3.3044 \(\int \frac{1}{(a+b (c x^n)^{3/n})^2} \, dx\)

Optimal. Leaf size=210 \[ -\frac{x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{2 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{2 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )} \]

[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))

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Rubi [A]  time = 0.0948812, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{2 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{2 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )} \]

Antiderivative was successfully verified.

[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 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 )^2} \, dx &=\left (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 )\\ &=\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 ) \operatorname{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 ) \operatorname{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 ) \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 )}{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 ) \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 )}{3 a^{4/3}}-\frac{\left (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 )}{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 ) \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 )}{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]  time = 0.223498, size = 158, normalized size = 0.75 \[ \frac{x \left (\frac{1}{a+b \left (c x^n\right )^{3/n}}-\frac{\left (c x^n\right )^{-1/n} \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*((a + b*(c*x^n)^(3/n))^(-1) - (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)])/(3*a
^(2/3)*b^(1/3)*(c*x^n)^n^(-1))))/(3*a)

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Maple [F]  time = 2.388, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( c{x}^{n} \right ) ^{3\,{n}^{-1}} \right ) ^{-2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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Fricas [A]  time = 1.39486, size = 1413, normalized size = 6.73 \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))^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))]

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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 )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)