\(\int \frac {(a+b x^3)^{2/3}}{x^7 (a d-b d x^3)} \, dx\) [806]

Optimal result

Integrand size = 28, antiderivative size = 284 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^7 \left (a d-b d x^3\right )} \, dx=-\frac {5 b \left (a+b x^3\right )^{2/3}}{18 a^2 d x^3}-\frac {\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6}+\frac {14 b^2 \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} d}-\frac {2^{2/3} b^2 \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3} d}-\frac {7 b^2 \log (x)}{9 a^{7/3} d}+\frac {b^2 \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{7/3} d}+\frac {7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{9 a^{7/3} d}-\frac {b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{7/3} d} \] Output:

-5/18*b*(b*x^3+a)^(2/3)/a^2/d/x^3-1/6*(b*x^3+a)^(5/3)/a^2/d/x^6+14/27*b^2* 
arctan(1/3*(a^(1/3)+2*(b*x^3+a)^(1/3))*3^(1/2)/a^(1/3))*3^(1/2)/a^(7/3)/d- 
1/3*2^(2/3)*b^2*arctan(1/3*(a^(1/3)+2^(2/3)*(b*x^3+a)^(1/3))*3^(1/2)/a^(1/ 
3))*3^(1/2)/a^(7/3)/d-7/9*b^2*ln(x)/a^(7/3)/d+1/6*b^2*ln(-b*x^3+a)*2^(2/3) 
/a^(7/3)/d+7/9*b^2*ln(a^(1/3)-(b*x^3+a)^(1/3))/a^(7/3)/d-1/2*b^2*ln(2^(1/3 
)*a^(1/3)-(b*x^3+a)^(1/3))*2^(2/3)/a^(7/3)/d
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^7 \left (a d-b d x^3\right )} \, dx=\frac {-9 a^{4/3} \left (a+b x^3\right )^{2/3}-24 \sqrt [3]{a} b x^3 \left (a+b x^3\right )^{2/3}+28 \sqrt {3} b^2 x^6 \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-18\ 2^{2/3} \sqrt {3} b^2 x^6 \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+28 b^2 x^6 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )-18\ 2^{2/3} b^2 x^6 \log \left (-2 \sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}\right )-14 b^2 x^6 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )+9\ 2^{2/3} b^2 x^6 \log \left (2 a^{2/3}+2^{2/3} \sqrt [3]{a} \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{54 a^{7/3} d x^6} \] Input:

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

Output:

(-9*a^(4/3)*(a + b*x^3)^(2/3) - 24*a^(1/3)*b*x^3*(a + b*x^3)^(2/3) + 28*Sq 
rt[3]*b^2*x^6*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 18*2^( 
2/3)*Sqrt[3]*b^2*x^6*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt 
[3]] + 28*b^2*x^6*Log[-a^(1/3) + (a + b*x^3)^(1/3)] - 18*2^(2/3)*b^2*x^6*L 
og[-2*a^(1/3) + 2^(2/3)*(a + b*x^3)^(1/3)] - 14*b^2*x^6*Log[a^(2/3) + a^(1 
/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)] + 9*2^(2/3)*b^2*x^6*Log[2*a^(2/ 
3) + 2^(2/3)*a^(1/3)*(a + b*x^3)^(1/3) + 2^(1/3)*(a + b*x^3)^(2/3)])/(54*a 
^(7/3)*d*x^6)
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {948, 27, 114, 27, 166, 27, 174, 67, 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.

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {\frac {b \left (\frac {\int \frac {4 a b \left (2 b x^3+7 a\right )}{3 x^3 \left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx^3}{a}-\frac {5 \left (a+b x^3\right )^{2/3}}{x^3}\right )}{6 a^2}-\frac {\left (a+b x^3\right )^{5/3}}{2 a^2 x^6}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {b \left (\frac {4}{3} b \int \frac {2 b x^3+7 a}{x^3 \left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx^3-\frac {5 \left (a+b x^3\right )^{2/3}}{x^3}\right )}{6 a^2}-\frac {\left (a+b x^3\right )^{5/3}}{2 a^2 x^6}}{3 d}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {b \left (\frac {4}{3} b \left (7 \int \frac {1}{x^3 \sqrt [3]{b x^3+a}}dx^3+9 b \int \frac {1}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx^3\right )-\frac {5 \left (a+b x^3\right )^{2/3}}{x^3}\right )}{6 a^2}-\frac {\left (a+b x^3\right )^{5/3}}{2 a^2 x^6}}{3 d}\)

