\(\int \frac {(a+b x^3)^{4/3}}{x^7 (c+d x^3)} \, dx\) [716]

Optimal result

Integrand size = 24, antiderivative size = 403 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^7 \left (c+d x^3\right )} \, dx=\frac {b (2 b c-3 a d) \sqrt [3]{a+b x^3}}{9 a c^2}-\frac {(b c-6 a d) \left (a+b x^3\right )^{4/3}}{18 a c^2 x^3}-\frac {\left (a+b x^3\right )^{7/3}}{6 a c x^6}-\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{2/3} c^3}+\frac {d^{2/3} (b c-a d)^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c^3}-\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \log (x)}{18 a^{2/3} c^3}+\frac {d^{2/3} (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^3}+\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{2/3} c^3}-\frac {d^{2/3} (b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^3} \] Output:

1/9*b*(-3*a*d+2*b*c)*(b*x^3+a)^(1/3)/a/c^2-1/18*(-6*a*d+b*c)*(b*x^3+a)^(4/ 
3)/a/c^2/x^3-1/6*(b*x^3+a)^(7/3)/a/c/x^6-1/27*(9*a^2*d^2-12*a*b*c*d+2*b^2* 
c^2)*arctan(1/3*(a^(1/3)+2*(b*x^3+a)^(1/3))*3^(1/2)/a^(1/3))*3^(1/2)/a^(2/ 
3)/c^3+1/3*d^(2/3)*(-a*d+b*c)^(4/3)*arctan(1/3*(1-2*d^(1/3)*(b*x^3+a)^(1/3 
)/(-a*d+b*c)^(1/3))*3^(1/2))*3^(1/2)/c^3-1/18*(9*a^2*d^2-12*a*b*c*d+2*b^2* 
c^2)*ln(x)/a^(2/3)/c^3+1/6*d^(2/3)*(-a*d+b*c)^(4/3)*ln(d*x^3+c)/c^3+1/18*( 
9*a^2*d^2-12*a*b*c*d+2*b^2*c^2)*ln(a^(1/3)-(b*x^3+a)^(1/3))/a^(2/3)/c^3-1/ 
2*d^(2/3)*(-a*d+b*c)^(4/3)*ln((-a*d+b*c)^(1/3)+d^(1/3)*(b*x^3+a)^(1/3))/c^ 
3
 

Mathematica [A] (verified)

Time = 1.58 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^7 \left (c+d x^3\right )} \, dx=\frac {\frac {3 c \sqrt [3]{a+b x^3} \left (-3 a c-7 b c x^3+6 a d x^3\right )}{x^6}-\frac {2 \sqrt {3} \left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}+18 \sqrt {3} d^{2/3} (b c-a d)^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )+\frac {2 \left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )}{a^{2/3}}-18 d^{2/3} (b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )+\frac {\left (-2 b^2 c^2+12 a b c d-9 a^2 d^2\right ) \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{a^{2/3}}+9 d^{2/3} (b c-a d)^{4/3} \log \left ((b c-a d)^{2/3}-\sqrt [3]{d} \sqrt [3]{b c-a d} \sqrt [3]{a+b x^3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )}{54 c^3} \] Input:

Integrate[(a + b*x^3)^(4/3)/(x^7*(c + d*x^3)),x]
 

Output:

((3*c*(a + b*x^3)^(1/3)*(-3*a*c - 7*b*c*x^3 + 6*a*d*x^3))/x^6 - (2*Sqrt[3] 
*(2*b^2*c^2 - 12*a*b*c*d + 9*a^2*d^2)*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^ 
(1/3))/Sqrt[3]])/a^(2/3) + 18*Sqrt[3]*d^(2/3)*(b*c - a*d)^(4/3)*ArcTan[(1 
- (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] + (2*(2*b^2*c^ 
2 - 12*a*b*c*d + 9*a^2*d^2)*Log[-a^(1/3) + (a + b*x^3)^(1/3)])/a^(2/3) - 1 
8*d^(2/3)*(b*c - a*d)^(4/3)*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1 
/3)] + ((-2*b^2*c^2 + 12*a*b*c*d - 9*a^2*d^2)*Log[a^(2/3) + a^(1/3)*(a + b 
*x^3)^(1/3) + (a + b*x^3)^(2/3)])/a^(2/3) + 9*d^(2/3)*(b*c - a*d)^(4/3)*Lo 
g[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3 
)*(a + b*x^3)^(2/3)])/(54*c^3)
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.97, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {948, 114, 27, 166, 27, 174, 60, 69, 16, 70, 16, 1082, 217}

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.

