3.510 \(\int \frac{x^2}{(a+b x^2)^{5/2}} \, dx\)

Optimal. Leaf size=21 \[ \frac{x^3}{3 a \left (a+b x^2\right )^{3/2}} \]

[Out]

x^3/(3*a*(a + b*x^2)^(3/2))

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Rubi [A]  time = 0.0045606, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ \frac{x^3}{3 a \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^2)^(5/2),x]

[Out]

x^3/(3*a*(a + b*x^2)^(3/2))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^2}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac{x^3}{3 a \left (a+b x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0049984, size = 21, normalized size = 1. \[ \frac{x^3}{3 a \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x^2)^(5/2),x]

[Out]

x^3/(3*a*(a + b*x^2)^(3/2))

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Maple [A]  time = 0.004, size = 18, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x^2+a)^(5/2),x)

[Out]

1/3*x^3/a/(b*x^2+a)^(3/2)

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Maxima [A]  time = 3.02234, size = 46, normalized size = 2.19 \begin{align*} -\frac{x}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} + \frac{x}{3 \, \sqrt{b x^{2} + a} a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

-1/3*x/((b*x^2 + a)^(3/2)*b) + 1/3*x/(sqrt(b*x^2 + a)*a*b)

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Fricas [B]  time = 1.2413, size = 77, normalized size = 3.67 \begin{align*} \frac{\sqrt{b x^{2} + a} x^{3}}{3 \,{\left (a b^{2} x^{4} + 2 \, a^{2} b x^{2} + a^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.707849, size = 44, normalized size = 2.1 \begin{align*} \frac{x^{3}}{3 a^{\frac{5}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{3}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b*x**2+a)**(5/2),x)

[Out]

x**3/(3*a**(5/2)*sqrt(1 + b*x**2/a) + 3*a**(3/2)*b*x**2*sqrt(1 + b*x**2/a))

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Giac [A]  time = 2.04626, size = 23, normalized size = 1.1 \begin{align*} \frac{x^{3}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^2+a)^(5/2),x, algorithm="giac")

[Out]

1/3*x^3/((b*x^2 + a)^(3/2)*a)