Integrand size = 35, antiderivative size = 741 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx=\frac {27 a c \sqrt {a+b x^3}}{20 x^4}+\frac {a d \sqrt {a+b x^3}}{x^3}+\frac {27 a e \sqrt {a+b x^3}}{10 x^2}-\frac {27 (7 b c+8 a f) \sqrt {a+b x^3}}{56 x}+\frac {27 \sqrt [3]{b} (7 b c+8 a f) \sqrt {a+b x^3}}{56 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 a \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{105 x^5}+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}-\frac {1}{3} \sqrt {a} (3 b d+2 a g) \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \sqrt [3]{b} (7 b c+8 a f) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{112 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \sqrt [3]{b} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (7 b c+8 a f)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{280 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
2/315*(b*x^3+a)^(3/2)*(35*g*x^5+45*f*x^4+63*e*x^3+105*d*x^2+315*c*x)/x^5-1 /3*(2*a*g+3*b*d)*arctanh((b*x^3+a)^(1/2)/a^(1/2))*a^(1/2)+27/20*a*c*(b*x^3 +a)^(1/2)/x^4+a*d*(b*x^3+a)^(1/2)/x^3+27/10*a*e*(b*x^3+a)^(1/2)/x^2-27/56* (8*a*f+7*b*c)*(b*x^3+a)^(1/2)/x-2/105*a*(-35*g*x^5-135*f*x^4+189*e*x^3+105 *d*x^2+189*c*x)*(b*x^3+a)^(1/2)/x^5+27/56*b^(1/3)*(8*a*f+7*b*c)*(b*x^3+a)^ (1/2)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))-27/112*3^(1/4)*a^(1/3)*b^(1/3)*(8*a* f+7*b*c)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-3^(1/2)))/(b^ (1/3)*x+a^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)-1/2*2^(1/2))*((a^ (2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))^2)^(1 /2)/(b*x^3+a)^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+3^( 1/2)))^2)^(1/2)+9/280*3^(3/4)*a^(1/3)*b^(1/3)*(a^(1/3)+b^(1/3)*x)*Elliptic F((b^(1/3)*x+a^(1/3)*(1-3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1+3^(1/2))),I*3^(1/2 )+2*I)*(28*a^(2/3)*b^(1/3)*e-5*(8*a*f+7*b*c)*(1-3^(1/2)))*(1/2*6^(1/2)+1/2 *2^(1/2))*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+3 ^(1/2)))^2)^(1/2)/(b*x^3+a)^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+ a^(1/3)*(1+3^(1/2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.60 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.33 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx=\frac {-45 a^3 c \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {4}{3},-\frac {1}{3},-\frac {b x^3}{a}\right )-90 a^3 e x^2 \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {2}{3},\frac {1}{3},-\frac {b x^3}{a}\right )+4 x^3 \left (-45 a^3 f \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{3},\frac {2}{3},-\frac {b x^3}{a}\right )+2 x \sqrt {1+\frac {b x^3}{a}} \left (5 a^2 g \left (\sqrt {a+b x^3} \left (4 a+b x^3\right )-3 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\right )+3 b d \left (a+b x^3\right )^{5/2} \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},1+\frac {b x^3}{a}\right )\right )\right )}{180 a^2 x^4 \sqrt {1+\frac {b x^3}{a}}} \]
(-45*a^3*c*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, -4/3, -1/3, -((b*x^3)/a )] - 90*a^3*e*x^2*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, -2/3, 1/3, -((b* x^3)/a)] + 4*x^3*(-45*a^3*f*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, -1/3, 2/3, -((b*x^3)/a)] + 2*x*Sqrt[1 + (b*x^3)/a]*(5*a^2*g*(Sqrt[a + b*x^3]*(4* a + b*x^3) - 3*a^(3/2)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]) + 3*b*d*(a + b*x^ 3)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^3)/a])))/(180*a^2*x^4*Sqr t[1 + (b*x^3)/a])
