Integrand size = 25, antiderivative size = 541 \[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^4} \, dx=-\frac {d \sqrt {c+d x} \sqrt {a-b x^2}}{4 x^2}+\frac {\left (\frac {8 b c}{a}-\frac {3 d^2}{c}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{24 x}-\frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{3 x^3}-\frac {\sqrt {b} \left (8 b c^2-3 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{24 \sqrt {a} c \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {\sqrt {b} \left (8 b c^2+31 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{24 \sqrt {a} \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {d \left (12 b c^2+a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{8 c \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-1/4*d*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/x^2+1/24*(8*b*c/a-3*d^2/c)*(d*x+c)^( 1/2)*(-b*x^2+a)^(1/2)/x-1/3*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/x^3-1/24*b^(1/2 )*(-3*a*d^2+8*b*c^2)*(d*x+c)^(1/2)*(1-b*x^2/a)^(1/2)*EllipticE(1/2*(1-b^(1 /2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/ 2))/a^(1/2)/c/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/ 2)+1/24*b^(1/2)*(31*a*d^2+8*b*c^2)*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d)) ^(1/2)*(1-b*x^2/a)^(1/2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2) ,2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(1/2)/(d*x+c)^(1/2)/(- b*x^2+a)^(1/2)+1/8*d*(a*d^2+12*b*c^2)*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)* d))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticPi(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)* 2^(1/2),2,2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/c/(d*x+c)^(1/2) /(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.80 (sec) , antiderivative size = 1062, normalized size of antiderivative = 1.96 \[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^4} \, dx =\text {Too large to display} \] Input:
Integrate[((c + d*x)^(3/2)*Sqrt[a - b*x^2])/x^4,x]
Output:
(Sqrt[a - b*x^2]*(((c + d*x)*(8*b*c^2*x^2 - a*(8*c^2 + 14*c*d*x + 3*d^2*x^ 2)))/(a*c*x^3) + (-8*b^2*c^5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 11*a*b*c^3*d ^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 3*a^2*c*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt [b]] + 16*b^2*c^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) - 6*a*b*c^2*d^2 *Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) - 8*b^2*c^3*Sqrt[-c + (Sqrt[a]*d )/Sqrt[b]]*(c + d*x)^2 + 3*a*b*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d *x)^2 + I*Sqrt[b]*c*(8*b^(3/2)*c^3 - 8*Sqrt[a]*b*c^2*d - 3*a*Sqrt[b]*c*d^2 + 3*a^(3/2)*d^3)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[ a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[ -c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b] *c - Sqrt[a]*d)] + I*Sqrt[a]*d*(8*b^(3/2)*c^3 - 2*Sqrt[a]*b*c^2*d - 3*a*Sq rt[b]*c*d^2 - 3*a^(3/2)*d^3)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqr t[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*Ar cSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]* d)/(Sqrt[b]*c - Sqrt[a]*d)] + (36*I)*a*b*c^2*d^2*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^ (3/2)*EllipticPi[(Sqrt[b]*c)/(Sqrt[b]*c - Sqrt[a]*d), I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + (3*I)*a^2*d^4*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt [-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticPi[(...
Time = 1.93 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.11, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {628, 2352, 2352, 25, 2351, 600, 509, 508, 327, 512, 511, 321, 633, 632, 186, 413, 412}
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.
| \(\displaystyle \int \frac {\sqrt {a-b x^2} (c+d x)^{3/2}}{x^4} \, dx\) |
| \(\Big \downarrow \) 628 |
| \(\displaystyle \frac {1}{6} \int \frac {-6 b d^2 x^3-9 b c d x^2-2 \left (b c^2-3 a d^2\right ) x+7 a c d}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 2352 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\int \frac {22 a b d x c^2+17 a b d^2 x^2 c+a \left (8 b c^2-3 a d^2\right ) c}{x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 2352 |
| \(\displaystyle \frac {1}{6} \left (-\frac {-\frac {\int -\frac {3 c d \left (12 b c^2+a d^2\right ) a^2+34 b c^2 d^2 x a^2-b c d \left (8 b c^2-3 a d^2\right ) x^2 a}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right )}{x}}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {\int \frac {3 c d \left (12 b c^2+a d^2\right ) a^2+34 b c^2 d^2 x a^2-b c d \left (8 b c^2-3 a d^2\right ) x^2 a}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right )}{x}}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {3 a^2 c d \left (a d^2+12 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\int \frac {34 a^2 b c^2 d^2-a b c d \left (8 b c^2-3 a d^2\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right )}{x}}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {3 a^2 c d \left (a d^2+12 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a b c^2 \left (31 a d^2+8 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-a b c \left (8 b c^2-3 a d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right )}{x}}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {3 a^2 c d \left (a d^2+12 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a b c^2 \left (31 a d^2+8 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {a b c \sqrt {1-\frac {b x^2}{a}} \left (8 b c^2-3 a d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right )}{x}}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {\frac {2 