3.1.1 \(\int (b x)^p (c x)^m \, dx\) [1]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {(b x)^{p+1} (c x)^m}{b (m+p+1)} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 18, normalized size = 0.82 \begin {gather*} \frac {x (b x)^p (c x)^m}{1+m+p} \end {gather*}

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 = 0.09, size = 19, normalized size = 0.86

method	result	size
gosper	\(\frac {x \left (b x \right )^{p} \left (c x \right )^{m}}{1+m +p}\)	\(19\)
norman	\(\frac {x \,{\mathrm e}^{m \ln \left (c x \right )} {\mathrm e}^{p \ln \left (b x \right )}}{1+m +p}\)	\(23\)
risch	\(\frac {x \,{\mathrm e}^{-\frac {i \mathrm {csgn}\left (i c x \right )^{3} \pi m}{2}+\frac {i \mathrm {csgn}\left (i c x \right )^{2} \mathrm {csgn}\left (i c \right ) \pi m}{2}+\frac {i \mathrm {csgn}\left (i c x \right )^{2} \mathrm {csgn}\left (i x \right ) \pi m}{2}-\frac {i \mathrm {csgn}\left (i c x \right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x \right ) \pi m}{2}+m \ln \left (x \right )+m \ln \left (c \right )+\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i b x \right )^{2} p}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i b x \right ) \mathrm {csgn}\left (i b \right ) p}{2}-\frac {i \pi \mathrm {csgn}\left (i b x \right )^{3} p}{2}+\frac {i \pi \mathrm {csgn}\left (i b x \right )^{2} \mathrm {csgn}\left (i b \right ) p}{2}+p \ln \left (x \right )+p \ln \left (b \right )}}{1+m +p}\)	\(167\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x)^p*(c*x)^m,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 24, normalized size = 1.09 \begin {gather*} \frac {b^{p} c^{m} x e^{\left (m \log \left (x\right ) + p \log \left (x\right )\right )}}{m + p + 1} \end {gather*}

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 = 0.35, size = 29, normalized size = 1.32 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (17) = 34\).
time = 2.51, size = 56, normalized size = 2.55 \begin {gather*} \begin {cases} \frac {x \left (b x\right )^{p} \left (c x\right )^{m}}{m + p + 1} & \text {for}\: m \neq - p - 1 \\\begin {cases} \frac {b^{p} c^{- p} \log {\left (x \right )}}{c} & \text {for}\: \left |{x}\right | < 1 \\- \frac {b^{p} c^{- p} {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )}}{c} + \frac {b^{p} c^{- p} {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )}}{c} & \text {otherwise} \end {cases} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)**p*(c*x)**m,x)

[Out]

Piecewise((x*(b*x)**p*(c*x)**m/(m + p + 1), Ne(m, -p - 1)), (Piecewise((b**p*log(x)/(c*c**p), Abs(x) < 1), (-b
**p*meijerg(((), (1, 1)), ((0, 0), ()), x)/(c*c**p) + b**p*meijerg(((1, 1), ()), ((), (0, 0)), x)/(c*c**p), Tr
ue)), True))

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Giac [A]
time = 1.94, size = 26, normalized size = 1.18 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 1.06, size = 18, normalized size = 0.82 \begin {gather*} \frac {x\,{\left (b\,x\right )}^p\,{\left (c\,x\right )}^m}{m+p+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x)^p*(c*x)^m,x)

[Out]

(x*(b*x)^p*(c*x)^m)/(m + p + 1)

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