Time = 1.70 (sec) , antiderivative size = 772, normalized size of antiderivative = 1.04, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.543, Rules used = {2365, 27, 2365, 27, 2374, 25, 2374, 27, 2374, 27, 2374, 25, 2371, 798, 73, 221, 2417, 759, 2416}
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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx\) |
| \(\Big \downarrow \) 2365 |
| \(\displaystyle \frac {9}{2} a \int \frac {2 \sqrt {b x^3+a} \left (35 g x^4+45 f x^3+63 e x^2+105 d x+315 c\right )}{315 x^5}dx+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} a \int \frac {\sqrt {b x^3+a} \left (35 g x^4+45 f x^3+63 e x^2+105 d x+315 c\right )}{x^5}dx+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 2365 |
| \(\displaystyle \frac {1}{35} a \left (\frac {3}{2} a \int -\frac {2 \left (-35 g x^4-135 f x^3+189 e x^2+105 d x+189 c\right )}{3 x^5 \sqrt {b x^3+a}}dx-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} a \left (-a \int \frac {-35 g x^4-135 f x^3+189 e x^2+105 d x+189 c}{x^5 \sqrt {b x^3+a}}dx-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {1}{35} a \left (-a \left (-\frac {\int -\frac {-280 a g x^3-135 (7 b c+8 a f) x^2+1512 a e x+840 a d}{x^4 \sqrt {b x^3+a}}dx}{8 a}-\frac {189 c \sqrt {a+b x^3}}{4 a x^4}\right )-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{35} a \left (-a \left (\frac {\int \frac {-280 a g x^3-135 (7 b c+8 a f) x^2+1512 a e x+840 a d}{x^4 \sqrt {b x^3+a}}dx}{8 a}-\frac {189 c \sqrt {a+b x^3}}{4 a x^4}\right )-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {1}{35} a \left (-a \left (\frac {-\frac {\int -\frac {6 \left (1512 e a^2-140 (3 b d+2 a g) x^2 a-135 (7 b c+8 a f) x a\right )}{x^3 \sqrt {b x^3+a}}dx}{6 a}-\frac {280 d \sqrt {a+b x^3}}{x^3}}{8 a}-\frac {189 c \sqrt {a+b x^3}}{4 a x^4}\right )-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} a \left (-a \left (\frac {\frac {\int \frac {1512 e a^2-140 (3 b d+2 a g) x^2 a-135 (7 b c+8 a f) x a}{x^3 \sqrt {b x^3+a}}dx}{a}-\frac {280 d \sqrt {a+b x^3}}{x^3}}{8 a}-\frac {189 c \sqrt {a+b x^3}}{4 a x^4}\right )-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {1}{35} a \left (-a \left (\frac {\frac {-\frac {\int \frac {4 \left (378 b e x^2 a^2+135 (7 b c+8 a f) a^2+140 (3 b d+2 a g) x a^2\right )}{x^2 \sqrt {b x^3+a}}dx}{4 a}-\frac {756 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {280 d \sqrt {a+b x^3}}{x^3}}{8 a}-\frac {189 c \sqrt {a+b x^3}}{4 a x^4}\right )-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} a \left (-a \left (\frac {\frac {-\frac {\int \frac {378 b e x^2 a^2+135 (7 b c+8 a f) a^2+140 (3 b d+2 a g) x a^2}{x^2 \sqrt {b x^3+a}}dx}{a}-\frac {756 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {280 d \sqrt {a+b x^3}}{x^3}}{8 a}-\frac {189 c \sqrt {a+b x^3}}{4 a x^4}\right )-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {1}{35} a \left (-a \left (\frac {\frac {-\frac {-\frac {\int -\frac {280 (3 b d+2 a g) a^3+756 b e x a^3+135 b (7 b c+8 a f) x^2 a^2}{x \sqrt {b x^3+a}}dx}{2 a}-\frac {135 a \sqrt {a+b x^3} (8 a f+7 b c)}{x}}{a}-\frac {756 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {280 d \sqrt {a+b