\(\int \frac {(a+b x^3)^{2/3}}{x^7 (c+d x^3)} \, dx\) [698]

Optimal result

Integrand size = 24, antiderivative size = 370 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^7 \left (c+d x^3\right )} \, dx=\frac {(b c+6 a d) \left (a+b x^3\right )^{2/3}}{18 a c^2 x^3}-\frac {\left (a+b x^3\right )^{5/3}}{6 a c x^6}-\frac {\left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} c^3}-\frac {d^{4/3} (b c-a d)^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c^3}+\frac {\left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) \log (x)}{18 a^{4/3} c^3}+\frac {d^{4/3} (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^3}-\frac {\left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{4/3} c^3}-\frac {d^{4/3} (b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^3} \] Output:

1/18*(6*a*d+b*c)*(b*x^3+a)^(2/3)/a/c^2/x^3-1/6*(b*x^3+a)^(5/3)/a/c/x^6-1/2 
7*(-9*a^2*d^2+6*a*b*c*d+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^(4/3)/c^3-1/3*d^(4/3)*(-a*d+b*c)^(2/3)*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+6*a*b*c*d+b^2*c^2)*ln(x)/a^(4/3)/c^3+1/6*d^(4/3)*(-a*d+b*c)^(2 
/3)*ln(d*x^3+c)/c^3-1/18*(-9*a^2*d^2+6*a*b*c*d+b^2*c^2)*ln(a^(1/3)-(b*x^3+ 
a)^(1/3))/a^(4/3)/c^3-1/2*d^(4/3)*(-a*d+b*c)^(2/3)*ln((-a*d+b*c)^(1/3)+d^( 
1/3)*(b*x^3+a)^(1/3))/c^3
 

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^7 \left (c+d x^3\right )} \, dx=\frac {\frac {3 c \left (a+b x^3\right )^{2/3} \left (-3 a c-2 b c x^3+6 a d x^3\right )}{a x^6}-\frac {2 \sqrt {3} \left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{4/3}}-18 \sqrt {3} d^{4/3} (b c-a d)^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )-\frac {2 \left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )}{a^{4/3}}-18 d^{4/3} (b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )+\frac {\left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{a^{4/3}}+9 d^{4/3} (b c-a d)^{2/3} \log \left ((b c-a d)^{2/3}-\sqrt [3]{d} \sqrt [3]{b c-a d} \sqrt [3]{a+b x^3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )}{54 c^3} \] Input:

Integrate[(a + b*x^3)^(2/3)/(x^7*(c + d*x^3)),x]
 

Output:

((3*c*(a + b*x^3)^(2/3)*(-3*a*c - 2*b*c*x^3 + 6*a*d*x^3))/(a*x^6) - (2*Sqr 
t[3]*(b^2*c^2 + 6*a*b*c*d - 9*a^2*d^2)*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a 
^(1/3))/Sqrt[3]])/a^(4/3) - 18*Sqrt[3]*d^(4/3)*(b*c - a*d)^(2/3)*ArcTan[(1 
 - (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - (2*(b^2*c^2 
 + 6*a*b*c*d - 9*a^2*d^2)*Log[-a^(1/3) + (a + b*x^3)^(1/3)])/a^(4/3) - 18* 
d^(4/3)*(b*c - a*d)^(2/3)*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3 
)] + ((b^2*c^2 + 6*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^(4/3) + 9*d^(4/3)*(b*c - a*d)^(2/3)*Log[(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 = 0.78 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.93, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {948, 114, 27, 166, 27, 174, 67, 16, 68, 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.

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

\(\Big \downarrow \) 948

\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^{2/3}}{x^9 \left (d x^3+c\right )}dx^3\)

