Integrand size = 24, antiderivative size = 799 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=-\frac {d \left (6 b^3 c d^2-48 a b c^2 d^2-8 b^4 e^2+57 a b^2 c e^2-84 a^2 c^2 e^2\right ) x \sqrt {a+b x^2+c x^4}}{105 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {3 \left (b^2-4 a c\right ) e \left (6 c d^2-b e^2\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^3}+\frac {d x \left (3 b^2 c d^2+30 a c^2 d^2-4 b^3 e^2+9 a b c e^2+3 c \left (3 b c d^2-4 b^2 e^2+14 a c e^2\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{105 c^2}+\frac {e \left (6 c d^2-b e^2\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac {d x \left (3 \left (c d^2+b e^2\right )+7 c e^2 x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{21 c}+\frac {e^3 \left (a+b x^2+c x^4\right )^{5/2}}{10 c}+\frac {3 \left (b^2-4 a c\right )^2 e \left (6 c d^2-b e^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{7/2}}+\frac {\sqrt [4]{a} d \left (6 b^3 c d^2-48 a b c^2 d^2-8 b^4 e^2+57 a b^2 c e^2-84 a^2 c^2 e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{105 c^{11/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} \left (b+2 \sqrt {a} \sqrt {c}\right ) d \left (6 b^2 c d^2-9 \sqrt {a} b c^{3/2} d^2-30 a c^2 d^2-8 b^3 e^2+12 \sqrt {a} b^2 \sqrt {c} e^2+33 a b c e^2-42 a^{3/2} c^{3/2} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{210 c^{11/4} \sqrt {a+b x^2+c x^4}} \] Output:
-1/105*d*(-84*a^2*c^2*e^2+57*a*b^2*c*e^2-48*a*b*c^2*d^2-8*b^4*e^2+6*b^3*c* d^2)*x*(c*x^4+b*x^2+a)^(1/2)/c^(5/2)/(a^(1/2)+c^(1/2)*x^2)-3/256*(-4*a*c+b ^2)*e*(-b*e^2+6*c*d^2)*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(1/2)/c^3+1/105*d*x*(3* b^2*c*d^2+30*a*c^2*d^2-4*b^3*e^2+9*a*b*c*e^2+3*c*(14*a*c*e^2-4*b^2*e^2+3*b *c*d^2)*x^2)*(c*x^4+b*x^2+a)^(1/2)/c^2+1/32*e*(-b*e^2+6*c*d^2)*(2*c*x^2+b) *(c*x^4+b*x^2+a)^(3/2)/c^2+1/21*d*x*(7*c*e^2*x^2+3*b*e^2+3*c*d^2)*(c*x^4+b *x^2+a)^(3/2)/c+1/10*e^3*(c*x^4+b*x^2+a)^(5/2)/c+3/512*(-4*a*c+b^2)^2*e*(- b*e^2+6*c*d^2)*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/c^(7 /2)+1/105*a^(1/4)*d*(-84*a^2*c^2*e^2+57*a*b^2*c*e^2-48*a*b*c^2*d^2-8*b^4*e ^2+6*b^3*c*d^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)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^ (1/2))^(1/2))/c^(11/4)/(c*x^4+b*x^2+a)^(1/2)-1/210*a^(1/4)*(b+2*a^(1/2)*c^ (1/2))*d*(6*b^2*c*d^2-9*a^(1/2)*b*c^(3/2)*d^2-30*a*c^2*d^2-8*b^3*e^2+12*a^ (1/2)*b^2*c^(1/2)*e^2+33*a*b*c*e^2-42*a^(3/2)*c^(3/2)*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*a rctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(11/4)/(c*x^4+ b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 12.68 (sec) , antiderivative size = 2681, normalized size of antiderivative = 3.36 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\text {Result too large to show} \] Input:
Integrate[(d + e*x)^3*(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(-2*Sqrt[c]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(a + b*x^2 + c*x^4)*(-315*b^4* e^3 + 2*b^3*c*e*(945*d^2 + 512*d*e*x + 105*e^2*x^2) - 12*b^2*c*(-175*a*e^3 + c*x*(64*d^3 + 105*d^2*e*x + 64*d*e^2*x^2 + 14*e^3*x^3)) - 8*b*c^2*(3*a* e*(525*d^2 + 256*d*e*x + 49*e^2*x^2) + 2*c*x^3*(384*d^3 + 945*d^2*e*x + 80 0*d*e^2*x^2 + 231*e^3*x^3)) - 16*c^2*(168*a^2*e^3 + 2*c^2*x^5*(120*d^3 + 3 15*d^2*e*x + 280*d*e^2*x^2 + 84*e^3*x^3) + a*c*x*(720*d^3 + 1575*d^2*e*x + 1232*d*e^2*x^2 + 336*e^3*x^3))) + (768*I)*Sqrt[2]*b^3*c^(3/2)*(b - Sqrt[b ^2 - 4*a*c])*d^3*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4* a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(EllipticE[I*ArcSinh[Sq rt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqr t[b^2 - 4*a*c])] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a* c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]) + (6144*I)*Sqrt [2]*a*b*c^(5/2)*(-b + Sqrt[b^2 - 4*a*c])*d^3*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a* c])]*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticF[I*ArcSinh[Sqrt[2] *Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]) + (7296*I)*Sqrt[2]*a*b^2*c^(3/2)*(b - Sqrt[b^2 - 4*a*c])*d*e^ 2*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c...