x^3}}{x^3}}{8 a}-\frac {189 c \sqrt {a+b x^3}}{4 a x^4}\right )-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{35} a \left (-a \left (\frac {\frac {-\frac {\frac {\int \frac {280 (3 b d+2 a g) a^3+756 b e x a^3+135 b (7 b c+8 a f) x^2 a^2}{x \sqrt {b x^3+a}}dx}{2 a}-\frac {135 a \sqrt {a+b x^3} (8 a f+7 b c)}{x}}{a}-\frac {756 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {280 d \sqrt {a+b x^3}}{x^3}}{8 a}-\frac {189 c \sqrt {a+b x^3}}{4 a x^4}\right )-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 2371 |
| \(\displaystyle \frac {1}{35} a \left (-a \left (\frac {\frac {-\frac {\frac {280 a^3 (2 a g+3 b d) \int \frac {1}{x \sqrt {b x^3+a}}dx+\int \frac {756 b e a^3+135 b (7 b c+8 a f) x a^2}{\sqrt {b x^3+a}}dx}{2 a}-\frac {135 a \sqrt {a+b x^3} (8 a f+7 b c)}{x}}{a}-\frac {756 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {280 d \sqrt {a+b x^3}}{x^3}}{8 a}-\frac {189 c \sqrt {a+b x^3}}{4 a x^4}\right )-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {1}{35} a \left (-a \left (\frac {\frac {-\frac {\frac {\frac {280}{3} a^3 (2 a g+3 b d) \int \frac {1}{x^3 \sqrt {b x^3+a}}dx^3+\int \frac {756 b e a^3+135 b (7 b c+8 a f) x a^2}{\sqrt {b x^3+a}}dx}{2 a}-\frac {135 a \sqrt {a+b x^3} (8 a f+7 b c)}{x}}{a}-\frac {756 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {280 d \sqrt {a+b x^3}}{x^3}}{8 a}-\frac {189 c \sqrt {a+b x^3}}{4 a x^4}\right )-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{35} a \left (-a \left (\frac {\frac {-\frac {\frac {\frac {560 a^3 (2 a g+3 b d) \int \frac {1}{\frac {x^6}{b}-\frac {a}{b}}d\sqrt {b x^3+a}}{3 b}+\int \frac {756 b e a^3+135 b (7 b c+8 a f) x a^2}{\sqrt {b x^3+a}}dx}{2 a}-\frac {135 a \sqrt {a+b x^3} (8 a f+7 b c)}{x}}{a}-\frac {756 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {280 d \sqrt {a+b x^3}}{x^3}}{8 a}-\frac {189 c \sqrt {a+b x^3}}{4 a x^4}\right )-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{35} a \left (-a \left (\frac {\frac {-\frac {\frac {\int \frac {756 b e a^3+135 b (7 b c+8 a f) x a^2}{\sqrt {b x^3+a}}dx-\frac {560}{3} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (2 a g+3 b d)}{2 a}-\frac {135 a \sqrt {a+b x^3} (8 a f+7 b c)}{x}}{a}-\frac {756 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {280 d \sqrt {a+b x^3}}{x^3}}{8 a}-\frac {189 c \sqrt {a+b x^3}}{4 a x^4}\right )-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {1}{35} a \left (-a \left (\frac {\frac {-\frac {\frac {27 a^{7/3} b^{2/3} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (8 a f+7 b c)\right ) \int \frac {1}{\sqrt {b x^3+a}}dx+135 a^2 b^{2/3} (8 a f+7 b c) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx-\frac {560}{3} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (2 a g+3 b d)}{2 a}-\frac {135 a \sqrt {a+b x^3} (8 a f+7 b c)}{x}}{a}-\frac {756 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {280 d \sqrt {a+b x^3}}{x^3}}{8 a}-\frac {189 c \sqrt {a+b x^3}}{4 a x^4}\right )-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {1}{35} a \left (-a \left (\frac {\frac {-\frac {\frac {135 a^2 b^{2/3} (8 a f+7 b c) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{7/3} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (8 a f+7 b c)\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {560}{3} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (2 