Integrand size = 35, antiderivative size = 692 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^4} \, dx=\frac {a c \sqrt {a+b x^3}}{x^3}+\frac {27 a d \sqrt {a+b x^3}}{10 x^2}-\frac {27 a e \sqrt {a+b x^3}}{7 x}+\frac {27 a \sqrt [3]{b} e \sqrt {a+b x^3}}{7 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 a \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{1155 x^4}+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}-\frac {1}{3} \sqrt {a} (3 b c+2 a f) \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} \sqrt [3]{b} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{14 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} a \left (77 b d-110 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+28 a g\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{770 \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
a*c*(b*x^3+a)^(1/2)/x^3+27/10*a*d*(b*x^3+a)^(1/2)/x^2-27/7*a*e*(b*x^3+a)^( 1/2)/x+27*a*b^(1/3)*e*(b*x^3+a)^(1/2)/(7*(1+3^(1/2))*a^(1/3)+7*b^(1/3)*x)- 2/1155*a*(b*x^3+a)^(1/2)*(-189*g*x^5-385*f*x^4-1485*e*x^3+2079*d*x^2+1155* c*x)/x^4+2/3465*(b*x^3+a)^(3/2)*(315*g*x^5+385*f*x^4+495*e*x^3+693*d*x^2+1 155*c*x)/x^4-1/3*a^(1/2)*(2*a*f+3*b*c)*arctanh((b*x^3+a)^(1/2)/a^(1/2))-27 /14*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*a^(4/3)*b^(1/3)*e*(a^(1/3)+b^(1/3)*x )*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x) ^2)^(1/2)*EllipticE(((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b ^(1/3)*x),I*3^(1/2)+2*I)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3) +b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)+9/770*3^(3/4)*(1/2*6^(1/2)+1/2*2^(1/2 ))*a*(77*b*d-110*(1-3^(1/2))*a^(1/3)*b^(2/3)*e+28*a*g)*(a^(1/3)+b^(1/3)*x) *((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^ 2)^(1/2)*EllipticF(((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^ (1/3)*x),I*3^(1/2)+2*I)/b^(1/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/((1+3^(1/2))* a^(1/3)+b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.11 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.35 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^4} \, dx=\frac {-45 a^3 d \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {2}{3},\frac {1}{3},-\frac {b x^3}{a}\right )-90 a^3 e x \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{3},\frac {2}{3},-\frac {b x^3}{a}\right )+90 a^3 g x^3 \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )+4 x^2 \sqrt {1+\frac {b x^3}{a}} \left (5 a^2 f \left (\sqrt {a+b x^3} \left (4 a+b x^3\right )-3 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\right )+3 b c \left (a+b x^3\right )^{5/2} \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},1+\frac {b x^3}{a}\right )\right )}{90 a^2 x^2 \sqrt {1+\frac {b x^3}{a}}} \] Input:
Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^4,x]
Output:
(-45*a^3*d*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, -2/3, 1/3, -((b*x^3)/a) ] - 90*a^3*e*x*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, -1/3, 2/3, -((b*x^3 )/a)] + 90*a^3*g*x^3*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, 1/3, 4/3, -(( b*x^3)/a)] + 4*x^2*Sqrt[1 + (b*x^3)/a]*(5*a^2*f*(Sqrt[a + b*x^3]*(4*a + b* x^3) - 3*a^(3/2)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]) + 3*b*c*(a + b*x^3)^(5/ 2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^3)/a]))/(90*a^2*x^2*Sqrt[1 + (b *x^3)/a])
Time = 2.52 (sec) , antiderivative size = 725, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {2365, 27, 2365, 27, 2374, 27, 2374, 27, 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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 2365 |
| \(\displaystyle \frac {9}{2} a \int \frac {2 \sqrt {b x^3+a} \left (315 g x^4+385 f x^3+495 e x^2+693 d x+1155 c\right )}{3465 x^4}dx+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{385} a \int \frac {\sqrt {b x^3+a} \left (315 g x^4+385 f x^3+495 e x^2+693 d x+1155 c\right )}{x^4}dx+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 2365 |
| \(\displaystyle \frac {1}{385} a \left (\frac {3}{2} a \int -\frac {2 \left (-189 g x^4-385 f x^3-1485 e x^2+2079 d x+1155 c\right )}{3 x^4 \sqrt {b x^3+a}}dx-\frac {2 \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{3 x^4}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{385} a \left (-a \int \frac {-189 g x^4-385 f x^3-1485 e x^2+2079 d x+1155 c}{x^4 \sqrt {b x^3+a}}dx-\frac {2 \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{3 x^4}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {1}{385} a \left (-a \left (-\frac {\int -\frac {3 \left (-378 a g x^3-385 (3 b c+2 a f) x^2-2970 a e x+4158 a d\right )}{x^3 \sqrt {b x^3+a}}dx}{6 a}-\frac {385 c \sqrt {a+b x^3}}{a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{3 x^4}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{385} a \left (-a \left (\frac {\int \frac {-378 a g x^3-385 (3 b c+2 a f) x^2-2970 a e x+4158 a d}{x^3 \sqrt {b x^3+a}}dx}{2 a}-\frac {385 c \sqrt {a+b x^3}}{a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{3 x^4}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {1}{385} a \left (-a \left (\frac {-\frac {\int \frac {2 \left (5940 e a^2+189 (11 b d+4 a g) x^2 a+770 (3 b c+2 a f) x a\right )}{x^2 \sqrt {b x^3+a}}dx}{4 a}-\frac {2079 d \sqrt {a+b x^3}}{x^2}}{2 a}-\frac {385 c \sqrt {a+b x^3}}{a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{3 x^4}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{385} a \left (-a \left (\frac {-\frac {\int \frac {5940 e a^2+189 (11 b d+4 a g) x^2 a+770 (3 b c+2 a f) x a}{x^2 \sqrt {b x^3+a}}dx}{2 a}-\frac {2079 d \sqrt {a+b x^3}}{x^2}}{2 a}-\frac {385 c \sqrt {a+b x^3}}{a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{3 x^4}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {1}{385} a \left (-a \left (\frac {-\frac {-\frac {\int -\frac {2 \left (2970 b e x^2 a^2+770 (3 b c+2 a f) a^2+189 (11 b d+4 a g) x a^2\right )}{x \sqrt {b x^3+a}}dx}{2 a}-\frac {5940 a e \sqrt {a+b x^3}}{x}}{2 a}-\frac {2079 d \sqrt {a+b x^3}}{x^2}}{2 a}-\frac {385 c \sqrt {a+b x^3}}{a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{3 x^4}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{385} a \left (-a \left (\frac {-\frac {\frac {\int \frac {2970 b e x^2 a^2+770 (3 b c+2 a f) a^2+189 (11 b d+4 a g) x a^2}{x \sqrt {b x^3+a}}dx}{a}-\frac {5940 a e \sqrt {a+b x^3}}{x}}{2 a}-\frac {2079 d \sqrt {a+b x^3}}{x^2}}{2 a}-\frac {385 c \sqrt {a+b x^3}}{a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{3 x^4}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 2371 |
| \(\displaystyle \frac {1}{385} a \left (-a \left (\frac {-\frac {\frac {770 a^2 (2 a f+3 b c) \int \frac {1}{x \sqrt {b x^3+a}}dx+\int \frac {189 (11 b d+4 a g) a^2+2970 b e x a^2}{\sqrt {b x^3+a}}dx}{a}-\frac {5940 a e \sqrt {a+b x^3}}{x}}{2 a}-\frac {2079 d \sqrt {a+b x^3}}{x^2}}{2 a}-\frac {385 c \sqrt {a+b x^3}}{a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{3 x^4}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {1}{385} a \left (-a \left (\frac {-\frac {\frac {\frac {770}{3} a^2 (2 a f+3 b c) \int \frac {1}{x^3 \sqrt {b x^3+a}}dx^3+\int \frac {189 (11 b d+4 a g) a^2+2970 b e x a^2}{\sqrt {b x^3+a}}dx}{a}-\frac {5940 a e \sqrt {a+b x^3}}{x}}{2 a}-\frac {2079 d \sqrt {a+b x^3}}{x^2}}{2 a}-\frac {385 c \sqrt {a+b x^3}}{a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{3 x^4}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{385} a \left (-a \left (\frac {-\frac {\frac {\frac {1540 a^2 (2 a f+3 b c) \int \frac {1}{\frac {x^6}{b}-\frac {a}{b}}d\sqrt {b x^3+a}}{3 b}+\int \frac {189 (11 b d+4 a g) a^2+2970 b e x a^2}{\sqrt {b x^3+a}}dx}{a}-\frac {5940 a e \sqrt {a+b x^3}}{x}}{2 a}-\frac {2079 d \sqrt {a+b x^3}}{x^2}}{2 a}-\frac {385 c \sqrt {a+b x^3}}{a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{3 x^4}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{385} a \left (-a \left (\frac {-\frac {\frac {\int \frac {189 (11 b d+4 a g) a^2+2970 b e x a^2}{\sqrt {b x^3+a}}dx-\frac {1540}{3} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (2 a f+3 b c)}{a}-\frac {5940 a e \sqrt {a+b x^3}}{x}}{2 a}-\frac {2079 d \sqrt {a+b x^3}}{x^2}}{2 a}-\frac {385 c \sqrt {a+b x^3}}{a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{3 x^4}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {1}{385} a \left (-a \left (\frac {-\frac {\frac {27 a^2 \left (-110 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+28 a g+77 b d\right ) \int \frac {1}{\sqrt {b x^3+a}}dx+2970 a^2 b^{2/3} e \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx-\frac {1540}{3} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (2 a f+3 b c)}{a}-\frac {5940 a e \sqrt {a+b x^3}}{x}}{2 a}-\frac {2079 d \sqrt {a+b x^3}}{x^2}}{2 a}-\frac {385 c \sqrt {a+b x^3}}{a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{3 x^4}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {1}{385} a \left (-a \left (\frac {-\frac {\frac {2970 a^2 b^{2/3} e \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx-\frac {1540}{3} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (2 a f+3 b c)+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (-110 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+28 a g+77 b d\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}}{a}-\frac {5940 a e \sqrt {a+b x^3}}{x}}{2 a}-\frac {2079 d \sqrt {a+b x^3}}{x^2}}{2 a}-\frac {385 c \sqrt {a+b x^3}}{a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{3 x^4}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {1}{385} a \left (-a \left (\frac {-\frac {\frac {-\frac {1540}{3} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (2 a f+3 b c)+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (-110 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+28 a g+77 b d\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+2970 a^2 b^{2/3} e \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{a}-\frac {5940 a e \sqrt {a+b x^3}}{x}}{2 a}-\frac {2079 d \sqrt {a+b x^3}}{x^2}}{2 a}-\frac {385 c \sqrt {a+b x^3}}{a x^3}\right )-\frac {2 \sqrt {a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{3 x^4}\right )+\frac {2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}\) |
Input:
Int[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^4,x]
Output:
(2*(a + b*x^3)^(3/2)*(1155*c*x + 693*d*x^2 + 495*e*x^3 + 385*f*x^4 + 315*g *x^5))/(3465*x^4) + (a*((-2*Sqrt[a + b*x^3]*(1155*c*x + 2079*d*x^2 - 1485* e*x^3 - 385*f*x^4 - 189*g*x^5))/(3*x^4) - a*((-385*c*Sqrt[a + b*x^3])/(a*x ^3) + ((-2079*d*Sqrt[a + b*x^3])/x^2 - ((-5940*a*e*Sqrt[a + b*x^3])/x + (( -1540*a^(3/2)*(3*b*c + 2*a*f)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/3 + 2970*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*(77*b*d - 110*(1 - Sqrt[3])*a^(1/3)*b^(2/3)* e + 28*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 - Sqr t[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sq rt[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)/(2*a))/(2*a))))/385
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 = 1.46 (sec) , antiderivative size = 920, normalized size of antiderivative = 1.33
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(920\) |
| default | \(\text {Expression too large to display}\) | \(1193\) |
| risch | \(\text {Expression too large to display}\) | \(2513\) |
Input:
int((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^4,x,method=_RETURNVERBOSE)
Output:
-1/3*a*c*(b*x^3+a)^(1/2)/x^3-1/2*a*d*(b*x^3+a)^(1/2)/x^2-a*e*(b*x^3+a)^(1/ 2)/x+2/11*b*g*x^4*(b*x^3+a)^(1/2)+2/9*f*b*x^3*(b*x^3+a)^(1/2)+2/7*b*e*x^2* (b*x^3+a)^(1/2)+2/5*(14/11*a*b*g+b^2*d)/b*x*(b*x^3+a)^(1/2)+2/3*(4/3*a*b*f +b^2*c)/b*(b*x^3+a)^(1/2)-2/3*I*(a^2*g+7/4*d*a*b-2/5*(14/11*a*b*g+b^2*d)/b *a)*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))- 9/7*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)*((-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*(...