\(\displaystyle \int \frac {\left (a+b x^3\right )^{4/3}}{x^7 \left (c+d x^3\right )} \, dx\)

\(\Big \downarrow \) 948

\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^{4/3}}{x^9 \left (d x^3+c\right )}dx^3\)

\(\Big \downarrow \) 114

\(\displaystyle \frac {1}{3} \left (-\frac {\int -\frac {\left (b x^3+a\right )^{4/3} \left (b d x^3+b c-6 a d\right )}{3 x^6 \left (d x^3+c\right )}dx^3}{2 a c}-\frac {\left (a+b x^3\right )^{7/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {\int \frac {\left (b x^3+a\right )^{4/3} \left (b d x^3+b c-6 a d\right )}{x^6 \left (d x^3+c\right )}dx^3}{6 a c}-\frac {\left (a+b x^3\right )^{7/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{3} \left (\frac {\frac {\int \frac {2 \sqrt [3]{b x^3+a} \left (b d (2 b c-3 a d) x^3+2 b^2 c^2+9 a^2 d^2-12 a b c d\right )}{3 x^3 \left (d x^3+c\right )}dx^3}{c}-\frac {\left (a+b x^3\right )^{4/3} (b c-6 a d)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{7/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {\frac {2 \int \frac {\sqrt [3]{b x^3+a} \left (b d (2 b c-3 a d) x^3+2 b^2 c^2+9 a^2 d^2-12 a b c d\right )}{x^3 \left (d x^3+c\right )}dx^3}{3 c}-\frac {\left (a+b x^3\right )^{4/3} (b c-6 a d)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{7/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{3} \left (\frac {\frac {2 \left (\frac {\left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right ) \int \frac {\sqrt [3]{b x^3+a}}{x^3}dx^3}{c}+\frac {9 a d^2 (b c-a d) \int \frac {\sqrt [3]{b x^3+a}}{d x^3+c}dx^3}{c}\right )}{3 c}-\frac {\left (a+b x^3\right )^{4/3} (b c-6 a d)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{7/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{3} \left (\frac {\frac {2 \left (\frac {\left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right ) \left (a \int \frac {1}{x^3 \left (b x^3+a\right )^{2/3}}dx^3+3 \sqrt [3]{a+b x^3}\right )}{c}+\frac {9 a d^2 (b c-a d) \left (\frac {3 \sqrt [3]{a+b x^3}}{d}-\frac {(b c-a d) \int \frac {1}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx^3}{d}\right )}{c}\right )}{3 c}-\frac {\left (a+b x^3\right )^{4/3} (b c-6 a d)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{7/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 69

\(\displaystyle \frac {1}{3} \left (\frac {\frac {2 \left (\frac {\left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right ) \left (a \left (-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 a^{2/3}}-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )}{c}+\frac {9 a d^2 (b c-a d) \left (\frac {3 \sqrt [3]{a+b x^3}}{d}-\frac {(b c-a d) \int \frac {1}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx^3}{d}\right )}{c}\right )}{3 c}-\frac {\left (a+b x^3\right )^{4/3} (b c-6 a d)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{7/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (\frac {\frac {2 \left (\frac {\left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right ) \left (a \left (-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )}{c}+\frac {9 a d^2 (b c-a d) \left (\frac {3 \sqrt [3]{a+b x^3}}{d}-\frac {(b c-a d) \int \frac {1}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx^3}{d}\right )}{c}\right )}{3 c}-\frac {\left (a+b x^3\right )^{4/3} (b c-6 a d)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{7/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 70