\(\Big \downarrow \) 114

\(\displaystyle \frac {1}{3} \left (-\frac {\int \frac {\left (b x^3+a\right )^{2/3} \left (b d x^3+b c+6 a d\right )}{3 x^6 \left (d x^3+c\right )}dx^3}{2 a c}-\frac {\left (a+b x^3\right )^{5/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (-\frac {\int \frac {\left (b x^3+a\right )^{2/3} \left (b d x^3+b c+6 a d\right )}{x^6 \left (d x^3+c\right )}dx^3}{6 a c}-\frac {\left (a+b x^3\right )^{5/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {\int \frac {2 \left (b d (b c-3 a d) x^3+b^2 c^2-9 a^2 d^2+6 a b c d\right )}{3 x^3 \sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx^3}{c}-\frac {\left (a+b x^3\right )^{2/3} (6 a d+b c)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{5/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {2 \int \frac {b d (b c-3 a d) x^3+b^2 c^2-9 a^2 d^2+6 a b c d}{x^3 \sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx^3}{3 c}-\frac {\left (a+b x^3\right )^{2/3} (6 a d+b c)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{5/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {2 \left (\frac {\left (-9 a^2 d^2+6 a b c d+b^2 c^2\right ) \int \frac {1}{x^3 \sqrt [3]{b x^3+a}}dx^3}{c}-\frac {9 a d^2 (b c-a d) \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx^3}{c}\right )}{3 c}-\frac {\left (a+b x^3\right )^{2/3} (6 a d+b c)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{5/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 67

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {2 \left (\frac {\left (-9 a^2 d^2+6 a b c d+b^2 c^2\right ) \left (\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )}{c}-\frac {9 a d^2 (b c-a d) \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx^3}{c}\right )}{3 c}-\frac {\left (a+b x^3\right )^{2/3} (6 a d+b c)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{5/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {2 \left (\frac {\left (-9 a^2 d^2+6 a b c d+b^2 c^2\right ) \left (\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )}{c}-\frac {9 a d^2 (b c-a d) \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx^3}{c}\right )}{3 c}-\frac {\left (a+b x^3\right )^{2/3} (6 a d+b c)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{5/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 68

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {2 \left (\frac {\left (-9 a^2 d^2+6 a b c d+b^2 c^2\right ) \left (\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )}{c}-\frac {9 a d^2 (b c-a d) \left (-\frac {3 \int \frac {1}{\frac {\sqrt [3]{b c-a d}}{\sqrt [3]{d}}+\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 d^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \int \frac {1}{x^6+\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} \sqrt [3]{b x^3+a}}{\sqrt [3]{d}}}d\sqrt [3]{b x^3+a}}{2 d}+\frac {\log \left (c+d x^3\right )}{2 d^{2/3} \sqrt [3]{b c-a d}}\right )}{c}\right )}{3 c}-\frac {\left (a+b x^3\right )^{2/3} (6 a d+b c)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{5/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {2 \left (\frac {\left (-9 a^2 d^2+6 a b c d+b^2 c^2\right ) \left (\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )}{c}-\frac {9 a d^2 (b c-a d) \left (\frac {3 \int \frac {1}{x^6+\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} \sqrt [3]{b x^3+a}}{\sqrt [3]{d}}}d\sqrt [3]{b x^3+a}}{2 d}+\frac {\log \left (c+d x^3\right )}{2 d^{2/3} \sqrt [3]{b c-a d}}-\frac {3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{2/3} \sqrt [3]{b c-a d}}\right )}{c}\right )}{3 c}-\frac {\left (a+b x^3\right )^{2/3} (6 a d+b c)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{5/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {2 \left (\frac {\left (-9 a^2 d^2+6 a b c d+b^2 c^2\right ) \left (-\frac {3 \int \frac {1}{-x^6-3}d\left (\frac {2 \sqrt [3]{b x^3+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )}{c}-\frac {9 a d^2 (b c-a d) \left (\frac {3 \int \frac {1}{-x^6-3}d\left (1-\frac {2 \sqrt [3]{d} \sqrt [3]{b x^3+a}}{\sqrt [3]{b c-a d}}\right )}{d^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{2 d^{2/3} \sqrt [3]{b c-a d}}-\frac {3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{2/3} \sqrt [3]{b c-a d}}\right )}{c}\right )}{3 c}-\frac {\left (a+b x^3\right )^{2/3} (6 a d+b c)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{5/3}}{2 a c x^6}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {2 \left (\frac {\left (-9 a^2 d^2+6 a b c d+b^2 c^2\right ) \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )}{c}-\frac {9 a d^2 (b c-a d) \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{d^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{2 d^{2/3} \sqrt [3]{b c-a d}}-\frac {3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{2/3} \sqrt [3]{b c-a d}}\right )}{c}\right )}{3 c}-\frac {\left (a+b x^3\right )^{2/3} (6 a d+b c)}{c x^3}}{6 a c}-\frac {\left (a+b x^3\right )^{5/3}}{2 a c x^6}\right )\)

