Integrand size = 35, antiderivative size = 694 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx=\frac {27 a c \sqrt {a+b x^3}}{10 x^2}-\frac {27 a d \sqrt {a+b x^3}}{7 x}+\frac {27 a (13 b d+2 a g) \sqrt {a+b x^3}}{91 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 a \sqrt {a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{15015 x^3}+\frac {2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}-\frac {2}{3} a^{3/2} e \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} (13 b d+2 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{182 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} a \left (91 \sqrt [3]{b} (11 b c+4 a f)-110 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (13 b d+2 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{10010 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
27/10*a*c*(b*x^3+a)^(1/2)/x^2-27/7*a*d*(b*x^3+a)^(1/2)/x+27/91*a*(2*a*g+13 *b*d)*(b*x^3+a)^(1/2)/b^(2/3)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)-2/15015*a*(b *x^3+a)^(1/2)*(-1485*g*x^5-2457*f*x^4-5005*e*x^3-19305*d*x^2+27027*c*x)/x^ 3+2/45045*(b*x^3+a)^(3/2)*(3465*g*x^5+4095*f*x^4+5005*e*x^3+6435*d*x^2+900 9*c*x)/x^3-2/3*a^(3/2)*e*arctanh((b*x^3+a)^(1/2)/a^(1/2))-27/182*3^(1/4)*( 1/2*6^(1/2)-1/2*2^(1/2))*a^(4/3)*(2*a*g+13*b*d)*(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)/b^(2/3)/(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)+9/10010*3^(3/4)*(1/2*6^(1/2)+1/2*2^(1 /2))*a*(91*b^(1/3)*(4*a*f+11*b*c)-110*(1-3^(1/2))*a^(1/3)*(2*a*g+13*b*d))* (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^(2/3)/(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 = 9.67 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.33 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx=\frac {4 e x^2 \sqrt {1+\frac {b x^3}{a}} \left (\sqrt {a+b x^3} \left (4 a+b x^3\right )-3 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\right )-9 a c \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {2}{3},\frac {1}{3},-\frac {b x^3}{a}\right )-18 a d x \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{3},\frac {2}{3},-\frac {b x^3}{a}\right )+18 a f x^3 \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )+9 a g x^4 \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{18 x^2 \sqrt {1+\frac {b x^3}{a}}} \] Input:
Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^3,x]
Output:
(4*e*x^2*Sqrt[1 + (b*x^3)/a]*(Sqrt[a + b*x^3]*(4*a + b*x^3) - 3*a^(3/2)*Ar cTanh[Sqrt[a + b*x^3]/Sqrt[a]]) - 9*a*c*Sqrt[a + b*x^3]*Hypergeometric2F1[ -3/2, -2/3, 1/3, -((b*x^3)/a)] - 18*a*d*x*Sqrt[a + b*x^3]*Hypergeometric2F 1[-3/2, -1/3, 2/3, -((b*x^3)/a)] + 18*a*f*x^3*Sqrt[a + b*x^3]*Hypergeometr ic2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)] + 9*a*g*x^4*Sqrt[a + b*x^3]*Hypergeome tric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)])/(18*x^2*Sqrt[1 + (b*x^3)/a])
Time = 2.26 (sec) , antiderivative size = 708, normalized size of antiderivative = 1.02, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2365, 27, 2365, 27, 2374, 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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 2365 |
| \(\displaystyle \frac {9}{2} a \int \frac {2 \sqrt {b x^3+a} \left (3465 g x^4+4095 f x^3+5005 e x^2+6435 d x+9009 c\right )}{45045 x^3}dx+\frac {2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \int \frac {\sqrt {b x^3+a} \left (3465 g x^4+4095 f x^3+5005 e x^2+6435 d x+9009 c\right )}{x^3}dx}{5005}+\frac {2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}\) |
