frac {\frac {2}{5} c^2 x^3 \sqrt {c x^4+b x^2+a} e^4+\frac {\frac {1}{3} \left (\left (15 b c d^4+15 b^2 e^2 d^2+20 a c e^2 d^2+33 a b e^4-15 \sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )+\frac {2 \sqrt {a} \left (15 c^2 d^4+20 b c e^2 d^2+3 b^2 e^4+21 a c e^4\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\frac {2 \sqrt {a} \left (15 c^2 d^4+20 b c e^2 d^2+3 b^2 e^4+21 a c e^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right ) c^2+\frac {2}{3} e^2 \left (5 c d^2+6 b e^2\right ) x \sqrt {c x^4+b x^2+a} c^2}{5 c}}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {16 c \left (c d^4+b e^2 d^2+a e^4\right )^{3/2} \text {arctanh}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b e^2 d^2+a e^4} \sqrt {c x^4+b x^2+a}}\right )}{e^2}-\frac {\left (2 c d^2+b e^2\right ) \left (8 c^2 d^4+8 b c e^2 d^2-b^2 e^4+12 a c e^4\right ) \text {arctanh}\left (\frac {2 c x^2+b}{2 \sqrt {c} \sqrt {c x^4+b x^2+a}}\right )}{\sqrt {c} e^2}}{8 c e^4}-\frac {\left (8 c^2 d^4+10 b c e^2 d^2+b^2 e^4+8 a c e^4+2 c e^2 \left (2 c d^2+b e^2\right ) x^2\right ) \sqrt {c x^4+b x^2+a}}{4 c e^4}}{2 e^2}-\frac {\left (c x^4+b x^2+a\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\left (b-\sqrt {b^2-4 a c}\right )^{5/2} \left (2 c d^2+\left (b+\sqrt {b^2-4 a c}\right ) e^2\right ) \left (c d^4+b e^2 d^2+a e^4\right ) \left (x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}\right ) \operatorname {EllipticPi}\left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) e^2}{2 c d^2}+1,\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ),\frac {1}{2} \left (4-\frac {b \left (b-\sqrt {b^2-4 a c}\right )}{a c}\right )\right )}{8 a c^{3/2} d^2 e^6 \sqrt {\frac {\left (b-\sqrt {b^2-4 a c}\right )^2 \left (x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}\right )}{a \left (2 c x^2+b-\sqrt {b^2-4 a c}\right )}} \sqrt {c x^4+b x^2+a}}-\frac {\frac {2}{5} c^2 x^3 \sqrt {c x^4+b x^2+a} e^4+\frac {\frac {1}{3} \left (\left (15 b c d^4+15 b^2 e^2 d^2+20 a c e^2 d^2+33 a b e^4-15 \sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )+\frac {2 \sqrt {a} \left (15 c^2 d^4+20 b c e^2 d^2+3 b^2 e^4+21 a c e^4\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\frac {2 \left (15 c^2 d^4+20 b c e^2 d^2+3 b^2 e^4+21 a c e^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right ) c^2+\frac {2}{3} e^2 \left (5 c d^2+6 b e^2\right ) x \sqrt {c x^4+b x^2+a} c^2}{5 c}}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {16 c \left (c d^4+b e^2 d^2+a e^4\right )^{3/2} \text {arctanh}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b e^2 d^2+a e^4} \sqrt {c x^4+b x^2+a}}\right )}{e^2}-\frac {\left (2 c d^2+b e^2\right ) \left (8 c^2 d^4+8 b c e^2 d^2-b^2 e^4+12 a c e^4\right ) \text {arctanh}\left (\frac {2 c x^2+b}{2 \sqrt {c} \sqrt {c x^4+b x^2+a}}\right )}{\sqrt {c} e^2}}{8 c e^4}-\frac {\left (8 c^2 d^4+10 b c e^2 d^2+b^2 e^4+8 a c e^4+2 c e^2 \left (2 c d^2+b e^2\right ) x^2\right ) \sqrt {c x^4+b x^2+a}}{4 c e^4}}{2 e^2}-\frac {\left (c x^4+b x^2+a\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle d \left (\frac {\left (b-\sqrt {b^2-4 a c}\right )^{5/2} \left (2 c d^2+\left (b+\sqrt {b^2-4 a c}\right ) e^2\right ) \left (c d^4+b e^2 d^2+a e^4\right ) \left (x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}\right ) \operatorname {EllipticPi}\left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) e^2}{2 