Integrand size = 35, antiderivative size = 689 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^6} \, dx=\frac {27 b c \sqrt {a+b x^3}}{20 x^2}-\frac {27 b d \sqrt {a+b x^3}}{8 x}+\frac {27 \sqrt [3]{b} (7 b d+8 a g) \sqrt {a+b x^3}}{56 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{60} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (252 c x-315 d x^2-140 e x^3-126 f x^4-180 g x^5\right )}{140 x^3}-\sqrt {a} b e \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \sqrt [3]{b} (7 b d+8 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{112 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt [3]{b} \left (14 \sqrt [3]{b} (b c+2 a f)-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (7 b d+8 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{280 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
-1/60*(12*c/x^5+15*d/x^4+20*e/x^3+30*f/x^2+60*g/x)*(b*x^3+a)^(3/2)-b*e*arc tanh((b*x^3+a)^(1/2)/a^(1/2))*a^(1/2)+27/20*b*c*(b*x^3+a)^(1/2)/x^2-27/8*b *d*(b*x^3+a)^(1/2)/x-1/140*b*(-180*g*x^5-126*f*x^4-140*e*x^3-315*d*x^2+252 *c*x)*(b*x^3+a)^(1/2)/x^3+27/56*b^(1/3)*(8*a*g+7*b*d)*(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*g+7*b*d)*( 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)*b^(1/3)*(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)*(14*b^(1/3) *(2*a*f+b*c)-5*a^(1/3)*(8*a*g+7*b*d)*(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 = 10.24 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.28 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^6} \, dx=\frac {\sqrt {a+b x^3} \left (-12 a^3 c \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},-\frac {3}{2},-\frac {2}{3},-\frac {b x^3}{a}\right )-15 a^3 d x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {4}{3},-\frac {1}{3},-\frac {b x^3}{a}\right )-30 a^3 f x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {2}{3},\frac {1}{3},-\frac {b x^3}{a}\right )-60 a^3 g x^4 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{3},\frac {2}{3},-\frac {b x^3}{a}\right )+8 b e x^5 \left (a+b x^3\right )^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},1+\frac {b x^3}{a}\right )\right )}{60 a^2 x^5 \sqrt {1+\frac {b x^3}{a}}} \]
(Sqrt[a + b*x^3]*(-12*a^3*c*Hypergeometric2F1[-5/3, -3/2, -2/3, -((b*x^3)/ a)] - 15*a^3*d*x*Hypergeometric2F1[-3/2, -4/3, -1/3, -((b*x^3)/a)] - 30*a^ 3*f*x^3*Hypergeometric2F1[-3/2, -2/3, 1/3, -((b*x^3)/a)] - 60*a^3*g*x^4*Hy pergeometric2F1[-3/2, -1/3, 2/3, -((b*x^3)/a)] + 8*b*e*x^5*(a + b*x^3)^2*S qrt[1 + (b*x^3)/a]*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^3)/a]))/(60*a^2 *x^5*Sqrt[1 + (b*x^3)/a])
Time = 1.35 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.03, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2364, 27, 2365, 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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^6} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -\frac {9}{2} b \int -\frac {\sqrt {b x^3+a} \left (60 g x^4+30 f x^3+20 e x^2+15 d x+12 c\right )}{60 x^3}dx-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{40} b \int \frac {\sqrt {b x^3+a} \left (60 g x^4+30 f x^3+20 e x^2+15 d x+12 c\right )}{x^3}dx-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right )\) |
| \(\Big \downarrow \) 2365 |
| \(\displaystyle \frac {3}{40} b \left (\frac {3}{2} a \int -\frac {2 \left (-180 g x^4-126 f x^3-140 e x^2-315 d x+252 c\right )}{21 x^3 \sqrt {b x^3+a}}dx-\frac {2 \sqrt {a+b x^3} \left (252 c x-315 d x^2-140 e x^3-126 f x^4-180 g x^5\right )}{21 x^3}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{40} b \left (-\frac {1}{7} a \int \frac {-180 g x^4-126 f x^3-140 e x^2-315 d x+252 c}{x^3 \sqrt {b x^3+a}}dx-\frac {2 \sqrt {a+b x^3} \left (252 c x-315 d x^2-140 e x^3-126 f x^4-180 g x^5\right )}{21 x^3}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right )\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {3}{40} b \left (-\frac {1}{7} a \left (-\frac {\int \frac {4 \left (180 a g x^3+63 (b c+2 a f) x^2+140 a e x+315 a d\right )}{x^2 \sqrt {b x^3+a}}dx}{4 a}-\frac {126 c \sqrt {a+b x^3}}{a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (252 c x-315 d x^2-140 e x^3-126 f x^4-180 g x^5\right )}{21 x^3}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{40} b \left (-\frac {1}{7} a \left (-\frac {\int \frac {180 a g x^3+63 (b c+2 a f) x^2+140 a e x+315 a d}{x^2 \sqrt {b x^3+a}}dx}{a}-\frac {126 c \sqrt {a+b x^3}}{a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (252 c x-315 d x^2-140 e x^3-126 f x^4-180 g x^5\right )}{21 x^3}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right )\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {3}{40} b \left (-\frac {1}{7} a \left (-\frac {-\frac {\int -\frac {280 e a^2+45 (7 b d+8 a g) x^2 a+126 (b c+2 a f) x a}{x \sqrt {b x^3+a}}dx}{2 a}-\frac {315 d \sqrt {a+b x^3}}{x}}{a}-\frac {126 c \sqrt {a+b x^3}}{a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (252 c x-315 d x^2-140 e x^3-126 f x^4-180 g x^5\right )}{21 x^3}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{40} b \left (-\frac {1}{7} a \left (-\frac {\frac {\int \frac {280 e a^2+45 (7 b d+8 a g) x^2 a+126 (b c+2 a f) x a}{x \sqrt {b x^3+a}}dx}{2 a}-\frac {315 d \sqrt {a+b x^3}}{x}}{a}-\frac {126 c \sqrt {a+b x^3}}{a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (252 c x-315 d x^2-140 e x^3-126 f x^4-180 g x^5\right )}{21 x^3}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right )\) |
| \(\Big \downarrow \) 2371 |
| \(\displaystyle \frac {3}{40} b \left (-\frac {1}{7} a \left (-\frac {\frac {280 a^2 e \int \frac {1}{x \sqrt {b x^3+a}}dx+\int \frac {126 a (b c+2 a f)+45 a (7 b d+8 a g) x}{\sqrt {b x^3+a}}dx}{2 a}-\frac {315 d \sqrt {a+b x^3}}{x}}{a}-\frac {126 c \sqrt {a+b x^3}}{a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (252 c x-315 d x^2-140 e x^3-126 f x^4-180 g x^5\right )}{21 x^3}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right )\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {3}{40} b \left (-\frac {1}{7} a \left (-\frac {\frac {\frac {280}{3} a^2 e \int \frac {1}{x^3 \sqrt {b x^3+a}}dx^3+\int \frac {126 a (b c+2 a f)+45 a (7 b d+8 a g) x}{\sqrt {b x^3+a}}dx}{2 a}-\frac {315 d \sqrt {a+b x^3}}{x}}{a}-\frac {126 c \sqrt {a+b x^3}}{a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (252 c x-315 d x^2-140 e x^3-126 f x^4-180 g x^5\right )}{21 x^3}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {3}{40} b \left (-\frac {1}{7} a \left (-\frac {\frac {\frac {560 a^2 e \int \frac {1}{\frac {x^6}{b}-\frac {a}{b}}d\sqrt {b x^3+a}}{3 b}+\int \frac {126 a (b c+2 a f)+45 a (7 b d+8 a g) x}{\sqrt {b x^3+a}}dx}{2 a}-\frac {315 d \sqrt {a+b x^3}}{x}}{a}-\frac {126 c \sqrt {a+b x^3}}{a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (252 c x-315 d x^2-140 e x^3-126 f x^4-180 g x^5\right )}{21 x^3}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {3}{40} b \left (-\frac {1}{7} a \left (-\frac {\frac {\int \frac {126 a (b c+2 a f)+45 a (7 b d+8 a g) x}{\sqrt {b x^3+a}}dx-\frac {560}{3} a^{3/2} e \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{2 a}-\frac {315 d \sqrt {a+b x^3}}{x}}{a}-\frac {126 c \sqrt {a+b x^3}}{a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (252 c x-315 d x^2-140 e x^3-126 f x^4-180 g x^5\right )}{21 x^3}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right )\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {3}{40} b \left (-\frac {1}{7} a \left (-\frac {\frac {9 a \left (14 (2 a f+b c)-\frac {5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (8 a g+7 b d)}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt {b x^3+a}}dx+\frac {45 a (8 a g+7 b d) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-\frac {560}{3} a^{3/2} e \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{2 a}-\frac {315 d \sqrt {a+b x^3}}{x}}{a}-\frac {126 c \sqrt {a+b x^3}}{a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (252 c x-315 d x^2-140 e x^3-126 f x^4-180 g x^5\right )}{21 x^3}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right )\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {3}{40} b \left (-\frac {1}{7} a \left (-\frac {\frac {\frac {45 a (8 a g+7 b d) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}+\frac {6\ 3^{3/4} \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (14 (2 a f+b c)-\frac {5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (8 a g+7 b d)}{\sqrt [3]{b}}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {560}{3} a^{3/2} e \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{2 a}-\frac {315 d \sqrt {a+b x^3}}{x}}{a}-\frac {126 c \sqrt {a+b x^3}}{a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (252 c x-315 d x^2-140 e x^3-126 f x^4-180 g x^5\right )}{21 x^3}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right )\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {3}{40} b \left (-\frac {1}{7} a \left (-\frac {\frac {\frac {6\ 3^{3/4} \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (14 (2 a f+b c)-\frac {5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (8 a g+7 b d)}{\sqrt [3]{b}}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {45 a (8 a g+7 b d) \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{\sqrt [3]{b}}-\frac {560}{3} a^{3/2} e \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{2 a}-\frac {315 d \sqrt {a+b x^3}}{x}}{a}-\frac {126 c \sqrt {a+b x^3}}{a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (252 c x-315 d x^2-140 e x^3-126 f x^4-180 g x^5\right )}{21 x^3}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}+\frac {60 g}{x}\right )\) |
-1/60*(((12*c)/x^5 + (15*d)/x^4 + (20*e)/x^3 + (30*f)/x^2 + (60*g)/x)*(a + b*x^3)^(3/2)) + (3*b*((-2*Sqrt[a + b*x^3]*(252*c*x - 315*d*x^2 - 140*e*x^ 3 - 126*f*x^4 - 180*g*x^5))/(21*x^3) - (a*((-126*c*Sqrt[a + b*x^3])/(a*x^2 ) - ((-315*d*Sqrt[a + b*x^3])/x + ((-560*a^(3/2)*e*ArcTanh[Sqrt[a + b*x^3] /Sqrt[a]])/3 + (45*a*(7*b*d + 8*a*g)*((2*Sqrt[a + b*x^3])/(b^(1/3)*((1 + S qrt[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)*Sq rt[(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])))/b^(1/3) + (6*3^(3/4)*Sqrt[2 + Sqrt[3]]*a*(14*(b*c + 2*a *f) - (5*(1 - Sqrt[3])*a^(1/3)*(7*b*d + 8*a*g))/b^(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]*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]))/(2*a))/a))/7))/40
3.5.67.3.1 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_)*((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 = 1.96 (sec) , antiderivative size = 920, normalized size of antiderivative = 1.34
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(920\) |
| default | \(\text {Expression too large to display}\) | \(1606\) |
| risch | \(\text {Expression too large to display}\) | \(2289\) |
-1/5*a*c*(b*x^3+a)^(1/2)/x^5-1/4*a*d*(b*x^3+a)^(1/2)/x^4-1/3*a*e*(b*x^3+a) ^(1/2)/x^3-1/2*(a*f+13/10*b*c)*(b*x^3+a)^(1/2)/x^2-(a*g+11/8*b*d)*(b*x^3+a )^(1/2)/x+2/7*g*b*x^2*(b*x^3+a)^(1/2)+2/5*b*f*x*(b*x^3+a)^(1/2)+2/3*b*e*(b *x^3+a)^(1/2)-2/3*I*(8/5*a*f*b+b^2*c-1/40*b*(10*a*f+13*b*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) *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*(10/7*a*b*g +b^2*d+1/16*b*(8*a*g+11*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)*((-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)^(...