| \(\Big \downarrow \) 2365 |
| \(\displaystyle \frac {a \left (\frac {3}{2} a \int -\frac {2 \left (-1485 g x^4-2457 f x^3-5005 e x^2-19305 d x+27027 c\right )}{3 x^3 \sqrt {b x^3+a}}dx-\frac {2 \sqrt {a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{3 x^3}\right )}{5005}+\frac {2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (-a \int \frac {-1485 g x^4-2457 f x^3-5005 e x^2-19305 d x+27027 c}{x^3 \sqrt {b x^3+a}}dx-\frac {2 \sqrt {a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{3 x^3}\right )}{5005}+\frac {2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {a \left (-a \left (-\frac {\int \frac {5940 a g x^3+2457 (11 b c+4 a f) x^2+20020 a e x+77220 a d}{x^2 \sqrt {b x^3+a}}dx}{4 a}-\frac {27027 c \sqrt {a+b x^3}}{2 a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{3 x^3}\right )}{5005}+\frac {2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {a \left (-a \left (-\frac {-\frac {\int -\frac {2 \left (20020 e a^2+2970 (13 b d+2 a g) x^2 a+2457 (11 b c+4 a f) x a\right )}{x \sqrt {b x^3+a}}dx}{2 a}-\frac {77220 d \sqrt {a+b x^3}}{x}}{4 a}-\frac {27027 c \sqrt {a+b x^3}}{2 a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{3 x^3}\right )}{5005}+\frac {2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (-a \left (-\frac {\frac {\int \frac {20020 e a^2+2970 (13 b d+2 a g) x^2 a+2457 (11 b c+4 a f) x a}{x \sqrt {b x^3+a}}dx}{a}-\frac {77220 d \sqrt {a+b x^3}}{x}}{4 a}-\frac {27027 c \sqrt {a+b x^3}}{2 a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{3 x^3}\right )}{5005}+\frac {2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}\) |
| \(\Big \downarrow \) 2371 |
| \(\displaystyle \frac {a \left (-a \left (-\frac {\frac {20020 a^2 e \int \frac {1}{x \sqrt {b x^3+a}}dx+\int \frac {2457 a (11 b c+4 a f)+2970 a (13 b d+2 a g) x}{\sqrt {b x^3+a}}dx}{a}-\frac {77220 d \sqrt {a+b x^3}}{x}}{4 a}-\frac {27027 c \sqrt {a+b x^3}}{2 a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{3 x^3}\right )}{5005}+\frac {2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {a \left (-a \left (-\frac {\frac {\frac {20020}{3} a^2 e \int \frac {1}{x^3 \sqrt {b x^3+a}}dx^3+\int \frac {2457 a (11 b c+4 a f)+2970 a (13 b d+2 a g) x}{\sqrt {b x^3+a}}dx}{a}-\frac {77220 d \sqrt {a+b x^3}}{x}}{4 a}-\frac {27027 c \sqrt {a+b x^3}}{2 a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{3 x^3}\right )}{5005}+\frac {2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {a \left (-a \left (-\frac {\frac {\frac {40040 a^2 e \int \frac {1}{\frac {x^6}{b}-\frac {a}{b}}d\sqrt {b x^3+a}}{3 b}+\int \frac {2457 a (11 b c+4 a f)+2970 a (13 b d+2 a g) x}{\sqrt {b x^3+a}}dx}{a}-\frac {77220 d \sqrt {a+b x^3}}{x}}{4 a}-\frac {27027 c \sqrt {a+b x^3}}{2 a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{3 x^3}\right )}{5005}+\frac {2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a \left (-a \left (-\frac {\frac {\int \frac {2457 a (11 b c+4 a f)+2970 a (13 b d+2 a g) x}{\sqrt {b x^3+a}}dx-\frac {40040}{3} a^{3/2} e \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{a}-\frac {77220 d \sqrt {a+b x^3}}{x}}{4 a}-\frac {27027 c \sqrt {a+b x^3}}{2 a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{3 x^3}\right )}{5005}+\frac {2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {a \left (-a \left (-\frac {\frac {27 a \left (91 (4 a f+11 b c)-\frac {110 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (2 a