Time = 2.16 (sec) , antiderivative size = 749, normalized size of antiderivative = 0.94, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2202, 1490, 27, 1490, 25, 1511, 27, 1416, 1509, 1576, 27, 1160, 1087, 1087, 1092, 219}
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.
| \(\displaystyle \int (d+e x)^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \left (d^3+3 e^2 x^2 d\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\int x \left (x^2 e^3+3 d^2 e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {\int 3 d \left (\left (3 b c d^2-4 b^2 e^2+14 a c e^2\right ) x^2+a \left (6 c d^2-b e^2\right )\right ) \sqrt {c x^4+b x^2+a}dx}{21 c}+\int x \left (x^2 e^3+3 d^2 e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\frac {d x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+c d^2\right )+7 c e^2 x^2\right )}{21 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \int \left (\left (3 b c d^2-4 b^2 e^2+14 a c e^2\right ) x^2+a \left (6 c d^2-b e^2\right )\right ) \sqrt {c x^4+b x^2+a}dx}{7 c}+\int x \left (x^2 e^3+3 d^2 e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\frac {d x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+c d^2\right )+7 c e^2 x^2\right )}{21 c}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {d \left (\frac {\int -\frac {\left (-8 e^2 b^4+6 c d^2 b^3+57 a c e^2 b^2-48 a c^2 d^2 b-84 a^2 c^2 e^2\right ) x^2+a \left (-4 e^2 b^3+3 c d^2 b^2+24 a c e^2 b-60 a c^2 d^2\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c}+\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+3 b c d^2\right )+9 a b c e^2+30 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )}{15 c}\right )}{7 c}+\int x \left (x^2 e^3+3 d^2 e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\frac {d x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+c d^2\right )+7 c e^2 x^2\right )}{21 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+3 b c d^2\right )+9 a b c e^2+30 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )}{15 c}-\frac {\int \frac {\left (-8 e^2 b^4+6 c d^2 b^3+57 a c e^2 b^2-48 a c^2 d^2 b-84 a^2 c^2 e^2\right ) x^2+a \left (-4 e^2 b^3+3 c d^2 b^2+24 a c e^2 b-60 a c^2 d^2\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c}\right )}{7 c}+\int x \left (x^2 e^3+3 d^2 e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\frac {d x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+c d^2\right )+7 c e^2 x^2\right )}{21 c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {d \left (\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+3 b c d^2\right )+9 a b c e^2+30 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )}{15 c}-\frac {\frac {\sqrt {a} \left (-84 a^2 c^2 e^2+57 a b^2 c e^2+\sqrt {a} \sqrt {c} \left (24 a b c e^2-60 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \left (-84 a^2 c^2 e^2+57 a b^2 c e^2-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}\right )}{7 c}+\int x \left (x^2 e^3+3 d^2 e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\frac {d x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+c d^2\right )+7 c e^2 x^2\right )}{21 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+3 b c d^2\right )+9 a b c e^2+30 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )}{15 c}-\frac {\frac {\sqrt {a} \left (-84 a^2 c^2 e^2+57 a b^2 c e^2+\sqrt {a} \sqrt {c} \left (24 a b c e^2-60 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (-84 a^2 c^2 e^2+57 a b^2 c e^2-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}\right )}{7 c}+\int x \left (x^2 e^3+3 d^2 e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\frac {d x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+c d^2\right )+7 c e^2 x^2\right )}{21 c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {d \left (\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+3 b c d^2\right )+9 a b c e^2+30 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 e^2+57 a b^2 c e^2+\sqrt {a} \sqrt {c} \left (24 a b c e^2-60 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 e^2+57 a b^2 c e^2-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}\right )}{7 c}+\int x \left (x^2 e^3+3 d^2 e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\frac {d x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+c d^2\right )+7 c e^2 x^2\right )}{21 c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \int x \left (x^2 e^3+3 d^2 e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\frac {d \left (\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+3 b c d^2\right )+9 a b c e^2+30 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 e^2+57 a b^2 c e^2+\sqrt {a} \sqrt {c} \left (24 a b c e^2-60 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 e^2+57 a b^2 c e^2-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}\right )}{7 c}+\frac {d x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+c d^2\right )+7 c e^2 x^2\right )}{21 c}\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle \frac {1}{2} \int e \left (3 d^2+e^2 x^2\right ) \left (c x^4+b x^2+a\right )^{3/2}dx^2+\frac {d \left (\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+3 b c d^2\right )+9 a b c e^2+30 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 e^2+57 a b^2 c e^2+\sqrt {a} \sqrt {c} \left (24 a b c e^2-60 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 e^2+57 a b^2 c e^2-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}\right )}{7 c}+\frac {d x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+c d^2\right )+7 c e^2 x^2\right )}{21 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} e \int \left (3 d^2+e^2 x^2\right ) \left (c x^4+b x^2+a\right )^{3/2}dx^2+\frac {d \left (\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+3 b c d^2\right )+9 a b c e^2+30 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 e^2+57 a b^2 c e^2+\sqrt {a} \sqrt {c} \left (24 a b c e^2-60 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 e^2+57 a b^2 c e^2-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}\right )}{7 c}+\frac {d x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+c d^2\right )+7 c e^2 x^2\right )}{21 c}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{2} e \left (\frac {\left (6 c d^2-b e^2\right ) \int \left (c x^4+b x^2+a\right )^{3/2}dx^2}{2 c}+\frac {e^2 \left (a+b x^2+c x^4\right )^{5/2}}{5 c}\right )+\frac {d \left (\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+3 b c d^2\right )+9 a b c e^2+30 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 e^2+57 a b^2 c e^2+\sqrt {a} \sqrt {c} \left (24 a b c e^2-60 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 e^2+57 a b^2 c e^2-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}\right )}{7 c}+\frac {d x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+c d^2\right )+7 c e^2 x^2\right )}{21 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{2} e \left (\frac {\left (6 c d^2-b e^2\right ) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^4+b x^2+a}dx^2}{16 c}\right )}{2 c}+\frac {e^2 \left (a+b x^2+c x^4\right )^{5/2}}{5 c}\right )+\frac {d \left (\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+3 b c d^2\right )+9 a b c e^2+30 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 e^2+57 a b^2 c e^2+\sqrt {a} \sqrt {c} \left (24 a b c e^2-60 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 e^2+57 a b^2 c e^2-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}\right )}{7 c}+\frac {d x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+c d^2\right )+7 c e^2 x^2\right )}{21 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{2} e \left (\frac {\left (6 c d^2-b e^2\right ) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{8 c}\right )}{16 c}\right )}{2 c}+\frac {e^2 \left (a+b x^2+c x^4\right )^{5/2}}{5 c}\right )+\frac {d \left (\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+3 b c d^2\right )+9 a b c e^2+30 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 e^2+57 a b^2 c e^2+\sqrt {a} \sqrt {c} \left (24 a b c e^2-60 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 e^2+57 a b^2 c e^2-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}\right )}{7 c}+\frac {d x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+c d^2\right )+7 c e^2 x^2\right )}{21 c}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{2} e \left (\frac {\left (6 c d^2-b e^2\right ) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{4 c}\right )}{16 c}\right )}{2 c}+\frac {e^2 \left (a+b x^2+c x^4\right )^{5/2}}{5 c}\right )+\frac {d \left (\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+3 b c d^2\right )+9 a b c e^2+30 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 e^2+57 a b^2 c e^2+\sqrt {a} \sqrt {c} \left (24 a b c e^2-60 