Time = 0.38 (sec) , antiderivative size = 444, normalized size of antiderivative = 0.64 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^4} \, dx=\left [-\frac {53460 \, a b^{\frac {3}{2}} e x^{3} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - 1155 \, {\left (3 \, b^{2} c + 2 \, a b f\right )} \sqrt {a} x^{3} \log \left (-\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 3402 \, {\left (11 \, a b d + 4 \, a^{2} g\right )} \sqrt {b} x^{3} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 2 \, {\left (1260 \, b^{2} g x^{7} + 1540 \, b^{2} f x^{6} + 1980 \, b^{2} e x^{5} - 6930 \, a b e x^{2} + 252 \, {\left (11 \, b^{2} d + 14 \, a b g\right )} x^{4} - 3465 \, a b d x + 1540 \, {\left (3 \, b^{2} c + 4 \, a b f\right )} x^{3} - 2310 \, a b c\right )} \sqrt {b x^{3} + a}}{13860 \, b x^{3}}, -\frac {26730 \, a b^{\frac {3}{2}} e x^{3} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - 1155 \, {\left (3 \, b^{2} c + 2 \, a b f\right )} \sqrt {-a} x^{3} \arctan \left (\frac {{\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {-a}}{2 \, {\left (a b x^{3} + a^{2}\right )}}\right ) - 1701 \, {\left (11 \, a b d + 4 \, a^{2} g\right )} \sqrt {b} x^{3} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - {\left (1260 \, b^{2} g x^{7} + 1540 \, b^{2} f x^{6} + 1980 \, b^{2} e x^{5} - 6930 \, a b e x^{2} + 252 \, {\left (11 \, b^{2} d + 14 \, a b g\right )} x^{4} - 3465 \, a b d x + 1540 \, {\left (3 \, b^{2} c + 4 \, a b f\right )} x^{3} - 2310 \, a b c\right )} \sqrt {b x^{3} + a}}{6930 \, b x^{3}}\right ] \] Input:
integrate((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^4,x, algorithm="fric as")
Output:
[-1/13860*(53460*a*b^(3/2)*e*x^3*weierstrassZeta(0, -4*a/b, weierstrassPIn verse(0, -4*a/b, x)) - 1155*(3*b^2*c + 2*a*b*f)*sqrt(a)*x^3*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) - 3402 *(11*a*b*d + 4*a^2*g)*sqrt(b)*x^3*weierstrassPInverse(0, -4*a/b, x) - 2*(1 260*b^2*g*x^7 + 1540*b^2*f*x^6 + 1980*b^2*e*x^5 - 6930*a*b*e*x^2 + 252*(11 *b^2*d + 14*a*b*g)*x^4 - 3465*a*b*d*x + 1540*(3*b^2*c + 4*a*b*f)*x^3 - 231 0*a*b*c)*sqrt(b*x^3 + a))/(b*x^3), -1/6930*(26730*a*b^(3/2)*e*x^3*weierstr assZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) - 1155*(3*b^2*c + 2* a*b*f)*sqrt(-a)*x^3*arctan(1/2*(b*x^3 + 2*a)*sqrt(b*x^3 + a)*sqrt(-a)/(a*b *x^3 + a^2)) - 1701*(11*a*b*d + 4*a^2*g)*sqrt(b)*x^3*weierstrassPInverse(0 , -4*a/b, x) - (1260*b^2*g*x^7 + 1540*b^2*f*x^6 + 1980*b^2*e*x^5 - 6930*a* b*e*x^2 + 252*(11*b^2*d + 14*a*b*g)*x^4 - 3465*a*b*d*x + 1540*(3*b^2*c + 4 *a*b*f)*x^3 - 2310*a*b*c)*sqrt(b*x^3 + a))/(b*x^3)]
Time = 6.03 (sec) , antiderivative size = 484, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^4} \, dx =\text {Too large to display} \] Input:
integrate((b*x**3+a)**(3/2)*(g*x**4+f*x**3+e*x**2+d*x+c)/x**4,x)
Output:
a**(3/2)*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)) + a**(3/2)*e*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)*f*asinh(sqrt(a)/ (sqrt(b)*x**(3/2)))/3 + a**(3/2)*g*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/3)) - sqrt(a)*b*c*asinh(sqrt(a)/(sqr t(b)*x**(3/2))) + sqrt(a)*b*d*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), b*x* *3*exp_polar(I*pi)/a)/(3*gamma(4/3)) + sqrt(a)*b*e*x**2*gamma(2/3)*hyper(( -1/2, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(5/3)) + sqrt(a)*b*g *x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*g amma(7/3)) + 2*a**2*f/(3*sqrt(b)*x**(3/2)*sqrt(a/(b*x**3) + 1)) - a*sqrt(b )*c*sqrt(a/(b*x**3) + 1)/(3*x**(3/2)) + 2*a*sqrt(b)*c/(3*x**(3/2)*sqrt(a/( b*x**3) + 1)) + 2*a*sqrt(b)*f*x**(3/2)/(3*sqrt(a/(b*x**3) + 1)) + 2*b**(3/ 2)*c*x**(3/2)/(3*sqrt(a/(b*x**3) + 1)) + b*f*Piecewise((sqrt(a)*x**3/3, Eq (b, 0)), (2*(a + b*x**3)**(3/2)/(9*b), True))
