3.1.12 \(\int x^2 (b x^2)^{3/2} \, dx\) [12]

Optimal. Leaf size=17 \[ \frac {1}{6} b x^5 \sqrt {b x^2} \]

[Out]

1/6*b*x^5*(b*x^2)^(1/2)

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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \begin {gather*} \frac {1}{6} b x^5 \sqrt {b x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x^5*Sqrt[b*x^2])/6

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 x^2 \left (b x^2\right )^{3/2} \, dx &=\frac {\left (b \sqrt {b x^2}\right ) \int x^5 \, dx}{x}\\ &=\frac {1}{6} b x^5 \sqrt {b x^2}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 0.94 \begin {gather*} \frac {1}{6} x^3 \left (b x^2\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(b*x^2)^(3/2))/6

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Maple [A]
time = 0.02, size = 13, normalized size = 0.76

method	result	size
gosper	\(\frac {x^{3} \left (b \,x^{2}\right )^{\frac {3}{2}}}{6}\)	\(13\)
default	\(\frac {x^{3} \left (b \,x^{2}\right )^{\frac {3}{2}}}{6}\)	\(13\)
risch	\(\frac {b \,x^{5} \sqrt {b \,x^{2}}}{6}\)	\(14\)
trager	\(\frac {b \left (x^{5}+x^{4}+x^{3}+x^{2}+x +1\right ) \left (x -1\right ) \sqrt {b \,x^{2}}}{6 x}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/6*x^3*(b*x^2)^(3/2)

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Maxima [A]
time = 0.27, size = 13, normalized size = 0.76 \begin {gather*} \frac {\left (b x^{2}\right )^{\frac {5}{2}} x}{6 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b*x^2)^(5/2)*x/b

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Fricas [A]
time = 0.35, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, \sqrt {b x^{2}} b x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(b*x^2)*b*x^5

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Sympy [A]
time = 0.15, size = 12, normalized size = 0.71 \begin {gather*} \frac {x^{3} \left (b x^{2}\right )^{\frac {3}{2}}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3*(b*x**2)**(3/2)/6

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Giac [A]
time = 1.76, size = 10, normalized size = 0.59 \begin {gather*} \frac {1}{6} \, b^{\frac {3}{2}} x^{6} \mathrm {sgn}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^(3/2)*x^6*sgn(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*(b*x^2)^(3/2), x)

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