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

Optimal. Leaf size=29 \[ \frac{b \sqrt{b x^2} (c x)^{m+4}}{c^4 (m+4) x} \]

[Out]

(b*(c*x)^(4 + m)*Sqrt[b*x^2])/(c^4*(4 + m)*x)

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Rubi [A]  time = 0.0098853, antiderivative size = 29, 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} \[ \frac{b \sqrt{b x^2} (c x)^{m+4}}{c^4 (m+4) x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(c*x)^(4 + m)*Sqrt[b*x^2])/(c^4*(4 + 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 \left (b x^2\right )^{3/2} \, dx &=\frac{\left (b \sqrt{b x^2}\right ) \int x^3 (c x)^m \, dx}{x}\\ &=\frac{\left (b \sqrt{b x^2}\right ) \int (c x)^{3+m} \, dx}{c^3 x}\\ &=\frac{b (c x)^{4+m} \sqrt{b x^2}}{c^4 (4+m) x}\\ \end{align*}

Mathematica [A]  time = 0.0044416, size = 21, normalized size = 0.72 \[ \frac{x \left (b x^2\right )^{3/2} (c x)^m}{m+4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(c*x)^m*(b*x^2)^(3/2))/(4 + m)

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Maple [A]  time = 0.003, size = 20, normalized size = 0.7 \begin{align*}{\frac{x \left ( cx \right ) ^{m}}{4+m} \left ( b{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(4+m)*(c*x)^m*(b*x^2)^(3/2)

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Maxima [A]  time = 1.05888, size = 24, normalized size = 0.83 \begin{align*} \frac{b^{\frac{3}{2}} c^{m} x^{4} x^{m}}{m + 4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^(3/2)*c^m*x^4*x^m/(m + 4)

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Fricas [A]  time = 1.50792, size = 47, normalized size = 1.62 \begin{align*} \frac{\sqrt{b x^{2}} \left (c x\right )^{m} b x^{3}}{m + 4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2)*(c*x)^m*b*x^3/(m + 4)

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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)**(3/2),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)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError