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 = 0.0071705, 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, Rules used = {20, 32} \[ \frac{(b x)^{p+1} (c x)^m}{b (m+p+1)} \]

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))

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int (b x)^p (c x)^m \, dx &=\left ((b x)^{-m} (c x)^m\right ) \int (b x)^{m+p} \, dx\\ &=\frac{(b x)^{1+p} (c x)^m}{b (1+m+p)}\\ \end{align*}

Mathematica [A]  time = 0.0039337, 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 \begin{align*}{\frac{x \left ( bx \right ) ^{p} \left ( cx \right ) ^{m}}{1+m+p}} \end{align*}

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 = 1.06032, size = 32, normalized size = 1.45 \begin{align*} \frac{b^{p} c^{m} x e^{\left (m \log \left (x\right ) + p \log \left (x\right )\right )}}{m + p + 1} \end{align*}

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 = 1.38174, size = 69, normalized size = 3.14 \begin{align*} \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} \end{align*}

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. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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 [A]  time = 1.18008, size = 35, normalized size = 1.59 \begin{align*} \frac{x e^{\left (p \log \left (b\right ) + m \log \left (c\right ) + m \log \left (x\right ) + p \log \left (x\right )\right )}}{m + p + 1} \end{align*}

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