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

Optimal. Leaf size=54 \[ -\frac{a^2}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac{2 a}{b^3 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.0314975, antiderivative size = 54, 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} \[ -\frac{a^2}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac{2 a}{b^3 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x^2)^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.0173704, size = 39, normalized size = 0.72 \[ \frac{8 a^2+12 a b x^2+3 b^2 x^4}{3 b^3 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x^2)^(5/2),x]

[Out]

(8*a^2 + 12*a*b*x^2 + 3*b^2*x^4)/(3*b^3*(a + b*x^2)^(3/2))

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Maple [A]  time = 0.003, size = 36, normalized size = 0.7 \begin{align*}{\frac{3\,{b}^{2}{x}^{4}+12\,ab{x}^{2}+8\,{a}^{2}}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*b^2*x^4+12*a*b*x^2+8*a^2)/(b*x^2+a)^(3/2)/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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.27047, size = 119, normalized size = 2.2 \begin{align*} \frac{{\left (3 \, b^{2} x^{4} + 12 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*b^2*x^4 + 12*a*b*x^2 + 8*a^2)*sqrt(b*x^2 + a)/(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)

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Sympy [A]  time = 1.07568, size = 138, normalized size = 2.56 \begin{align*} \begin{cases} \frac{8 a^{2}}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} + \frac{12 a b x^{2}}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} + \frac{3 b^{2} x^{4}}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 a^{\frac{5}{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)**(5/2),x)

[Out]

Piecewise((8*a**2/(3*a*b**3*sqrt(a + b*x**2) + 3*b**4*x**2*sqrt(a + b*x**2)) + 12*a*b*x**2/(3*a*b**3*sqrt(a +
b*x**2) + 3*b**4*x**2*sqrt(a + b*x**2)) + 3*b**2*x**4/(3*a*b**3*sqrt(a + b*x**2) + 3*b**4*x**2*sqrt(a + b*x**2
)), Ne(b, 0)), (x**6/(6*a**(5/2)), True))

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Giac [A]  time = 2.35319, size = 58, normalized size = 1.07 \begin{align*} \frac{3 \, \sqrt{b x^{2} + a} + \frac{6 \,{\left (b x^{2} + a\right )} a - a^{2}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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