3.3006 \(\int \frac{x^3}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx\)

Optimal. Leaf size=114 \[ \frac{a^3 x^4 \left (c x^n\right )^{-4/n}}{b^4 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{3 a^2 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^4}-\frac{2 a x^4 \left (c x^n\right )^{-3/n}}{b^3}+\frac{x^4 \left (c x^n\right )^{-2/n}}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.103353, antiderivative size = 114, 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 \[ \frac{a^3 x^4 \left (c x^n\right )^{-4/n}}{b^4 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{3 a^2 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^4}-\frac{2 a x^4 \left (c x^n\right )^{-3/n}}{b^3}+\frac{x^4 \left (c x^n\right )^{-2/n}}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{3} x^{4} \left (c x^{n}\right )^{- \frac{4}{n}}}{b^{4} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )} + \frac{3 a^{2} x^{4} \left (c x^{n}\right )^{- \frac{4}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{b^{4}} - \frac{2 a x^{4} \left (c x^{n}\right )^{- \frac{3}{n}}}{b^{3}} + \frac{x^{4} \left (c x^{n}\right )^{- \frac{4}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} x\, dx}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 4.38211, size = 0, normalized size = 0. \[ \int \frac{x^3}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [C]  time = 0.06, size = 662, normalized size = 5.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^4/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))-3*a/(c^(1/n))^3/b^3*x*exp(-3/2*(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*c
sgn(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/a/(c^(1/n))/b*x^3*exp(-1/2*(I*Pi*csgn(I*x^n)*cs
gn(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)+3/2/(c^(1/n))^2/b^2*x^2*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)+3*a^2/(c^
(1/n))^4/b^4*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)*exp(-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)

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Maxima [A]  time = 22.7162, size = 136, normalized size = 1.19 \[ \frac{x^{4}}{a b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2}} + \frac{3 \, a^{2} c^{-\frac{4}{n}} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{4}} - \frac{{\left (2 \, b^{2} c^{\frac{2}{n}} x^{3} - 3 \, a b c^{\left (\frac{1}{n}\right )} x^{2} + 6 \, a^{2} x\right )} c^{-\frac{3}{n}}}{2 \, a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((c*x^n)^(1/n)*b + a)^2,x, algorithm="maxima")

[Out]

x^4/(a*b*c^(1/n)*(x^n)^(1/n) + a^2) + 3*a^2*c^(-4/n)*log(b*c^(1/n)*x + a)/b^4 -
1/2*(2*b^2*c^(2/n)*x^3 - 3*a*b*c^(1/n)*x^2 + 6*a^2*x)*c^(-3/n)/(a*b^3)

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Fricas [A]  time = 0.232186, size = 142, normalized size = 1.25 \[ \frac{b^{3} c^{\frac{3}{n}} x^{3} - 3 \, a b^{2} c^{\frac{2}{n}} x^{2} - 4 \, a^{2} b c^{\left (\frac{1}{n}\right )} x + 2 \, a^{3} + 6 \,{\left (a^{2} b c^{\left (\frac{1}{n}\right )} x + a^{3}\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{2 \,{\left (b^{5} c^{\frac{5}{n}} x + a b^{4} c^{\frac{4}{n}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((c*x^n)^(1/n)*b + a)^2,x, algorithm="fricas")

[Out]

1/2*(b^3*c^(3/n)*x^3 - 3*a*b^2*c^(2/n)*x^2 - 4*a^2*b*c^(1/n)*x + 2*a^3 + 6*(a^2*
b*c^(1/n)*x + a^3)*log(b*c^(1/n)*x + a))/(b^5*c^(5/n)*x + a*b^4*c^(4/n))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((c*x^n)^(1/n)*b + a)^2,x, algorithm="giac")

[Out]

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