Integrand size = 35, antiderivative size = 714 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx=-\frac {b \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right ) \sqrt {a+b x^3}}{1680}-\frac {b^2 c \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 d \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 e \sqrt {a+b x^3}}{112 a x}+\frac {27 b^{7/3} e \sqrt {a+b x^3}}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac {b^2 (b c-6 a f) \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{24 a^{3/2}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{224 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{5/3} \left (7 b d+20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-112 a g\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{2240 a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
-1/2520*(280*c/x^9+315*d/x^8+360*e/x^7+420*f/x^6+504*g/x^5)*(b*x^3+a)^(3/2 )+1/24*b^2*(-6*a*f+b*c)*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(3/2)-1/1680*b* (140*c/x^6+189*d/x^5+270*e/x^4+420*f/x^3+756*g/x^2)*(b*x^3+a)^(1/2)-1/24*b ^2*c*(b*x^3+a)^(1/2)/a/x^3-27/320*b^2*d*(b*x^3+a)^(1/2)/a/x^2-27/112*b^2*e *(b*x^3+a)^(1/2)/a/x+27/112*b^(7/3)*e*(b*x^3+a)^(1/2)/a/(b^(1/3)*x+a^(1/3) *(1+3^(1/2)))-27/224*3^(1/4)*b^(7/3)*e*(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)/a^(2/3)/(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)-9/2240*3^(3/4)*b^(5/ 3)*(a^(1/3)+b^(1/3)*x)*EllipticF((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)*(7*b*d-112*a*g+20*a^(1/3)*b^(2/3)*e* (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)/a/(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 = 10.70 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.32 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx=-\frac {\sqrt {a+b x^3} \left (105 a^5 d \operatorname {Hypergeometric2F1}\left (-\frac {8}{3},-\frac {3}{2},-\frac {5}{3},-\frac {b x^3}{a}\right )+2 x \left (60 a^5 e \operatorname {Hypergeometric2F1}\left (-\frac {7}{3},-\frac {3}{2},-\frac {4}{3},-\frac {b x^3}{a}\right )+7 x \left (5 a^3 f \left (a \left (2 a+5 b x^3\right ) \sqrt {1+\frac {b x^3}{a}}+3 b^2 x^6 \text {arctanh}\left (\sqrt {1+\frac {b x^3}{a}}\right )\right )+12 a^5 g x \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},-\frac {3}{2},-\frac {2}{3},-\frac {b x^3}{a}\right )-8 b^3 c x^6 \left (a+b x^3\right )^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},4,\frac {7}{2},1+\frac {b x^3}{a}\right )\right )\right )\right )}{840 a^4 x^8 \sqrt {1+\frac {b x^3}{a}}} \]
-1/840*(Sqrt[a + b*x^3]*(105*a^5*d*Hypergeometric2F1[-8/3, -3/2, -5/3, -(( b*x^3)/a)] + 2*x*(60*a^5*e*Hypergeometric2F1[-7/3, -3/2, -4/3, -((b*x^3)/a )] + 7*x*(5*a^3*f*(a*(2*a + 5*b*x^3)*Sqrt[1 + (b*x^3)/a] + 3*b^2*x^6*ArcTa nh[Sqrt[1 + (b*x^3)/a]]) + 12*a^5*g*x*Hypergeometric2F1[-5/3, -3/2, -2/3, -((b*x^3)/a)] - 8*b^3*c*x^6*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*Hypergeometr ic2F1[5/2, 4, 7/2, 1 + (b*x^3)/a]))))/(a^4*x^8*Sqrt[1 + (b*x^3)/a])
Time = 1.58 (sec) , antiderivative size = 726, normalized size of antiderivative = 1.02, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {2364, 27, 2364, 27, 2374, 27, 2374, 25, 2374, 27, 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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -\frac {9}{2} b \int -\frac {\sqrt {b x^3+a} \left (504 g x^4+420 f x^3+360 e x^2+315 d x+280 c\right )}{2520 x^7}dx-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{560} b \int \frac {\sqrt {b x^3+a} \left (504 g x^4+420 f x^3+360 e x^2+315 d x+280 c\right )}{x^7}dx-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle \frac {1}{560} b \left (-\frac {3}{2} b \int -\frac {756 g x^4+420 f x^3+270 e x^2+189 