Integrand size = 13, antiderivative size = 140 \[ \int \frac {x^6}{\left (a+b x^3\right )^2} \, dx=\frac {x}{b^2}+\frac {a x}{3 b^2 \left (a+b x^3\right )}+\frac {4 \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} b^{7/3}}-\frac {4 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{7/3}}+\frac {2 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 b^{7/3}} \] Output:
x/b^2+1/3*a*x/b^2/(b*x^3+a)+4/9*a^(1/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3 ^(1/2)/a^(1/3))*3^(1/2)/b^(7/3)-4/9*a^(1/3)*ln(a^(1/3)+b^(1/3)*x)/b^(7/3)+ 2/9*a^(1/3)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(7/3)
Time = 0.06 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int \frac {x^6}{\left (a+b x^3\right )^2} \, dx=\frac {9 \sqrt [3]{b} x+\frac {3 a \sqrt [3]{b} x}{a+b x^3}+4 \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-4 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 b^{7/3}} \] Input:
Integrate[x^6/(a + b*x^3)^2,x]
Output:
(9*b^(1/3)*x + (3*a*b^(1/3)*x)/(a + b*x^3) + 4*Sqrt[3]*a^(1/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 4*a^(1/3)*Log[a^(1/3) + b^(1/3)*x] + 2* a^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(9*b^(7/3))
Time = 0.49 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {817, 843, 750, 16, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {x^6}{\left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {4 \int \frac {x^3}{b x^3+a}dx}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {4 \left (\frac {x}{b}-\frac {a \int \frac {1}{b x^3+a}dx}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\) |
Input:
Int[x^6/(a + b*x^3)^2,x]
Output:
-1/3*x^4/(b*(a + b*x^3)) + (4*(x/b - (a*(Log[a^(1/3) + b^(1/3)*x]/(3*a^(2/ 3)*b^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^( 1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(2*b^(1/3)))/(3*a^( 2/3))))/b))/(3*b)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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_)^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])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
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]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.44 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.36
| method | result | size |
| risch | \(\frac {x}{b^{2}}+\frac {a x}{3 b^{2} \left (b \,x^{3}+a \right )}-\frac {4 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{9 b^{3}}\) | \(50\) |
| default | \(\frac {x}{b^{2}}-\frac {a \left (-\frac {x}{3 \left (b \,x^{3}+a \right )}+\frac {4 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {2 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{b^{2}}\) | \(115\) |
Input:
int(x^6/(b*x^3+a)^2,x,method=_RETURNVERBOSE)
Output:
x/b^2+1/3*a*x/b^2/(b*x^3+a)-4/9/b^3*a*sum(1/_R^2*ln(x-_R),_R=RootOf(_Z^3*b +a))
Time = 0.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.04 \[ \int \frac {x^6}{\left (a+b x^3\right )^2} \, dx=\frac {9 \, b x^{4} + 4 \, \sqrt {3} {\left (b x^{3} + a\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - 2 \, {\left (b x^{3} + a\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 4 \, {\left (b x^{3} + a\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 12 \, a x}{9 \, {\left (b^{3} x^{3} + a b^{2}\right )}} \] Input:
integrate(x^6/(b*x^3+a)^2,x, algorithm="fricas")
Output:
1/9*(9*b*x^4 + 4*sqrt(3)*(b*x^3 + a)*(-a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*b* x*(-a/b)^(2/3) - sqrt(3)*a)/a) - 2*(b*x^3 + a)*(-a/b)^(1/3)*log(x^2 + x*(- a/b)^(1/3) + (-a/b)^(2/3)) + 4*(b*x^3 + a)*(-a/b)^(1/3)*log(x - (-a/b)^(1/ 3)) + 12*a*x)/(b^3*x^3 + a*b^2)
Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.34 \[ \int \frac {x^6}{\left (a+b x^3\right )^2} \, dx=\frac {a x}{3 a b^{2} + 3 b^{3} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} b^{7} + 64 a, \left ( t \mapsto t \log {\left (- \frac {9 t b^{2}}{4} + x \right )} \right )\right )} + \frac {x}{b^{2}} \] Input:
integrate(x**6/(b*x**3+a)**2,x)
Output:
a*x/(3*a*b**2 + 3*b**3*x**3) + RootSum(729*_t**3*b**7 + 64*a, Lambda(_t, _ t*log(-9*_t*b**2/4 + x))) + x/b**2
Time = 0.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.89 \[ \int \frac {x^6}{\left (a+b x^3\right )^2} \, dx=\frac {a x}{3 \, {\left (b^{3} x^{3} + a b^{2}\right )}} + \frac {x}{b^{2}} - \frac {4 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, a \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {4 \, a \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(x^6/(b*x^3+a)^2,x, algorithm="maxima")
Output:
1/3*a*x/(b^3*x^3 + a*b^2) + x/b^2 - 4/9*sqrt(3)*a*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^3*(a/b)^(2/3)) + 2/9*a*log(x^2 - x*(a/b)^(1 /3) + (a/b)^(2/3))/(b^3*(a/b)^(2/3)) - 4/9*a*log(x + (a/b)^(1/3))/(b^3*(a/ b)^(2/3))
Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int \frac {x^6}{\left (a+b x^3\right )^2} \, dx=\frac {4 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, b^{2}} + \frac {a x}{3 \, {\left (b x^{3} + a\right )} b^{2}} + \frac {x}{b^{2}} - \frac {4 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{3}} - \frac {2 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, b^{3}} \] Input:
integrate(x^6/(b*x^3+a)^2,x, algorithm="giac")
Output:
4/9*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/b^2 + 1/3*a*x/((b*x^3 + a)*b^2 ) + x/b^2 - 4/9*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1 /3))/(-a/b)^(1/3))/b^3 - 2/9*(-a*b^2)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a /b)^(2/3))/b^3
Time = 0.38 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94 \[ \int \frac {x^6}{\left (a+b x^3\right )^2} \, dx=\frac {x}{b^2}+\frac {4\,{\left (-a\right )}^{1/3}\,\ln \left ({\left (-a\right )}^{4/3}+a\,b^{1/3}\,x\right )}{9\,b^{7/3}}+\frac {a\,x}{3\,\left (b^3\,x^3+a\,b^2\right )}-\frac {4\,{\left (-a\right )}^{1/3}\,\ln \left (4\,a\,x-\frac {4\,{\left (-a\right )}^{4/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{1/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,b^{7/3}}+\frac {{\left (-a\right )}^{1/3}\,\ln \left (4\,a\,x+\frac {9\,{\left (-a\right )}^{4/3}\,\left (-\frac {2}{9}+\frac {\sqrt {3}\,2{}\mathrm {i}}{9}\right )}{b^{1/3}}\right )\,\left (-\frac {2}{9}+\frac {\sqrt {3}\,2{}\mathrm {i}}{9}\right )}{b^{7/3}} \] Input:
int(x^6/(a + b*x^3)^2,x)
Output:
x/b^2 + (4*(-a)^(1/3)*log((-a)^(4/3) + a*b^(1/3)*x))/(9*b^(7/3)) + (a*x)/( 3*(a*b^2 + b^3*x^3)) - (4*(-a)^(1/3)*log(4*a*x - (4*(-a)^(4/3)*((3^(1/2)*1 i)/2 + 1/2))/b^(1/3))*((3^(1/2)*1i)/2 + 1/2))/(9*b^(7/3)) + ((-a)^(1/3)*lo g(4*a*x + (9*(-a)^(4/3)*((3^(1/2)*2i)/9 - 2/9))/b^(1/3))*((3^(1/2)*2i)/9 - 2/9))/b^(7/3)
Time = 0.27 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.26 \[ \int \frac {x^6}{\left (a+b x^3\right )^2} \, dx=\frac {4 a^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right )+4 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b \,x^{3}+2 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )+2 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b \,x^{3}-4 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )-4 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b \,x^{3}+12 b^{\frac {1}{3}} a x +9 b^{\frac {4}{3}} x^{4}}{9 b^{\frac {7}{3}} \left (b \,x^{3}+a \right )} \] Input:
int(x^6/(b*x^3+a)^2,x)
Output:
(4*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a + 4*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b*x **3 + 2*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a + 2 *a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b*x**3 - 4*a **(1/3)*log(a**(1/3) + b**(1/3)*x)*a - 4*a**(1/3)*log(a**(1/3) + b**(1/3)* x)*b*x**3 + 12*b**(1/3)*a*x + 9*b**(1/3)*b*x**4)/(9*b**(1/3)*b**2*(a + b*x **3))