c d^2}+1,\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ),\frac {1}{2} \left (4-\frac {b \left (b-\sqrt {b^2-4 a c}\right )}{a c}\right )\right )}{8 a c^{3/2} d^2 e^6 \sqrt {\frac {\left (b-\sqrt {b^2-4 a c}\right )^2 \left (x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}\right )}{a \left (2 c x^2+b-\sqrt {b^2-4 a c}\right )}} \sqrt {c x^4+b x^2+a}}-\frac {\frac {2}{5} c^2 x^3 \sqrt {c x^4+b x^2+a} e^4+\frac {\frac {1}{3} \left (\frac {\left (15 b c d^4+15 b^2 e^2 d^2+20 a c e^2 d^2+33 a b e^4-15 \sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )+\frac {2 \sqrt {a} \left (15 c^2 d^4+20 b c e^2 d^2+3 b^2 e^4+21 a c e^4\right )}{\sqrt {c}}\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {c x^4+b x^2+a}}-\frac {2 \left (15 c^2 d^4+20 b c e^2 d^2+3 b^2 e^4+21 a c e^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right ) c^2+\frac {2}{3} e^2 \left (5 c d^2+6 b e^2\right ) x \sqrt {c x^4+b x^2+a} c^2}{5 c}}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {16 c \left (c d^4+b e^2 d^2+a e^4\right )^{3/2} \text {arctanh}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b e^2 d^2+a e^4} \sqrt {c x^4+b x^2+a}}\right )}{e^2}-\frac {\left (2 c d^2+b e^2\right ) \left (8 c^2 d^4+8 b c e^2 d^2-b^2 e^4+12 a c e^4\right ) \text {arctanh}\left (\frac {2 c x^2+b}{2 \sqrt {c} \sqrt {c x^4+b x^2+a}}\right )}{\sqrt {c} e^2}}{8 c e^4}-\frac {\left (8 c^2 d^4+10 b c e^2 d^2+b^2 e^4+8 a c e^4+2 c e^2 \left (2 c d^2+b e^2\right ) x^2\right ) \sqrt {c x^4+b x^2+a}}{4 c e^4}}{2 e^2}-\frac {\left (c x^4+b x^2+a\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle d \left (\frac {\left (b-\sqrt {b^2-4 a c}\right )^{5/2} \left (2 c d^2+\left (b+\sqrt {b^2-4 a c}\right ) e^2\right ) \left (c d^4+b e^2 d^2+a e^4\right ) \left (x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}\right ) \operatorname {EllipticPi}\left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) e^2}{2 c d^2}+1,\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ),\frac {1}{2} \left (4-\frac {b \left (b-\sqrt {b^2-4 a c}\right )}{a c}\right )\right )}{8 a c^{3/2} d^2 e^6 \sqrt {\frac {\left (b-\sqrt {b^2-4 a c}\right )^2 \left (x^2+\frac {2 a}{b-\sqrt {b^2-4 a c}}\right )}{a \left (2 c x^2+b-\sqrt {b^2-4 a c}\right )}} \sqrt {c x^4+b x^2+a}}-\frac {\frac {2}{5} c^2 x^3 \sqrt {c x^4+b x^2+a} e^4+\frac {\frac {1}{3} \left (\frac {\left (15 b c d^4+15 b^2 e^2 d^2+20 a c e^2 d^2+33 a b e^4-15 \sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )+\frac {2 \sqrt {a} \left (15 c^2 d^4+20 b c e^2 d^2+3 b^2 e^4+21 a c e^4\right )}{\sqrt {c}}\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {c x^4+b x^2+a}}-\frac {2 \left (15 c^2 d^4+20 b c e^2 d^2+3 b^2 e^4+21 a c e^4\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {c x^4+b x^2+a}}-\frac {x \sqrt {c x^4+b x^2+a}}{\sqrt {c} x^2+\sqrt {a}}\right )}{\sqrt {c}}\right ) c^2+\frac {2}{3} e^2 \left (5 c d^2+6 b e^2\right ) x \sqrt {c x^4+b x^2+a} c^2}{5 c}}{2 c e^6}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {16 c \left (c d^4+b e^2 d^2+a e^4\right )^{3/2} \text {arctanh}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b e^2 d^2+a e^4} \sqrt {c x^4+b x^2+a}}\right )}{e^2}-\frac {\left (2 c d^2+b e^2\right ) \left (8 c^2 d^4+8 b c e^2 d^2-b^2 e^4+12 a c e^4\right ) \text {arctanh}\left (\frac {2 c x^2+b}{2 \sqrt {c} \sqrt {c x^4+b x^2+a}}\right )}{\sqrt {c} e^2}}{8 c e^4}-\frac {\left (8 c^2 d^4+10 b c e^2 d^2+b^2 e^4+8 a c e^4+2 c e^2 \left (2 c d^2+b e^2\right ) x^2\right ) \sqrt {c x^4+b x^2+a}}{4 c e^4}}{2 e^2}-\frac {\left (c x^4+b x^2+a\right )^{3/2}}{3 e^2}\right )\) |