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 e^2+57 a b^2 c e^2-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}\right )}{7 c}+\frac {d x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+c d^2\right )+7 c e^2 x^2\right )}{21 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {d \left (\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+3 b c d^2\right )+9 a b c e^2+30 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 e^2+57 a b^2 c e^2+\sqrt {a} \sqrt {c} \left (24 a b c e^2-60 a c^2 d^2-4 b^3 e^2+3 b^2 c d^2\right )-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 e^2+57 a b^2 c e^2-48 a b c^2 d^2-8 b^4 e^2+6 b^3 c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}\right )}{7 c}+\frac {1}{2} e \left (\frac {\left (6 c d^2-b e^2\right ) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{2 c}+\frac {e^2 \left (a+b x^2+c x^4\right )^{5/2}}{5 c}\right )+\frac {d x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+c d^2\right )+7 c e^2 x^2\right )}{21 c}\) |
Input:
Int[(d + e*x)^3*(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(d*x*(3*(c*d^2 + b*e^2) + 7*c*e^2*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(21*c) + (e*((e^2*(a + b*x^2 + c*x^4)^(5/2))/(5*c) + ((6*c*d^2 - b*e^2)*(((b + 2*c *x^2)*(a + b*x^2 + c*x^4)^(3/2))/(8*c) - (3*(b^2 - 4*a*c)*(((b + 2*c*x^2)* Sqrt[a + b*x^2 + c*x^4])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/(2*S qrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(8*c^(3/2))))/(16*c)))/(2*c)))/2 + (d*(( x*(3*b^2*c*d^2 + 30*a*c^2*d^2 - 4*b^3*e^2 + 9*a*b*c*e^2 + 3*c*(3*b*c*d^2 - 4*b^2*e^2 + 14*a*c*e^2)*x^2)*Sqrt[a + b*x^2 + c*x^4])/(15*c) - (-(((6*b^3 *c*d^2 - 48*a*b*c^2*d^2 - 8*b^4*e^2 + 57*a*b^2*c*e^2 - 84*a^2*c^2*e^2)*(-( (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]*Elliptic E[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sq rt[a + b*x^2 + c*x^4])))/Sqrt[c]) + (a^(1/4)*(6*b^3*c*d^2 - 48*a*b*c^2*d^2 - 8*b^4*e^2 + 57*a*b^2*c*e^2 - 84*a^2*c^2*e^2 + Sqrt[a]*Sqrt[c]*(3*b^2*c* d^2 - 60*a*c^2*d^2 - 4*b^3*e^2 + 24*a*b*c*e^2))*(Sqrt[a] + Sqrt[c]*x^2)*Sq rt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1 /4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(3/4)*Sqrt[a + b*x^2 + c*x^4]))/(15*c)))/(7*c)
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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
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[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
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)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c *x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), 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] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
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[(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[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Time = 4.58 (sec) , antiderivative size = 1300, normalized size of antiderivative = 1.63
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1300\) |
| default | \(\text {Expression too large to display}\) | \(1592\) |
| elliptic | \(\text {Expression too large to display}\) | \(1671\) |
Input:
int((e*x+d)^3*(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/26880/c^3*(2688*c^4*e^3*x^8+8960*c^4*d*e^2*x^7+3696*b*c^3*e^3*x^6+10080* c^4*d^2*e*x^6+12800*b*c^3*d*e^2*x^5+3840*c^4*d^3*x^5+5376*a*c^3*e^3*x^4+16 8*b^2*c^2*e^3*x^4+15120*b*c^3*d^2*e*x^4+19712*a*c^3*d*e^2*x^3+768*b^2*c^2* d*e^2*x^3+6144*b*c^3*d^3*x^3+1176*a*b*c^2*e^3*x^2+25200*a*c^3*d^2*e*x^2-21 0*b^3*c*e^3*x^2+1260*b^2*c^2*d^2*e*x^2+6144*a*b*c^2*d*e^2*x+11520*a*c^3*d^ 3*x-1024*b^3*c*d*e^2*x+768*b^2*c^2*d^3*x+2688*a^2*c^2*e^3-2100*a*b^2*c*e^3 +12600*a*b*c^2*d^2*e+315*b^4*e^3-1890*b^3*c*d^2*e)*(c*x^4+b*x^2+a)^(1/2)-1 /26880/c^3*(128*c*d*(84*a^2*c^2*e^2-57*a*b^2*c*e^2+48*a*b*c^2*d^2+8*b^4*e^ 2-6*b^3*c*d^2)*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^(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))-Ell ipticE(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)))+315/2*e*(16*a^2*b*c^2*e^2-96*a^2*c^3*d^2-8*a* b^3*c*e^2+48*a*b^2*c^2*d^2+b^5*e^2-6*b^4*c*d^2)*ln((1/2*b+c*x^2)/c^(1/2)+( c*x^4+b*x^2+a)^(1/2))/c^(1/2)-3840*a^2*c^3*d^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 ))+192*a*b^2*c^2*d^3*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b...
