3.3037 \(\int \frac{1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx\)

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 = 0.191393, 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 \[ -\frac{x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{2 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{2 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )} \]

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))

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Rubi in Sympy [A]  time = 32.9159, size = 185, normalized size = 0.88 \[ \frac{x}{3 a \left (a + b \left (c x^{n}\right )^{\frac{3}{n}}\right )} + \frac{2 x \left (c x^{n}\right )^{- \frac{1}{n}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} - \frac{x \left (c x^{n}\right )^{- \frac{1}{n}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \left (c x^{n}\right )^{\frac{1}{n}} + b^{\frac{2}{3}} \left (c x^{n}\right )^{\frac{2}{n}} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} - \frac{2 \sqrt{3} x \left (c x^{n}\right )^{- \frac{1}{n}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \left (c x^{n}\right )^{\frac{1}{n}}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*(c*x**n)**(3/n))**2,x)

[Out]

x/(3*a*(a + b*(c*x**n)**(3/n))) + 2*x*(c*x**n)**(-1/n)*log(a**(1/3) + b**(1/3)*(
c*x**n)**(1/n))/(9*a**(5/3)*b**(1/3)) - x*(c*x**n)**(-1/n)*log(a**(2/3) - a**(1/
3)*b**(1/3)*(c*x**n)**(1/n) + b**(2/3)*(c*x**n)**(2/n))/(9*a**(5/3)*b**(1/3)) -
2*sqrt(3)*x*(c*x**n)**(-1/n)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*(c*x**n)**(1/
n)/3)/a**(1/3))/(9*a**(5/3)*b**(1/3))

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Mathematica [A]  time = 4.26296, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.658, size = 0, normalized size = 0. \[ \int \left ( a+b \left ( c{x}^{n} \right ) ^{3\,{n}^{-1}} \right ) ^{-2}\, dx \]

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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(3/n)*b + a)^(-2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238991, size = 294, normalized size = 1.4 \[ \frac{6 \,{\left (b c^{\frac{3}{n}} x^{3} + a\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) -{\left (\sqrt{3} b c^{\frac{3}{n}} x^{3} + \sqrt{3} a\right )} \log \left (\left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} a x + a^{2}\right ) + 2 \,{\left (\sqrt{3} b c^{\frac{3}{n}} x^{3} + \sqrt{3} a\right )} \log \left (\left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} x + a\right ) + 3 \, \sqrt{3} \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} x}{9 \,{\left (\sqrt{3} a b c^{\frac{3}{n}} x^{3} + \sqrt{3} a^{2}\right )} \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(3/n)*b + a)^(-2),x, algorithm="fricas")

[Out]

1/9*(6*(b*c^(3/n)*x^3 + a)*arctan(1/3*(2*sqrt(3)*(a^2*b*c^(3/n))^(1/3)*x - sqrt(
3)*a)/a) - (sqrt(3)*b*c^(3/n)*x^3 + sqrt(3)*a)*log((a^2*b*c^(3/n))^(2/3)*x^2 - (
a^2*b*c^(3/n))^(1/3)*a*x + a^2) + 2*(sqrt(3)*b*c^(3/n)*x^3 + sqrt(3)*a)*log((a^2
*b*c^(3/n))^(1/3)*x + a) + 3*sqrt(3)*(a^2*b*c^(3/n))^(1/3)*x)/((sqrt(3)*a*b*c^(3
/n)*x^3 + sqrt(3)*a^2)*(a^2*b*c^(3/n))^(1/3))

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(3/n)*b + a)^(-2),x, algorithm="giac")

[Out]

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