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

Optimal. Leaf size=21 \[ \frac{x^{m+1} \left (b x^n\right )^p}{m+n p+1} \]

[Out]

(x^(1 + m)*(b*x^n)^p)/(1 + m + n*p)

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Rubi [A]  time = 0.0157806, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x^{m+1} \left (b x^n\right )^p}{m+n p+1} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(b*x^n)^p,x]

[Out]

(x^(1 + m)*(b*x^n)^p)/(1 + m + n*p)

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Rubi in Sympy [A]  time = 3.3874, size = 26, normalized size = 1.24 \[ \frac{x^{- n p} x^{m + n p + 1} \left (b x^{n}\right )^{p}}{m + n p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x**n)**p,x)

[Out]

x**(-n*p)*x**(m + n*p + 1)*(b*x**n)**p/(m + n*p + 1)

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Mathematica [A]  time = 0.0079071, size = 21, normalized size = 1. \[ \frac{x^{m+1} \left (b x^n\right )^p}{m+n p+1} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(b*x^n)^p,x]

[Out]

(x^(1 + m)*(b*x^n)^p)/(1 + m + n*p)

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Maple [A]  time = 0.003, size = 22, normalized size = 1.1 \[{\frac{{x}^{1+m} \left ( b{x}^{n} \right ) ^{p}}{np+m+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^n)^p,x)

[Out]

x^(1+m)*(b*x^n)^p/(n*p+m+1)

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Maxima [A]  time = 1.45394, size = 34, normalized size = 1.62 \[ \frac{b^{p} x e^{\left (m \log \left (x\right ) + p \log \left (x^{n}\right )\right )}}{n p + m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^p*x*e^(m*log(x) + p*log(x^n))/(n*p + m + 1)

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Fricas [A]  time = 0.235055, size = 32, normalized size = 1.52 \[ \frac{x x^{m} e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*x^m*e^(n*p*log(x) + p*log(b))/(n*p + m + 1)

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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**m*(b*x**n)**p,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.230086, size = 34, normalized size = 1.62 \[ \frac{x e^{\left (n p{\rm ln}\left (x\right ) + p{\rm ln}\left (b\right ) + m{\rm ln}\left (x\right )\right )}}{n p + m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*e^(n*p*ln(x) + p*ln(b) + m*ln(x))/(n*p + m + 1)