Input:

Int[(a + b*x^3)^(2/3)/(x^7*(c + d*x^3)),x]
 

Output:

(-1/2*(a + b*x^3)^(5/3)/(a*c*x^6) - (-(((b*c + 6*a*d)*(a + b*x^3)^(2/3))/( 
c*x^3)) + (2*(((b^2*c^2 + 6*a*b*c*d - 9*a^2*d^2)*((Sqrt[3]*ArcTan[(1 + (2* 
(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]])/a^(1/3) - Log[x^3]/(2*a^(1/3)) + (3* 
Log[a^(1/3) - (a + b*x^3)^(1/3)])/(2*a^(1/3))))/c - (9*a*d^2*(b*c - a*d)*( 
-((Sqrt[3]*ArcTan[(1 - (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sq 
rt[3]])/(d^(2/3)*(b*c - a*d)^(1/3))) + Log[c + d*x^3]/(2*d^(2/3)*(b*c - a* 
d)^(1/3)) - (3*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)])/(2*d^(2 
/3)*(b*c - a*d)^(1/3))))/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 67

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   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 68

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[-(b*c - a*d)/b, 3]}, Simp[Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   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 = 2.00 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.85

method	result	size
pseudoelliptic	\(\frac {\left (-\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} c \left (-6 a d \,x^{3}+2 x^{3} b c +3 a c \right ) a^{\frac {4}{3}}}{6}+a \left (a^{2} d^{2}-\frac {2}{3} a b c d -\frac {1}{9} b^{2} c^{2}\right ) \left (\arctan \left (\frac {2 \sqrt {3}\, \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{3 a^{\frac {1}{3}}}+\frac {\sqrt {3}}{3}\right ) \sqrt {3}+\ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )-\frac {\ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2}\right ) x^{6}\right ) \left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}+\frac {d \left (-2 \arctan \left (\frac {2 \sqrt {3}\, \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{3 \left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}}{3}\right ) \sqrt {3}+\ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {2}{3}}\right )-2 \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}\right )\right ) \left (a d -b c \right ) a^{\frac {7}{3}} x^{6}}{2}}{3 \left (\frac {a d -b c}{d}\right )^{\frac {1}{3}} c^{3} x^{6} a^{\frac {7}{3}}}\)	\(314\)

Input:

int((b*x^3+a)^(2/3)/x^7/(d*x^3+c),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 1.21 (sec) , antiderivative size = 1151, normalized size of antiderivative = 3.11 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^7 \left (c+d x^3\right )} \, dx=\text {Too large to display} \] Input:

integrate((b*x^3+a)^(2/3)/x^7/(d*x^3+c),x, algorithm="fricas")
 

Output:

