3.67 \(\int (c x)^m \left (b x^2\right )^p \, dx\)

Optimal. Leaf size=22 \[ \frac{x \left (b x^2\right )^p (c x)^m}{m+2 p+1} \]

[Out]

(x*(c*x)^m*(b*x^2)^p)/(1 + m + 2*p)

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Rubi [A]  time = 0.01594, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{x \left (b x^2\right )^p (c x)^m}{m+2 p+1} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m*(b*x^2)^p,x]

[Out]

(x*(c*x)^m*(b*x^2)^p)/(1 + m + 2*p)

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Rubi in Sympy [A]  time = 4.52286, size = 34, normalized size = 1.55 \[ \frac{x^{- m} x^{- 2 p} x^{m + 2 p + 1} \left (b x^{2}\right )^{p} \left (c x\right )^{m}}{m + 2 p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**m*(b*x**2)**p,x)

[Out]

x**(-m)*x**(-2*p)*x**(m + 2*p + 1)*(b*x**2)**p*(c*x)**m/(m + 2*p + 1)

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Mathematica [A]  time = 0.00617215, size = 22, normalized size = 1. \[ \frac{x \left (b x^2\right )^p (c x)^m}{m+2 p+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^m*(b*x^2)^p,x]

[Out]

(x*(c*x)^m*(b*x^2)^p)/(1 + m + 2*p)

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Maple [A]  time = 0.001, size = 23, normalized size = 1.1 \[{\frac{x \left ( cx \right ) ^{m} \left ( b{x}^{2} \right ) ^{p}}{1+m+2\,p}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m*(b*x^2)^p,x)

[Out]

x*(c*x)^m*(b*x^2)^p/(1+m+2*p)

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Maxima [A]  time = 1.45728, size = 36, normalized size = 1.64 \[ \frac{b^{p} c^{m} x e^{\left (m \log \left (x\right ) + 2 \, p \log \left (x\right )\right )}}{m + 2 \, p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2)^p*(c*x)^m,x, algorithm="maxima")

[Out]

b^p*c^m*x*e^(m*log(x) + 2*p*log(x))/(m + 2*p + 1)

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Fricas [A]  time = 0.238384, size = 43, normalized size = 1.95 \[ \frac{\left (c x\right )^{m} x e^{\left (2 \, p \log \left (c x\right ) + p \log \left (\frac{b}{c^{2}}\right )\right )}}{m + 2 \, p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2)^p*(c*x)^m,x, algorithm="fricas")

[Out]

(c*x)^m*x*e^(2*p*log(c*x) + p*log(b/c^2))/(m + 2*p + 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**2)**p,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.218912, size = 39, normalized size = 1.77 \[ \frac{x e^{\left (p{\rm ln}\left (b\right ) + m{\rm ln}\left (c\right ) + m{\rm ln}\left (x\right ) + 2 \, p{\rm ln}\left (x\right )\right )}}{m + 2 \, p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2)^p*(c*x)^m,x, algorithm="giac")

[Out]

x*e^(p*ln(b) + m*ln(c) + m*ln(x) + 2*p*ln(x))/(m + 2*p + 1)