3.63 \(\int (c x)^m \sqrt{b x^2} \, dx\)

Optimal. Leaf size=28 \[ \frac{\sqrt{b x^2} (c x)^{m+2}}{c^2 (m+2) x} \]

[Out]

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

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Rubi [A]  time = 0.0090569, antiderivative size = 28, 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, 16, 32} \[ \frac{\sqrt{b x^2} (c x)^{m+2}}{c^2 (m+2) x} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*Sqrt[b*x^2],x]

[Out]

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

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 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0033506, size = 21, normalized size = 0.75 \[ \frac{x \sqrt{b x^2} (c x)^m}{m+2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m*Sqrt[b*x^2],x]

[Out]

(x*(c*x)^m*Sqrt[b*x^2])/(2 + m)

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Maple [A]  time = 0., size = 20, normalized size = 0.7 \begin{align*}{\frac{x \left ( cx \right ) ^{m}}{2+m}\sqrt{b{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03814, size = 24, normalized size = 0.86 \begin{align*} \frac{\sqrt{b} c^{m} x^{2} x^{m}}{m + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*c^m*x^2*x^m/(m + 2)

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Fricas [A]  time = 1.90814, size = 42, normalized size = 1.5 \begin{align*} \frac{\sqrt{b x^{2}} \left (c x\right )^{m} x}{m + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2)*(c*x)^m*x/(m + 2)

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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)**(1/2),x)

[Out]

Exception raised: TypeError

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Giac [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)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError