3.67 \(\int (c x)^m (b x^2)^p \, dx\)

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

[Out]

(x*(c*x)^m*(b*x^2)^p)/(1 + m + 2*p)

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Rubi [A]  time = 0.0054969, antiderivative size = 22, normalized size of antiderivative = 1., 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} \[ \frac{x \left (b x^2\right )^p (c x)^m}{m+2 p+1} \]

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 = 0.0029196, size = 22, normalized size = 1. \[ \frac{x \left (b x^2\right )^p (c x)^m}{m+2 p+1} \]

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)

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Maple [A]  time = 0., size = 23, normalized size = 1.1 \begin{align*}{\frac{x \left ( cx \right ) ^{m} \left ( b{x}^{2} \right ) ^{p}}{1+m+2\,p}} \end{align*}

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/(1+m+2*p)

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Maxima [A]  time = 1.09021, size = 36, normalized size = 1.64 \begin{align*} \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{align*}

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)

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Fricas [A]  time = 2.06913, size = 77, normalized size = 3.5 \begin{align*} \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{align*}

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)

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.17046, size = 39, normalized size = 1.77 \begin{align*} \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{align*}

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)