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)*(b*x^2)^(1/2)/c^4/(4+m)/x

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Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.69, size = 22, normalized size = 0.76 \[ \frac {\sqrt {b x^{2}} \left (c x\right )^{m} b x^{3}}{m + 4} \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]Undef/Unsigned Inf encountered in limitLimit: Max order reached or unable to make series expansion Error:
Bad Argument Value

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maple [A]  time = 0.00, size = 20, normalized size = 0.69 \[ \frac {\left (b \,x^{2}\right )^{\frac {3}{2}} x \left (c x \right )^{m}}{m +4} \]

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 = 1.39, size = 18, normalized size = 0.62 \[ \frac {b^{\frac {3}{2}} c^{m} x^{4} x^{m}}{m + 4} \]

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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mupad [B]  time = 0.96, size = 22, normalized size = 0.76 \[ \frac {b^{3/2}\,x^3\,{\left (c\,x\right )}^m\,\sqrt {x^2}}{m+4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^(3/2)*x^3*(c*x)^m*(x^2)^(1/2))/(m + 4)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {b^{\frac {3}{2}} c^{m} x x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m + 4} & \text {for}\: m \neq -4 \\\frac {\int \frac {\left (b x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx}{c^{4}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((b**(3/2)*c**m*x*x**m*(x**2)**(3/2)/(m + 4), Ne(m, -4)), (Integral((b*x**2)**(3/2)/x**4, x)/c**4, Tr
ue))

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