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

Optimal result

Integrand size = 11, antiderivative size = 22 \[ \int (b x)^p (c x)^m \, dx=\frac {(b x)^{1+p} (c x)^m}{b (1+m+p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {20, 32} \[ \int (b x)^p (c x)^m \, dx=\frac {(b x)^{p+1} (c x)^m}{b (m+p+1)} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int (b x)^p (c x)^m \, dx=\frac {x (b x)^p (c x)^m}{1+m+p} \]

[In]

Integrate[(b*x)^p*(c*x)^m,x]

[Out]

(x*(b*x)^p*(c*x)^m)/(1 + m + p)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86

method	result	size
gosper	\(\frac {x \left (b x \right )^{p} \left (c x \right )^{m}}{1+m +p}\)	\(19\)
parallelrisch	\(\frac {x \left (b x \right )^{p} \left (c x \right )^{m}}{1+m +p}\)	\(19\)
norman	\(\frac {x \,{\mathrm e}^{m \ln \left (c x \right )} {\mathrm e}^{p \ln \left (b x \right )}}{1+m +p}\)	\(23\)
risch	\(\frac {b^{p} x^{p} c^{m} x^{m} x \,{\mathrm e}^{\frac {i \pi \left (-\operatorname {csgn}\left (i c x \right )^{3} m +\operatorname {csgn}\left (i c x \right )^{2} \operatorname {csgn}\left (i c \right ) m +\operatorname {csgn}\left (i c x \right )^{2} \operatorname {csgn}\left (i x \right ) m -\operatorname {csgn}\left (i c x \right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x \right ) m +\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i b x \right )^{2} p -\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i b x \right ) \operatorname {csgn}\left (i b \right ) p -\operatorname {csgn}\left (i b x \right )^{3} p +\operatorname {csgn}\left (i b x \right )^{2} \operatorname {csgn}\left (i b \right ) p \right )}{2}}}{1+m +p}\)	\(147\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int (b x)^p (c x)^m \, dx=\frac {\left (b x\right )^{p} x e^{\left (m \log \left (b x\right ) + m \log \left (\frac {c}{b}\right )\right )}}{m + p + 1} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (17) = 34\).

Time = 0.81 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.77 \[ \int (b x)^p (c x)^m \, dx=\begin {cases} \frac {x \left (b x\right )^{p} \left (c x\right )^{m}}{m + p + 1} & \text {for}\: m \neq - p - 1 \\\begin {cases} b^{p} c^{- p - 1} \log {\left (x \right )} & \text {for}\: \left |{x}\right | < 1 \\- b^{p} c^{- p - 1} {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} + b^{p} c^{- p - 1} {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} & \text {otherwise} \end {cases} & \text {otherwise} \end {cases} \]

[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*c**(-p - 1)*log(x), Abs(x) < 1),
(-b**p*c**(-p - 1)*meijerg(((), (1, 1)), ((0, 0), ()), x) + b**p*c**(-p - 1)*meijerg(((1, 1), ()), ((), (0, 0)
), x), True)), True))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (b x)^p (c x)^m \, dx=\frac {b^{p} c^{m} x e^{\left (m \log \left (x\right ) + p \log \left (x\right )\right )}}{m + p + 1} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int (b x)^p (c x)^m \, dx=\frac {x e^{\left (p \log \left (b\right ) + m \log \left (c\right ) + m \log \left (x\right ) + p \log \left (x\right )\right )}}{m + p + 1} \]

[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)

Mupad [B] (verification not implemented)

Time = 5.78 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int (b x)^p (c x)^m \, dx=\frac {x\,{\left (b\,x\right )}^p\,{\left (c\,x\right )}^m}{m+p+1} \]

[In]

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

[Out]

(x*(b*x)^p*(c*x)^m)/(m + p + 1)