[-1/54*(18*sqrt(3)*(-b^2*c^2*d + 2*a*b*c*d^2 - a^2*d^3)^(1/3)*a^2*d*x^6*ar 
ctan(-1/3*(sqrt(3)*(b*c - a*d) + 2*sqrt(3)*(-b^2*c^2*d + 2*a*b*c*d^2 - a^2 
*d^3)^(1/3)*(b*x^3 + a)^(1/3))/(b*c - a*d)) + 9*(-b^2*c^2*d + 2*a*b*c*d^2 
- a^2*d^3)^(1/3)*a^2*d*x^6*log(-(b*x^3 + a)^(2/3)*(b*c*d - a*d^2) + (-b^2* 
c^2*d + 2*a*b*c*d^2 - a^2*d^3)^(1/3)*(b*c - a*d) + (-b^2*c^2*d + 2*a*b*c*d 
^2 - a^2*d^3)^(2/3)*(b*x^3 + a)^(1/3)) - 18*(-b^2*c^2*d + 2*a*b*c*d^2 - a^ 
2*d^3)^(1/3)*a^2*d*x^6*log(-(b*x^3 + a)^(1/3)*(b*c*d - a*d^2) - (-b^2*c^2* 
d + 2*a*b*c*d^2 - a^2*d^3)^(2/3)) + 3*sqrt(1/3)*(a*b^2*c^2 + 6*a^2*b*c*d - 
 9*a^3*d^2)*x^6*sqrt(-1/a^(2/3))*log((2*b*x^3 + 3*sqrt(1/3)*(2*(b*x^3 + a) 
^(2/3)*a^(2/3) - (b*x^3 + a)^(1/3)*a - a^(4/3))*sqrt(-1/a^(2/3)) - 3*(b*x^ 
3 + a)^(1/3)*a^(2/3) + 3*a)/x^3) - (b^2*c^2 + 6*a*b*c*d - 9*a^2*d^2)*a^(2/ 
3)*x^6*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3)) + 2*(b 
^2*c^2 + 6*a*b*c*d - 9*a^2*d^2)*a^(2/3)*x^6*log((b*x^3 + a)^(1/3) - a^(1/3 
)) + 3*(3*a^2*c^2 + 2*(a*b*c^2 - 3*a^2*c*d)*x^3)*(b*x^3 + a)^(2/3))/(a^2*c 
^3*x^6), -1/54*(18*sqrt(3)*(-b^2*c^2*d + 2*a*b*c*d^2 - a^2*d^3)^(1/3)*a^2* 
d*x^6*arctan(-1/3*(sqrt(3)*(b*c - a*d) + 2*sqrt(3)*(-b^2*c^2*d + 2*a*b*c*d 
^2 - a^2*d^3)^(1/3)*(b*x^3 + a)^(1/3))/(b*c - a*d)) + 9*(-b^2*c^2*d + 2*a* 
b*c*d^2 - a^2*d^3)^(1/3)*a^2*d*x^6*log(-(b*x^3 + a)^(2/3)*(b*c*d - a*d^2) 
+ (-b^2*c^2*d + 2*a*b*c*d^2 - a^2*d^3)^(1/3)*(b*c - a*d) + (-b^2*c^2*d + 2 
*a*b*c*d^2 - a^2*d^3)^(2/3)*(b*x^3 + a)^(1/3)) - 18*(-b^2*c^2*d + 2*a*b...
 

Sympy [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^7 \left (c+d x^3\right )} \, dx=\int \frac {\left (a + b x^{3}\right )^{\frac {2}{3}}}{x^{7} \left (c + d x^{3}\right )}\, dx \] Input:

integrate((b*x**3+a)**(2/3)/x**7/(d*x**3+c),x)
 

Output:

Integral((a + b*x**3)**(2/3)/(x**7*(c + d*x**3)), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^7 \left (c+d x^3\right )} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )} x^{7}} \,d x } \] Input:

integrate((b*x^3+a)^(2/3)/x^7/(d*x^3+c),x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a)^(2/3)/((d*x^3 + c)*x^7), x)
 

Giac [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^7 \left (c+d x^3\right )} \, dx=-\frac {{\left (b c d^{2} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} - a d^{3} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{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 {2}{3}} \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 {2}{3}} \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 (b^{2} c^{2} + 6 \, 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 {4}{3}} c^{3}} + \frac {{\left (b^{2} c^{2} + 6 \, 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 {4}{3}} c^{3}} - \frac {{\left (a^{\frac {1}{3}} b^{2} c^{2} + 6 \, a^{\frac {4}{3}} b c d - 9 \, a^{\frac {7}{3}} d^{2}\right )} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{27 \, a^{\frac {5}{3}} c^{3}} - \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} b^{2} c + {\left (b x^{3} + a\right )}^{\frac {2}{3}} a b^{2} c - 6 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a b d + 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{2} b d}{18 \, a b^{2} c^{2} x^{6}} \] Input:

integrate((b*x^3+a)^(2/3)/x^7/(d*x^3+c),x, algorithm="giac")
 

Output:

-1/3*(b*c*d^2*(-(b*c - a*d)/d)^(1/3) - a*d^3*(-(b*c - a*d)/d)^(1/3))*(-(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)^(2/3)*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)^(2/3)*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)*(b^2*c^ 
2 + 6*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^(4/3)*c^3) + 1/54*(b^2*c^2 + 6*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^(4/3)*c^3) - 1/2 
7*(a^(1/3)*b^2*c^2 + 6*a^(4/3)*b*c*d - 9*a^(7/3)*d^2)*log(abs((b*x^3 + a)^ 
(1/3) - a^(1/3)))/(a^(5/3)*c^3) - 1/18*(2*(b*x^3 + a)^(5/3)*b^2*c + (b*x^3 
 + a)^(2/3)*a*b^2*c - 6*(b*x^3 + a)^(5/3)*a*b*d + 6*(b*x^3 + a)^(2/3)*a^2* 
b*d)/(a*b^2*c^2*x^6)
 