g+13 b d)}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt {b x^3+a}}dx+\frac {2970 a (2 a g+13 b d) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-\frac {40040}{3} a^{3/2} e \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{a}-\frac {77220 d \sqrt {a+b x^3}}{x}}{4 a}-\frac {27027 c \sqrt {a+b x^3}}{2 a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{3 x^3}\right )}{5005}+\frac {2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {a \left (-a \left (-\frac {\frac {\frac {2970 a (2 a g+13 b d) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (91 (4 a f+11 b c)-\frac {110 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (2 a g+13 b d)}{\sqrt [3]{b}}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {40040}{3} a^{3/2} e \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{a}-\frac {77220 d \sqrt {a+b x^3}}{x}}{4 a}-\frac {27027 c \sqrt {a+b x^3}}{2 a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{3 x^3}\right )}{5005}+\frac {2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {a \left (-a \left (-\frac {\frac {\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (91 (4 a f+11 b c)-\frac {110 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (2 a g+13 b d)}{\sqrt [3]{b}}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2970 a (2 a g+13 b d) \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{\sqrt [3]{b}}-\frac {40040}{3} a^{3/2} e \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{a}-\frac {77220 d \sqrt {a+b x^3}}{x}}{4 a}-\frac {27027 c \sqrt {a+b x^3}}{2 a x^2}\right )-\frac {2 \sqrt {a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{3 x^3}\right )}{5005}+\frac {2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}\) |
Input:
Int[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^3,x]
Output:
(2*(a + b*x^3)^(3/2)*(9009*c*x + 6435*d*x^2 + 5005*e*x^3 + 4095*f*x^4 + 34 65*g*x^5))/(45045*x^3) + (a*((-2*Sqrt[a + b*x^3]*(27027*c*x - 19305*d*x^2 - 5005*e*x^3 - 2457*f*x^4 - 1485*g*x^5))/(3*x^3) - a*((-27027*c*Sqrt[a + b *x^3])/(2*a*x^2) - ((-77220*d*Sqrt[a + b*x^3])/x + ((-40040*a^(3/2)*e*ArcT anh[Sqrt[a + b*x^3]/Sqrt[a]])/3 + (2970*a*(13*b*d + 2*a*g)*((2*Sqrt[a + b* x^3])/(b^(1/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (3^(1/4)*Sqrt[2 - Sq rt[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])))/b^(1/3) + (18*3^(3/4)*Sqrt[2 + Sq rt[3]]*a*(91*(11*b*c + 4*a*f) - (110*(1 - Sqrt[3])*a^(1/3)*(13*b*d + 2*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 - 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)/(4*a))))/5005
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 = 1.93 (sec) , antiderivative size = 941, normalized size of antiderivative = 1.36
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(941\) |
| default | \(\text {Expression too large to display}\) | \(1613\) |
| risch | \(\text {Expression too large to display}\) | \(3858\) |
Input:
int((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/2*a*c*(b*x^3+a)^(1/2)/x^2-a*d*(b*x^3+a)^(1/2)/x+2/13*b*g*x^5*(b*x^3+a)^ (1/2)+2/11*f*b*x^4*(b*x^3+a)^(1/2)+2/9*b*e*x^3*(b*x^3+a)^(1/2)+2/7*(16/13* a*b*g+b^2*d)/b*x^2*(b*x^3+a)^(1/2)+2/5*(14/11*a*b*f+b^2*c)/b*x*(b*x^3+a)^( 1/2)+8/9*a*e*(b*x^3+a)^(1/2)-2/3*I*(f*a^2+7/4*a*b*c-2/5*(14/11*a*b*f+b^2*c )/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 ))-2/3*I*(a^2*g+5/2*d*a*b-4/7*(16/13*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)*((-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/...