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+3 a^2 c d \left (a d^2+12 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a b c^2 \left (31 a d^2+8 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right )}{x}}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {3 a^2 c d \left (a d^2+12 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a b c^2 \left (31 a d^2+8 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {2 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right )}{x}}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {3 a^2 c d \left (a d^2+12 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {a b c^2 \sqrt {1-\frac {b x^2}{a}} \left (31 a d^2+8 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}+\frac {2 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right )}{x}}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {-\frac {2 a^{3/2} \sqrt {b} c^2 \sqrt {1-\frac {b x^2}{a}} \left (31 a d^2+8 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {c+d x}}+3 a^2 c d \left (a d^2+12 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {2 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right )}{x}}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {3 a^2 c d \left (a d^2+12 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 a^{3/2} \sqrt {b} c^2 \sqrt {1-\frac {b x^2}{a}} \left (31 a d^2+8 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right )}{x}}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 633 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {\frac {3 a^2 c d \sqrt {1-\frac {b x^2}{a}} \left (a d^2+12 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\frac {2 a^{3/2} \sqrt {b} c^2 \sqrt {1-\frac {b x^2}{a}} \left (31 a d^2+8 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right )}{x}}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {\frac {3 a^2 c d \sqrt {1-\frac {b x^2}{a}} \left (a d^2+12 b c^2\right ) \int \frac {1}{x \sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}} \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+d x}}dx}{\sqrt {a-b x^2}}-\frac {2 a^{3/2} \sqrt {b} c^2 \sqrt {1-\frac {b x^2}{a}} \left (31 a d^2+8 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right )}{x}}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {-\frac {6 a^2 c d \sqrt {1-\frac {b x^2}{a}} \left (a d^2+12 b c^2\right ) \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2}}-\frac {2 a^{3/2} \sqrt {b} c^2 \sqrt {1-\frac {b x^2}{a}} \left (31 a d^2+8 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right )}{x}}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {-\frac {6 a^2 c d \sqrt {1-\frac {b x^2}{a}} \left (a d^2+12 b c^2\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 a^{3/2} \sqrt {b} c^2 \sqrt {1-\frac {b x^2}{a}} \left (31 a d^2+8 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right )}{x}}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {-\frac {2 a^{3/2} \sqrt {b} c^2 \sqrt {1-\frac {b x^2}{a}} \left (31 a d^2+8 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {6 a^2 c d \sqrt {1-\frac {b x^2}{a}} \left (a d^2+12 b c^2\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (8 b c^2-3 a d^2\right )}{x}}{4 a c}-\frac {7 d \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{3 x^3}\) |
Input:
Int[((c + d*x)^(3/2)*Sqrt[a - b*x^2])/x^4,x]
Output:
-1/3*(c*Sqrt[c + d*x]*Sqrt[a - b*x^2])/x^3 + ((-7*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(2*x^2) - (-(((8*b*c^2 - 3*a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/ x) + ((2*a^(3/2)*Sqrt[b]*c*(8*b*c^2 - 3*a*d^2)*Sqrt[c + d*x]*Sqrt[1 - (b*x ^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sq rt[b]*c)/Sqrt[a] + d)])/(Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)] *Sqrt[a - b*x^2]) - (2*a^(3/2)*Sqrt[b]*c^2*(8*b*c^2 + 31*a*d^2)*Sqrt[(Sqrt [b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcS in[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d) ])/(Sqrt[c + d*x]*Sqrt[a - b*x^2]) - (6*a^2*c*d*(12*b*c^2 + a*d^2)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (Sqrt[a]*d*(1 - (Sqrt[b]*x)/Sqrt[a]))/(Sqrt[b]*c + S qrt[a]*d)]*EllipticPi[2, ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2 *Sqrt[a]*d)/(Sqrt[b]*c + Sqrt[a]*d)])/(Sqrt[a - b*x^2]*Sqrt[c + (Sqrt[a]*d )/Sqrt[b] - (Sqrt[a]*d*(1 - (Sqrt[b]*x)/Sqrt[a]))/Sqrt[b]]))/(2*a*c))/(4*a *c))/6
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*Sqrt[(a_) + (b_.)*(x_)^2], x _Symbol] :> Simp[c^(n - 1/2)*(e*x)^(m + 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2]/( e*(m + 1))), x] - Simp[1/(2*e*(m + 1)) Int[((e*x)^(m + 1)/(Sqrt[c + d*x]* Sqrt[a + b*x^2]))*ExpandToSum[(2*a*c^(n + 1/2)*(m + 1) + a*c^(n - 1/2)*d*(2 *m + 3)*x + 2*b*c^(n + 1/2)*(m + 2)*x^2 + b*c^(n - 1/2)*d*(2*m + 5)*x^3 - 2 *a*(m + 1)*(c + d*x)^(n + 1/2) - 2*b*(m + 1)*x^2*(c + d*x)^(n + 1/2))/x, x] , x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n + 3/2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > Simp[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 + b*(x^2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, x, x] Int[(c + d*x)^n*((a + b*x^2)^p/x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
Int[((Px_)*((e_.)*(x_))^(m_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x _)^2]), x_Symbol] :> With[{Px0 = Coefficient[Px, x, 0]}, Simp[Px0*(e*x)^(m + 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2]/(a*c*e*(m + 1))), x] + Simp[1/(2*a*c*e* (m + 1)) Int[((e*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[ 2*a*c*(m + 1)*((Px - Px0)/x) - Px0*(a*d*(2*m + 3) + 2*b*c*(m + 2)*x + b*d*( 2*m + 5)*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(912\) vs. \(2(440)=880\).