Mupad [B] (verification not implemented)

Time = 15.02 (sec) , antiderivative size = 2788, normalized size of antiderivative = 7.54 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^7 \left (c+d x^3\right )} \, dx=\text {Too large to display} \] Input:

int((a + b*x^3)^(2/3)/(x^7*(c + d*x^3)),x)
 

Output:

log(((((27*a*b^4*c^4*d^3*(2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)*(-(d^4*(a*d - b 
*c)^2)/c^9)^(2/3) - (b^4*d^3*(a + b*x^3)^(1/3)*(a*d - b*c)^2*(162*a^4*d^4 
+ b^4*c^4 + 18*a^2*b^2*c^2*d^2 + 12*a*b^3*c^3*d - 108*a^3*b*c*d^3))/(3*a^2 
*c^2))*(-(d^4*(a*d - b*c)^2)/c^9)^(1/3))/3 - (b^5*d^4*(729*a^6*d^6 + b^6*c 
^6 + 63*a^2*b^4*c^4*d^2 - 918*a^3*b^3*c^3*d^3 + 2295*a^4*b^2*c^2*d^4 + 17* 
a*b^5*c^5*d - 2187*a^5*b*c*d^5))/(81*a^2*c^4))*(-(d^4*(a*d - b*c)^2)/c^9)^ 
(2/3))/9 + (2*b^5*d^7*(a + b*x^3)^(1/3)*(6*a*d - 5*b*c)*(9*a^3*d^3 + b^3*c 
^3 + 5*a*b^2*c^2*d - 15*a^2*b*c*d^2)^2)/(729*a^2*c^10))*(-(a^2*d^6 + b^2*c 
^2*d^4 - 2*a*b*c*d^5)/(27*c^9))^(1/3) + log((((((a*b^4*c^4*d^3*(2*a^2*d^2 
+ b^2*c^2 - 3*a*b*c*d)*(-(b^2*c^2 - 9*a^2*d^2 + 6*a*b*c*d)^3/(a^4*c^9))^(2 
/3))/3 - (b^4*d^3*(a + b*x^3)^(1/3)*(a*d - b*c)^2*(162*a^4*d^4 + b^4*c^4 + 
 18*a^2*b^2*c^2*d^2 + 12*a*b^3*c^3*d - 108*a^3*b*c*d^3))/(3*a^2*c^2))*(-(b 
^2*c^2 - 9*a^2*d^2 + 6*a*b*c*d)^3/(a^4*c^9))^(1/3))/27 - (b^5*d^4*(729*a^6 
*d^6 + b^6*c^6 + 63*a^2*b^4*c^4*d^2 - 918*a^3*b^3*c^3*d^3 + 2295*a^4*b^2*c 
^2*d^4 + 17*a*b^5*c^5*d - 2187*a^5*b*c*d^5))/(81*a^2*c^4))*(-(b^2*c^2 - 9* 
a^2*d^2 + 6*a*b*c*d)^3/(a^4*c^9))^(2/3))/729 + (2*b^5*d^7*(a + b*x^3)^(1/3 
)*(6*a*d - 5*b*c)*(9*a^3*d^3 + b^3*c^3 + 5*a*b^2*c^2*d - 15*a^2*b*c*d^2)^2 
)/(729*a^2*c^10))*(-(b^6*c^6 - 729*a^6*d^6 + 81*a^2*b^4*c^4*d^2 - 108*a^3* 
b^3*c^3*d^3 - 729*a^4*b^2*c^2*d^4 + 18*a*b^5*c^5*d + 1458*a^5*b*c*d^5)/(19 
683*a^4*c^9))^(1/3) - (((a + b*x^3)^(2/3)*(b^2*c + 6*a*b*d))/(18*c^2) -...
 

Reduce [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^7 \left (c+d x^3\right )} \, dx=\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{d \,x^{10}+c \,x^{7}}d x \] Input:

int((b*x^3+a)^(2/3)/x^7/(d*x^3+c),x)
 

Output:

int((a + b*x**3)**(2/3)/(c*x**7 + d*x**10),x)