\(\displaystyle \frac {1}{3} \left (\frac {\frac {2 \left (\frac {\left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right ) \left (a \left (-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )}{c}+\frac {9 a d^2 (b c-a d) \left (\frac {3 \sqrt [3]{a+b x^3}}{d}-\frac {(b c-a d) \left (\frac {3 \int \frac {1}{x^6+\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} \sqrt [3]{b x^3+a}}{\sqrt [3]{d}}}d\sqrt [3]{b x^3+a}}{2 d^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \int \frac {1}{\frac {\sqrt [3]{b c-a d}}{\sqrt [3]{d}}+\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{d} (b c-a d)^{2/3}}-\frac {\log \left (c+d x^3\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}\right )}{d}\right )}{c}\right )}{3 c}-\frac {\left (a+b x^3\right )^{4/3} (b c-6 a d)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{7/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (\frac {\frac {2 \left (\frac {\left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right ) \left (a \left (-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )}{c}+\frac {9 a d^2 (b c-a d) \left (\frac {3 \sqrt [3]{a+b x^3}}{d}-\frac {(b c-a d) \left (\frac {3 \int \frac {1}{x^6+\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} \sqrt [3]{b x^3+a}}{\sqrt [3]{d}}}d\sqrt [3]{b x^3+a}}{2 d^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (c+d x^3\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}\right )}{d}\right )}{c}\right )}{3 c}-\frac {\left (a+b x^3\right )^{4/3} (b c-6 a d)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{7/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{3} \left (\frac {\frac {2 \left (\frac {\left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right ) \left (a \left (\frac {3 \int \frac {1}{-x^6-3}d\left (\frac {2 \sqrt [3]{b x^3+a}}{\sqrt [3]{a}}+1\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )}{c}+\frac {9 a d^2 (b c-a d) \left (\frac {3 \sqrt [3]{a+b x^3}}{d}-\frac {(b c-a d) \left (\frac {3 \int \frac {1}{-x^6-3}d\left (1-\frac {2 \sqrt [3]{d} \sqrt [3]{b x^3+a}}{\sqrt [3]{b c-a d}}\right )}{\sqrt [3]{d} (b c-a d)^{2/3}}-\frac {\log \left (c+d x^3\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}\right )}{d}\right )}{c}\right )}{3 c}-\frac {\left (a+b x^3\right )^{4/3} (b c-6 a d)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{7/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{3} \left (\frac {\frac {2 \left (\frac {\left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right ) \left (a \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )}{c}+\frac {9 a d^2 (b c-a d) \left (\frac {3 \sqrt [3]{a+b x^3}}{d}-\frac {(b c-a d) \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt [3]{d} (b c-a d)^{2/3}}-\frac {\log \left (c+d x^3\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}\right )}{d}\right )}{c}\right )}{3 c}-\frac {\left (a+b x^3\right )^{4/3} (b c-6 a d)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{7/3}}{2 a c x^6}\right )\)

Input:

Int[(a + b*x^3)^(4/3)/(x^7*(c + d*x^3)),x]
 

Output:

(-1/2*(a + b*x^3)^(7/3)/(a*c*x^6) + (-(((b*c - 6*a*d)*(a + b*x^3)^(4/3))/( 
c*x^3)) + (2*(((2*b^2*c^2 - 12*a*b*c*d + 9*a^2*d^2)*(3*(a + b*x^3)^(1/3) + 
 a*(-((Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]])/a^(2/3 
)) - Log[x^3]/(2*a^(2/3)) + (3*Log[a^(1/3) - (a + b*x^3)^(1/3)])/(2*a^(2/3 
)))))/c + (9*a*d^2*(b*c - a*d)*((3*(a + b*x^3)^(1/3))/d - ((b*c - a*d)*(-( 
(Sqrt[3]*ArcTan[(1 - (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt 
[3]])/(d^(1/3)*(b*c - a*d)^(2/3))) - Log[c + d*x^3]/(2*d^(1/3)*(b*c - a*d) 
^(2/3)) + (3*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)])/(2*d^(1/3 
)*(b*c - a*d)^(2/3))))/d))/c))/(3*c))/(6*a*c))/3
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 69

