Integrand size = 35, antiderivative size = 692 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx=\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {27 b e \sqrt {a+b x^3}}{8 x}+\frac {27 b^{4/3} e \sqrt {a+b x^3}}{8 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac {b (b c+4 a f) \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} b^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{16 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \left (2 b d-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+4 a g\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{40 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
-1/60*(10*c/x^6+12*d/x^5+15*e/x^4+20*f/x^3+30*g/x^2)*(b*x^3+a)^(3/2)-1/4*b *(4*a*f+b*c)*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(1/2)+1/4*b*c*(b*x^3+a)^(1 /2)/x^3+27/20*b*d*(b*x^3+a)^(1/2)/x^2-27/8*b*e*(b*x^3+a)^(1/2)/x-1/20*b*(- 18*g*x^5-20*f*x^4-45*e*x^3+36*d*x^2+10*c*x)*(b*x^3+a)^(1/2)/x^4+27/8*b^(4/ 3)*e*(b*x^3+a)^(1/2)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))-27/16*3^(1/4)*a^(1/3) *b^(4/3)*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)/(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)+9/40*3^(3/4)*b^(2/3)*(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)* (2*b*d+4*a*g-5*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)/(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 = 10.56 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.35 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx=\frac {-\frac {15 b c \left (a+b x^3\right )}{x^3}-\frac {10 c \left (a+b x^3\right )^2}{x^6}-15 b^2 c \sqrt {1+\frac {b x^3}{a}} \text {arctanh}\left (\sqrt {1+\frac {b x^3}{a}}\right )-\frac {12 a^2 d \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},-\frac {3}{2},-\frac {2}{3},-\frac {b x^3}{a}\right )}{x^5}-\frac {15 a^2 e \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {4}{3},-\frac {1}{3},-\frac {b x^3}{a}\right )}{x^4}-\frac {30 a^2 g \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {2}{3},\frac {1}{3},-\frac {b x^3}{a}\right )}{x^2}+\frac {8 b f \left (a+b x^3\right )^3 \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},1+\frac {b x^3}{a}\right )}{a^2}}{60 \sqrt {a+b x^3}} \]
((-15*b*c*(a + b*x^3))/x^3 - (10*c*(a + b*x^3)^2)/x^6 - 15*b^2*c*Sqrt[1 + (b*x^3)/a]*ArcTanh[Sqrt[1 + (b*x^3)/a]] - (12*a^2*d*Sqrt[1 + (b*x^3)/a]*Hy pergeometric2F1[-5/3, -3/2, -2/3, -((b*x^3)/a)])/x^5 - (15*a^2*e*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-3/2, -4/3, -1/3, -((b*x^3)/a)])/x^4 - (30*a^ 2*g*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-3/2, -2/3, 1/3, -((b*x^3)/a)])/ x^2 + (8*b*f*(a + b*x^3)^3*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^3)/a])/ a^2)/(60*Sqrt[a + b*x^3])
Time = 1.48 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.04, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {2364, 27, 2365, 27, 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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -\frac {9}{2} b \int -\frac {\sqrt {b x^3+a} \left (30 g x^4+20 f x^3+15 e x^2+12 d x+10 c\right )}{60 x^4}dx-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{40} b \int \frac {\sqrt {b x^3+a} \left (30 g x^4+20 f x^3+15 e x^2+12 d x+10 c\right )}{x^4}dx-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 2365 |
