\(\int \frac {1}{(a+b (c x^n)^{2/n})^3} \, dx\) [3033]

Optimal result

Integrand size = 17, antiderivative size = 98 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx=\frac {x}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2}+\frac {3 x}{8 a^2 \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac {3 x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b}} \]

[Out]

1/4*x/a/(a+b*(c*x^n)^(2/n))^2+3/8*x/a^2/(a+b*(c*x^n)^(2/n))+3/8*x*arctan((c*x^n)^(1/n)*b^(1/2)/a^(1/2))/a^(5/2
)/((c*x^n)^(1/n))/b^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {260, 205, 211} \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx=\frac {3 x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b}}+\frac {3 x}{8 a^2 \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac {x}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2} \]

[In]

Int[(a + b*(c*x^n)^(2/n))^(-3),x]

[Out]

x/(4*a*(a + b*(c*x^n)^(2/n))^2) + (3*x)/(8*a^2*(a + b*(c*x^n)^(2/n))) + (3*x*ArcTan[(Sqrt[b]*(c*x^n)^n^(-1))/S
qrt[a]])/(8*a^(5/2)*Sqrt[b]*(c*x^n)^n^(-1))

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 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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)]

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^2\right )^3} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \frac {x}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2}+\frac {\left (3 x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{4 a} \\ & = \frac {x}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2}+\frac {3 x}{8 a^2 \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac {\left (3 x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{8 a^2} \\ & = \frac {x}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2}+\frac {3 x}{8 a^2 \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac {3 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx=\frac {x \left (\frac {\sqrt {a} \left (5 a+3 b \left (c x^n\right )^{2/n}\right )}{\left (a+b \left (c x^n\right )^{2/n}\right )^2}+\frac {3 \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{\sqrt {b}}\right )}{8 a^{5/2}} \]

[In]

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

[Out]

(x*((Sqrt[a]*(5*a + 3*b*(c*x^n)^(2/n)))/(a + b*(c*x^n)^(2/n))^2 + (3*ArcTan[(Sqrt[b]*(c*x^n)^n^(-1))/Sqrt[a]])
/(Sqrt[b]*(c*x^n)^n^(-1))))/(8*a^(5/2))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 6.68 (sec) , antiderivative size = 378, normalized size of antiderivative = 3.86

method	result	size
risch	\(\frac {x \left (3 b \,c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}+5 a \right )}{8 a^{2} \left (a +b \,c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}\right )^{2}}+\frac {3 \arctan \left (\frac {b \left (x^{n}\right )^{\frac {2}{n}} c^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{x \sqrt {\frac {a b \left (x^{n}\right )^{\frac {2}{n}} c^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{x^{2}}}}\right )}{8 a^{2} \sqrt {\frac {a b \left (x^{n}\right )^{\frac {2}{n}} c^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{x^{2}}}}\)	\(378\)

[In]

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

[Out]

1/8*x*(3*b*c^(2/n)*(x^n)^(2/n)*exp(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
)+5*a)/a^2/(a+b*c^(2/n)*(x^n)^(2/n)*exp(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+3/8/a^2/(a*b/x^2*(x^n)^(2/n)*c^(2/n)*exp(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))^(1/2)*arctan(b/x*(x^n)^(2/n)*c^(2/n)*exp(I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(c
sgn(I*c)-csgn(I*c*x^n))/n)/(a*b/x^2*(x^n)^(2/n)*c^(2/n)*exp(I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(c
sgn(I*c)-csgn(I*c*x^n))/n))^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 328, normalized size of antiderivative = 3.35 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx=\left [\frac {6 \, a b^{2} c^{\frac {4}{n}} x^{3} + 10 \, a^{2} b c^{\frac {2}{n}} x - 3 \, {\left (b^{2} c^{\frac {4}{n}} x^{4} + 2 \, a b c^{\frac {2}{n}} x^{2} + a^{2}\right )} \sqrt {-a b c^{\frac {2}{n}}} \log \left (\frac {b c^{\frac {2}{n}} x^{2} - 2 \, \sqrt {-a b c^{\frac {2}{n}}} x - a}{b c^{\frac {2}{n}} x^{2} + a}\right )}{16 \, {\left (a^{3} b^{3} c^{\frac {6}{n}} x^{4} + 2 \, a^{4} b^{2} c^{\frac {4}{n}} x^{2} + a^{5} b c^{\frac {2}{n}}\right )}}, \frac {3 \, a b^{2} c^{\frac {4}{n}} x^{3} + 5 \, a^{2} b c^{\frac {2}{n}} x + 3 \, {\left (b^{2} c^{\frac {4}{n}} x^{4} + 2 \, a b c^{\frac {2}{n}} x^{2} + a^{2}\right )} \sqrt {a b c^{\frac {2}{n}}} \arctan \left (\frac {\sqrt {a b c^{\frac {2}{n}}} x}{a}\right )}{8 \, {\left (a^{3} b^{3} c^{\frac {6}{n}} x^{4} + 2 \, a^{4} b^{2} c^{\frac {4}{n}} x^{2} + a^{5} b c^{\frac {2}{n}}\right )}}\right ] \]

[In]

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

[Out]

[1/16*(6*a*b^2*c^(4/n)*x^3 + 10*a^2*b*c^(2/n)*x - 3*(b^2*c^(4/n)*x^4 + 2*a*b*c^(2/n)*x^2 + a^2)*sqrt(-a*b*c^(2
/n))*log((b*c^(2/n)*x^2 - 2*sqrt(-a*b*c^(2/n))*x - a)/(b*c^(2/n)*x^2 + a)))/(a^3*b^3*c^(6/n)*x^4 + 2*a^4*b^2*c
^(4/n)*x^2 + a^5*b*c^(2/n)), 1/8*(3*a*b^2*c^(4/n)*x^3 + 5*a^2*b*c^(2/n)*x + 3*(b^2*c^(4/n)*x^4 + 2*a*b*c^(2/n)
*x^2 + a^2)*sqrt(a*b*c^(2/n))*arctan(sqrt(a*b*c^(2/n))*x/a))/(a^3*b^3*c^(6/n)*x^4 + 2*a^4*b^2*c^(4/n)*x^2 + a^
5*b*c^(2/n))]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/8*(3*b*c^(2/n)*x*(x^n)^(2/n) + 5*a*x)/(a^2*b^2*c^(4/n)*(x^n)^(4/n) + 2*a^3*b*c^(2/n)*(x^n)^(2/n) + a^4) + 3*
integrate(1/8/(a^2*b*c^(2/n)*(x^n)^(2/n) + a^3), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(1/(a + b*(c*x^n)^(2/n))^3,x)

[Out]

int(1/(a + b*(c*x^n)^(2/n))^3, x)