Time = 0.32 (sec) , antiderivative size = 981, normalized size of antiderivative = 1.23 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^3*(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
-1/107520*(512*sqrt(1/2)*((6*(b^3*c^2 - 8*a*b*c^3)*d^3 - (8*b^4*c - 57*a*b ^2*c^2 + 84*a^2*c^3)*d*e^2)*x*sqrt((b^2 - 4*a*c)/c^2) - (6*(b^4*c - 8*a*b^ 2*c^2)*d^3 - (8*b^5 - 57*a*b^3*c + 84*a^2*b*c^2)*d*e^2)*x)*sqrt(c)*sqrt((c *sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_e(arcsin(sqrt(1/2)*sqrt((c*sqrt( (b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2* a*c)/(a*c)) - 512*sqrt(1/2)*((3*(2*b^3*c^2 + 20*a*c^4 - (16*a*b + b^2)*c^3 )*d^3 - (8*b^4*c + 12*(7*a^2 + 2*a*b)*c^3 - (57*a*b^2 + 4*b^3)*c^2)*d*e^2) *x*sqrt((b^2 - 4*a*c)/c^2) - (3*(2*b^4*c - 20*a*b*c^3 - (16*a*b^2 - b^3)*c ^2)*d^3 - (8*b^5 + 12*(7*a^2*b - 2*a*b^2)*c^2 - (57*a*b^3 - 4*b^4)*c)*d*e^ 2)*x)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_f(arcsin(sq rt(1/2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4 *a*c)/c^2) + b^2 - 2*a*c)/(a*c)) + 315*(6*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^ 3)*d^2*e - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^3)*sqrt(c)*x*log(8*c^2*x^4 + 8*b*c*x^2 + b^2 - 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) + 4*a*c ) - 4*(2688*c^5*e^3*x^9 + 8960*c^5*d*e^2*x^8 + 336*(30*c^5*d^2*e + 11*b*c^ 4*e^3)*x^7 + 1280*(3*c^5*d^3 + 10*b*c^4*d*e^2)*x^6 + 168*(90*b*c^4*d^2*e + (b^2*c^3 + 32*a*c^4)*e^3)*x^5 + 256*(24*b*c^4*d^3 + (3*b^2*c^3 + 77*a*c^4 )*d*e^2)*x^4 - 1536*(b^3*c^2 - 8*a*b*c^3)*d^3 + 256*(8*b^4*c - 57*a*b^2*c^ 2 + 84*a^2*c^3)*d*e^2 + 42*(30*(b^2*c^3 + 20*a*c^4)*d^2*e - (5*b^3*c^2 - 2 8*a*b*c^3)*e^3)*x^3 + 256*(3*(b^2*c^3 + 15*a*c^4)*d^3 - 4*(b^3*c^2 - 6*...
\[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int \left (d + e x\right )^{3} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((e*x+d)**3*(c*x**4+b*x**2+a)**(3/2),x)
Output:
Integral((d + e*x)**3*(a + b*x**2 + c*x**4)**(3/2), x)
\[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{3} \,d x } \] Input:
integrate((e*x+d)^3*(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)*(e*x + d)^3, x)
\[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{3} \,d x } \] Input:
integrate((e*x+d)^3*(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)*(e*x + d)^3, x)
Timed out. \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int {\left (d+e\,x\right )}^3\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2} \,d x \] Input:
int((d + e*x)^3*(a + b*x^2 + c*x^4)^(3/2),x)
Output:
int((d + e*x)^3*(a + b*x^2 + c*x^4)^(3/2), x)
\[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int \left (e x +d \right )^{3} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}d x \] Input:
int((e*x+d)^3*(c*x^4+b*x^2+a)^(3/2),x)
Output:
int((e*x+d)^3*(c*x^4+b*x^2+a)^(3/2),x)