3.2999 \(\int \frac{x^3}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{a^3 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^4}+\frac{a^2 x^4 \left (c x^n\right )^{-3/n}}{b^3}-\frac{a x^4 \left (c x^n\right )^{-2/n}}{2 b^2}+\frac{x^4 \left (c x^n\right )^{-1/n}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0918268, antiderivative size = 101, 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 \[ -\frac{a^3 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^4}+\frac{a^2 x^4 \left (c x^n\right )^{-3/n}}{b^3}-\frac{a x^4 \left (c x^n\right )^{-2/n}}{2 b^2}+\frac{x^4 \left (c x^n\right )^{-1/n}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*x^4)/(b^3*(c*x^n)^(3/n)) - (a*x^4)/(2*b^2*(c*x^n)^(2/n)) + x^4/(3*b*(c*x^n)
^n^(-1)) - (a^3*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. \[ - \frac{a^{3} x^{4} \left (c x^{n}\right )^{- \frac{4}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{b^{4}} - \frac{a x^{4} \left (c x^{n}\right )^{- \frac{4}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} x\, dx}{b^{2}} + \frac{x^{4} \left (c x^{n}\right )^{- \frac{1}{n}}}{3 b} + \frac{x^{4} \left (c x^{n}\right )^{- \frac{4}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} a^{2}\, dx}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 4.82499, size = 0, normalized size = 0. \[ \int \frac{x^3}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [C]  time = 0.279, size = 553, normalized size = 5.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b*(c*x^n)^(1/n)),x)

[Out]

1/(c^(1/n))^3/b^3*a^2*x*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)+1/3/(c^(1/n))/b*x^3*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)-1/2*a/(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)-1/(c^(1/n))^4/
b^4*a^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*csgn(I*c)*csgn(I*c*x^n)^2-I*Pi*csgn(I*c*x^n)^3+2*ln(c)+2*l
n(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*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 = 22.5987, size = 99, normalized size = 0.98 \[ -\frac{a^{3} c^{-\frac{4}{n}} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{4}} + \frac{{\left (2 \, b^{2} c^{\frac{2}{n}} x^{3} - 3 \, a b c^{\left (\frac{1}{n}\right )} x^{2} + 6 \, a^{2} x\right )} c^{-\frac{3}{n}}}{6 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.256682, size = 100, normalized size = 0.99 \[ \frac{2 \, b^{3} c^{\frac{3}{n}} x^{3} - 3 \, a b^{2} c^{\frac{2}{n}} x^{2} + 6 \, a^{2} b c^{\left (\frac{1}{n}\right )} x - 6 \, a^{3} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{6 \, b^{4} c^{\frac{4}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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