\(\Big \downarrow \) 67

\(\displaystyle \frac {\frac {b \left (\frac {4}{3} b \left (7 \left (\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )+9 b \left (-\frac {3 \int \frac {1}{x^6+2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 b}+\frac {3 \int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{2} \sqrt [3]{a} b}+\frac {\log \left (a-b x^3\right )}{2 \sqrt [3]{2} \sqrt [3]{a} b}\right )\right )-\frac {5 \left (a+b x^3\right )^{2/3}}{x^3}\right )}{6 a^2}-\frac {\left (a+b x^3\right )^{5/3}}{2 a^2 x^6}}{3 d}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {\frac {b \left (\frac {4}{3} b \left (7 \left (\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )+9 b \left (-\frac {3 \int \frac {1}{x^6+2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 b}+\frac {\log \left (a-b x^3\right )}{2 \sqrt [3]{2} \sqrt [3]{a} b}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} b}\right )\right )-\frac {5 \left (a+b x^3\right )^{2/3}}{x^3}\right )}{6 a^2}-\frac {\left (a+b x^3\right )^{5/3}}{2 a^2 x^6}}{3 d}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {b \left (\frac {4}{3} b \left (7 \left (-\frac {3 \int \frac {1}{-x^6-3}d\left (\frac {2 \sqrt [3]{b x^3+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )+9 b \left (\frac {3 \int \frac {1}{-x^6-3}d\left (\frac {2^{2/3} \sqrt [3]{b x^3+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{2} \sqrt [3]{a} b}+\frac {\log \left (a-b x^3\right )}{2 \sqrt [3]{2} \sqrt [3]{a} b}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} b}\right )\right )-\frac {5 \left (a+b x^3\right )^{2/3}}{x^3}\right )}{6 a^2}-\frac {\left (a+b x^3\right )^{5/3}}{2 a^2 x^6}}{3 d}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {b \left (\frac {4}{3} b \left (7 \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )+9 b \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a} b}+\frac {\log \left (a-b x^3\right )}{2 \sqrt [3]{2} \sqrt [3]{a} b}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} b}\right )\right )-\frac {5 \left (a+b x^3\right )^{2/3}}{x^3}\right )}{6 a^2}-\frac {\left (a+b x^3\right )^{5/3}}{2 a^2 x^6}}{3 d}\)

Input:

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

Output:

(-1/2*(a + b*x^3)^(5/3)/(a^2*x^6) + (b*((-5*(a + b*x^3)^(2/3))/x^3 + (4*b* 
(7*((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))) 
 + 9*b*(-((Sqrt[3]*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3 
]])/(2^(1/3)*a^(1/3)*b)) + Log[a - b*x^3]/(2*2^(1/3)*a^(1/3)*b) - (3*Log[2 
^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)])/(2*2^(1/3)*a^(1/3)*b))))/3))/(6*a^2)) 
/(3*d)
 

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 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 = 1.78 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.89

method	result	size
pseudoelliptic	\(\frac {-18 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\left (a^{\frac {1}{3}}+2^{\frac {2}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) b^{2} x^{6}-18 \,2^{\frac {2}{3}} \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right ) b^{2} x^{6}+9 \,2^{\frac {2}{3}} \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right ) b^{2} x^{6}+28 \sqrt {3}\, \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) b^{2} x^{6}+28 \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) b^{2} x^{6}-14 \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) b^{2} x^{6}-24 b \,x^{3} \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{\frac {1}{3}}-9 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{\frac {4}{3}}}{54 a^{\frac {7}{3}} x^{6} d}\)	\(254\)

Input:

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

Output:

