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

Optimal. Leaf size=56 \[ \frac{a^2 \sqrt{a+b x^2}}{b^3}+\frac{\left (a+b x^2\right )^{5/2}}{5 b^3}-\frac{2 a \left (a+b x^2\right )^{3/2}}{3 b^3} \]

[Out]

(a^2*Sqrt[a + b*x^2])/b^3 - (2*a*(a + b*x^2)^(3/2))/(3*b^3) + (a + b*x^2)^(5/2)/(5*b^3)

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Rubi [A]  time = 0.0307414, antiderivative size = 56, 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} \[ \frac{a^2 \sqrt{a+b x^2}}{b^3}+\frac{\left (a+b x^2\right )^{5/2}}{5 b^3}-\frac{2 a \left (a+b x^2\right )^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/Sqrt[a + b*x^2],x]

[Out]

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

Mathematica [A]  time = 0.0182842, size = 39, normalized size = 0.7 \[ \frac{\sqrt{a+b x^2} \left (8 a^2-4 a b x^2+3 b^2 x^4\right )}{15 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/Sqrt[a + b*x^2],x]

[Out]

(Sqrt[a + b*x^2]*(8*a^2 - 4*a*b*x^2 + 3*b^2*x^4))/(15*b^3)

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Maple [A]  time = 0.004, size = 36, normalized size = 0.6 \begin{align*}{\frac{3\,{b}^{2}{x}^{4}-4\,ab{x}^{2}+8\,{a}^{2}}{15\,{b}^{3}}\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)^(1/2),x)

[Out]

1/15*(b*x^2+a)^(1/2)*(3*b^2*x^4-4*a*b*x^2+8*a^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)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.27504, size = 78, normalized size = 1.39 \begin{align*} \frac{{\left (3 \, b^{2} x^{4} - 4 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{b x^{2} + a}}{15 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

1/15*(3*b^2*x^4 - 4*a*b*x^2 + 8*a^2)*sqrt(b*x^2 + a)/b^3

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Sympy [A]  time = 0.781171, size = 68, normalized size = 1.21 \begin{align*} \begin{cases} \frac{8 a^{2} \sqrt{a + b x^{2}}}{15 b^{3}} - \frac{4 a x^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{x^{4} \sqrt{a + b x^{2}}}{5 b} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 \sqrt{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x**2+a)**(1/2),x)

[Out]

Piecewise((8*a**2*sqrt(a + b*x**2)/(15*b**3) - 4*a*x**2*sqrt(a + b*x**2)/(15*b**2) + x**4*sqrt(a + b*x**2)/(5*
b), Ne(b, 0)), (x**6/(6*sqrt(a)), True))

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Giac [A]  time = 2.49782, size = 58, normalized size = 1.04 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 10 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x^{2} + a} a^{2}}{15 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

1/15*(3*(b*x^2 + a)^(5/2) - 10*(b*x^2 + a)^(3/2)*a + 15*sqrt(b*x^2 + a)*a^2)/b^3