d x+140 c}{3 x^4 \sqrt {b x^3+a}}dx-\frac {1}{3} \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )\right )-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{560} b \left (\frac {1}{2} b \int \frac {756 g x^4+420 f x^3+270 e x^2+189 d x+140 c}{x^4 \sqrt {b x^3+a}}dx-\frac {1}{3} \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )\right )-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {1}{560} b \left (\frac {1}{2} b \left (-\frac {\int -\frac {6 \left (756 a g x^3-70 (b c-6 a f) x^2+270 a e x+189 a d\right )}{x^3 \sqrt {b x^3+a}}dx}{6 a}-\frac {140 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {1}{3} \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )\right )-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{560} b \left (\frac {1}{2} b \left (\frac {\int \frac {756 a g x^3-70 (b c-6 a f) x^2+270 a e x+189 a d}{x^3 \sqrt {b x^3+a}}dx}{a}-\frac {140 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {1}{3} \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )\right )-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {1}{560} b \left (\frac {1}{2} b \left (\frac {-\frac {\int -\frac {1080 e a^2-189 (b d-16 a g) x^2 a-280 (b c-6 a f) x a}{x^2 \sqrt {b x^3+a}}dx}{4 a}-\frac {189 d \sqrt {a+b x^3}}{2 x^2}}{a}-\frac {140 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {1}{3} \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )\right )-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{560} b \left (\frac {1}{2} b \left (\frac {\frac {\int \frac {1080 e a^2-189 (b d-16 a g) x^2 a-280 (b c-6 a f) x a}{x^2 \sqrt {b x^3+a}}dx}{4 a}-\frac {189 d \sqrt {a+b x^3}}{2 x^2}}{a}-\frac {140 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {1}{3} \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )\right )-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {1}{560} b \left (\frac {1}{2} b \left (\frac {\frac {-\frac {\int \frac {2 \left (-540 b e x^2 a^2+280 (b c-6 a f) a^2+189 (b d-16 a g) x a^2\right )}{x \sqrt {b x^3+a}}dx}{2 a}-\frac {1080 a e \sqrt {a+b x^3}}{x}}{4 a}-\frac {189 d \sqrt {a+b x^3}}{2 x^2}}{a}-\frac {140 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {1}{3} \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )\right )-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{560} b \left (\frac {1}{2} b \left (\frac {\frac {-\frac {\int \frac {-540 b e x^2 a^2+280 (b c-6 a f) a^2+189 (b d-16 a g) x a^2}{x \sqrt {b x^3+a}}dx}{a}-\frac {1080 a e \sqrt {a+b x^3}}{x}}{4 a}-\frac {189 d \sqrt {a+b x^3}}{2 x^2}}{a}-\frac {140 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {1}{3} \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )\right )-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 2371 |
| \(\displaystyle \frac {1}{560} b \left (\frac {1}{2} b \left (\frac {\frac {-\frac {280 a^2 (b c-6 a f) \int \frac {1}{x \sqrt {b x^3+a}}dx+\int \frac {189 a^2 (b d-16 a g)-540 a^2 b e x}{\sqrt {b x^3+a}}dx}{a}-\frac {1080 a e \sqrt {a+b x^3}}{x}}{4 a}-\frac {189 d \sqrt {a+b x^3}}{2 x^2}}{a}-\frac {140 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {1}{3} \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )\right )-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {1}{560} b \left (\frac {1}{2} b \left (\frac {\frac {-\frac {\frac {280}{3} a^2 (b c-6 a f) \int \frac {1}{x^3 \sqrt {b x^3+a}}dx^3+\int \frac {189 a^2 (b d-16 a g)-540 a^2 b e x}{\sqrt {b x^3+a}}dx}{a}-\frac {1080 a e \sqrt {a+b x^3}}{x}}{4 a}-\frac {189 d \sqrt {a+b x^3}}{2 x^2}}{a}-\frac {140 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {1}{3} \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )\right )-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{560} b \left (\frac {1}{2} b \left (\frac {\frac {-\frac {\frac {560 a^2 (b c-6 a f) \int \frac {1}{\frac {x^6}{b}-\frac {a}{b}}d\sqrt {b x^3+a}}{3 b}+\int \frac {189 a^2 (b d-16 a g)-540 a^2 b e x}{\sqrt {b x^3+a}}dx}{a}-\frac {1080 a e \sqrt {a+b x^3}}{x}}{4 a}-\frac {189 d \sqrt {a+b x^3}}{2 x^2}}{a}-\frac {140 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {1}{3} \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )\right )-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{560} b \left (\frac {1}{2} b \left (\frac {\frac {-\frac {\int \frac {189 a^2 (b d-16 a g)-540 a^2 b e x}{\sqrt {b x^3+a}}dx-\frac {560}{3} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (b c-6 a f)}{a}-\frac {1080 a e \sqrt {a+b x^3}}{x}}{4 a}-\frac {189 d \sqrt {a+b x^3}}{2 x^2}}{a}-\frac {140 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {1}{3} \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )\right )-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {1}{560} b \left (\frac {1}{2} b \left (\frac {\frac {-\frac {27 a^2 \left (20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-112 a g+7 b d\right ) \int \frac {1}{\sqrt {b x^3+a}}dx-540 a^2 b^{2/3} e \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx-\frac {560}{3} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (b c-6 a f)}{a}-\frac {1080 a e \sqrt {a+b x^3}}{x}}{4 a}-\frac {189 d \sqrt {a+b x^3}}{2 x^2}}{a}-\frac {140 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {1}{3} \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )\right )-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {1}{560} b \left (\frac {1}{2} b \left (\frac {\frac {-\frac {-540 a^2 b^{2/3} e \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx-\frac {560}{3} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (b c-6 a f)+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-112 a g+7 b d\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}}{a}-\frac {1080 a e \sqrt {a+b x^3}}{x}}{4 a}-\frac {189 d \sqrt {a+b x^3}}{2 x^2}}{a}-\frac {140 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {1}{3} \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )\right )-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {1}{560} b \left (\frac {1}{2} b \left (\frac {\frac {-\frac {-\frac {560}{3} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (b c-6 a f)+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \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 (20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-112 a g+7 b d\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}}-540 a^2 b^{2/3} e \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 )}{a}-\frac {1080 a e \sqrt {a+b x^3}}{x}}{4 a}-\frac {189 d \sqrt {a+b x^3}}{2 x^2}}{a}-\frac {140 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {1}{3} \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )\right )-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520}\) |
-1/2520*(((280*c)/x^9 + (315*d)/x^8 + (360*e)/x^7 + (420*f)/x^6 + (504*g)/ x^5)*(a + b*x^3)^(3/2)) + (b*(-1/3*(((140*c)/x^6 + (189*d)/x^5 + (270*e)/x ^4 + (420*f)/x^3 + (756*g)/x^2)*Sqrt[a + b*x^3]) + (b*((-140*c*Sqrt[a + b* x^3])/(3*a*x^3) + ((-189*d*Sqrt[a + b*x^3])/(2*x^2) + ((-1080*a*e*Sqrt[a + b*x^3])/x - ((-560*a^(3/2)*(b*c - 6*a*f)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]] )/3 - 540*a^2*b^(2/3)*e*((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^2*(7*b*d + 20*(1 - Sqrt[3])*a^(1/3) *b^(2/3)*e - 112*a*g)*(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]])/(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]))/a)/(4*a))/a))/2))/560
