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

Optimal. Leaf size=26 \[ \frac{(c x)^{m+1} \left (b x^n\right )^p}{c (m+n p+1)} \]

[Out]

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

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Rubi [A]  time = 0.016426, antiderivative size = 22, normalized size of antiderivative = 0.85, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{x (c x)^m \left (b x^n\right )^p}{m+n p+1} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m*(b*x^n)^p,x]

[Out]

(x*(c*x)^m*(b*x^n)^p)/(1 + m + n*p)

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Rubi in Sympy [A]  time = 4.97659, size = 34, normalized size = 1.31 \[ \frac{x^{- m} x^{- n p} x^{m + n p + 1} \left (b x^{n}\right )^{p} \left (c x\right )^{m}}{m + n p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0072297, size = 22, normalized size = 0.85 \[ \frac{x (c x)^m \left (b x^n\right )^p}{m+n p+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^m*(b*x^n)^p,x]

[Out]

(x*(c*x)^m*(b*x^n)^p)/(1 + m + n*p)

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Maple [A]  time = 0.001, size = 23, normalized size = 0.9 \[{\frac{x \left ( cx \right ) ^{m} \left ( b{x}^{n} \right ) ^{p}}{np+m+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m*(b*x^n)^p,x)

[Out]

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

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Maxima [A]  time = 1.44685, size = 38, normalized size = 1.46 \[ \frac{b^{p} c^{m} x e^{\left (m \log \left (x\right ) + p \log \left (x^{n}\right )\right )}}{n p + m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.252716, size = 39, normalized size = 1.5 \[ \frac{x e^{\left (n p \log \left (x\right ) + p \log \left (b\right ) + m \log \left (c\right ) + m \log \left (x\right )\right )}}{n p + m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.231311, size = 39, normalized size = 1.5 \[ \frac{x e^{\left (n p{\rm ln}\left (x\right ) + p{\rm ln}\left (b\right ) + m{\rm ln}\left (c\right ) + m{\rm ln}\left (x\right )\right )}}{n p + m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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