3.162 \(\int (c x)^m (b x^n)^{5/2} \, dx\)

Optimal. Leaf size=31 \[ \frac{2 \left (b x^n\right )^{5/2} (c x)^{m+1}}{c (2 m+5 n+2)} \]

[Out]

(2*(c*x)^(1 + m)*(b*x^n)^(5/2))/(c*(2 + 2*m + 5*n))

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Rubi [A]  time = 0.0133699, antiderivative size = 36, normalized size of antiderivative = 1.16, 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} \[ \frac{2 b^2 x^{2 n+1} \sqrt{b x^n} (c x)^m}{2 m+5 n+2} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*(b*x^n)^(5/2),x]

[Out]

(2*b^2*x^(1 + 2*n)*(c*x)^m*Sqrt[b*x^n])/(2 + 2*m + 5*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 )^{5/2} \, dx &=\left (b^2 x^{-n/2} \sqrt{b x^n}\right ) \int x^{5 n/2} (c x)^m \, dx\\ &=\left (b^2 x^{-m-\frac{n}{2}} (c x)^m \sqrt{b x^n}\right ) \int x^{m+\frac{5 n}{2}} \, dx\\ &=\frac{2 b^2 x^{1+2 n} (c x)^m \sqrt{b x^n}}{2+2 m+5 n}\\ \end{align*}

Mathematica [A]  time = 0.0095282, size = 26, normalized size = 0.84 \[ \frac{x \left (b x^n\right )^{5/2} (c x)^m}{m+\frac{5 n}{2}+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m*(b*x^n)^(5/2),x]

[Out]

(x*(c*x)^m*(b*x^n)^(5/2))/(1 + m + (5*n)/2)

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Maple [A]  time = 0.004, size = 26, normalized size = 0.8 \begin{align*} 2\,{\frac{x \left ( cx \right ) ^{m} \left ( b{x}^{n} \right ) ^{5/2}}{2+2\,m+5\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(b*x^n)^(5/2),x)

[Out]

2*x/(2+2*m+5*n)*(c*x)^m*(b*x^n)^(5/2)

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Maxima [A]  time = 1.03179, size = 36, normalized size = 1.16 \begin{align*} \frac{2 \, b^{\frac{5}{2}} c^{m} x x^{m}{\left (x^{n}\right )}^{\frac{5}{2}}}{2 \, m + 5 \, n + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(b*x^n)^(5/2),x, algorithm="maxima")

[Out]

2*b^(5/2)*c^m*x*x^m*(x^n)^(5/2)/(2*m + 5*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)^(5/2),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)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.15526, size = 42, normalized size = 1.35 \begin{align*} \frac{2 \, b^{\frac{5}{2}} x x^{\frac{5}{2} \, n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{2 \, m + 5 \, n + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(b*x^n)^(5/2),x, algorithm="giac")

[Out]

2*b^(5/2)*x*x^(5/2*n)*e^(m*log(c) + m*log(x))/(2*m + 5*n + 2)