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), 
 x] + (-Simp[3/(2*b*q)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] - Simp[3/(2*b*q^2)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], 
 x])] /; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

rule 70

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[-(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2) 
, x] + (Simp[3/(2*b*q)   Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] + Simp[3/(2*b*q^2)   Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], 
 x])] /; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 166

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ 
p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ 
[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

Maple [A] (verified)

Time = 2.08 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.11

method	result	size
pseudoelliptic	\(\frac {\frac {x^{6} \left (a^{\frac {8}{3}} d^{2}-2 a^{\frac {5}{3}} b c d +b^{2} c^{2} a^{\frac {2}{3}}\right ) \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, x^{6} \left (a^{\frac {8}{3}} d^{2}-2 a^{\frac {5}{3}} b c d +b^{2} c^{2} a^{\frac {2}{3}}\right ) \arctan \left (\frac {2 \sqrt {3}\, \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{3 \left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}}{3}\right )-\frac {\left (\frac {a d -b c}{d}\right )^{\frac {2}{3}} x^{6} \left (a^{2} d^{2}-\frac {4}{3} a b c d +\frac {2}{9} b^{2} c^{2}\right ) \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2}-x^{6} \left (a^{\frac {8}{3}} d^{2}-2 a^{\frac {5}{3}} b c d +b^{2} c^{2} a^{\frac {2}{3}}\right ) \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}\right )+\left (-\sqrt {3}\, x^{6} \left (a^{2} d^{2}-\frac {4}{3} a b c d +\frac {2}{9} b^{2} c^{2}\right ) \arctan \left (\frac {2 \sqrt {3}\, \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{3 a^{\frac {1}{3}}}+\frac {\sqrt {3}}{3}\right )+x^{6} \left (a^{2} d^{2}-\frac {4}{3} a b c d +\frac {2}{9} b^{2} c^{2}\right ) \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )-\frac {7 c \left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (\frac {3 \left (-2 d \,x^{3}+c \right ) a^{\frac {5}{3}}}{7}+b c \,x^{3} a^{\frac {2}{3}}\right )}{6}\right ) \left (\frac {a d -b c}{d}\right )^{\frac {2}{3}}}{3 \left (\frac {a d -b c}{d}\right )^{\frac {2}{3}} a^{\frac {2}{3}} c^{3} x^{6}}\)	\(448\)

Input:

int((b*x^3+a)^(4/3)/x^7/(d*x^3+c),x,method=_RETURNVERBOSE)
 

Output:

1/3/((a*d-b*c)/d)^(2/3)*(1/2*x^6*(a^(8/3)*d^2-2*a^(5/3)*b*c*d+b^2*c^2*a^(2 
/3))*ln((b*x^3+a)^(2/3)+((a*d-b*c)/d)^(1/3)*(b*x^3+a)^(1/3)+((a*d-b*c)/d)^ 
(2/3))+3^(1/2)*x^6*(a^(8/3)*d^2-2*a^(5/3)*b*c*d+b^2*c^2*a^(2/3))*arctan(2/ 
3*3^(1/2)/((a*d-b*c)/d)^(1/3)*(b*x^3+a)^(1/3)+1/3*3^(1/2))-1/2*((a*d-b*c)/ 
d)^(2/3)*x^6*(a^2*d^2-4/3*a*b*c*d+2/9*b^2*c^2)*ln((b*x^3+a)^(2/3)+a^(1/3)* 
(b*x^3+a)^(1/3)+a^(2/3))-x^6*(a^(8/3)*d^2-2*a^(5/3)*b*c*d+b^2*c^2*a^(2/3)) 
*ln((b*x^3+a)^(1/3)-((a*d-b*c)/d)^(1/3))+(-3^(1/2)*x^6*(a^2*d^2-4/3*a*b*c* 
d+2/9*b^2*c^2)*arctan(2/3*3^(1/2)/a^(1/3)*(b*x^3+a)^(1/3)+1/3*3^(1/2))+x^6 
*(a^2*d^2-4/3*a*b*c*d+2/9*b^2*c^2)*ln((b*x^3+a)^(1/3)-a^(1/3))-7/6*c*(b*x^ 
3+a)^(1/3)*(3/7*(-2*d*x^3+c)*a^(5/3)+b*c*x^3*a^(2/3)))*((a*d-b*c)/d)^(2/3) 
)/a^(2/3)/c^3/x^6
 

