3.84 \(\int (b x^2)^{3/2} \, dx\)

Optimal. Leaf size=17 \[ \frac {1}{4} b x^3 \sqrt {b x^2} \]

[Out]

1/4*b*x^3*(b*x^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \[ \frac {1}{4} b x^3 \sqrt {b x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x^3*Sqrt[b*x^2])/4

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \left (b x^2\right )^{3/2} \, dx &=\frac {\left (b \sqrt {b x^2}\right ) \int x^3 \, dx}{x}\\ &=\frac {1}{4} b x^3 \sqrt {b x^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 14, normalized size = 0.82 \[ \frac {1}{4} x \left (b x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b*x^2)^(3/2))/4

________________________________________________________________________________________

fricas [A]  time = 0.72, size = 13, normalized size = 0.76 \[ \frac {1}{4} \, \sqrt {b x^{2}} b x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(b*x^2)*b*x^3

________________________________________________________________________________________

giac [A]  time = 0.17, size = 10, normalized size = 0.59 \[ \frac {1}{4} \, b^{\frac {3}{2}} x^{4} \mathrm {sgn}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(3/2),x, algorithm="giac")

[Out]

1/4*b^(3/2)*x^4*sgn(x)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 11, normalized size = 0.65 \[ \frac {\left (b \,x^{2}\right )^{\frac {3}{2}} x}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b*x^2)^(3/2)*x

________________________________________________________________________________________

maxima [A]  time = 1.31, size = 10, normalized size = 0.59 \[ \frac {1}{4} \, \left (b x^{2}\right )^{\frac {3}{2}} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b*x^2)^(3/2)*x

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int {\left (b\,x^2\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((b*x^2)^(3/2), x)

________________________________________________________________________________________

sympy [A]  time = 0.39, size = 14, normalized size = 0.82 \[ \frac {b^{\frac {3}{2}} x \left (x^{2}\right )^{\frac {3}{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**(3/2)*x*(x**2)**(3/2)/4

________________________________________________________________________________________