3.1.67 \(\int (c x)^m (b x^2)^p \, dx\) [67]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {15, 20, 30} \begin {gather*} \frac {x \left (b x^2\right )^p (c x)^m}{m+2 p+1} \end {gather*}

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)

Rule 15

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

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int (c x)^m \left (b x^2\right )^p \, dx &=\left (x^{-2 p} \left (b x^2\right )^p\right ) \int x^{2 p} (c x)^m \, dx\\ &=\left (x^{-m-2 p} (c x)^m \left (b x^2\right )^p\right ) \int x^{m+2 p} \, dx\\ &=\frac {x (c x)^m \left (b x^2\right )^p}{1+m+2 p}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} \frac {x (c x)^m \left (b x^2\right )^p}{1+m+2 p} \end {gather*}

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)

________________________________________________________________________________________

Maple [A]
time = 0.08, size = 23, normalized size = 1.05

method	result	size
gosper	\(\frac {x \left (c x \right )^{m} \left (b \,x^{2}\right )^{p}}{1+m +2 p}\)	\(23\)
norman	\(\frac {x \,{\mathrm e}^{m \ln \left (c x \right )} {\mathrm e}^{p \ln \left (b \,x^{2}\right )}}{1+m +2 p}\)	\(27\)
risch	\(\frac {x \,{\mathrm e}^{-\frac {i \mathrm {csgn}\left (i c x \right )^{3} \pi m}{2}+\frac {i \mathrm {csgn}\left (i c x \right )^{2} \mathrm {csgn}\left (i c \right ) \pi m}{2}+\frac {i \mathrm {csgn}\left (i c x \right )^{2} \mathrm {csgn}\left (i x \right ) \pi m}{2}-\frac {i \mathrm {csgn}\left (i c x \right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x \right ) \pi m}{2}+m \ln \left (c \right )+m \ln \left (x \right )-\frac {i \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2} \pi p}{2}+i \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right ) \pi p -\frac {i \mathrm {csgn}\left (i x^{2}\right )^{3} \pi p}{2}+\frac {i \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i b \,x^{2}\right )^{2} \pi p}{2}-\frac {i \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i b \,x^{2}\right ) \mathrm {csgn}\left (i b \right ) \pi p}{2}-\frac {i \mathrm {csgn}\left (i b \,x^{2}\right )^{3} \pi p}{2}+\frac {i \mathrm {csgn}\left (i b \,x^{2}\right )^{2} \mathrm {csgn}\left (i b \right ) \pi p}{2}+p \ln \left (b \right )+2 p \ln \left (x \right )}}{1+m +2 p}\)	\(234\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(b*x^2)^p,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 27, normalized size = 1.23 \begin {gather*} \frac {b^{p} c^{m} x e^{\left (m \log \left (x\right ) + 2 \, p \log \left (x\right )\right )}}{m + 2 \, p + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 32, normalized size = 1.45 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {x \left (b x^{2}\right )^{p} \left (c x\right )^{m}}{m + 2 p + 1} & \text {for}\: m \neq - 2 p - 1 \\\int \left (b x^{2}\right )^{p} \left (c x\right )^{- 2 p - 1}\, dx & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**m*(b*x**2)**p,x)

[Out]

Piecewise((x*(b*x**2)**p*(c*x)**m/(m + 2*p + 1), Ne(m, -2*p - 1)), (Integral((b*x**2)**p*(c*x)**(-2*p - 1), x)
, True))

________________________________________________________________________________________

Giac [A]
time = 2.00, size = 29, normalized size = 1.32 \begin {gather*} \frac {x e^{\left (p \log \left (b\right ) + m \log \left (c\right ) + m \log \left (x\right ) + 2 \, p \log \left (x\right )\right )}}{m + 2 \, p + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.96, size = 22, normalized size = 1.00 \begin {gather*} \frac {x\,{\left (c\,x\right )}^m\,{\left (b\,x^2\right )}^p}{m+2\,p+1} \end {gather*}

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)/(m + 2*p + 1)

________________________________________________________________________________________