3.163 \(\int (c x)^m (b x^n)^{3/2} \, dx\)

Optimal. Leaf size=32 \[ \frac{2 b x^{n+1} \sqrt{b x^n} (c x)^m}{2 m+3 n+2} \]

[Out]

(2*b*x^(1 + n)*(c*x)^m*Sqrt[b*x^n])/(2 + 2*m + 3*n)

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Rubi [A]  time = 0.0068314, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {15, 20, 30} \[ \frac{2 b x^{n+1} \sqrt{b x^n} (c x)^m}{2 m+3 n+2} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*(b*x^n)^(3/2),x]

[Out]

(2*b*x^(1 + n)*(c*x)^m*Sqrt[b*x^n])/(2 + 2*m + 3*n)

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^n\right )^{3/2} \, dx &=\left (b x^{-n/2} \sqrt{b x^n}\right ) \int x^{3 n/2} (c x)^m \, dx\\ &=\left (b x^{-m-\frac{n}{2}} (c x)^m \sqrt{b x^n}\right ) \int x^{m+\frac{3 n}{2}} \, dx\\ &=\frac{2 b x^{1+n} (c x)^m \sqrt{b x^n}}{2+2 m+3 n}\\ \end{align*}

Mathematica [A]  time = 0.0058748, size = 26, normalized size = 0.81 \[ \frac{x \left (b x^n\right )^{3/2} (c x)^m}{m+\frac{3 n}{2}+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m*(b*x^n)^(3/2),x]

[Out]

(x*(c*x)^m*(b*x^n)^(3/2))/(1 + m + (3*n)/2)

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Maple [A]  time = 0.002, size = 26, normalized size = 0.8 \begin{align*} 2\,{\frac{x \left ( cx \right ) ^{m} \left ( b{x}^{n} \right ) ^{3/2}}{2+2\,m+3\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(b*x^n)^(3/2),x)

[Out]

2*x/(2+2*m+3*n)*(c*x)^m*(b*x^n)^(3/2)

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Maxima [A]  time = 1.03371, size = 36, normalized size = 1.12 \begin{align*} \frac{2 \, b^{\frac{3}{2}} c^{m} x x^{m}{\left (x^{n}\right )}^{\frac{3}{2}}}{2 \, m + 3 \, n + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(b*x^n)^(3/2),x, algorithm="maxima")

[Out]

2*b^(3/2)*c^m*x*x^m*(x^n)^(3/2)/(2*m + 3*n + 2)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(b*x^n)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**m*(b*x**n)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.16829, size = 42, normalized size = 1.31 \begin{align*} \frac{2 \, b^{\frac{3}{2}} x x^{\frac{3}{2} \, n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{2 \, m + 3 \, n + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(b*x^n)^(3/2),x, algorithm="giac")

[Out]

2*b^(3/2)*x*x^(3/2*n)*e^(m*log(c) + m*log(x))/(2*m + 3*n + 2)