\(\int \frac {1}{(a+b (c x^n)^{3/n})^2} \, dx\) [3044]

Optimal result

Integrand size = 17, antiderivative size = 210 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx=\frac {x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}-\frac {2 x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \sqrt [3]{b}}+\frac {2 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac {x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{9 a^{5/3} \sqrt [3]{b}} \]

[Out]

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)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {260, 205, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx=-\frac {2 x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \sqrt [3]{b}}-\frac {x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac {2 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac {x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )} \]

[In]

Int[(a + b*(c*x^n)^(3/n))^(-2),x]

[Out]

x/(3*a*(a + b*(c*x^n)^(3/n))) - (2*x*ArcTan[(a^(1/3) - 2*b^(1/3)*(c*x^n)^n^(-1))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3
]*a^(5/3)*b^(1/3)*(c*x^n)^n^(-1)) + (2*x*Log[a^(1/3) + b^(1/3)*(c*x^n)^n^(-1)])/(9*a^(5/3)*b^(1/3)*(c*x^n)^n^(
-1)) - (x*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c*x^n)^n^(-1) + b^(2/3)*(c*x^n)^(2/n)])/(9*a^(5/3)*b^(1/3)*(c*x^n)^n^
(-1))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(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] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[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]

Rule 210

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 260

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_.), x_Symbol] :> Dist[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)]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{\left (a+b x^3\right )^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \frac {x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac {\left (2 x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{3 a} \\ & = \frac {x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac {\left (2 x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{9 a^{5/3}}+\frac {\left (2 x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{9 a^{5/3}} \\ & = \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} \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 {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{4/3}}-\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{9 a^{5/3} \sqrt [3]{b}} \\ & = \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} \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}}+\frac {\left (2 x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}\right )}{3 a^{5/3} \sqrt [3]{b}} \\ & = \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} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}}{\sqrt {3}}\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx=\frac {x \left (c x^n\right )^{-1/n} \left (\frac {3 a^{2/3} \left (c x^n\right )^{\frac {1}{n}}}{a+b \left (c x^n\right )^{3/n}}-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{\sqrt [3]{b}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{\sqrt [3]{b}}\right )}{9 a^{5/3}} \]

[In]

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

[Out]

(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))

Maple [C] (warning: unable to verify)

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

method	result	size
risch	\(\text {Expression too large to display}\)	\(908\)

[In]

int(1/(a+b*(c*x^n)^(3/n))^2,x,method=_RETURNVERBOSE)

[Out]

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)-csgn(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*c
sgn(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^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*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-1))

Fricas [A] (verification not implemented)

none

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 ] \]

[In]

integrate(1/(a+b*(c*x^n)^(3/n))^2,x, algorithm="fricas")

[Out]

[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)*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))]

Sympy [F]

\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx=\int \frac {1}{\left (a + b \left (c x^{n}\right )^{\frac {3}{n}}\right )^{2}}\, dx \]

[In]

integrate(1/(a+b*(c*x**n)**(3/n))**2,x)

[Out]

Integral((a + b*(c*x**n)**(3/n))**(-2), x)

Maxima [F]

\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\frac {3}{n}} b + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(a+b*(c*x^n)^(3/n))^2,x, algorithm="maxima")

[Out]

1/3*x/(a*b*c^(3/n)*(x^n)^(3/n) + a^2) + 2*integrate(1/3/(a*b*c^(3/n)*(x^n)^(3/n) + a^2), x)

Giac [F]

\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\frac {3}{n}} b + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(a+b*(c*x^n)^(3/n))^2,x, algorithm="giac")

[Out]

integrate(((c*x^n)^(3/n)*b + a)^(-2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx=\int \frac {1}{{\left (a+b\,{\left (c\,x^n\right )}^{3/n}\right )}^2} \,d x \]

[In]

int(1/(a + b*(c*x^n)^(3/n))^2,x)

[Out]

int(1/(a + b*(c*x^n)^(3/n))^2, x)