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

Optimal. Leaf size=94 \[ -\frac{a^4}{7 b^5 \left (a+b x^2\right )^{7/2}}+\frac{4 a^3}{5 b^5 \left (a+b x^2\right )^{5/2}}-\frac{2 a^2}{b^5 \left (a+b x^2\right )^{3/2}}+\frac{4 a}{b^5 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^5} \]

[Out]

-a^4/(7*b^5*(a + b*x^2)^(7/2)) + (4*a^3)/(5*b^5*(a + b*x^2)^(5/2)) - (2*a^2)/(b^5*(a + b*x^2)^(3/2)) + (4*a)/(
b^5*Sqrt[a + b*x^2]) + Sqrt[a + b*x^2]/b^5

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Rubi [A]  time = 0.0531119, antiderivative size = 94, 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} \[ -\frac{a^4}{7 b^5 \left (a+b x^2\right )^{7/2}}+\frac{4 a^3}{5 b^5 \left (a+b x^2\right )^{5/2}}-\frac{2 a^2}{b^5 \left (a+b x^2\right )^{3/2}}+\frac{4 a}{b^5 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^9/(a + b*x^2)^(9/2),x]

[Out]

-a^4/(7*b^5*(a + b*x^2)^(7/2)) + (4*a^3)/(5*b^5*(a + b*x^2)^(5/2)) - (2*a^2)/(b^5*(a + b*x^2)^(3/2)) + (4*a)/(
b^5*Sqrt[a + b*x^2]) + Sqrt[a + b*x^2]/b^5

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^9}{\left (a+b x^2\right )^{9/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{(a+b x)^{9/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^4}{b^4 (a+b x)^{9/2}}-\frac{4 a^3}{b^4 (a+b x)^{7/2}}+\frac{6 a^2}{b^4 (a+b x)^{5/2}}-\frac{4 a}{b^4 (a+b x)^{3/2}}+\frac{1}{b^4 \sqrt{a+b x}}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^4}{7 b^5 \left (a+b x^2\right )^{7/2}}+\frac{4 a^3}{5 b^5 \left (a+b x^2\right )^{5/2}}-\frac{2 a^2}{b^5 \left (a+b x^2\right )^{3/2}}+\frac{4 a}{b^5 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.0281393, size = 61, normalized size = 0.65 \[ \frac{560 a^2 b^2 x^4+448 a^3 b x^2+128 a^4+280 a b^3 x^6+35 b^4 x^8}{35 b^5 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(a + b*x^2)^(9/2),x]

[Out]

(128*a^4 + 448*a^3*b*x^2 + 560*a^2*b^2*x^4 + 280*a*b^3*x^6 + 35*b^4*x^8)/(35*b^5*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.004, size = 58, normalized size = 0.6 \begin{align*}{\frac{35\,{x}^{8}{b}^{4}+280\,a{x}^{6}{b}^{3}+560\,{a}^{2}{x}^{4}{b}^{2}+448\,{a}^{3}{x}^{2}b+128\,{a}^{4}}{35\,{b}^{5}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35*(35*b^4*x^8+280*a*b^3*x^6+560*a^2*b^2*x^4+448*a^3*b*x^2+128*a^4)/(b*x^2+a)^(7/2)/b^5

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.34429, size = 217, normalized size = 2.31 \begin{align*} \frac{{\left (35 \, b^{4} x^{8} + 280 \, a b^{3} x^{6} + 560 \, a^{2} b^{2} x^{4} + 448 \, a^{3} b x^{2} + 128 \, a^{4}\right )} \sqrt{b x^{2} + a}}{35 \,{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35*(35*b^4*x^8 + 280*a*b^3*x^6 + 560*a^2*b^2*x^4 + 448*a^3*b*x^2 + 128*a^4)*sqrt(b*x^2 + a)/(b^9*x^8 + 4*a*b
^8*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5)