a g+3 b d)}{2 a}-\frac {135 a \sqrt {a+b x^3} (8 a f+7 b c)}{x}}{a}-\frac {756 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {280 d \sqrt {a+b x^3}}{x^3}}{8 a}-\frac {189 c \sqrt {a+b x^3}}{4 a x^4}\right )-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {1}{35} a \left (-a \left (\frac {\frac {-\frac {\frac {\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{7/3} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (8 a f+7 b c)\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {560}{3} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (2 a g+3 b d)+135 a^2 b^{2/3} (8 a f+7 b c) \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{2 a}-\frac {135 a \sqrt {a+b x^3} (8 a f+7 b c)}{x}}{a}-\frac {756 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {280 d \sqrt {a+b x^3}}{x^3}}{8 a}-\frac {189 c \sqrt {a+b x^3}}{4 a x^4}\right )-\frac {2 \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{3 x^5}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}\) |
(2*(a + b*x^3)^(3/2)*(315*c*x + 105*d*x^2 + 63*e*x^3 + 45*f*x^4 + 35*g*x^5 ))/(315*x^5) + (a*((-2*Sqrt[a + b*x^3]*(189*c*x + 105*d*x^2 + 189*e*x^3 - 135*f*x^4 - 35*g*x^5))/(3*x^5) - a*((-189*c*Sqrt[a + b*x^3])/(4*a*x^4) + ( (-280*d*Sqrt[a + b*x^3])/x^3 + ((-756*a*e*Sqrt[a + b*x^3])/x^2 - ((-135*a* (7*b*c + 8*a*f)*Sqrt[a + b*x^3])/x + ((-560*a^(5/2)*(3*b*d + 2*a*g)*ArcTan h[Sqrt[a + b*x^3]/Sqrt[a]])/3 + 135*a^2*b^(2/3)*(7*b*c + 8*a*f)*((2*Sqrt[a + b*x^3])/(b^(1/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)* x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticE[ArcSin[( (1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], - 7 - 4*Sqrt[3]])/(b^(1/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3 ])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])) + (18*3^(3/4)*Sqrt[2 + Sqrt[3 ]]*a^(7/3)*b^(1/3)*(28*a^(2/3)*b^(1/3)*e - 5*(1 - Sqrt[3])*(7*b*c + 8*a*f) )*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/( (1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1 /3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(S qrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2] *Sqrt[a + b*x^3]))/(2*a))/a)/a)/(8*a))))/35
3.5.66.3.1 Defintions of rubi rules used
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* ((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & & PosQ[a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> M odule[{q = Expon[Pq, x], i}, Simp[(c*x)^m*(a + b*x^n)^p*Sum[Coeff[Pq, x, i] *(x^(i + 1)/(m + n*p + i + 1)), {i, 0, q}], x] + Simp[a*n*p Int[(c*x)^m*( a + b*x^n)^(p - 1)*Sum[Coeff[Pq, x, i]*(x^i/(m + n*p + i + 1)), {i, 0, q}], x], x]] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[(n - 1)/2, 0] && GtQ[p, 0]
Int[(Pq_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Simp[Coeff[Pq, x, 0] Int[1/(x*Sqrt[a + b*x^n]), x], x] + Int[ExpandToSum[(Pq - Coeff[Pq, x, 0])/x, x]/Sqrt[a + b*x^n], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IG tQ[n, 0] && NeQ[Coeff[Pq, x, 0], 0]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wit h[{Pq0 = Coeff[Pq, x, 0]}, Simp[Pq0*(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c *(m + 1))), x] + Simp[1/(2*a*c*(m + 1)) Int[(c*x)^(m + 1)*ExpandToSum[2*a *(m + 1)*((Pq - Pq0)/x) - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b* x^n)^p, x], x] /; NeQ[Pq0, 0]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt [3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) *s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(c*r - (1 - Sqrt[3])*d*s)/r Int[1/Sqrt[a + b*x^3], x], x] + Simp[d/r Int[((1 - Sqrt[3])*s + r*x)/Sq rt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Time = 1.99 (sec) , antiderivative size = 900, normalized size of antiderivative = 1.21