Time = 0.22 (sec) , antiderivative size = 443, normalized size of antiderivative = 0.64 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx=\left [\frac {15015 \, a^{\frac {3}{2}} b e x^{2} \log \left (-\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 22113 \, {\left (11 \, a b c + 4 \, a^{2} f\right )} \sqrt {b} x^{2} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 26730 \, {\left (13 \, a b d + 2 \, a^{2} g\right )} \sqrt {b} x^{2} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (13860 \, b^{2} g x^{7} + 16380 \, b^{2} f x^{6} + 20020 \, b^{2} e x^{5} + 80080 \, a b e x^{2} + 1980 \, {\left (13 \, b^{2} d + 16 \, a b g\right )} x^{4} - 90090 \, a b d x + 3276 \, {\left (11 \, b^{2} c + 14 \, a b f\right )} x^{3} - 45045 \, a b c\right )} \sqrt {b x^{3} + a}}{90090 \, b x^{2}}, \frac {30030 \, \sqrt {-a} a b e x^{2} \arctan \left (\frac {{\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {-a}}{2 \, {\left (a b x^{3} + a^{2}\right )}}\right ) + 22113 \, {\left (11 \, a b c + 4 \, a^{2} f\right )} \sqrt {b} x^{2} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 26730 \, {\left (13 \, a b d + 2 \, a^{2} g\right )} \sqrt {b} x^{2} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (13860 \, b^{2} g x^{7} + 16380 \, b^{2} f x^{6} + 20020 \, b^{2} e x^{5} + 80080 \, a b e x^{2} + 1980 \, {\left (13 \, b^{2} d + 16 \, a b g\right )} x^{4} - 90090 \, a b d x + 3276 \, {\left (11 \, b^{2} c + 14 \, a b f\right )} x^{3} - 45045 \, a b c\right )} \sqrt {b x^{3} + a}}{90090 \, b x^{2}}\right ] \] Input:
integrate((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^3,x, algorithm="fric as")
Output:
[1/90090*(15015*a^(3/2)*b*e*x^2*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) + 22113*(11*a*b*c + 4*a^2*f)*sqrt( b)*x^2*weierstrassPInverse(0, -4*a/b, x) - 26730*(13*a*b*d + 2*a^2*g)*sqrt (b)*x^2*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (1 3860*b^2*g*x^7 + 16380*b^2*f*x^6 + 20020*b^2*e*x^5 + 80080*a*b*e*x^2 + 198 0*(13*b^2*d + 16*a*b*g)*x^4 - 90090*a*b*d*x + 3276*(11*b^2*c + 14*a*b*f)*x ^3 - 45045*a*b*c)*sqrt(b*x^3 + a))/(b*x^2), 1/90090*(30030*sqrt(-a)*a*b*e* x^2*arctan(1/2*(b*x^3 + 2*a)*sqrt(b*x^3 + a)*sqrt(-a)/(a*b*x^3 + a^2)) + 2 2113*(11*a*b*c + 4*a^2*f)*sqrt(b)*x^2*weierstrassPInverse(0, -4*a/b, x) - 26730*(13*a*b*d + 2*a^2*g)*sqrt(b)*x^2*weierstrassZeta(0, -4*a/b, weierstr assPInverse(0, -4*a/b, x)) + (13860*b^2*g*x^7 + 16380*b^2*f*x^6 + 20020*b^ 2*e*x^5 + 80080*a*b*e*x^2 + 1980*(13*b^2*d + 16*a*b*g)*x^4 - 90090*a*b*d*x + 3276*(11*b^2*c + 14*a*b*f)*x^3 - 45045*a*b*c)*sqrt(b*x^3 + a))/(b*x^2)]
Time = 4.88 (sec) , antiderivative size = 462, normalized size of antiderivative = 0.67 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx =\text {Too large to display} \] Input:
integrate((b*x**3+a)**(3/2)*(g*x**4+f*x**3+e*x**2+d*x+c)/x**3,x)
Output:
