3.522 \(\int \frac{x^5}{(a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=59 \[ -\frac{a^2}{7 b^3 \left (a+b x^2\right )^{7/2}}+\frac{2 a}{5 b^3 \left (a+b x^2\right )^{5/2}}-\frac{1}{3 b^3 \left (a+b x^2\right )^{3/2}} \]

[Out]

-a^2/(7*b^3*(a + b*x^2)^(7/2)) + (2*a)/(5*b^3*(a + b*x^2)^(5/2)) - 1/(3*b^3*(a + b*x^2)^(3/2))

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Rubi [A]  time = 0.033334, antiderivative size = 59, 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} \[ -\frac{a^2}{7 b^3 \left (a+b x^2\right )^{7/2}}+\frac{2 a}{5 b^3 \left (a+b x^2\right )^{5/2}}-\frac{1}{3 b^3 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x^2)^(9/2),x]

[Out]

-a^2/(7*b^3*(a + b*x^2)^(7/2)) + (2*a)/(5*b^3*(a + b*x^2)^(5/2)) - 1/(3*b^3*(a + b*x^2)^(3/2))

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 )^{9/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{9/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{9/2}}-\frac{2 a}{b^2 (a+b x)^{7/2}}+\frac{1}{b^2 (a+b x)^{5/2}}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^2}{7 b^3 \left (a+b x^2\right )^{7/2}}+\frac{2 a}{5 b^3 \left (a+b x^2\right )^{5/2}}-\frac{1}{3 b^3 \left (a+b x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0179678, size = 39, normalized size = 0.66 \[ \frac{-8 a^2-28 a b x^2-35 b^2 x^4}{105 b^3 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x^2)^(9/2),x]

[Out]

(-8*a^2 - 28*a*b*x^2 - 35*b^2*x^4)/(105*b^3*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.006, size = 36, normalized size = 0.6 \begin{align*} -{\frac{35\,{b}^{2}{x}^{4}+28\,ab{x}^{2}+8\,{a}^{2}}{105\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^2+a)^(9/2),x)

[Out]

-1/105*(35*b^2*x^4+28*a*b*x^2+8*a^2)/(b*x^2+a)^(7/2)/b^3

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Maxima [A]  time = 1.4914, size = 72, normalized size = 1.22 \begin{align*} -\frac{x^{4}}{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{4 \, a x^{2}}{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} - \frac{8 \, a^{2}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

-1/3*x^4/((b*x^2 + a)^(7/2)*b) - 4/15*a*x^2/((b*x^2 + a)^(7/2)*b^2) - 8/105*a^2/((b*x^2 + a)^(7/2)*b^3)

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Fricas [A]  time = 1.31278, size = 167, normalized size = 2.83 \begin{align*} -\frac{{\left (35 \, b^{2} x^{4} + 28 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{b x^{2} + a}}{105 \,{\left (b^{7} x^{8} + 4 \, a b^{6} x^{6} + 6 \, a^{2} b^{5} x^{4} + 4 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

-1/105*(35*b^2*x^4 + 28*a*b*x^2 + 8*a^2)*sqrt(b*x^2 + a)/(b^7*x^8 + 4*a*b^6*x^6 + 6*a^2*b^5*x^4 + 4*a^3*b^4*x^
2 + a^4*b^3)

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Sympy [A]  time = 5.39027, size = 272, normalized size = 4.61 \begin{align*} \begin{cases} - \frac{8 a^{2}}{105 a^{3} b^{3} \sqrt{a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt{a + b x^{2}} + 315 a b^{5} x^{4} \sqrt{a + b x^{2}} + 105 b^{6} x^{6} \sqrt{a + b x^{2}}} - \frac{28 a b x^{2}}{105 a^{3} b^{3} \sqrt{a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt{a + b x^{2}} + 315 a b^{5} x^{4} \sqrt{a + b x^{2}} + 105 b^{6} x^{6} \sqrt{a + b x^{2}}} - \frac{35 b^{2} x^{4}}{105 a^{3} b^{3} \sqrt{a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt{a + b x^{2}} + 315 a b^{5} x^{4} \sqrt{a + b x^{2}} + 105 b^{6} x^{6} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 a^{\frac{9}{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)**(9/2),x)

[Out]

Piecewise((-8*a**2/(105*a**3*b**3*sqrt(a + b*x**2) + 315*a**2*b**4*x**2*sqrt(a + b*x**2) + 315*a*b**5*x**4*sqr
t(a + b*x**2) + 105*b**6*x**6*sqrt(a + b*x**2)) - 28*a*b*x**2/(105*a**3*b**3*sqrt(a + b*x**2) + 315*a**2*b**4*
x**2*sqrt(a + b*x**2) + 315*a*b**5*x**4*sqrt(a + b*x**2) + 105*b**6*x**6*sqrt(a + b*x**2)) - 35*b**2*x**4/(105
*a**3*b**3*sqrt(a + b*x**2) + 315*a**2*b**4*x**2*sqrt(a + b*x**2) + 315*a*b**5*x**4*sqrt(a + b*x**2) + 105*b**
6*x**6*sqrt(a + b*x**2)), Ne(b, 0)), (x**6/(6*a**(9/2)), True))

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Giac [A]  time = 2.4579, size = 55, normalized size = 0.93 \begin{align*} -\frac{35 \,{\left (b x^{2} + a\right )}^{2} - 42 \,{\left (b x^{2} + a\right )} a + 15 \, a^{2}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

-1/105*(35*(b*x^2 + a)^2 - 42*(b*x^2 + a)*a + 15*a^2)/((b*x^2 + a)^(7/2)*b^3)