3.496 \(\int \frac{x^5}{(a+b x^2)^{3/2}} \, dx\)

Optimal. Leaf size=55 \[ -\frac{a^2}{b^3 \sqrt{a+b x^2}}-\frac{2 a \sqrt{a+b x^2}}{b^3}+\frac{\left (a+b x^2\right )^{3/2}}{3 b^3} \]

[Out]

-(a^2/(b^3*Sqrt[a + b*x^2])) - (2*a*Sqrt[a + b*x^2])/b^3 + (a + b*x^2)^(3/2)/(3*b^3)

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Rubi [A]  time = 0.0327409, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{a^2}{b^3 \sqrt{a+b x^2}}-\frac{2 a \sqrt{a+b x^2}}{b^3}+\frac{\left (a+b x^2\right )^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x^2)^(3/2),x]

[Out]

-(a^2/(b^3*Sqrt[a + b*x^2])) - (2*a*Sqrt[a + b*x^2])/b^3 + (a + b*x^2)^(3/2)/(3*b^3)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^5}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{3/2}}-\frac{2 a}{b^2 \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b^2}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^2}{b^3 \sqrt{a+b x^2}}-\frac{2 a \sqrt{a+b x^2}}{b^3}+\frac{\left (a+b x^2\right )^{3/2}}{3 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0159502, size = 38, normalized size = 0.69 \[ \frac{-8 a^2-4 a b x^2+b^2 x^4}{3 b^3 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x^2)^(3/2),x]

[Out]

(-8*a^2 - 4*a*b*x^2 + b^2*x^4)/(3*b^3*Sqrt[a + b*x^2])

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Maple [A]  time = 0.004, size = 36, normalized size = 0.7 \begin{align*} -{\frac{-{b}^{2}{x}^{4}+4\,ab{x}^{2}+8\,{a}^{2}}{3\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^2+a)^(3/2),x)

[Out]

-1/3*(-b^2*x^4+4*a*b*x^2+8*a^2)/(b*x^2+a)^(1/2)/b^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.29551, size = 93, normalized size = 1.69 \begin{align*} \frac{{\left (b^{2} x^{4} - 4 \, a b x^{2} - 8 \, a^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (b^{4} x^{2} + a b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

1/3*(b^2*x^4 - 4*a*b*x^2 - 8*a^2)*sqrt(b*x^2 + a)/(b^4*x^2 + a*b^3)

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Sympy [A]  time = 0.943373, size = 68, normalized size = 1.24 \begin{align*} \begin{cases} - \frac{8 a^{2}}{3 b^{3} \sqrt{a + b x^{2}}} - \frac{4 a x^{2}}{3 b^{2} \sqrt{a + b x^{2}}} + \frac{x^{4}}{3 b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x**2+a)**(3/2),x)

[Out]

Piecewise((-8*a**2/(3*b**3*sqrt(a + b*x**2)) - 4*a*x**2/(3*b**2*sqrt(a + b*x**2)) + x**4/(3*b*sqrt(a + b*x**2)
), Ne(b, 0)), (x**6/(6*a**(3/2)), True))

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Giac [A]  time = 1.95599, size = 55, normalized size = 1. \begin{align*} \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}} - 6 \, \sqrt{b x^{2} + a} a - \frac{3 \, a^{2}}{\sqrt{b x^{2} + a}}}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

1/3*((b*x^2 + a)^(3/2) - 6*sqrt(b*x^2 + a)*a - 3*a^2/sqrt(b*x^2 + a))/b^3