Input:
Int[(a + b*x^2 + c*x^4)^(3/2)/(d + e*x),x]
Output:
-1/2*(e*(-1/3*(a + b*x^2 + c*x^4)^(3/2)/e^2 + (-1/4*((8*c^2*d^4 + 10*b*c*d ^2*e^2 + b^2*e^4 + 8*a*c*e^4 + 2*c*e^2*(2*c*d^2 + b*e^2)*x^2)*Sqrt[a + b*x ^2 + c*x^4])/(c*e^4) + (-(((2*c*d^2 + b*e^2)*(8*c^2*d^4 + 8*b*c*d^2*e^2 - b^2*e^4 + 12*a*c*e^4)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c* x^4])])/(Sqrt[c]*e^2)) + (16*c*(c*d^4 + b*d^2*e^2 + a*e^4)^(3/2)*ArcTanh[( b*d^2 + 2*a*e^2 + (2*c*d^2 + b*e^2)*x^2)/(2*Sqrt[c*d^4 + b*d^2*e^2 + a*e^4 ]*Sqrt[a + b*x^2 + c*x^4])])/e^2)/(8*c*e^4))/(2*e^2))) + d*(-1/2*((2*c^2*e ^4*x^3*Sqrt[a + b*x^2 + c*x^4])/5 + ((2*c^2*e^2*(5*c*d^2 + 6*b*e^2)*x*Sqrt [a + b*x^2 + c*x^4])/3 + (c^2*((-2*(15*c^2*d^4 + 20*b*c*d^2*e^2 + 3*b^2*e^ 4 + 21*a*c*e^4)*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[ c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c ]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/Sqrt[c] + ((15*b*c*d^4 + 15*b^ 2*d^2*e^2 + 20*a*c*d^2*e^2 + 33*a*b*e^4 - 15*Sqrt[b^2 - 4*a*c]*(c*d^4 + b* d^2*e^2 + a*e^4) + (2*Sqrt[a]*(15*c^2*d^4 + 20*b*c*d^2*e^2 + 3*b^2*e^4 + 2 1*a*c*e^4))/Sqrt[c])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqr t[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sq rt[a]*Sqrt[c]))/4])/(2*a^(1/4)*c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/3)/(5*c) )/(c*e^6) + ((b - Sqrt[b^2 - 4*a*c])^(5/2)*(2*c*d^2 + (b + Sqrt[b^2 - 4*a* c])*e^2)*(c*d^4 + b*d^2*e^2 + a*e^4)*((2*a)/(b - Sqrt[b^2 - 4*a*c]) + x...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[c*(Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), ArcTan[ Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ [d/c]
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x ] - Simp[p/(e*(m + 2*p + 1)) Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x ] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(2*c*d - e*(b + q)))*((c*d^2 - b*d*e + a*e^2)^(p - 1/2)/(4*c*e^(2*p))) Int[(b - q + 2*c*x^2)/((d + e*x^ 2)*Sqrt[a + b*x^2 + c*x^4]), x], x] + Simp[1/(4*c*e^(2*p)) Int[(1/Sqrt[a + b*x^2 + c*x^4])*ExpandToSum[(4*c*e^(2*p)*(a + b*x^2 + c*x^4)^(p + 1/2) + (2*c*d - e*(b + q))*(c*d^2 - b*d*e + a*e^2)^(p - 1/2)*(b - q + 2*c*x^2))/(d + e*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p - 1/2, 0] && PosQ[b^2 - 4*a*c] & & PosQ[c/a]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + ( b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(a + b*x^n + c*x ^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]) Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x], x] /; Fre eQ[{a, b, c, d, e, f, g, n, p, q, r}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c , 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{n = Expon[Px, x^2], e = Coeff[Px, x^2, Expon[Px, x^2]]}, Simp[e*x^(2*n - 3)*(( a + b*x^2 + c*x^4)^(p + 1)/(c*(2*n + 4*p + 1))), x] + Simp[1/(c*(2*n + 4*p + 1)) Int[(a + b*x^2 + c*x^4)^p*ExpandToSum[c*(2*n + 4*p + 1)*Px - a*e*(2 *n - 3)*x^(2*n - 4) - b*e*(2*n + 2*p - 1)*x^(2*n - 2) - c*e*(2*n + 4*p + 1) *x^(2*n), x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Px, x^2] && Expon[ Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && !LtQ[p, -1]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.)/((d_) + (e_.)*(x_)), x_Symbo l] :> Simp[d Int[(a + b*x^2 + c*x^4)^p/(d^2 - e^2*x^2), x], x] - Simp[e Int[x*((a + b*x^2 + c*x^4)^p/(d^2 - e^2*x^2)), x], x] /; FreeQ[{a, b, c, d , e}, x] && IntegerQ[p + 1/2]