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(900\) |
| default | \(\text {Expression too large to display}\) | \(1342\) |
| risch | \(\text {Expression too large to display}\) | \(2048\) |
-1/4*a*c*(b*x^3+a)^(1/2)/x^4-1/3*a*d*(b*x^3+a)^(1/2)/x^3-1/2*a*e*(b*x^3+a) ^(1/2)/x^2-(a*f+11/8*b*c)*(b*x^3+a)^(1/2)/x+2/9*g*b*x^3*(b*x^3+a)^(1/2)+2/ 7*b*f*x^2*(b*x^3+a)^(1/2)+2/5*b*e*x*(b*x^3+a)^(1/2)+2/3*(4/3*a*b*g+b^2*d)/ b*(b*x^3+a)^(1/2)-9/10*I*a*e*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^( 1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1 /b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))) ^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2) *b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b *(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^ (1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*( -a*b^2)^(1/3)))^(1/2))-2/3*I*(27/14*a*f*b+27/16*b^2*c)*3^(1/2)/b*(-a*b^2)^ (1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b /(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2* I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/ 2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*((-3/ 2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*EllipticE(1/3*3^(1/2)*( I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^ 2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3 ^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+1/b*(-a*b^2)^(1/3)*EllipticF(1/3*3^(1/2)* (I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.82 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.52 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx=\left [\frac {6804 \, a \sqrt {b} e x^{4} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 210 \, {\left (3 \, b d + 2 \, a g\right )} \sqrt {a} x^{4} \log \left (-\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 1215 \, {\left (7 \, b c + 8 \, a f\right )} \sqrt {b} x^{4} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (560 \, b g x^{7} + 720 \, b f x^{6} + 1008 \, b e x^{5} + 560 \, {\left (3 \, b d + 4 \, a g\right )} x^{4} - 1260 \, a e x^{2} - 315 \, {\left (11 \, b c + 8 \, a f\right )} x^{3} - 840 \, a d x - 630 \, a c\right )} \sqrt {b x^{3} + a}}{2520 \, x^{4}}, \frac {6804 \, a \sqrt {b} e x^{4} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 420 \, {\left (3 \, b d + 2 \, a g\right )} \sqrt {-a} x^{4} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-a}}{b x^{3} + 2 \, a}\right ) - 1215 \, {\left (7 \, b c + 8 \, a f\right )} \sqrt {b} x^{4} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (560 \, b g x^{7} + 720 \, b f x^{6} + 1008 \, b e x^{5} + 560 \, {\left (3 \, b d + 4 \, a g\right )} x^{4} - 1260 \, a e x^{2} - 315 \, {\left (11 \, b c + 8 \, a f\right )} x^{3} - 840 \, a d x - 630 \, a c\right )} \sqrt {b x^{3} + a}}{2520 \, x^{4}}\right ] \]
