3.3000 \(\int \frac{x^2}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx\)

Optimal. Leaf size=77 \[ \frac{a^2 x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^3}-\frac{a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac{x^3 \left (c x^n\right )^{-1/n}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0705028, antiderivative size = 77, 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 \[ \frac{a^2 x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^3}-\frac{a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac{x^3 \left (c x^n\right )^{-1/n}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

-((a*x^3)/(b^2*(c*x^n)^(2/n))) + x^3/(2*b*(c*x^n)^n^(-1)) + (a^2*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. \[ \frac{a^{2} x^{3} \left (c x^{n}\right )^{- \frac{3}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{b^{3}} + \frac{x^{3} \left (c x^{n}\right )^{- \frac{3}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} x\, dx}{b} - \frac{x^{3} \left (c x^{n}\right )^{- \frac{3}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} a\, dx}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 4.74079, size = 0, normalized size = 0. \[ \int \frac{x^2}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [C]  time = 0.099, size = 439, normalized size = 5.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b*(c*x^n)^(1/n)),x)

[Out]

1/2/(c^(1/n))/b*x^2*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*n*
ln(x)+2*ln(x^n))/n)-1/(c^(1/n))^2/b^2*a*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)*cs
gn(I*c*x^n)^2-2*n*ln(x)+2*ln(x^n))/n)+1/(c^(1/n))^3/b^3*a^2*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*csgn(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)*ex
p(-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 = 23.0289, size = 72, normalized size = 0.94 \[ \frac{a^{2} c^{-\frac{3}{n}} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{3}} + \frac{{\left (b c^{\left (\frac{1}{n}\right )} x^{2} - 2 \, a x\right )} c^{-\frac{2}{n}}}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.255823, size = 74, normalized size = 0.96 \[ \frac{b^{2} c^{\frac{2}{n}} x^{2} - 2 \, a b c^{\left (\frac{1}{n}\right )} x + 2 \, a^{2} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{2 \, b^{3} c^{\frac{3}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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