| \(\displaystyle \frac {3}{40} b \left (\frac {3}{2} a \int -\frac {2 \left (-18 g x^4-20 f x^3-45 e x^2+36 d x+10 c\right )}{3 x^4 \sqrt {b x^3+a}}dx-\frac {2 \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{3 x^4}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{40} b \left (-a \int \frac {-18 g x^4-20 f x^3-45 e x^2+36 d x+10 c}{x^4 \sqrt {b x^3+a}}dx-\frac {2 \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{3 x^4}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {3}{40} b \left (-a \left (-\frac {\int -\frac {6 \left (-18 a g x^3-5 (b c+4 a f) x^2-45 a e x+36 a d\right )}{x^3 \sqrt {b x^3+a}}dx}{6 a}-\frac {10 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{3 x^4}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{40} b \left (-a \left (\frac {\int \frac {-18 a g x^3-5 (b c+4 a f) x^2-45 a e x+36 a d}{x^3 \sqrt {b x^3+a}}dx}{a}-\frac {10 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{3 x^4}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {3}{40} b \left (-a \left (\frac {-\frac {\int \frac {4 \left (45 e a^2+9 (b d+2 a g) x^2 a+5 (b c+4 a f) x a\right )}{x^2 \sqrt {b x^3+a}}dx}{4 a}-\frac {18 d \sqrt {a+b x^3}}{x^2}}{a}-\frac {10 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{3 x^4}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{40} b \left (-a \left (\frac {-\frac {\int \frac {45 e a^2+9 (b d+2 a g) x^2 a+5 (b c+4 a f) x a}{x^2 \sqrt {b x^3+a}}dx}{a}-\frac {18 d \sqrt {a+b x^3}}{x^2}}{a}-\frac {10 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{3 x^4}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {3}{40} b \left (-a \left (\frac {-\frac {-\frac {\int -\frac {45 b e x^2 a^2+10 (b c+4 a f) a^2+18 (b d+2 a g) x a^2}{x \sqrt {b x^3+a}}dx}{2 a}-\frac {45 a e \sqrt {a+b x^3}}{x}}{a}-\frac {18 d \sqrt {a+b x^3}}{x^2}}{a}-\frac {10 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{3 x^4}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{40} b \left (-a \left (\frac {-\frac {\frac {\int \frac {45 b e x^2 a^2+10 (b c+4 a f) a^2+18 (b d+2 a g) x a^2}{x \sqrt {b x^3+a}}dx}{2 a}-\frac {45 a e \sqrt {a+b x^3}}{x}}{a}-\frac {18 d \sqrt {a+b x^3}}{x^2}}{a}-\frac {10 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{3 x^4}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 2371 |
| \(\displaystyle \frac {3}{40} b \left (-a \left (\frac {-\frac {\frac {10 a^2 (4 a f+b c) \int \frac {1}{x \sqrt {b x^3+a}}dx+\int \frac {18 (b d+2 a g) a^2+45 b e x a^2}{\sqrt {b x^3+a}}dx}{2 a}-\frac {45 a e \sqrt {a+b x^3}}{x}}{a}-\frac {18 d \sqrt {a+b x^3}}{x^2}}{a}-\frac {10 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{3 x^4}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {3}{40} b \left (-a \left (\frac {-\frac {\frac {\frac {10}{3} a^2 (4 a f+b c) \int \frac {1}{x^3 \sqrt {b x^3+a}}dx^3+\int \frac {18 (b d+2 a g) a^2+45 b e x a^2}{\sqrt {b x^3+a}}dx}{2 a}-\frac {45 a e \sqrt {a+b x^3}}{x}}{a}-\frac {18 d \sqrt {a+b x^3}}{x^2}}{a}-\frac {10 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{3 x^4}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {3}{40} b \left (-a \left (\frac {-\frac {\frac {\frac {20 a^2 (4 a f+b c) \int \frac {1}{\frac {x^6}{b}-\frac {a}{b}}d\sqrt {b x^3+a}}{3 b}+\int \frac {18 (b d+2 a g) a^2+45 b e x a^2}{\sqrt {b x^3+a}}dx}{2 a}-\frac {45 a e \sqrt {a+b x^3}}{x}}{a}-\frac {18 d \sqrt {a+b x^3}}{x^2}}{a}-\frac {10 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{3 x^4}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {3}{40} b \left (-a \left (\frac {-\frac {\frac {\int \frac {18 (b d+2 a g) a^2+45 b e x a^2}{\sqrt {b x^3+a}}dx-\frac {20}{3} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (4 a f+b c)}{2 a}-\frac {45 a e \sqrt {a+b x^3}}{x}}{a}-\frac {18 d \sqrt {a+b x^3}}{x^2}}{a}-\frac {10 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{3 x^4}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {3}{40} b \left (-a \left (\frac {-\frac {\frac {9 a^2 \left (-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+4 a g+2 b d\right ) \int \frac {1}{\sqrt {b x^3+a}}dx+45 a^2 b^{2/3} e \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx-\frac {20}{3} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (4 a f+b c)}{2 a}-\frac {45 a e \sqrt {a+b x^3}}{x}}{a}-\frac {18 d \sqrt {a+b x^3}}{x^2}}{a}-\frac {10 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{3 x^4}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {3}{40} b \left (-a \left (\frac {-\frac {\frac {45 a^2 b^{2/3} e \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx-\frac {20}{3} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (4 a f+b c)+\frac {6\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+4 a g+2 b d\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}}{2 a}-\frac {45 a e \sqrt {a+b x^3}}{x}}{a}-\frac {18 d \sqrt {a+b x^3}}{x^2}}{a}-\frac {10 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{3 x^4}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {3}{40} b \left (-a \left (\frac {-\frac {\frac {-\frac {20}{3} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (4 a f+b c)+\frac {6\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+4 a g+2 b d\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+45 a^2 b^{2/3} e \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{2 a}-\frac {45 a e \sqrt {a+b x^3}}{x}}{a}-\frac {18 d \sqrt {a+b x^3}}{x^2}}{a}-\frac {10 c \sqrt {a+b x^3}}{3 a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{3 x^4}\right )-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )\) |
-1/60*(((10*c)/x^6 + (12*d)/x^5 + (15*e)/x^4 + (20*f)/x^3 + (30*g)/x^2)*(a + b*x^3)^(3/2)) + (3*b*((-2*Sqrt[a + b*x^3]*(10*c*x + 36*d*x^2 - 45*e*x^3 - 20*f*x^4 - 18*g*x^5))/(3*x^4) - a*((-10*c*Sqrt[a + b*x^3])/(3*a*x^3) + ((-18*d*Sqrt[a + b*x^3])/x^2 - ((-45*a*e*Sqrt[a + b*x^3])/x + ((-20*a^(3/2 )*(b*c + 4*a*f)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/3 + 45*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[A rcSin[((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])) + (6*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*(2*b*d - 5*(1 - Sqrt[3])*a^(1/3)*b^(2/3)*e + 4*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]*Sq rt[a + b*x^3]))/(2*a))/a)/a)))/40
3.5.68.3.1 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 = 1.86 (sec) , antiderivative size = 903, normalized size of antiderivative = 1.30
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(903\) |
| default | \(\text {Expression too large to display}\) | \(1196\) |
| risch | \(\text {Expression too large to display}\) | \(1411\) |
-1/6*a*c*(b*x^3+a)^(1/2)/x^6-1/5*a*d*(b*x^3+a)^(1/2)/x^5-1/4*a*e*(b*x^3+a) ^(1/2)/x^4-1/3*(a*f+5/4*b*c)*(b*x^3+a)^(1/2)/x^3-1/2*(a*g+13/10*b*d)*(b*x^ 3+a)^(1/2)/x^2-11/8*b*e*(b*x^3+a)^(1/2)/x+2/5*g*b*x*(b*x^3+a)^(1/2)+2/3*b* f*(b*x^3+a)^(1/2)-2/3*I*(8/5*a*b*g+b^2*d-1/40*b*(10*a*g+13*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))-9/8*I*b*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))*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)*Elliptic F(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)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.39 (sec) , antiderivative size = 430, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx=\left [-\frac {810 \, a b^{\frac {3}{2}} e x^{6} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - 15 \, {\left (b^{2} c + 4 \, a b f\right )} \sqrt {a} x^{6} \log \left (-\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 324 \, {\left (a b d + 2 \, a^{2} g\right )} \sqrt {b} x^{6} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 2 \, {\left (48 \, a b g x^{7} + 80 \, a b f x^{6} - 165 \, a b e x^{5} - 30 \, a^{2} e x^{2} - 6 \, {\left (13 \, a b d + 10 \, a^{2} g\right )} x^{4} - 24 \, a^{2} d x - 10 \, {\left (5 \, a b c + 4 \, a^{2} f\right )} x^{3} - 20 \, a^{2} c\right )} \sqrt {b x^{3} + a}}{240 \, a x^{6}}, -\frac {405 \, a b^{\frac {3}{2}} e x^{6} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - 15 \, {\left (b^{2} c + 4 \, a b f\right )} \sqrt {-a} x^{6} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-a}}{b x^{3} + 2 \, a}\right ) - 162 \, {\left (a b d + 2 \, a^{2} g\right )} \sqrt {b} x^{6} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - {\left (48 \, a b g x^{7} + 80 \, a b f x^{6} - 165 \, a b e x^{5} - 30 \, a^{2} e x^{2} - 6 \, {\left (13 \, a b d + 10 \, a^{2} g\right )} x^{4} - 24 \, a^{2} d x - 10 \, {\left (5 \, a b c + 4 \, a^{2} f\right )} x^{3} - 20 \, a^{2} c\right )} \sqrt {b x^{3} + a}}{120 \, a x^{6}}\right ] \]