Fricas [A] (verification not implemented)

Time = 1.11 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^7 \left (c+d x^3\right )} \, dx=\frac {18 \, \sqrt {3} {\left (a^{2} b c - a^{3} d\right )} {\left (b c d^{2} - a d^{3}\right )}^{\frac {1}{3}} x^{6} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b c d^{2} - a d^{3}\right )}^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \sqrt {3} {\left (b c d - a d^{2}\right )}}{3 \, {\left (b c d - a d^{2}\right )}}\right ) - 6 \, \sqrt {\frac {1}{3}} {\left (2 \, a b^{2} c^{2} - 12 \, a^{2} b c d + 9 \, a^{3} d^{2}\right )} {\left (a^{2}\right )}^{\frac {1}{6}} x^{6} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (a^{2}\right )}^{\frac {1}{6}} {\left ({\left (a^{2}\right )}^{\frac {1}{3}} a + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {2}{3}}\right )}}{a^{2}}\right ) - {\left (2 \, b^{2} c^{2} - 12 \, a b c d + 9 \, a^{2} d^{2}\right )} {\left (a^{2}\right )}^{\frac {2}{3}} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} a + {\left (a^{2}\right )}^{\frac {1}{3}} a + {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + 2 \, {\left (2 \, b^{2} c^{2} - 12 \, a b c d + 9 \, a^{2} d^{2}\right )} {\left (a^{2}\right )}^{\frac {2}{3}} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} a - {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + 9 \, {\left (a^{2} b c - a^{3} d\right )} {\left (b c d^{2} - a d^{3}\right )}^{\frac {1}{3}} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} d^{2} - {\left (b c d^{2} - a d^{3}\right )}^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} d + {\left (b c d^{2} - a d^{3}\right )}^{\frac {2}{3}}\right ) - 18 \, {\left (a^{2} b c - a^{3} d\right )} {\left (b c d^{2} - a d^{3}\right )}^{\frac {1}{3}} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} d + {\left (b c d^{2} - a d^{3}\right )}^{\frac {1}{3}}\right ) - 3 \, {\left (3 \, a^{3} c^{2} + {\left (7 \, a^{2} b c^{2} - 6 \, a^{3} c d\right )} x^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{54 \, a^{2} c^{3} x^{6}} \] Input:

integrate((b*x^3+a)^(4/3)/x^7/(d*x^3+c),x, algorithm="fricas")
 

Output:

1/54*(18*sqrt(3)*(a^2*b*c - a^3*d)*(b*c*d^2 - a*d^3)^(1/3)*x^6*arctan(-1/3 
*(2*sqrt(3)*(b*c*d^2 - a*d^3)^(2/3)*(b*x^3 + a)^(1/3) - sqrt(3)*(b*c*d - a 
*d^2))/(b*c*d - a*d^2)) - 6*sqrt(1/3)*(2*a*b^2*c^2 - 12*a^2*b*c*d + 9*a^3* 
d^2)*(a^2)^(1/6)*x^6*arctan(sqrt(1/3)*(a^2)^(1/6)*((a^2)^(1/3)*a + 2*(b*x^ 
3 + a)^(1/3)*(a^2)^(2/3))/a^2) - (2*b^2*c^2 - 12*a*b*c*d + 9*a^2*d^2)*(a^2 
)^(2/3)*x^6*log((b*x^3 + a)^(2/3)*a + (a^2)^(1/3)*a + (b*x^3 + a)^(1/3)*(a 
^2)^(2/3)) + 2*(2*b^2*c^2 - 12*a*b*c*d + 9*a^2*d^2)*(a^2)^(2/3)*x^6*log((b 
*x^3 + a)^(1/3)*a - (a^2)^(2/3)) + 9*(a^2*b*c - a^3*d)*(b*c*d^2 - a*d^3)^( 
1/3)*x^6*log((b*x^3 + a)^(2/3)*d^2 - (b*c*d^2 - a*d^3)^(1/3)*(b*x^3 + a)^( 
1/3)*d + (b*c*d^2 - a*d^3)^(2/3)) - 18*(a^2*b*c - a^3*d)*(b*c*d^2 - a*d^3) 
^(1/3)*x^6*log((b*x^3 + a)^(1/3)*d + (b*c*d^2 - a*d^3)^(1/3)) - 3*(3*a^3*c 
^2 + (7*a^2*b*c^2 - 6*a^3*c*d)*x^3)*(b*x^3 + a)^(1/3))/(a^2*c^3*x^6)
 

