\(\int \frac {1}{a+b (c x^n)^{3/n}} \, dx\) [3043]

Optimal result

Integrand size = 17, antiderivative size = 183 \[ \int \frac {1}{a+b \left (c x^n\right )^{3/n}} \, dx=-\frac {x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b}}+\frac {x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{6 a^{2/3} \sqrt [3]{b}} \]

[Out]

1/3*x*ln(a^(1/3)+b^(1/3)*(c*x^n)^(1/n))/a^(2/3)/b^(1/3)/((c*x^n)^(1/n))-1/6*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^(2/3)/b^(1/3)/((c*x^n)^(1/n))-1/3*x*arctan(1/3*(a^(1/3)-2*b^(1/3)*(c*x^n)^(1
/n))/a^(1/3)*3^(1/2))/a^(2/3)/b^(1/3)/((c*x^n)^(1/n))*3^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {260, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{a+b \left (c x^n\right )^{3/n}} \, dx=-\frac {x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b}}-\frac {x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac {x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}} \]

[In]

Int[(a + b*(c*x^n)^(3/n))^(-1),x]

[Out]

-((x*ArcTan[(a^(1/3) - 2*b^(1/3)*(c*x^n)^n^(-1))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*b^(1/3)*(c*x^n)^n^(-1)))
+ (x*Log[a^(1/3) + b^(1/3)*(c*x^n)^n^(-1)])/(3*a^(2/3)*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)])/(6*a^(2/3)*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 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}{a+b x^3} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \frac {\left (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 )}{3 a^{2/3}}+\frac {\left (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 )}{3 a^{2/3}} \\ & = \frac {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 )}{3 a^{2/3} \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 )}{2 \sqrt [3]{a}}-\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 )}{6 a^{2/3} \sqrt [3]{b}} \\ & = \frac {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 )}{3 a^{2/3} \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 )}{6 a^{2/3} \sqrt [3]{b}}+\frac {\left (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 )}{a^{2/3} \sqrt [3]{b}} \\ & = -\frac {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 )}{\sqrt {3} a^{2/3} \sqrt [3]{b}}+\frac {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 )}{3 a^{2/3} \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 )}{6 a^{2/3} \sqrt [3]{b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.73 \[ \int \frac {1}{a+b \left (c x^n\right )^{3/n}} \, dx=-\frac {x \left (c x^n\right )^{-1/n} \left (2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )+\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )\right )}{6 a^{2/3} \sqrt [3]{b}} \]

[In]

Integrate[(a + b*(c*x^n)^(3/n))^(-1),x]

[Out]

-1/6*(x*(2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*(c*x^n)^n^(-1))/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) + b^(1/3)*(c*x^n)^
n^(-1)] + Log[a^(2/3) - a^(1/3)*b^(1/3)*(c*x^n)^n^(-1) + b^(2/3)*(c*x^n)^(2/n)]))/(a^(2/3)*b^(1/3)*(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 = 6.22 (sec) , antiderivative size = 821, normalized size of antiderivative = 4.49

method	result	size
risch	\(\text {Expression too large to display}\)	\(821\)

[In]

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

[Out]

1/3/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))-1/6/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^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))+1/3/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)-cs
gn(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))*(csg
n(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*c
sgn(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 = 0.30 (sec) , antiderivative size = 549, normalized size of antiderivative = 3.00 \[ \int \frac {1}{a+b \left (c x^n\right )^{3/n}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b c^{\frac {3}{n}} \sqrt {-\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}} \log \left (\frac {2 \, a b c^{\frac {3}{n}} x^{3} - 3 \, \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b c^{\frac {3}{n}} x^{2} + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}}}{b c^{\frac {3}{n}} x^{3} + a}\right ) - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x^{2} - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b c^{\frac {3}{n}}}, \frac {6 \, \sqrt {\frac {1}{3}} a b c^{\frac {3}{n}} \sqrt {\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}}}{a^{2}}\right ) - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x^{2} - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b c^{\frac {3}{n}}}\right ] \]

[In]

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

[Out]

[1/6*(3*sqrt(1/3)*a*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)) - (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*(a^2*b*c^(3/n))^(2/3)*log(a*b*c^(3/n)*x + (a^2*b*c^(3/n))^(2
/3)))/(a^2*b*c^(3/n)), 1/6*(6*sqrt(1/3)*a*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) - (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*(a^2*b*c^(3/n))^(2/3)
*log(a*b*c^(3/n)*x + (a^2*b*c^(3/n))^(2/3)))/(a^2*b*c^(3/n))]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(1/((c*x^n)^(3/n)*b + a), x)

Giac [F]

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

[In]

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

[Out]

integrate(1/((c*x^n)^(3/n)*b + a), x)

Mupad [F(-1)]

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

[In]

int(1/(a + b*(c*x^n)^(3/n)),x)

[Out]

int(1/(a + b*(c*x^n)^(3/n)), x)