a**(3/2)*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)) + a**(3/2)*d*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)*e*asinh(sqrt(a)/ (sqrt(b)*x**(3/2)))/3 + a**(3/2)*f*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/3)) + a**(3/2)*g*x**2*gamma(2/3)*hyp er((-1/2, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(5/3)) + sqrt(a) *b*c*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*d*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*f*x**4*gamma(4/3)*hyper((- 1/2, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(7/3)) + sqrt(a)*b*g* x**5*gamma(5/3)*hyper((-1/2, 5/3), (8/3,), b*x**3*exp_polar(I*pi)/a)/(3*ga mma(8/3)) + 2*a**2*e/(3*sqrt(b)*x**(3/2)*sqrt(a/(b*x**3) + 1)) + 2*a*sqrt( b)*e*x**(3/2)/(3*sqrt(a/(b*x**3) + 1)) + b*e*Piecewise((sqrt(a)*x**3/3, Eq (b, 0)), (2*(a + b*x**3)**(3/2)/(9*b), True))
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{3}} \,d x } \] Input:
integrate((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^3,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^3, x)
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{3}} \,d x } \] Input:
integrate((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^3,x, algorithm="giac ")
Output:
integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^3, x)
Timed out. \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^3} \,d x \] Input:
int(((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^3,x)
Output:
int(((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^3, x)
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx=\frac {-44226 \sqrt {b \,x^{3}+a}\, a^{2} f +26730 \sqrt {b \,x^{3}+a}\, a^{2} g x -144144 \sqrt {b \,x^{3}+a}\, a b c +128700 \sqrt {b \,x^{3}+a}\, a b d x +40040 \sqrt {b \,x^{3}+a}\, a b e \,x^{2}+22932 \sqrt {b \,x^{3}+a}\, a b f \,x^{3}+15840 \sqrt {b \,x^{3}+a}\, a b g \,x^{4}+18018 \sqrt {b \,x^{3}+a}\, b^{2} c \,x^{3}+12870 \sqrt {b \,x^{3}+a}\, b^{2} d \,x^{4}+10010 \sqrt {b \,x^{3}+a}\, b^{2} e \,x^{5}+8190 \sqrt {b \,x^{3}+a}\, b^{2} f \,x^{6}+6930 \sqrt {b \,x^{3}+a}\, b^{2} g \,x^{7}+15015 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {a}\right ) a b e \,x^{2}-15015 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {a}\right ) a b e \,x^{2}-88452 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{6}+a \,x^{3}}d x \right ) a^{3} f \,x^{2}-243243 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{6}+a \,x^{3}}d x \right ) a^{2} b c \,x^{2}+26730 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{5}+a \,x^{2}}d x \right ) a^{3} g \,x^{2}+173745 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{5}+a \,x^{2}}d x \right ) a^{2} b d \,x^{2}}{45045 b \,x^{2}} \] Input:
int((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^3,x)
Output:
( - 44226*sqrt(a + b*x**3)*a**2*f + 26730*sqrt(a + b*x**3)*a**2*g*x - 1441 44*sqrt(a + b*x**3)*a*b*c + 128700*sqrt(a + b*x**3)*a*b*d*x + 40040*sqrt(a + b*x**3)*a*b*e*x**2 + 22932*sqrt(a + b*x**3)*a*b*f*x**3 + 15840*sqrt(a + b*x**3)*a*b*g*x**4 + 18018*sqrt(a + b*x**3)*b**2*c*x**3 + 12870*sqrt(a + b*x**3)*b**2*d*x**4 + 10010*sqrt(a + b*x**3)*b**2*e*x**5 + 8190*sqrt(a + b *x**3)*b**2*f*x**6 + 6930*sqrt(a + b*x**3)*b**2*g*x**7 + 15015*sqrt(a)*log (sqrt(a + b*x**3) - sqrt(a))*a*b*e*x**2 - 15015*sqrt(a)*log(sqrt(a + b*x** 3) + sqrt(a))*a*b*e*x**2 - 88452*int(sqrt(a + b*x**3)/(a*x**3 + b*x**6),x) *a**3*f*x**2 - 243243*int(sqrt(a + b*x**3)/(a*x**3 + b*x**6),x)*a**2*b*c*x **2 + 26730*int(sqrt(a + b*x**3)/(a*x**2 + b*x**5),x)*a**3*g*x**2 + 173745 *int(sqrt(a + b*x**3)/(a*x**2 + b*x**5),x)*a**2*b*d*x**2)/(45045*b*x**2)