3.1 \(\int (b x)^p (c x)^m \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0227751, antiderivative size = 22, 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 \[ \frac{(b x)^{p+1} (c x)^m}{b (m+p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(b*x)^p*(c*x)^m,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.4361, size = 24, normalized size = 1.09 \[ \frac{\left (b x\right )^{- m} \left (b x\right )^{m + p + 1} \left (c x\right )^{m}}{b \left (m + p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00695899, size = 18, normalized size = 0.82 \[ \frac{x (b x)^p (c x)^m}{m+p+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x)^p*(c*x)^m,x]

[Out]

(x*(b*x)^p*(c*x)^m)/(1 + m + p)

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Maple [A]  time = 0.002, size = 19, normalized size = 0.9 \[{\frac{x \left ( bx \right ) ^{p} \left ( cx \right ) ^{m}}{1+m+p}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x)^p*(c*x)^m,x)

[Out]

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

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Maxima [A]  time = 1.44001, size = 32, normalized size = 1.45 \[ \frac{b^{p} c^{m} x e^{\left (m \log \left (x\right ) + p \log \left (x\right )\right )}}{m + p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.2681, size = 39, normalized size = 1.77 \[ \frac{\left (b x\right )^{p} x e^{\left (m \log \left (b x\right ) + m \log \left (\frac{c}{b}\right )\right )}}{m + p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.210412, size = 35, normalized size = 1.59 \[ \frac{x e^{\left (p{\rm ln}\left (b\right ) + m{\rm ln}\left (c\right ) + m{\rm ln}\left (x\right ) + p{\rm ln}\left (x\right )\right )}}{m + p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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