Integrand size = 17, antiderivative size = 210 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx=\frac {x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}-\frac {2 x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \sqrt [3]{b}}+\frac {2 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac {x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{9 a^{5/3} \sqrt [3]{b}} \]
1/3*x/a/(a+b*(c*x^n)^(3/n))+2/9*x*ln(a^(1/3)+b^(1/3)*(c*x^n)^(1/n))/a^(5/3 )/b^(1/3)/((c*x^n)^(1/n))-1/9*x*ln(a^(2/3)-a^(1/3)*b^(1/3)*(c*x^n)^(1/n)+b ^(2/3)*(c*x^n)^(2/n))/a^(5/3)/b^(1/3)/((c*x^n)^(1/n))-2/9*x*arctan(1/3*(a^ (1/3)-2*b^(1/3)*(c*x^n)^(1/n))/a^(1/3)*3^(1/2))/a^(5/3)/b^(1/3)/((c*x^n)^( 1/n))*3^(1/2)
Time = 0.56 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx=\frac {x \left (c x^n\right )^{-1/n} \left (\frac {3 a^{2/3} \left (c x^n\right )^{\frac {1}{n}}}{a+b \left (c x^n\right )^{3/n}}-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{\sqrt [3]{b}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{\sqrt [3]{b}}\right )}{9 a^{5/3}} \]
(x*((3*a^(2/3)*(c*x^n)^n^(-1))/(a + b*(c*x^n)^(3/n)) - (2*Sqrt[3]*ArcTan[( 1 - (2*b^(1/3)*(c*x^n)^n^(-1))/a^(1/3))/Sqrt[3]])/b^(1/3) + (2*Log[a^(1/3) + b^(1/3)*(c*x^n)^n^(-1)])/b^(1/3) - Log[a^(2/3) - a^(1/3)*b^(1/3)*(c*x^n )^n^(-1) + b^(2/3)*(c*x^n)^(2/n)]/b^(1/3)))/(9*a^(5/3)*(c*x^n)^n^(-1))
Time = 0.33 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.94, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {786, 749, 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.
| \(\displaystyle \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx\) |
| \(\Big \downarrow \) 786 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \int \frac {1}{\left (b \left (c x^n\right )^{3/n}+a\right )^2}d\left (c x^n\right )^{\frac {1}{n}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {2 \int \frac {1}{b \left (c x^n\right )^{3/n}+a}d\left (c x^n\right )^{\frac {1}{n}}}{3 a}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}\right )\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {2 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+\sqrt [3]{a}}d\left (c x^n\right )^{\frac {1}{n}}}{3 a^{2/3}}\right )}{3 a}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {2 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}+\frac {3 \int \frac {1}{-\left (c x^n\right )^{2/n}-3}d\left (1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {2 \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}\right )\) |
(x*((c*x^n)^n^(-1)/(3*a*(a + b*(c*x^n)^(3/n))) + (2*(Log[a^(1/3) + b^(1/3) *(c*x^n)^n^(-1)]/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)* (c*x^n)^n^(-1))/a^(1/3))/Sqrt[3]])/b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3 )*(c*x^n)^n^(-1) + b^(2/3)*(c*x^n)^(2/n)]/(2*b^(1/3)))/(3*a^(2/3))))/(3*a) ))/(c*x^n)^n^(-1)
3.31.44.3.1 Defintions of rubi rules used
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_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1)) Int[(a + b*x^ n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
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[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_), x_Symbol] :> Simp[x/(c*x^q )^(1/q) Subst[Int[(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{ a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]
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 = 6.37 (sec) , antiderivative size = 908, normalized size of antiderivative = 4.32