1/54*(-18*3^(1/2)*2^(2/3)*arctan(1/3*(a^(1/3)+2^(2/3)*(b*x^3+a)^(1/3))*3^( 
1/2)/a^(1/3))*b^2*x^6-18*2^(2/3)*ln((b*x^3+a)^(1/3)-2^(1/3)*a^(1/3))*b^2*x 
^6+9*2^(2/3)*ln((b*x^3+a)^(2/3)+2^(1/3)*a^(1/3)*(b*x^3+a)^(1/3)+2^(2/3)*a^ 
(2/3))*b^2*x^6+28*3^(1/2)*arctan(1/3*(a^(1/3)+2*(b*x^3+a)^(1/3))*3^(1/2)/a 
^(1/3))*b^2*x^6+28*ln((b*x^3+a)^(1/3)-a^(1/3))*b^2*x^6-14*ln((b*x^3+a)^(2/ 
3)+a^(1/3)*(b*x^3+a)^(1/3)+a^(2/3))*b^2*x^6-24*b*x^3*(b*x^3+a)^(2/3)*a^(1/ 
3)-9*(b*x^3+a)^(2/3)*a^(4/3))/a^(7/3)/x^6/d
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 660, normalized size of antiderivative = 2.32 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^7 \left (a d-b d x^3\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/54*(18*4^(1/3)*sqrt(3)*a*b^2*x^6*(-1/a)^(1/3)*arctan(1/3*4^(1/3)*sqrt( 
3)*(b*x^3 + a)^(1/3)*(-1/a)^(1/3) - 1/3*sqrt(3)) - 42*sqrt(1/3)*a*b^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) + 9*4^(1/3)*a*b^2*x^6*(-1/a)^(1/3)*log(4^(2/3)*(b*x^3 + a 
)^(1/3)*a*(-1/a)^(2/3) - 2*4^(1/3)*a*(-1/a)^(1/3) + 2*(b*x^3 + a)^(2/3)) - 
 18*4^(1/3)*a*b^2*x^6*(-1/a)^(1/3)*log(-4^(2/3)*a*(-1/a)^(2/3) + 2*(b*x^3 
+ a)^(1/3)) + 14*a^(2/3)*b^2*x^6*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3) 
*a^(1/3) + a^(2/3)) - 28*a^(2/3)*b^2*x^6*log((b*x^3 + a)^(1/3) - a^(1/3)) 
+ 3*(8*a*b*x^3 + 3*a^2)*(b*x^3 + a)^(2/3))/(a^3*d*x^6), -1/54*(18*4^(1/3)* 
sqrt(3)*a*b^2*x^6*(-1/a)^(1/3)*arctan(1/3*4^(1/3)*sqrt(3)*(b*x^3 + a)^(1/3 
)*(-1/a)^(1/3) - 1/3*sqrt(3)) + 9*4^(1/3)*a*b^2*x^6*(-1/a)^(1/3)*log(4^(2/ 
3)*(b*x^3 + a)^(1/3)*a*(-1/a)^(2/3) - 2*4^(1/3)*a*(-1/a)^(1/3) + 2*(b*x^3 
+ a)^(2/3)) - 18*4^(1/3)*a*b^2*x^6*(-1/a)^(1/3)*log(-4^(2/3)*a*(-1/a)^(2/3 
) + 2*(b*x^3 + a)^(1/3)) - 84*sqrt(1/3)*a^(2/3)*b^2*x^6*arctan(sqrt(1/3)*( 
2*(b*x^3 + a)^(1/3) + a^(1/3))/a^(1/3)) + 14*a^(2/3)*b^2*x^6*log((b*x^3 + 
a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3)) - 28*a^(2/3)*b^2*x^6*log(( 
b*x^3 + a)^(1/3) - a^(1/3)) + 3*(8*a*b*x^3 + 3*a^2)*(b*x^3 + a)^(2/3))/(a^ 
3*d*x^6)]
 

Sympy [F]

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

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

Output:

-Integral((a + b*x**3)**(2/3)/(-a*x**7 + b*x**10), x)/d
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^7 \left (a d-b d x^3\right )} \, dx=-\frac {\sqrt {3} 2^{\frac {2}{3}} b^{2} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {7}{3}} d} + \frac {2^{\frac {2}{3}} b^{2} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right )}{6 \, a^{\frac {7}{3}} d} - \frac {2^{\frac {2}{3}} b^{2} \log \left ({\left | -2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \right |}\right )}{3 \, a^{\frac {7}{3}} d} + \frac {14 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{27 \, a^{\frac {7}{3}} d} - \frac {7 \, b^{2} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{27 \, a^{\frac {7}{3}} d} + \frac {14 \, b^{2} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{27 \, a^{\frac {7}{3}} d} - \frac {8 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} b^{2} - 5 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a b^{2}}{18 \, a^{2} b^{2} d x^{6}} \] Input:

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

Output:

-1/3*sqrt(3)*2^(2/3)*b^2*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2*( 
b*x^3 + a)^(1/3))/a^(1/3))/(a^(7/3)*d) + 1/6*2^(2/3)*b^2*log(2^(2/3)*a^(2/ 
3) + 2^(1/3)*(b*x^3 + a)^(1/3)*a^(1/3) + (b*x^3 + a)^(2/3))/(a^(7/3)*d) - 
1/3*2^(2/3)*b^2*log(abs(-2^(1/3)*a^(1/3) + (b*x^3 + a)^(1/3)))/(a^(7/3)*d) 
 + 14/27*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + a^(1/3))/a^ 
(1/3))/(a^(7/3)*d) - 7/27*b^2*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^ 
(1/3) + a^(2/3))/(a^(7/3)*d) + 14/27*b^2*log(abs((b*x^3 + a)^(1/3) - a^(1/ 
3)))/(a^(7/3)*d) - 1/18*(8*(b*x^3 + a)^(5/3)*b^2 - 5*(b*x^3 + a)^(2/3)*a*b 
^2)/(a^2*b^2*d*x^6)
 

Mupad [B] (verification not implemented)