Time = 4.13 (sec) , antiderivative size = 1017, normalized size of antiderivative = 0.88
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1017\) |
| default | \(\text {Expression too large to display}\) | \(1295\) |
| elliptic | \(\text {Expression too large to display}\) | \(1295\) |
Input:
int((c*x^4+b*x^2+a)^(3/2)/(e*x+d),x,method=_RETURNVERBOSE)
Output:
1/240/c*(40*c^2*e^4*x^4-48*c^2*d*e^3*x^3+70*b*c*e^4*x^2+60*c^2*d^2*e^2*x^2 -96*b*c*d*e^3*x-80*c^2*d^3*e*x+160*a*c*e^4+15*b^2*e^4+150*b*c*d^2*e^2+120* c^2*d^4)*(c*x^4+b*x^2+a)^(1/2)/e^5-1/240/e^5/c*(-15/2/e^2*(12*a*b*c*e^6+24 *a*c^2*d^2*e^4-b^3*e^6+6*b^2*c*d^2*e^4+24*b*c^2*d^4*e^2+16*c^3*d^6)*ln((1/ 2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))/c^(1/2)+4*d*(24*a*b*e^6+25*a*c*d ^2*e^4+15*b^2*d^2*e^4+30*b*c*d^4*e^2+15*c^2*d^6)*c/e^3*2^(1/2)/((-b+(-4*a* c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+( -4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1 /2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a /c)^(1/2))-240*(a^2*e^8+2*a*b*d^2*e^6+2*a*c*d^4*e^4+b^2*d^4*e^4+2*b*c*d^6* e^2+c^2*d^8)*c/e^4*(-1/2/(c*d^4/e^4+b*d^2/e^2+a)^(1/2)*arctanh(1/2*(2*c*x^ 2*d^2/e^2+b*d^2/e^2+b*x^2+2*a)/(c*d^4/e^4+b*d^2/e^2+a)^(1/2)/(c*x^4+b*x^2+ a)^(1/2))+2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)/d*e*(1-1/2*(-b+(-4*a*c +b^2)^(1/2))/a*x^2)^(1/2)*(1+1/2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^ 4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2 ),2/(-b+(-4*a*c+b^2)^(1/2))*a/d^2*e^2,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2 )*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)))-8*c*d/e*(21*a*c*e^4+3*b^2*e^ 4+20*b*c*d^2*e^2+15*c^2*d^4)*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*( 4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2 )^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2*x*2...
Timed out. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{d+e x} \, dx=\text {Timed out} \] Input:
integrate((c*x^4+b*x^2+a)^(3/2)/(e*x+d),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{d + e x}\, dx \] Input:
integrate((c*x**4+b*x**2+a)**(3/2)/(e*x+d),x)
Output:
Integral((a + b*x**2 + c*x**4)**(3/2)/(d + e*x), x)
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{d+e x} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{e x + d} \,d x } \] Input:
integrate((c*x^4+b*x^2+a)^(3/2)/(e*x+d),x, algorithm="maxima")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)/(e*x + d), x)
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{d+e x} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{e x + d} \,d x } \] Input:
integrate((c*x^4+b*x^2+a)^(3/2)/(e*x+d),x, algorithm="giac")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)/(e*x + d), x)
Timed out. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{d+e x} \, dx=\int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{d+e\,x} \,d x \] Input:
int((a + b*x^2 + c*x^4)^(3/2)/(d + e*x),x)
Output:
int((a + b*x^2 + c*x^4)^(3/2)/(d + e*x), x)
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{e x +d}d x \] Input:
int((c*x^4+b*x^2+a)^(3/2)/(e*x+d),x)
Output:
int((c*x^4+b*x^2+a)^(3/2)/(e*x+d),x)