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^4} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:
integrate((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^4,x, algorithm="maxi ma")
Output:
integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^4, x)
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^4} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:
integrate((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^4,x, algorithm="giac ")
Output:
integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^4, x)
Timed out. \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^4} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^4} \,d x \] Input:
int(((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^4,x)
Output:
int(((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^4, x)
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^4} \, dx=\frac {-6804 \sqrt {b \,x^{3}+a}\, a^{2} g x -2310 \sqrt {b \,x^{3}+a}\, a b c -22176 \sqrt {b \,x^{3}+a}\, a b d x +19800 \sqrt {b \,x^{3}+a}\, a b e \,x^{2}+6160 \sqrt {b \,x^{3}+a}\, a b f \,x^{3}+3528 \sqrt {b \,x^{3}+a}\, a b g \,x^{4}+4620 \sqrt {b \,x^{3}+a}\, b^{2} c \,x^{3}+2772 \sqrt {b \,x^{3}+a}\, b^{2} d \,x^{4}+1980 \sqrt {b \,x^{3}+a}\, b^{2} e \,x^{5}+1540 \sqrt {b \,x^{3}+a}\, b^{2} f \,x^{6}+1260 \sqrt {b \,x^{3}+a}\, b^{2} g \,x^{7}+2310 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {a}\right ) a b f \,x^{3}+3465 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {a}\right ) b^{2} c \,x^{3}-2310 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {a}\right ) a b f \,x^{3}-3465 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {a}\right ) b^{2} c \,x^{3}-13608 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{6}+a \,x^{3}}d x \right ) a^{3} g \,x^{3}-37422 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{6}+a \,x^{3}}d x \right ) a^{2} b d \,x^{3}+26730 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{5}+a \,x^{2}}d x \right ) a^{2} b e \,x^{3}}{6930 b \,x^{3}} \] Input:
int((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^4,x)
Output:
( - 6804*sqrt(a + b*x**3)*a**2*g*x - 2310*sqrt(a + b*x**3)*a*b*c - 22176*s qrt(a + b*x**3)*a*b*d*x + 19800*sqrt(a + b*x**3)*a*b*e*x**2 + 6160*sqrt(a + b*x**3)*a*b*f*x**3 + 3528*sqrt(a + b*x**3)*a*b*g*x**4 + 4620*sqrt(a + b* x**3)*b**2*c*x**3 + 2772*sqrt(a + b*x**3)*b**2*d*x**4 + 1980*sqrt(a + b*x* *3)*b**2*e*x**5 + 1540*sqrt(a + b*x**3)*b**2*f*x**6 + 1260*sqrt(a + b*x**3 )*b**2*g*x**7 + 2310*sqrt(a)*log(sqrt(a + b*x**3) - sqrt(a))*a*b*f*x**3 + 3465*sqrt(a)*log(sqrt(a + b*x**3) - sqrt(a))*b**2*c*x**3 - 2310*sqrt(a)*lo g(sqrt(a + b*x**3) + sqrt(a))*a*b*f*x**3 - 3465*sqrt(a)*log(sqrt(a + b*x** 3) + sqrt(a))*b**2*c*x**3 - 13608*int(sqrt(a + b*x**3)/(a*x**3 + b*x**6),x )*a**3*g*x**3 - 37422*int(sqrt(a + b*x**3)/(a*x**3 + b*x**6),x)*a**2*b*d*x **3 + 26730*int(sqrt(a + b*x**3)/(a*x**2 + b*x**5),x)*a**2*b*e*x**3)/(6930 *b*x**3)