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Sympy [A]  time = 5.56295, size = 454, normalized size = 4.83 \begin{align*} \begin{cases} \frac{128 a^{4}}{35 a^{3} b^{5} \sqrt{a + b x^{2}} + 105 a^{2} b^{6} x^{2} \sqrt{a + b x^{2}} + 105 a b^{7} x^{4} \sqrt{a + b x^{2}} + 35 b^{8} x^{6} \sqrt{a + b x^{2}}} + \frac{448 a^{3} b x^{2}}{35 a^{3} b^{5} \sqrt{a + b x^{2}} + 105 a^{2} b^{6} x^{2} \sqrt{a + b x^{2}} + 105 a b^{7} x^{4} \sqrt{a + b x^{2}} + 35 b^{8} x^{6} \sqrt{a + b x^{2}}} + \frac{560 a^{2} b^{2} x^{4}}{35 a^{3} b^{5} \sqrt{a + b x^{2}} + 105 a^{2} b^{6} x^{2} \sqrt{a + b x^{2}} + 105 a b^{7} x^{4} \sqrt{a + b x^{2}} + 35 b^{8} x^{6} \sqrt{a + b x^{2}}} + \frac{280 a b^{3} x^{6}}{35 a^{3} b^{5} \sqrt{a + b x^{2}} + 105 a^{2} b^{6} x^{2} \sqrt{a + b x^{2}} + 105 a b^{7} x^{4} \sqrt{a + b x^{2}} + 35 b^{8} x^{6} \sqrt{a + b x^{2}}} + \frac{35 b^{4} x^{8}}{35 a^{3} b^{5} \sqrt{a + b x^{2}} + 105 a^{2} b^{6} x^{2} \sqrt{a + b x^{2}} + 105 a b^{7} x^{4} \sqrt{a + b x^{2}} + 35 b^{8} x^{6} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{10}}{10 a^{\frac{9}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(b*x**2+a)**(9/2),x)

[Out]

Piecewise((128*a**4/(35*a**3*b**5*sqrt(a + b*x**2) + 105*a**2*b**6*x**2*sqrt(a + b*x**2) + 105*a*b**7*x**4*sqr
t(a + b*x**2) + 35*b**8*x**6*sqrt(a + b*x**2)) + 448*a**3*b*x**2/(35*a**3*b**5*sqrt(a + b*x**2) + 105*a**2*b**
6*x**2*sqrt(a + b*x**2) + 105*a*b**7*x**4*sqrt(a + b*x**2) + 35*b**8*x**6*sqrt(a + b*x**2)) + 560*a**2*b**2*x*
*4/(35*a**3*b**5*sqrt(a + b*x**2) + 105*a**2*b**6*x**2*sqrt(a + b*x**2) + 105*a*b**7*x**4*sqrt(a + b*x**2) + 3
5*b**8*x**6*sqrt(a + b*x**2)) + 280*a*b**3*x**6/(35*a**3*b**5*sqrt(a + b*x**2) + 105*a**2*b**6*x**2*sqrt(a + b
*x**2) + 105*a*b**7*x**4*sqrt(a + b*x**2) + 35*b**8*x**6*sqrt(a + b*x**2)) + 35*b**4*x**8/(35*a**3*b**5*sqrt(a
+ b*x**2) + 105*a**2*b**6*x**2*sqrt(a + b*x**2) + 105*a*b**7*x**4*sqrt(a + b*x**2) + 35*b**8*x**6*sqrt(a + b*
x**2)), Ne(b, 0)), (x**10/(10*a**(9/2)), True))

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Giac [A]  time = 1.72733, size = 96, normalized size = 1.02 \begin{align*} \frac{35 \, \sqrt{b x^{2} + a} + \frac{140 \,{\left (b x^{2} + a\right )}^{3} a - 70 \,{\left (b x^{2} + a\right )}^{2} a^{2} + 28 \,{\left (b x^{2} + a\right )} a^{3} - 5 \, a^{4}}{{\left (b x^{2} + a\right )}^{\frac{7}{2}}}}{35 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35*(35*sqrt(b*x^2 + a) + (140*(b*x^2 + a)^3*a - 70*(b*x^2 + a)^2*a^2 + 28*(b*x^2 + a)*a^3 - 5*a^4)/(b*x^2 +
a)^(7/2))/b^5