3.5.71.3.1 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_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{u = IntHide[x^m*Pq, x]}, Simp[u*(a + b*x^n)^p, x] - Simp[b*n*p Int[x^(m + n)*(a + b*x^n)^(p - 1)*ExpandToSum[u/x^(m + 1), x], x], x]] /; FreeQ[{a, b} , x] && PolyQ[Pq, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m + Expon[Pq, x] + 1 , 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 = 1.83 (sec) , antiderivative size = 958, normalized size of antiderivative = 1.34
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(958\) |
| risch | \(\text {Expression too large to display}\) | \(1160\) |
| default | \(\text {Expression too large to display}\) | \(1273\) |
-1/9*a*c*(b*x^3+a)^(1/2)/x^9-1/8*a*d*(b*x^3+a)^(1/2)/x^8-1/7*a*e*(b*x^3+a) ^(1/2)/x^7-1/6*(a*f+7/6*b*c)*(b*x^3+a)^(1/2)/x^6-1/5*(a*g+19/16*b*d)*(b*x^ 3+a)^(1/2)/x^5-17/56*b*e*(b*x^3+a)^(1/2)/x^4-1/24*b*(10*a*f+b*c)/a*(b*x^3+ a)^(1/2)/x^3-1/320*b/a*(208*a*g+27*b*d)*(b*x^3+a)^(1/2)/x^2-27/112*b^2*e*( b*x^3+a)^(1/2)/a/x-2/3*I*(b^2*g-1/640*b^2/a*(208*a*g+27*b*d))*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 )*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))-9/112*I/a*b^2*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)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*Ellipt icE(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)*El...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.26 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx=\left [-\frac {4860 \, a b^{\frac {5}{2}} e x^{9} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + 210 \, {\left (b^{3} c - 6 \, a b^{2} f\right )} \sqrt {a} x^{9} \log \left (\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 1701 \, {\left (a b^{2} d - 16 \, a^{2} b g\right )} \sqrt {b} x^{9} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (4860 \, a b^{2} e x^{8} + 6120 \, a^{2} b e x^{5} + 63 \, {\left (27 \, a b^{2} d + 208 \, a^{2} b g\right )} x^{7} + 840 \, {\left (a b^{2} c + 10 \, a^{2} b f\right )} x^{6} + 2880 \, a^{3} e x^{2} + 2520 \, a^{3} d x + 252 \, {\left (19 \, a^{2} b d + 16 \, a^{3} g\right )} x^{4} + 2240 \, a^{3} c + 560 \, {\left (7 \, a^{2} b c + 6 \, a^{3} f\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{20160 \, a^{2} x^{9}}, -\frac {4860 \, a b^{\frac {5}{2}} e x^{9} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + 420 \, {\left (b^{3} c - 6 \, a b^{2} f\right )} \sqrt {-a} x^{9} \arctan \left (\frac {{\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {-a}}{2 \, {\left (a b x^{3} + a^{2}\right )}}\right ) + 1701 \, {\left (a b^{2} d - 16 \, a^{2} b g\right )} \sqrt {b} x^{9} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (4860 \, a b^{2} e x^{8} + 6120 \, a^{2} b e x^{5} + 63 \, {\left (27 \, a b^{2} d + 208 \, a^{2} b g\right )} x^{7} + 840 \, {\left (a b^{2} c + 10 \, a^{2} b f\right )} x^{6} + 2880 \, a^{3} e x^{2} + 2520 \, a^{3} d x + 252 \, {\left (19 \, a^{2} b d + 16 \, a^{3} g\right )} x^{4} + 2240 \, a^{3} c + 560 \, {\left (7 \, a^{2} b c + 6 \, a^{3} f\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{20160 \, a^{2} x^{9}}\right ] \]
[-1/20160*(4860*a*b^(5/2)*e*x^9*weierstrassZeta(0, -4*a/b, weierstrassPInv erse(0, -4*a/b, x)) + 210*(b^3*c - 6*a*b^2*f)*sqrt(a)*x^9*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) + 1701*(a *b^2*d - 16*a^2*b*g)*sqrt(b)*x^9*weierstrassPInverse(0, -4*a/b, x) + (4860 *a*b^2*e*x^8 + 6120*a^2*b*e*x^5 + 63*(27*a*b^2*d + 208*a^2*b*g)*x^7 + 840* (a*b^2*c + 10*a^2*b*f)*x^6 + 2880*a^3*e*x^2 + 2520*a^3*d*x + 252*(19*a^2*b *d + 16*a^3*g)*x^4 + 2240*a^3*c + 560*(7*a^2*b*c + 6*a^3*f)*x^3)*sqrt(b*x^ 3 + a))/(a^2*x^9), -1/20160*(4860*a*b^(5/2)*e*x^9*weierstrassZeta(0, -4*a/ b, weierstrassPInverse(0, -4*a/b, x)) + 420*(b^3*c - 6*a*b^2*f)*sqrt(-a)*x ^9*arctan(1/2*(b*x^3 + 2*a)*sqrt(b*x^3 + a)*sqrt(-a)/(a*b*x^3 + a^2)) + 17 01*(a*b^2*d - 16*a^2*b*g)*sqrt(b)*x^9*weierstrassPInverse(0, -4*a/b, x) + (4860*a*b^2*e*x^8 + 6120*a^2*b*e*x^5 + 63*(27*a*b^2*d + 208*a^2*b*g)*x^7 + 840*(a*b^2*c + 10*a^2*b*f)*x^6 + 2880*a^3*e*x^2 + 2520*a^3*d*x + 252*(19* a^2*b*d + 16*a^3*g)*x^4 + 2240*a^3*c + 560*(7*a^2*b*c + 6*a^3*f)*x^3)*sqrt (b*x^3 + a))/(a^2*x^9)]