[-1/240*(810*a*b^(3/2)*e*x^6*weierstrassZeta(0, -4*a/b, weierstrassPInvers e(0, -4*a/b, x)) - 15*(b^2*c + 4*a*b*f)*sqrt(a)*x^6*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) - 324*(a*b*d + 2*a^2*g)*sqrt(b)*x^6*weierstrassPInverse(0, -4*a/b, x) - 2*(48*a*b*g*x^7 + 80*a*b*f*x^6 - 165*a*b*e*x^5 - 30*a^2*e*x^2 - 6*(13*a*b*d + 10*a^2*g)*x^ 4 - 24*a^2*d*x - 10*(5*a*b*c + 4*a^2*f)*x^3 - 20*a^2*c)*sqrt(b*x^3 + a))/( a*x^6), -1/120*(405*a*b^(3/2)*e*x^6*weierstrassZeta(0, -4*a/b, weierstrass PInverse(0, -4*a/b, x)) - 15*(b^2*c + 4*a*b*f)*sqrt(-a)*x^6*arctan(2*sqrt( b*x^3 + a)*sqrt(-a)/(b*x^3 + 2*a)) - 162*(a*b*d + 2*a^2*g)*sqrt(b)*x^6*wei erstrassPInverse(0, -4*a/b, x) - (48*a*b*g*x^7 + 80*a*b*f*x^6 - 165*a*b*e* x^5 - 30*a^2*e*x^2 - 6*(13*a*b*d + 10*a^2*g)*x^4 - 24*a^2*d*x - 10*(5*a*b* c + 4*a^2*f)*x^3 - 20*a^2*c)*sqrt(b*x^3 + a))/(a*x^6)]
Time = 8.00 (sec) , antiderivative size = 524, normalized size of antiderivative = 0.76 \[ \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^7} \, dx=\frac {a^{\frac {3}{2}} 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 {a^{\frac {3}{2}} 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 {a^{\frac {3}{2}} 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 {\sqrt {a} b d \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 e \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 f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )} + \frac {\sqrt {a} b g 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 {a^{2} c}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} c}{4 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} f \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} + \frac {2 a \sqrt {b} f}{3 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {b^{\frac {3}{2}} c}{12 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 b^{\frac {3}{2}} f x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{2} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{4 \sqrt {a}} \]
a**(3/2)*d*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)*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)) + a**(3/2)*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 )) + sqrt(a)*b*d*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*e*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*f*asinh(sq rt(a)/(sqrt(b)*x**(3/2))) + sqrt(a)*b*g*x*gamma(1/3)*hyper((-1/2, 1/3), (4 /3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/3)) - a**2*c/(6*sqrt(b)*x**(15/ 2)*sqrt(a/(b*x**3) + 1)) - a*sqrt(b)*c/(4*x**(9/2)*sqrt(a/(b*x**3) + 1)) - a*sqrt(b)*f*sqrt(a/(b*x**3) + 1)/(3*x**(3/2)) + 2*a*sqrt(b)*f/(3*x**(3/2) *sqrt(a/(b*x**3) + 1)) - b**(3/2)*c*sqrt(a/(b*x**3) + 1)/(3*x**(3/2)) - b* *(3/2)*c/(12*x**(3/2)*sqrt(a/(b*x**3) + 1)) + 2*b**(3/2)*f*x**(3/2)/(3*sqr t(a/(b*x**3) + 1)) - b**2*c*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(4*sqrt(a))
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{7}} \,d x } \]
1/24*(3*b^2*log((sqrt(b*x^3 + a) - sqrt(a))/(sqrt(b*x^3 + a) + sqrt(a)))/s qrt(a) - 2*(5*(b*x^3 + a)^(3/2)*b^2 - 3*sqrt(b*x^3 + a)*a*b^2)/((b*x^3 + a )^2 - 2*(b*x^3 + a)*a + a^2))*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^6, x)
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{7}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^7} \,d x \]