3.190 \(\int x^2 \left (a \left (b x^n\right )^p\right )^q \, dx\)

Optimal. Leaf size=23 \[ \frac{x^3 \left (a \left (b x^n\right )^p\right )^q}{n p q+3} \]

[Out]

(x^3*(a*(b*x^n)^p)^q)/(3 + n*p*q)

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Rubi [A]  time = 0.082746, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^3 \left (a \left (b x^n\right )^p\right )^q}{n p q+3} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a*(b*x^n)^p)^q,x]

[Out]

(x^3*(a*(b*x^n)^p)^q)/(3 + n*p*q)

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Rubi in Sympy [A]  time = 7.21918, size = 31, normalized size = 1.35 \[ \frac{x^{- n p q} x^{n p q + 3} \left (a \left (b x^{n}\right )^{p}\right )^{q}}{n p q + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(a*(b*x**n)**p)**q,x)

[Out]

x**(-n*p*q)*x**(n*p*q + 3)*(a*(b*x**n)**p)**q/(n*p*q + 3)

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Mathematica [A]  time = 0.00874674, size = 23, normalized size = 1. \[ \frac{x^3 \left (a \left (b x^n\right )^p\right )^q}{n p q+3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a*(b*x^n)^p)^q,x]

[Out]

(x^3*(a*(b*x^n)^p)^q)/(3 + n*p*q)

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Maple [A]  time = 0.002, size = 24, normalized size = 1. \[{\frac{{x}^{3} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{npq+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a*(b*x^n)^p)^q,x)

[Out]

x^3*(a*(b*x^n)^p)^q/(n*p*q+3)

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Maxima [A]  time = 1.65175, size = 36, normalized size = 1.57 \[ \frac{a^{q}{\left (b^{p}\right )}^{q} x^{3}{\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{n p q + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^n)^p*a)^q*x^2,x, algorithm="maxima")

[Out]

a^q*(b^p)^q*x^3*((x^n)^p)^q/(n*p*q + 3)

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Fricas [A]  time = 0.24272, size = 39, normalized size = 1.7 \[ \frac{x^{3} e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{n p q + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^n)^p*a)^q*x^2,x, algorithm="fricas")

[Out]

x^3*e^(n*p*q*log(x) + p*q*log(b) + q*log(a))/(n*p*q + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(a*(b*x**n)**p)**q,x)

[Out]

Integral(x**2*(a*(b*x**n)**p)**q, x)

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GIAC/XCAS [A]  time = 0.22734, size = 39, normalized size = 1.7 \[ \frac{x^{3} e^{\left (n p q{\rm ln}\left (x\right ) + p q{\rm ln}\left (b\right ) + q{\rm ln}\left (a\right )\right )}}{n p q + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^n)^p*a)^q*x^2,x, algorithm="giac")

[Out]

x^3*e^(n*p*q*ln(x) + p*q*ln(b) + q*ln(a))/(n*p*q + 3)