Time = 4.62 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.81 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^7 \left (a d-b d x^3\right )} \, dx=\frac {\frac {5\,b^2\,{\left (b\,x^3+a\right )}^{2/3}}{18\,a}-\frac {4\,b^2\,{\left (b\,x^3+a\right )}^{5/3}}{9\,a^2}}{d\,{\left (b\,x^3+a\right )}^2+a^2\,d-2\,a\,d\,\left (b\,x^3+a\right )}+\ln \left (2\,b^4\,{\left (b\,x^3+a\right )}^{1/3}-2\,2^{1/3}\,a^5\,d^2\,{\left (-\frac {b^6}{a^7\,d^3}\right )}^{2/3}\right )\,{\left (-\frac {4\,b^6}{27\,a^7\,d^3}\right )}^{1/3}+\frac {14\,\ln \left (b^4\,{\left (b\,x^3+a\right )}^{1/3}-a^5\,d^2\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{2/3}\right )\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{1/3}}{27}-\ln \left (4\,b^4\,{\left (b\,x^3+a\right )}^{1/3}+2\,2^{1/3}\,a^5\,d^2\,{\left (-\frac {b^6}{a^7\,d^3}\right )}^{2/3}-2^{1/3}\,\sqrt {3}\,a^5\,d^2\,{\left (-\frac {b^6}{a^7\,d^3}\right )}^{2/3}\,2{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {4\,b^6}{27\,a^7\,d^3}\right )}^{1/3}+\ln \left (4\,b^4\,{\left (b\,x^3+a\right )}^{1/3}+2\,2^{1/3}\,a^5\,d^2\,{\left (-\frac {b^6}{a^7\,d^3}\right )}^{2/3}+2^{1/3}\,\sqrt {3}\,a^5\,d^2\,{\left (-\frac {b^6}{a^7\,d^3}\right )}^{2/3}\,2{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {4\,b^6}{27\,a^7\,d^3}\right )}^{1/3}-\frac {7\,\ln \left (2\,b^4\,{\left (b\,x^3+a\right )}^{1/3}+a^5\,d^2\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{2/3}-\sqrt {3}\,a^5\,d^2\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{2/3}\,1{}\mathrm {i}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{1/3}}{27}+\frac {7\,\ln \left (2\,b^4\,{\left (b\,x^3+a\right )}^{1/3}+a^5\,d^2\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{2/3}+\sqrt {3}\,a^5\,d^2\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{2/3}\,1{}\mathrm {i}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{1/3}}{27} \] Input:

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

Output:

((5*b^2*(a + b*x^3)^(2/3))/(18*a) - (4*b^2*(a + b*x^3)^(5/3))/(9*a^2))/(d* 
(a + b*x^3)^2 + a^2*d - 2*a*d*(a + b*x^3)) + log(2*b^4*(a + b*x^3)^(1/3) - 
 2*2^(1/3)*a^5*d^2*(-b^6/(a^7*d^3))^(2/3))*(-(4*b^6)/(27*a^7*d^3))^(1/3) + 
 (14*log(b^4*(a + b*x^3)^(1/3) - a^5*d^2*(b^6/(a^7*d^3))^(2/3))*(b^6/(a^7* 
d^3))^(1/3))/27 - log(4*b^4*(a + b*x^3)^(1/3) + 2*2^(1/3)*a^5*d^2*(-b^6/(a 
^7*d^3))^(2/3) - 2^(1/3)*3^(1/2)*a^5*d^2*(-b^6/(a^7*d^3))^(2/3)*2i)*((3^(1 
/2)*1i)/2 + 1/2)*(-(4*b^6)/(27*a^7*d^3))^(1/3) + log(4*b^4*(a + b*x^3)^(1/ 
3) + 2*2^(1/3)*a^5*d^2*(-b^6/(a^7*d^3))^(2/3) + 2^(1/3)*3^(1/2)*a^5*d^2*(- 
b^6/(a^7*d^3))^(2/3)*2i)*((3^(1/2)*1i)/2 - 1/2)*(-(4*b^6)/(27*a^7*d^3))^(1 
/3) - (7*log(2*b^4*(a + b*x^3)^(1/3) + a^5*d^2*(b^6/(a^7*d^3))^(2/3) - 3^( 
1/2)*a^5*d^2*(b^6/(a^7*d^3))^(2/3)*1i)*(3^(1/2)*1i + 1)*(b^6/(a^7*d^3))^(1 
/3))/27 + (7*log(2*b^4*(a + b*x^3)^(1/3) + a^5*d^2*(b^6/(a^7*d^3))^(2/3) + 
 3^(1/2)*a^5*d^2*(b^6/(a^7*d^3))^(2/3)*1i)*(3^(1/2)*1i - 1)*(b^6/(a^7*d^3) 
)^(1/3))/27
 

Reduce [F]

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

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

Output:

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