Sympy [F]

\[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^7 \left (c+d x^3\right )} \, dx=\int \frac {\left (a + b x^{3}\right )^{\frac {4}{3}}}{x^{7} \left (c + d x^{3}\right )}\, dx \] Input:

integrate((b*x**3+a)**(4/3)/x**7/(d*x**3+c),x)
 

Output:

Integral((a + b*x**3)**(4/3)/(x**7*(c + d*x**3)), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^7 \left (c+d x^3\right )} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{{\left (d x^{3} + c\right )} x^{7}} \,d x } \] Input:

integrate((b*x^3+a)^(4/3)/x^7/(d*x^3+c),x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a)^(4/3)/((d*x^3 + c)*x^7), x)
 

Giac [A] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^7 \left (c+d x^3\right )} \, dx=\frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{4} - a c^{3} d\right )}} - \frac {\sqrt {3} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{3 \, c^{3}} - \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, c^{3}} - \frac {\sqrt {3} {\left (2 \, b^{2} c^{2} - 12 \, a b c d + 9 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{27 \, a^{\frac {2}{3}} c^{3}} - \frac {{\left (2 \, b^{2} c^{2} - 12 \, a b c d + 9 \, a^{2} d^{2}\right )} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{54 \, a^{\frac {2}{3}} c^{3}} + \frac {{\left (2 \, b^{2} c^{2} - 12 \, a b c d + 9 \, a^{2} d^{2}\right )} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{27 \, a^{\frac {2}{3}} c^{3}} - \frac {7 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{2} c - 4 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a b^{2} c - 6 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a b d + 6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{2} b d}{18 \, b^{2} c^{2} x^{6}} \] Input:

integrate((b*x^3+a)^(4/3)/x^7/(d*x^3+c),x, algorithm="giac")
 

Output:

1/3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(-(b*c - a*d)/d)^(1/3)*log(abs((b* 
x^3 + a)^(1/3) - (-(b*c - a*d)/d)^(1/3)))/(b*c^4 - a*c^3*d) - 1/3*sqrt(3)* 
(-b*c*d^2 + a*d^3)^(1/3)*(b*c - a*d)*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/ 
3) + (-(b*c - a*d)/d)^(1/3))/(-(b*c - a*d)/d)^(1/3))/c^3 - 1/6*(-b*c*d^2 + 
 a*d^3)^(1/3)*(b*c - a*d)*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*(-(b*c 
 - a*d)/d)^(1/3) + (-(b*c - a*d)/d)^(2/3))/c^3 - 1/27*sqrt(3)*(2*b^2*c^2 - 
 12*a*b*c*d + 9*a^2*d^2)*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + a^(1/3) 
)/a^(1/3))/(a^(2/3)*c^3) - 1/54*(2*b^2*c^2 - 12*a*b*c*d + 9*a^2*d^2)*log(( 
b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3))/(a^(2/3)*c^3) + 1/ 
27*(2*b^2*c^2 - 12*a*b*c*d + 9*a^2*d^2)*log(abs((b*x^3 + a)^(1/3) - a^(1/3 
)))/(a^(2/3)*c^3) - 1/18*(7*(b*x^3 + a)^(4/3)*b^2*c - 4*(b*x^3 + a)^(1/3)* 
a*b^2*c - 6*(b*x^3 + a)^(4/3)*a*b*d + 6*(b*x^3 + a)^(1/3)*a^2*b*d)/(b^2*c^ 
2*x^6)
 

Mupad [B] (verification not implemented)