1/3*x/a/(a+b*c^(3/n)*(x^n)^(3/n)*exp(3/2*I*Pi*csgn(I*c*x^n)*(csgn(I*c)-csg n(I*c*x^n))*(-csgn(I*x^n)+csgn(I*c*x^n))/n))+2/9/a/b/(c^(3/n))/((x^n)^(3/n ))*x^3*exp(-3/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c) -csgn(I*c*x^n))/n)/(a/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*csgn(I*c *x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n))^(2/3)*ln( x+(a/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^ n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n))^(1/3))-1/9/a/b/(c^(3/n))/( (x^n)^(3/n))*x^3*exp(-3/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))* (csgn(I*c)-csgn(I*c*x^n))/n)/(a/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*P i*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)) ^(2/3)*ln(x^2-(a/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*csgn(I*c*x^n) *(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n))^(1/3)*x+(a/b/( c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn( I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n))^(2/3))+2/9/a/b/(c^(3/n))/((x^n)^(3 /n))*x^3*exp(-3/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I* c)-csgn(I*c*x^n))/n)/(a/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*csgn(I *c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n))^(2/3)*3 ^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*P i*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)) ^(1/3)*x-1))
Time = 0.39 (sec) , antiderivative size = 705, normalized size of antiderivative = 3.36 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx=\left [\frac {3 \, a^{2} b c^{\frac {3}{n}} x + 3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} c^{\frac {6}{n}} x^{3} + a^{2} b c^{\frac {3}{n}}\right )} \sqrt {-\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}} \log \left (\frac {2 \, a b c^{\frac {3}{n}} x^{3} - 3 \, \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b c^{\frac {3}{n}} x^{2} + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}}}{b c^{\frac {3}{n}} x^{3} + a}\right ) - {\left (b c^{\frac {3}{n}} x^{3} + a\right )} \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x^{2} - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right ) + 2 \, {\left (b c^{\frac {3}{n}} x^{3} + a\right )} \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}}\right )}{9 \, {\left (a^{3} b^{2} c^{\frac {6}{n}} x^{3} + a^{4} b c^{\frac {3}{n}}\right )}}, \frac {3 \, a^{2} b c^{\frac {3}{n}} x + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} c^{\frac {6}{n}} x^{3} + a^{2} b c^{\frac {3}{n}}\right )} \sqrt {\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}}}{a^{2}}\right ) - {\left (b c^{\frac {3}{n}} x^{3} + a\right )} \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x^{2} - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right ) + 2 \, {\left (b c^{\frac {3}{n}} x^{3} + a\right )} \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}}\right )}{9 \, {\left (a^{3} b^{2} c^{\frac {6}{n}} x^{3} + a^{4} b c^{\frac {3}{n}}\right )}}\right ] \]
[1/9*(3*a^2*b*c^(3/n)*x + 3*sqrt(1/3)*(a*b^2*c^(6/n)*x^3 + a^2*b*c^(3/n))* sqrt(-(a^2*b*c^(3/n))^(1/3)/(b*c^(3/n)))*log((2*a*b*c^(3/n)*x^3 - 3*(a^2*b *c^(3/n))^(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*c^(3/n)*x^2 + (a^2*b*c^(3/n ))^(2/3)*x - (a^2*b*c^(3/n))^(1/3)*a)*sqrt(-(a^2*b*c^(3/n))^(1/3)/(b*c^(3/ n))))/(b*c^(3/n)*x^3 + a)) - (b*c^(3/n)*x^3 + a)*(a^2*b*c^(3/n))^(2/3)*log (a*b*c^(3/n)*x^2 - (a^2*b*c^(3/n))^(2/3)*x + (a^2*b*c^(3/n))^(1/3)*a) + 2* (b*c^(3/n)*x^3 + a)*(a^2*b*c^(3/n))^(2/3)*log(a*b*c^(3/n)*x + (a^2*b*c^(3/ n))^(2/3)))/(a^3*b^2*c^(6/n)*x^3 + a^4*b*c^(3/n)), 1/9*(3*a^2*b*c^(3/n)*x + 6*sqrt(1/3)*(a*b^2*c^(6/n)*x^3 + a^2*b*c^(3/n))*sqrt((a^2*b*c^(3/n))^(1/ 3)/(b*c^(3/n)))*arctan(sqrt(1/3)*(2*(a^2*b*c^(3/n))^(2/3)*x - (a^2*b*c^(3/ n))^(1/3)*a)*sqrt((a^2*b*c^(3/n))^(1/3)/(b*c^(3/n)))/a^2) - (b*c^(3/n)*x^3 + a)*(a^2*b*c^(3/n))^(2/3)*log(a*b*c^(3/n)*x^2 - (a^2*b*c^(3/n))^(2/3)*x + (a^2*b*c^(3/n))^(1/3)*a) + 2*(b*c^(3/n)*x^3 + a)*(a^2*b*c^(3/n))^(2/3)*l og(a*b*c^(3/n)*x + (a^2*b*c^(3/n))^(2/3)))/(a^3*b^2*c^(6/n)*x^3 + a^4*b*c^ (3/n))]
\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx=\int \frac {1}{\left (a + b \left (c x^{n}\right )^{\frac {3}{n}}\right )^{2}}\, dx \]
\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\frac {3}{n}} b + a\right )}^{2}} \,d x } \]
\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\frac {3}{n}} b + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx=\int \frac {1}{{\left (a+b\,{\left (c\,x^n\right )}^{3/n}\right )}^2} \,d x \]