[1/2520*(6804*a*sqrt(b)*e*x^4*weierstrassPInverse(0, -4*a/b, x) + 210*(3*b *d + 2*a*g)*sqrt(a)*x^4*log(-(b^2*x^6 + 8*a*b*x^3 - 4*(b*x^3 + 2*a)*sqrt(b *x^3 + a)*sqrt(a) + 8*a^2)/x^6) - 1215*(7*b*c + 8*a*f)*sqrt(b)*x^4*weierst rassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (560*b*g*x^7 + 72 0*b*f*x^6 + 1008*b*e*x^5 + 560*(3*b*d + 4*a*g)*x^4 - 1260*a*e*x^2 - 315*(1 1*b*c + 8*a*f)*x^3 - 840*a*d*x - 630*a*c)*sqrt(b*x^3 + a))/x^4, 1/2520*(68 04*a*sqrt(b)*e*x^4*weierstrassPInverse(0, -4*a/b, x) + 420*(3*b*d + 2*a*g) *sqrt(-a)*x^4*arctan(2*sqrt(b*x^3 + a)*sqrt(-a)/(b*x^3 + 2*a)) - 1215*(7*b *c + 8*a*f)*sqrt(b)*x^4*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (560*b*g*x^7 + 720*b*f*x^6 + 1008*b*e*x^5 + 560*(3*b*d + 4*a *g)*x^4 - 1260*a*e*x^2 - 315*(11*b*c + 8*a*f)*x^3 - 840*a*d*x - 630*a*c)*s qrt(b*x^3 + a))/x^4]
Time = 6.22 (sec) , antiderivative size = 495, normalized size of antiderivative = 0.67 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx=\frac {a^{\frac {3}{2}} c \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, - \frac {1}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{3}\right )} + \frac {a^{\frac {3}{2}} e \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} + \frac {a^{\frac {3}{2}} f \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} - \frac {2 a^{\frac {3}{2}} g \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3} + \frac {\sqrt {a} b c \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} - \sqrt {a} b d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )} + \frac {\sqrt {a} b e x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {\sqrt {a} b f x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {2 a^{2} g}{3 \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} d \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} + \frac {2 a \sqrt {b} d}{3 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 a \sqrt {b} g x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 b^{\frac {3}{2}} d x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} + b g \left (\begin {cases} \frac {\sqrt {a} x^{3}}{3} & \text {for}\: b = 0 \\\frac {2 \left (a + b x^{3}\right )^{\frac {3}{2}}}{9 b} & \text {otherwise} \end {cases}\right ) \]
a**(3/2)*c*gamma(-4/3)*hyper((-4/3, -1/2), (-1/3,), b*x**3*exp_polar(I*pi) /a)/(3*x**4*gamma(-1/3)) + a**(3/2)*e*gamma(-2/3)*hyper((-2/3, -1/2), (1/3 ,), b*x**3*exp_polar(I*pi)/a)/(3*x**2*gamma(1/3)) + a**(3/2)*f*gamma(-1/3) *hyper((-1/2, -1/3), (2/3,), b*x**3*exp_polar(I*pi)/a)/(3*x*gamma(2/3)) - 2*a**(3/2)*g*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/3 + sqrt(a)*b*c*gamma(-1/3) *hyper((-1/2, -1/3), (2/3,), b*x**3*exp_polar(I*pi)/a)/(3*x*gamma(2/3)) - sqrt(a)*b*d*asinh(sqrt(a)/(sqrt(b)*x**(3/2))) + sqrt(a)*b*e*x*gamma(1/3)*h yper((-1/2, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/3)) + sqrt( a)*b*f*x**2*gamma(2/3)*hyper((-1/2, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a )/(3*gamma(5/3)) + 2*a**2*g/(3*sqrt(b)*x**(3/2)*sqrt(a/(b*x**3) + 1)) - a* sqrt(b)*d*sqrt(a/(b*x**3) + 1)/(3*x**(3/2)) + 2*a*sqrt(b)*d/(3*x**(3/2)*sq rt(a/(b*x**3) + 1)) + 2*a*sqrt(b)*g*x**(3/2)/(3*sqrt(a/(b*x**3) + 1)) + 2* b**(3/2)*d*x**(3/2)/(3*sqrt(a/(b*x**3) + 1)) + b*g*Piecewise((sqrt(a)*x**3 /3, Eq(b, 0)), (2*(a + b*x**3)**(3/2)/(9*b), True))
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{5}} \,d x } \]
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^5} \,d x \]