Time = 12.69 (sec) , antiderivative size = 2841, normalized size of antiderivative = 7.05 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^7 \left (c+d x^3\right )} \, dx=\text {Too large to display} \] Input:

int((a + b*x^3)^(4/3)/(x^7*(c + d*x^3)),x)
 

Output:

log(((((18*b^5*c^2*d^3*(a + b*x^3)^(1/3)*(a*d - b*c)^2*(6*a*d - b*c) + 9*a 
*b^4*c^4*d^3*(2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)*((9*a^2*d^2 + 2*b^2*c^2 - 1 
2*a*b*c*d)^3/(a^2*c^9))^(1/3))*((9*a^2*d^2 + 2*b^2*c^2 - 12*a*b*c*d)^3/(a^ 
2*c^9))^(2/3))/729 + (b^5*d^4*(8*b^6*c^6 - 729*a^6*d^6 - 3258*a^2*b^4*c^4* 
d^2 + 6939*a^3*b^3*c^3*d^3 - 7182*a^4*b^2*c^2*d^4 + 577*a*b^5*c^5*d + 3645 
*a^5*b*c*d^5))/(81*c^4))*((9*a^2*d^2 + 2*b^2*c^2 - 12*a*b*c*d)^3/(a^2*c^9) 
)^(1/3))/27 - (b^4*d^5*(a + b*x^3)^(1/3)*(a*d - b*c)^2*(1458*a^6*d^6 + 170 
*b^6*c^6 + 6561*a^2*b^4*c^4*d^2 - 12420*a^3*b^3*c^3*d^3 + 12798*a^4*b^2*c^ 
2*d^4 - 1764*a*b^5*c^5*d - 6804*a^5*b*c*d^5))/(243*c^8))*((729*a^6*d^6 + 8 
*b^6*c^6 + 972*a^2*b^4*c^4*d^2 - 3024*a^3*b^3*c^3*d^3 + 4374*a^4*b^2*c^2*d 
^4 - 144*a*b^5*c^5*d - 2916*a^5*b*c*d^5)/(19683*a^2*c^9))^(1/3) + log((((( 
18*b^5*c^2*d^3*(a + b*x^3)^(1/3)*(a*d - b*c)^2*(6*a*d - b*c) + 81*a*b^4*c^ 
4*d^3*(2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)*(-(d^2*(a*d - b*c)^4)/c^9)^(1/3))* 
(-(d^2*(a*d - b*c)^4)/c^9)^(2/3))/9 + (b^5*d^4*(8*b^6*c^6 - 729*a^6*d^6 - 
3258*a^2*b^4*c^4*d^2 + 6939*a^3*b^3*c^3*d^3 - 7182*a^4*b^2*c^2*d^4 + 577*a 
*b^5*c^5*d + 3645*a^5*b*c*d^5))/(81*c^4))*(-(d^2*(a*d - b*c)^4)/c^9)^(1/3) 
)/3 - (b^4*d^5*(a + b*x^3)^(1/3)*(a*d - b*c)^2*(1458*a^6*d^6 + 170*b^6*c^6 
 + 6561*a^2*b^4*c^4*d^2 - 12420*a^3*b^3*c^3*d^3 + 12798*a^4*b^2*c^2*d^4 - 
1764*a*b^5*c^5*d - 6804*a^5*b*c*d^5))/(243*c^8))*(-(a^4*d^6 + b^4*c^4*d^2 
- 4*a*b^3*c^3*d^3 + 6*a^2*b^2*c^2*d^4 - 4*a^3*b*c*d^5)/(27*c^9))^(1/3) ...
 

Reduce [F]

