3.3038 \(\int \frac{1}{\left (a+b \left (c x^n\right )^{3/n}\right )^3} \, dx\)

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

[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*Ar
cTan[(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 = 0.220745, 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 \[ -\frac{5 x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{54 a^{8/3} \sqrt [3]{b}}+\frac{5 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{5 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} \sqrt [3]{b}}+\frac{5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac{x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2} \]

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*Ar
cTan[(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 in Sympy [A]  time = 37.2975, size = 209, normalized size = 0.89 \[ \frac{x}{6 a \left (a + b \left (c x^{n}\right )^{\frac{3}{n}}\right )^{2}} + \frac{5 x}{18 a^{2} \left (a + b \left (c x^{n}\right )^{\frac{3}{n}}\right )} + \frac{5 x \left (c x^{n}\right )^{- \frac{1}{n}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{27 a^{\frac{8}{3}} \sqrt [3]{b}} - \frac{5 x \left (c x^{n}\right )^{- \frac{1}{n}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \left (c x^{n}\right )^{\frac{1}{n}} + b^{\frac{2}{3}} \left (c x^{n}\right )^{\frac{2}{n}} \right )}}{54 a^{\frac{8}{3}} \sqrt [3]{b}} - \frac{5 \sqrt{3} x \left (c x^{n}\right )^{- \frac{1}{n}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \left (c x^{n}\right )^{\frac{1}{n}}}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{8}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*(c*x**n)**(3/n))**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
*(c*x**n)**(-1/n)*log(a**(1/3) + b**(1/3)*(c*x**n)**(1/n))/(27*a**(8/3)*b**(1/3)
) - 5*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))/(54*a**(8/3)*b**(1/3)) - 5*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))/(27*a**(8/3)*b**(
1/3))

_______________________________________________________________________________________

Mathematica [A]  time = 4.36864, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b \left (c x^n\right )^{3/n}\right )^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int \left ( a+b \left ( c{x}^{n} \right ) ^{3\,{n}^{-1}} \right ) ^{-3}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.238946, size = 433, normalized size = 1.84 \[ \frac{30 \,{\left (b^{2} c^{\frac{6}{n}} x^{6} + 2 \, a b c^{\frac{3}{n}} x^{3} + a^{2}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 5 \,{\left (\sqrt{3} b^{2} c^{\frac{6}{n}} x^{6} + 2 \, \sqrt{3} a b c^{\frac{3}{n}} x^{3} + \sqrt{3} a^{2}\right )} \log \left (\left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} a x + a^{2}\right ) + 10 \,{\left (\sqrt{3} b^{2} c^{\frac{6}{n}} x^{6} + 2 \, \sqrt{3} a b c^{\frac{3}{n}} x^{3} + \sqrt{3} a^{2}\right )} \log \left (\left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} x + a\right ) + 3 \,{\left (5 \, \sqrt{3} b c^{\frac{3}{n}} x^{4} + 8 \, \sqrt{3} a x\right )} \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}}}{54 \,{\left (\sqrt{3} a^{2} b^{2} c^{\frac{6}{n}} x^{6} + 2 \, \sqrt{3} a^{3} b c^{\frac{3}{n}} x^{3} + \sqrt{3} a^{4}\right )} \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/54*(30*(b^2*c^(6/n)*x^6 + 2*a*b*c^(3/n)*x^3 + a^2)*arctan(1/3*(2*sqrt(3)*(a^2*
b*c^(3/n))^(1/3)*x - sqrt(3)*a)/a) - 5*(sqrt(3)*b^2*c^(6/n)*x^6 + 2*sqrt(3)*a*b*
c^(3/n)*x^3 + sqrt(3)*a^2)*log((a^2*b*c^(3/n))^(2/3)*x^2 - (a^2*b*c^(3/n))^(1/3)
*a*x + a^2) + 10*(sqrt(3)*b^2*c^(6/n)*x^6 + 2*sqrt(3)*a*b*c^(3/n)*x^3 + sqrt(3)*
a^2)*log((a^2*b*c^(3/n))^(1/3)*x + a) + 3*(5*sqrt(3)*b*c^(3/n)*x^4 + 8*sqrt(3)*a
*x)*(a^2*b*c^(3/n))^(1/3))/((sqrt(3)*a^2*b^2*c^(6/n)*x^6 + 2*sqrt(3)*a^3*b*c^(3/
n)*x^3 + sqrt(3)*a^4)*(a^2*b*c^(3/n))^(1/3))

_______________________________________________________________________________________

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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