3.3007 \(\int \frac{x^2}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx\)

Optimal. Leaf size=90 \[ -\frac{a^2 x^3 \left (c x^n\right )^{-3/n}}{b^3 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}-\frac{2 a x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^3}+\frac{x^3 \left (c x^n\right )^{-2/n}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0826087, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{a^2 x^3 \left (c x^n\right )^{-3/n}}{b^3 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}-\frac{2 a x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^3}+\frac{x^3 \left (c x^n\right )^{-2/n}}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*(c*x^n)^n^(-1))^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*x**3*(c*x**n)**(-3/n)/(b**3*(a + b*(c*x**n)**(1/n))) - 2*a*x**3*(c*x**n)**
(-3/n)*log(a + b*(c*x**n)**(1/n))/b**3 + x**3*(c*x**n)**(-3/n)*Integral(b**(-2),
(x, (c*x**n)**(1/n)))

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Mathematica [A]  time = 4.35264, size = 0, normalized size = 0. \[ \int \frac{x^2}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[x^2/(a + b*(c*x^n)^n^(-1))^2,x]

[Out]

Integrate[x^2/(a + b*(c*x^n)^n^(-1))^2, x]

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Maple [C]  time = 0.051, size = 548, normalized size = 6.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^3/a/(a+b*exp(1/2*(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*csgn(I*x^n)*csgn(I*c)*
csgn(I*c*x^n)-I*Pi*csgn(I*c*x^n)^3+I*Pi*csgn(I*c)*csgn(I*c*x^n)^2+2*ln(c)+2*ln(x
^n))/n))-1/a/(c^(1/n))/b*x^2*exp(-1/2*(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*csg
n(I*x^n)*csgn(I*c)*csgn(I*c*x^n)-I*Pi*csgn(I*c*x^n)^3+I*Pi*csgn(I*c)*csgn(I*c*x^
n)^2-2*n*ln(x)+2*ln(x^n))/n)+2/(c^(1/n))^2/b^2*x*exp(-(I*Pi*csgn(I*x^n)*csgn(I*c
*x^n)^2-I*Pi*csgn(I*x^n)*csgn(I*c)*csgn(I*c*x^n)-I*Pi*csgn(I*c*x^n)^3+I*Pi*csgn(
I*c)*csgn(I*c*x^n)^2-2*n*ln(x)+2*ln(x^n))/n)-2*a/(c^(1/n))^3/b^3*ln(b*exp(1/2*(-
I*Pi*csgn(I*x^n)*csgn(I*c)*csgn(I*c*x^n)+I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+I*Pi*c
sgn(I*c)*csgn(I*c*x^n)^2-I*Pi*csgn(I*c*x^n)^3+2*ln(c)+2*ln(x^n)-2*n*ln(x))/n)*x+
a)*exp(-3/2*(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*csgn(I*x^n)*csgn(I*c)*csgn(I*
c*x^n)-I*Pi*csgn(I*c*x^n)^3+I*Pi*csgn(I*c)*csgn(I*c*x^n)^2-2*n*ln(x)+2*ln(x^n))/
n)

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Maxima [A]  time = 22.8601, size = 108, normalized size = 1.2 \[ \frac{x^{3}}{a b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2}} - \frac{2 \, a c^{-\frac{3}{n}} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{3}} - \frac{{\left (b c^{\left (\frac{1}{n}\right )} x^{2} - 2 \, a x\right )} c^{-\frac{2}{n}}}{a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((c*x^n)^(1/n)*b + a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.230757, size = 112, normalized size = 1.24 \[ \frac{b^{2} c^{\frac{2}{n}} x^{2} + a b c^{\left (\frac{1}{n}\right )} x - a^{2} - 2 \,{\left (a b c^{\left (\frac{1}{n}\right )} x + a^{2}\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{4} c^{\frac{4}{n}} x + a b^{3} c^{\frac{3}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((c*x^n)^(1/n)*b + a)^2,x, algorithm="fricas")

[Out]

(b^2*c^(2/n)*x^2 + a*b*c^(1/n)*x - a^2 - 2*(a*b*c^(1/n)*x + a^2)*log(b*c^(1/n)*x
+ a))/(b^4*c^(4/n)*x + a*b^3*c^(3/n))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((c*x^n)^(1/n)*b + a)^2,x, algorithm="giac")

[Out]

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