3.2707 \(\int x \left (b x^n\right )^p \, dx\)

Optimal. Leaf size=18 \[ \frac{x^2 \left (b x^n\right )^p}{n p+2} \]

[Out]

(x^2*(b*x^n)^p)/(2 + n*p)

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Rubi [A]  time = 0.0164161, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{x^2 \left (b x^n\right )^p}{n p+2} \]

Antiderivative was successfully verified.

[In]  Int[x*(b*x^n)^p,x]

[Out]

(x^2*(b*x^n)^p)/(2 + n*p)

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Rubi in Sympy [A]  time = 2.78981, size = 22, normalized size = 1.22 \[ \frac{x^{- n p} x^{n p + 2} \left (b x^{n}\right )^{p}}{n p + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x**n)**p,x)

[Out]

x**(-n*p)*x**(n*p + 2)*(b*x**n)**p/(n*p + 2)

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Mathematica [A]  time = 0.00436617, size = 18, normalized size = 1. \[ \frac{x^2 \left (b x^n\right )^p}{n p+2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(b*x^n)^p,x]

[Out]

(x^2*(b*x^n)^p)/(2 + n*p)

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Maple [A]  time = 0.001, size = 19, normalized size = 1.1 \[{\frac{{x}^{2} \left ( b{x}^{n} \right ) ^{p}}{np+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^n)^p,x)

[Out]

x^2*(b*x^n)^p/(n*p+2)

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Maxima [A]  time = 1.43012, size = 26, normalized size = 1.44 \[ \frac{b^{p} x^{2}{\left (x^{n}\right )}^{p}}{n p + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^p*x^2*(x^n)^p/(n*p + 2)

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Fricas [A]  time = 0.232329, size = 30, normalized size = 1.67 \[ \frac{x^{2} e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^2*e^(n*p*log(x) + p*log(b))/(n*p + 2)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x**n)**p,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.218035, size = 30, normalized size = 1.67 \[ \frac{x^{2} e^{\left (n p{\rm ln}\left (x\right ) + p{\rm ln}\left (b\right )\right )}}{n p + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^2*e^(n*p*ln(x) + p*ln(b))/(n*p + 2)