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.26 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.55 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^6} \, dx=\left [\frac {210 \, \sqrt {a} b e x^{5} \log \left (-\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 1134 \, {\left (b c + 2 \, a f\right )} \sqrt {b} x^{5} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 405 \, {\left (7 \, b d + 8 \, a g\right )} \sqrt {b} x^{5} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (240 \, b g x^{7} + 336 \, b f x^{6} + 560 \, b e x^{5} - 105 \, {\left (11 \, b d + 8 \, a g\right )} x^{4} - 280 \, a e x^{2} - 42 \, {\left (13 \, b c + 10 \, a f\right )} x^{3} - 210 \, a d x - 168 \, a c\right )} \sqrt {b x^{3} + a}}{840 \, x^{5}}, \frac {420 \, \sqrt {-a} b e x^{5} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-a}}{b x^{3} + 2 \, a}\right ) + 1134 \, {\left (b c + 2 \, a f\right )} \sqrt {b} x^{5} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 405 \, {\left (7 \, b d + 8 \, a g\right )} \sqrt {b} x^{5} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (240 \, b g x^{7} + 336 \, b f x^{6} + 560 \, b e x^{5} - 105 \, {\left (11 \, b d + 8 \, a g\right )} x^{4} - 280 \, a e x^{2} - 42 \, {\left (13 \, b c + 10 \, a f\right )} x^{3} - 210 \, a d x - 168 \, a c\right )} \sqrt {b x^{3} + a}}{840 \, x^{5}}\right ] \]
[1/840*(210*sqrt(a)*b*e*x^5*log(-(b^2*x^6 + 8*a*b*x^3 - 4*(b*x^3 + 2*a)*sq rt(b*x^3 + a)*sqrt(a) + 8*a^2)/x^6) + 1134*(b*c + 2*a*f)*sqrt(b)*x^5*weier strassPInverse(0, -4*a/b, x) - 405*(7*b*d + 8*a*g)*sqrt(b)*x^5*weierstrass Zeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (240*b*g*x^7 + 336*b* f*x^6 + 560*b*e*x^5 - 105*(11*b*d + 8*a*g)*x^4 - 280*a*e*x^2 - 42*(13*b*c + 10*a*f)*x^3 - 210*a*d*x - 168*a*c)*sqrt(b*x^3 + a))/x^5, 1/840*(420*sqrt (-a)*b*e*x^5*arctan(2*sqrt(b*x^3 + a)*sqrt(-a)/(b*x^3 + 2*a)) + 1134*(b*c + 2*a*f)*sqrt(b)*x^5*weierstrassPInverse(0, -4*a/b, x) - 405*(7*b*d + 8*a* g)*sqrt(b)*x^5*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x )) + (240*b*g*x^7 + 336*b*f*x^6 + 560*b*e*x^5 - 105*(11*b*d + 8*a*g)*x^4 - 280*a*e*x^2 - 42*(13*b*c + 10*a*f)*x^3 - 210*a*d*x - 168*a*c)*sqrt(b*x^3 + a))/x^5]
Time = 5.91 (sec) , antiderivative size = 476, normalized size of antiderivative = 0.69 \[ \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^6} \, dx=\frac {a^{\frac {3}{2}} c \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 {a^{\frac {3}{2}} d \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}} f \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}} g \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 {\sqrt {a} b c \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 {\sqrt {a} b d \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 e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )} + \frac {\sqrt {a} b f 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 g 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 {a \sqrt {b} e \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} + \frac {2 a \sqrt {b} e}{3 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 b^{\frac {3}{2}} e x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} \]
a**(3/2)*c*gamma(-5/3)*hyper((-5/3, -1/2), (-2/3,), b*x**3*exp_polar(I*pi) /a)/(3*x**5*gamma(-2/3)) + a**(3/2)*d*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)*f*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)*g*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*c*gamma(-2/3)*hyper((-2/3, -1/2), (1/ 3,), b*x**3*exp_polar(I*pi)/a)/(3*x**2*gamma(1/3)) + sqrt(a)*b*d*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*e*asinh(sqrt(a)/(sqrt(b)*x**(3/2))) + sqrt(a)*b*f*x*gamma(1/3) *hyper((-1/2, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/3)) + sqr t(a)*b*g*x**2*gamma(2/3)*hyper((-1/2, 2/3), (5/3,), b*x**3*exp_polar(I*pi) /a)/(3*gamma(5/3)) - a*sqrt(b)*e*sqrt(a/(b*x**3) + 1)/(3*x**(3/2)) + 2*a*s qrt(b)*e/(3*x**(3/2)*sqrt(a/(b*x**3) + 1)) + 2*b**(3/2)*e*x**(3/2)/(3*sqrt (a/(b*x**3) + 1))
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^6} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{6}} \,d x } \]
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^6} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^6} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^6} \,d x \]