Time = 13.50 (sec) , antiderivative size = 573, normalized size of antiderivative = 0.80 \[ \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^{10}} \, dx=\frac {a^{\frac {3}{2}} d \Gamma \left (- \frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {8}{3}, - \frac {1}{2} \\ - \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{8} \Gamma \left (- \frac {5}{3}\right )} + \frac {a^{\frac {3}{2}} e \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, - \frac {1}{2} \\ - \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{7} \Gamma \left (- \frac {4}{3}\right )} + \frac {a^{\frac {3}{2}} g \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} + \frac {\sqrt {a} b d \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} + \frac {\sqrt {a} b e \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 {\sqrt {a} b g \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^{2} c}{9 \sqrt {b} x^{\frac {21}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a^{2} f}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {11 a \sqrt {b} c}{36 x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} f}{4 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {17 b^{\frac {3}{2}} c}{72 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{\frac {3}{2}} f \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {b^{\frac {3}{2}} f}{12 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{\frac {5}{2}} c}{24 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{2} f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{4 \sqrt {a}} + \frac {b^{3} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{24 a^{\frac {3}{2}}} \]
a**(3/2)*d*gamma(-8/3)*hyper((-8/3, -1/2), (-5/3,), b*x**3*exp_polar(I*pi) /a)/(3*x**8*gamma(-5/3)) + a**(3/2)*e*gamma(-7/3)*hyper((-7/3, -1/2), (-4/ 3,), b*x**3*exp_polar(I*pi)/a)/(3*x**7*gamma(-4/3)) + a**(3/2)*g*gamma(-5/ 3)*hyper((-5/3, -1/2), (-2/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**5*gamma(-2 /3)) + sqrt(a)*b*d*gamma(-5/3)*hyper((-5/3, -1/2), (-2/3,), b*x**3*exp_pol ar(I*pi)/a)/(3*x**5*gamma(-2/3)) + sqrt(a)*b*e*gamma(-4/3)*hyper((-4/3, -1 /2), (-1/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**4*gamma(-1/3)) + sqrt(a)*b*g *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**2*c/(9*sqrt(b)*x**(21/2)*sqrt(a/(b*x**3) + 1)) - a**2*f/ (6*sqrt(b)*x**(15/2)*sqrt(a/(b*x**3) + 1)) - 11*a*sqrt(b)*c/(36*x**(15/2)* sqrt(a/(b*x**3) + 1)) - a*sqrt(b)*f/(4*x**(9/2)*sqrt(a/(b*x**3) + 1)) - 17 *b**(3/2)*c/(72*x**(9/2)*sqrt(a/(b*x**3) + 1)) - b**(3/2)*f*sqrt(a/(b*x**3 ) + 1)/(3*x**(3/2)) - b**(3/2)*f/(12*x**(3/2)*sqrt(a/(b*x**3) + 1)) - b**( 5/2)*c/(24*a*x**(3/2)*sqrt(a/(b*x**3) + 1)) - b**2*f*asinh(sqrt(a)/(sqrt(b )*x**(3/2)))/(4*sqrt(a)) + b**3*c*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(24*a* *(3/2))
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{10}} \,d x } \]
-1/144*(3*b^3*log((sqrt(b*x^3 + a) - sqrt(a))/(sqrt(b*x^3 + a) + sqrt(a))) /a^(3/2) + 2*(3*(b*x^3 + a)^(5/2)*b^3 + 8*(b*x^3 + a)^(3/2)*a*b^3 - 3*sqrt (b*x^3 + a)*a^2*b^3)/((b*x^3 + a)^3*a - 3*(b*x^3 + a)^2*a^2 + 3*(b*x^3 + a )*a^3 - a^4))*c + integrate((b*g*x^6 + b*f*x^5 + b*e*x^4 + a*f*x^2 + (b*d + a*g)*x^3 + a*e*x + a*d)*sqrt(b*x^3 + a)/x^9, x)
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{10}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^{10}} \,d x \]