3.3045 \(\int \frac{1}{(a+b (c x^n)^{3/n})^3} \, dx\)

Optimal. Leaf size=235 \[ -\frac{5 x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{54 a^{8/3} \sqrt [3]{b}}+\frac{5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac{5 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{5 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} \sqrt [3]{b}}+\frac{x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2} \]

[Out]

x/(6*a*(a + b*(c*x^n)^(3/n))^2) + (5*x)/(18*a^2*(a + b*(c*x^n)^(3/n))) - (5*x*ArcTan[(a^(1/3) - 2*b^(1/3)*(c*x
^n)^n^(-1))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(1/3)*(c*x^n)^n^(-1)) + (5*x*Log[a^(1/3) + b^(1/3)*(c*x^n
)^n^(-1)])/(27*a^(8/3)*b^(1/3)*(c*x^n)^n^(-1)) - (5*x*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c*x^n)^n^(-1) + b^(2/3)*(
c*x^n)^(2/n)])/(54*a^(8/3)*b^(1/3)*(c*x^n)^n^(-1))

________________________________________________________________________________________

Rubi [A]  time = 0.115708, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {254, 199, 200, 31, 634, 617, 204, 628} \[ -\frac{5 x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{54 a^{8/3} \sqrt [3]{b}}+\frac{5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac{5 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{5 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} \sqrt [3]{b}}+\frac{x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x^n)^(3/n))^(-3),x]

[Out]

x/(6*a*(a + b*(c*x^n)^(3/n))^2) + (5*x)/(18*a^2*(a + b*(c*x^n)^(3/n))) - (5*x*ArcTan[(a^(1/3) - 2*b^(1/3)*(c*x
^n)^n^(-1))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(1/3)*(c*x^n)^n^(-1)) + (5*x*Log[a^(1/3) + b^(1/3)*(c*x^n
)^n^(-1)])/(27*a^(8/3)*b^(1/3)*(c*x^n)^n^(-1)) - (5*x*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c*x^n)^n^(-1) + b^(2/3)*(
c*x^n)^(2/n)])/(54*a^(8/3)*b^(1/3)*(c*x^n)^n^(-1))

Rule 254

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 199

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] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 200

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 31

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

Rule 634

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]

Rule 617

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

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]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \left (c x^n\right )^{3/n}\right )^3} \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^3\right )^3} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac{\left (5 x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^3\right )^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{6 a}\\ &=\frac{x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac{5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac{\left (5 x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^3} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{9 a^2}\\ &=\frac{x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac{5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac{\left (5 x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{27 a^{8/3}}+\frac{\left (5 x \left (c x^n\right )^{-1/n}\right ) \operatorname{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 )}{27 a^{8/3}}\\ &=\frac{x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac{5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac{5 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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac{\left (5 x \left (c x^n\right )^{-1/n}\right ) \operatorname{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 )}{18 a^{7/3}}-\frac{\left (5 x \left (c x^n\right )^{-1/n}\right ) \operatorname{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 )}{54 a^{8/3} \sqrt [3]{b}}\\ &=\frac{x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac{5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac{5 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac{5 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 )}{54 a^{8/3} \sqrt [3]{b}}+\frac{\left (5 x \left (c x^n\right )^{-1/n}\right ) \operatorname{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 )}{9 a^{8/3} \sqrt [3]{b}}\\ &=\frac{x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac{5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}-\frac{5 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 )}{9 \sqrt{3} a^{8/3} \sqrt [3]{b}}+\frac{5 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac{5 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 )}{54 a^{8/3} \sqrt [3]{b}}\\ \end{align*}

Mathematica [A]  time = 0.129573, size = 215, normalized size = 0.91 \[ \frac{x \left (-\frac{5 \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{\sqrt [3]{b}}+\frac{15 a^{2/3}}{a+b \left (c x^n\right )^{3/n}}+\frac{9 a^{5/3}}{\left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac{10 \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{\sqrt [3]{b}}-\frac{10 \sqrt{3} \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}\right )}{54 a^{8/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*((9*a^(5/3))/(a + b*(c*x^n)^(3/n))^2 + (15*a^(2/3))/(a + b*(c*x^n)^(3/n)) - (10*Sqrt[3]*ArcTan[(1 - (2*b^(1
/3)*(c*x^n)^n^(-1))/a^(1/3))/Sqrt[3]])/(b^(1/3)*(c*x^n)^n^(-1)) + (10*Log[a^(1/3) + b^(1/3)*(c*x^n)^n^(-1)])/(
b^(1/3)*(c*x^n)^n^(-1)) - (5*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)*(
c*x^n)^n^(-1))))/(54*a^(8/3))

________________________________________________________________________________________

Maple [F]  time = 0.411, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( c{x}^{n} \right ) ^{3\,{n}^{-1}} \right ) ^{-3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{5 \, b c^{\frac{3}{n}} x{\left (x^{n}\right )}^{\frac{3}{n}} + 8 \, a x}{18 \,{\left (a^{2} b^{2} c^{\frac{6}{n}}{\left (x^{n}\right )}^{\frac{6}{n}} + 2 \, a^{3} b c^{\frac{3}{n}}{\left (x^{n}\right )}^{\frac{3}{n}} + a^{4}\right )}} + 5 \, \int \frac{1}{9 \,{\left (a^{2} b c^{\frac{3}{n}}{\left (x^{n}\right )}^{\frac{3}{n}} + a^{3}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(5*b*c^(3/n)*x*(x^n)^(3/n) + 8*a*x)/(a^2*b^2*c^(6/n)*(x^n)^(6/n) + 2*a^3*b*c^(3/n)*(x^n)^(3/n) + a^4) + 5
*integrate(1/9/(a^2*b*c^(3/n)*(x^n)^(3/n) + a^3), x)

________________________________________________________________________________________

Fricas [B]  time = 1.4536, size = 1756, normalized size = 7.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/54*(15*a^2*b^2*c^(6/n)*x^4 + 24*a^3*b*c^(3/n)*x + 15*sqrt(1/3)*(a*b^3*c^(9/n)*x^6 + 2*a^2*b^2*c^(6/n)*x^3 +
a^3*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)) - 5*(b^2*c^(6/n)*x^6 + 2*a*b*c^(3/n)*x^3 + 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) + 10*(b^2*c^(6/n)*x^6 + 2*a*b*c^
(3/n)*x^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^4*b^3*c^(9/n)*x^6 + 2*a^
5*b^2*c^(6/n)*x^3 + a^6*b*c^(3/n)), 1/54*(15*a^2*b^2*c^(6/n)*x^4 + 24*a^3*b*c^(3/n)*x + 30*sqrt(1/3)*(a*b^3*c^
(9/n)*x^6 + 2*a^2*b^2*c^(6/n)*x^3 + a^3*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) - 5*(b^2*c^(6
/n)*x^6 + 2*a*b*c^(3/n)*x^3 + 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) + 10*(b^2*c^(6/n)*x^6 + 2*a*b*c^(3/n)*x^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^4*b^3*c^(9/n)*x^6 + 2*a^5*b^2*c^(6/n)*x^3 + a^6*b*c^(3/n))]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \left (c x^{n}\right )^{\frac{3}{n}}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (\left (c x^{n}\right )^{\frac{3}{n}} b + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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