3.164 \(\int (c x)^m \sqrt{b x^n} \, dx\)

Optimal. Leaf size=29 \[ \frac{2 \sqrt{b x^n} (c x)^{m+1}}{c (2 m+n+2)} \]

[Out]

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

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Rubi [A]  time = 0.005874, antiderivative size = 25, normalized size of antiderivative = 0.86, 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} \[ \frac{2 x \sqrt{b x^n} (c x)^m}{2 m+n+2} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*Sqrt[b*x^n],x]

[Out]

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

Mathematica [A]  time = 0.004673, size = 26, normalized size = 0.9 \[ \frac{x \sqrt{b x^n} (c x)^m}{m+\frac{n}{2}+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(c*x)^m*Sqrt[b*x^n])/(1 + m + n/2)

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Maple [A]  time = 0., size = 24, normalized size = 0.8 \begin{align*} 2\,{\frac{x \left ( cx \right ) ^{m}\sqrt{b{x}^{n}}}{2+2\,m+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(b*x^n)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.00467, size = 34, normalized size = 1.17 \begin{align*} \frac{2 \, \sqrt{b} c^{m} x x^{m} \sqrt{x^{n}}}{2 \, m + n + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(b*x^n)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(b)*c^m*x*x^m*sqrt(x^n)/(2*m + 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)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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**n)**(1/2),x)

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.18557, size = 39, normalized size = 1.34 \begin{align*} \frac{2 \, \sqrt{b} x x^{\frac{1}{2} \, n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{2 \, m + n + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(b*x^n)^(1/2),x, algorithm="giac")

[Out]

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