\[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^7 \left (c+d x^3\right )} \, dx=\frac {-\left (b \,x^{3}+a \right )^{\frac {1}{3}} a -36 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{6 a b \,d^{2} x^{10}+5 b^{2} c d \,x^{10}+6 a^{2} d^{2} x^{7}+11 a b c d \,x^{7}+5 b^{2} c^{2} x^{7}+6 a^{2} c d \,x^{4}+5 a b \,c^{2} x^{4}}d x \right ) a^{3} d^{2} x^{6}+12 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{6 a b \,d^{2} x^{10}+5 b^{2} c d \,x^{10}+6 a^{2} d^{2} x^{7}+11 a b c d \,x^{7}+5 b^{2} c^{2} x^{7}+6 a^{2} c d \,x^{4}+5 a b \,c^{2} x^{4}}d x \right ) a^{2} b c d \,x^{6}+35 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{6 a b \,d^{2} x^{10}+5 b^{2} c d \,x^{10}+6 a^{2} d^{2} x^{7}+11 a b c d \,x^{7}+5 b^{2} c^{2} x^{7}+6 a^{2} c d \,x^{4}+5 a b \,c^{2} x^{4}}d x \right ) a \,b^{2} c^{2} x^{6}-30 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{6 a b \,d^{2} x^{7}+5 b^{2} c d \,x^{7}+6 a^{2} d^{2} x^{4}+11 a b c d \,x^{4}+5 b^{2} c^{2} x^{4}+6 a^{2} c d x +5 a b \,c^{2} x}d x \right ) a^{2} b \,d^{2} x^{6}+11 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{6 a b \,d^{2} x^{7}+5 b^{2} c d \,x^{7}+6 a^{2} d^{2} x^{4}+11 a b c d \,x^{4}+5 b^{2} c^{2} x^{4}+6 a^{2} c d x +5 a b \,c^{2} x}d x \right ) a \,b^{2} c d \,x^{6}+30 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{6 a b \,d^{2} x^{7}+5 b^{2} c d \,x^{7}+6 a^{2} d^{2} x^{4}+11 a b c d \,x^{4}+5 b^{2} c^{2} x^{4}+6 a^{2} c d x +5 a b \,c^{2} x}d x \right ) b^{3} c^{2} x^{6}}{6 c \,x^{6}} \] Input:

int((b*x^3+a)^(4/3)/x^7/(d*x^3+c),x)
 

Output:

( - (a + b*x**3)**(1/3)*a - 36*int((a + b*x**3)**(1/3)/(6*a**2*c*d*x**4 + 
6*a**2*d**2*x**7 + 5*a*b*c**2*x**4 + 11*a*b*c*d*x**7 + 6*a*b*d**2*x**10 + 
5*b**2*c**2*x**7 + 5*b**2*c*d*x**10),x)*a**3*d**2*x**6 + 12*int((a + b*x** 
3)**(1/3)/(6*a**2*c*d*x**4 + 6*a**2*d**2*x**7 + 5*a*b*c**2*x**4 + 11*a*b*c 
*d*x**7 + 6*a*b*d**2*x**10 + 5*b**2*c**2*x**7 + 5*b**2*c*d*x**10),x)*a**2* 
b*c*d*x**6 + 35*int((a + b*x**3)**(1/3)/(6*a**2*c*d*x**4 + 6*a**2*d**2*x** 
7 + 5*a*b*c**2*x**4 + 11*a*b*c*d*x**7 + 6*a*b*d**2*x**10 + 5*b**2*c**2*x** 
7 + 5*b**2*c*d*x**10),x)*a*b**2*c**2*x**6 - 30*int((a + b*x**3)**(1/3)/(6* 
a**2*c*d*x + 6*a**2*d**2*x**4 + 5*a*b*c**2*x + 11*a*b*c*d*x**4 + 6*a*b*d** 
2*x**7 + 5*b**2*c**2*x**4 + 5*b**2*c*d*x**7),x)*a**2*b*d**2*x**6 + 11*int( 
(a + b*x**3)**(1/3)/(6*a**2*c*d*x + 6*a**2*d**2*x**4 + 5*a*b*c**2*x + 11*a 
*b*c*d*x**4 + 6*a*b*d**2*x**7 + 5*b**2*c**2*x**4 + 5*b**2*c*d*x**7),x)*a*b 
**2*c*d*x**6 + 30*int((a + b*x**3)**(1/3)/(6*a**2*c*d*x + 6*a**2*d**2*x**4 
 + 5*a*b*c**2*x + 11*a*b*c*d*x**4 + 6*a*b*d**2*x**7 + 5*b**2*c**2*x**4 + 5 
*b**2*c*d*x**7),x)*b**3*c**2*x**6)/(6*c*x**6)