Time = 3.01 (sec) , antiderivative size = 913, normalized size of antiderivative = 1.69
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {c \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 x^{3}}-\frac {7 d \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{12 x^{2}}-\frac {\left (3 a \,d^{2}-8 b \,c^{2}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{24 a c x}-\frac {17 b \,d^{2} \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{12 \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {b \left (3 a \,d^{2}-8 b \,c^{2}\right ) d \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{24 a c \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {\left (a \,d^{2}+12 b \,c^{2}\right ) d^{2} \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, -\frac {\left (-\frac {c}{d}+\frac {\sqrt {a b}}{b}\right ) d}{c}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{8 c^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(913\) |
| risch | \(-\frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, \left (3 a \,d^{2} x^{2}-8 b \,c^{2} x^{2}+14 a d x c +8 a \,c^{2}\right )}{24 x^{3} a c}-\frac {d \left (-\frac {8 b \,c^{2} \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {3 a \left (a \,d^{2}+12 b \,c^{2}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, 2, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {3 a \,d^{2} \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {34 a c d \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right ) \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}}{48 c a \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(924\) |
| default | \(\text {Expression too large to display}\) | \(1885\) |
Input:
int((d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/x^4,x,method=_RETURNVERBOSE)
Output:
((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(-1/3*c/x^3*(-b* d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-7/12*d/x^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/ 2)-1/24*(3*a*d^2-8*b*c^2)/a/c*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/x-17/12*b *d^2*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*( a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*( a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/( c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)) )^(1/2))-1/24*b*(3*a*d^2-8*b*c^2)*d/a/c*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/ d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/ 2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a* d*x+a*c)^(1/2)*((-c/d-1/b*(a*b)^(1/2))*EllipticE(((x+c/d)/(c/d-1/b*(a*b)^( 1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+1/b*(a *b)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b) ^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))+1/8*(a*d^2+12*b*c^2)*d^2/c^2*(c/d- 1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2) )/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2) ))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticPi(((x+c/d)/(c/d-1/b*( a*b)^(1/2)))^(1/2),-(-c/d+1/b*(a*b)^(1/2))/c*d,((-c/d+1/b*(a*b)^(1/2))/(-c /d-1/b*(a*b)^(1/2)))^(1/2)))
Timed out. \[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^4} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/x^4,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^4} \, dx=\int \frac {\sqrt {a - b x^{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{4}}\, dx \] Input:
integrate((d*x+c)**(3/2)*(-b*x**2+a)**(1/2)/x**4,x)
Output:
Integral(sqrt(a - b*x**2)*(c + d*x)**(3/2)/x**4, x)
\[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^4} \, dx=\int { \frac {\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:
integrate((d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/x^4,x, algorithm="maxima")
Output:
integrate(sqrt(-b*x^2 + a)*(d*x + c)^(3/2)/x^4, x)
\[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^4} \, dx=\int { \frac {\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:
integrate((d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/x^4,x, algorithm="giac")
Output:
integrate(sqrt(-b*x^2 + a)*(d*x + c)^(3/2)/x^4, x)
Timed out. \[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^4} \, dx=\int \frac {\sqrt {a-b\,x^2}\,{\left (c+d\,x\right )}^{3/2}}{x^4} \,d x \] Input:
int(((a - b*x^2)^(1/2)*(c + d*x)^(3/2))/x^4,x)
Output:
int(((a - b*x^2)^(1/2)*(c + d*x)^(3/2))/x^4, x)
\[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^4} \, dx=\int \frac {\left (d x +c \right )^{\frac {3}{2}} \sqrt {-b \,x^{2}+a}}{x^{4}}d x \] Input:
int((d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/x^4,x)
Output:
int((d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/x^4,x)