3.529 \(\int \frac{1}{x^2 (a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=100 \[ -\frac{128 b x}{35 a^5 \sqrt{a+b x^2}}-\frac{64 b x}{35 a^4 \left (a+b x^2\right )^{3/2}}-\frac{48 b x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{8 b x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac{1}{a x \left (a+b x^2\right )^{7/2}} \]

[Out]

-(1/(a*x*(a + b*x^2)^(7/2))) - (8*b*x)/(7*a^2*(a + b*x^2)^(7/2)) - (48*b*x)/(35*a^3*(a + b*x^2)^(5/2)) - (64*b
*x)/(35*a^4*(a + b*x^2)^(3/2)) - (128*b*x)/(35*a^5*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.0263844, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {271, 192, 191} \[ -\frac{128 b x}{35 a^5 \sqrt{a+b x^2}}-\frac{64 b x}{35 a^4 \left (a+b x^2\right )^{3/2}}-\frac{48 b x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{8 b x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac{1}{a x \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x^2)^(9/2)),x]

[Out]

-(1/(a*x*(a + b*x^2)^(7/2))) - (8*b*x)/(7*a^2*(a + b*x^2)^(7/2)) - (48*b*x)/(35*a^3*(a + b*x^2)^(5/2)) - (64*b
*x)/(35*a^4*(a + b*x^2)^(3/2)) - (128*b*x)/(35*a^5*Sqrt[a + b*x^2])

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 192

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx &=-\frac{1}{a x \left (a+b x^2\right )^{7/2}}-\frac{(8 b) \int \frac{1}{\left (a+b x^2\right )^{9/2}} \, dx}{a}\\ &=-\frac{1}{a x \left (a+b x^2\right )^{7/2}}-\frac{8 b x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac{(48 b) \int \frac{1}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a^2}\\ &=-\frac{1}{a x \left (a+b x^2\right )^{7/2}}-\frac{8 b x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac{48 b x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{(192 b) \int \frac{1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^3}\\ &=-\frac{1}{a x \left (a+b x^2\right )^{7/2}}-\frac{8 b x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac{48 b x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{64 b x}{35 a^4 \left (a+b x^2\right )^{3/2}}-\frac{(128 b) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{35 a^4}\\ &=-\frac{1}{a x \left (a+b x^2\right )^{7/2}}-\frac{8 b x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac{48 b x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{64 b x}{35 a^4 \left (a+b x^2\right )^{3/2}}-\frac{128 b x}{35 a^5 \sqrt{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0114345, size = 64, normalized size = 0.64 \[ \frac{-560 a^2 b^2 x^4-280 a^3 b x^2-35 a^4-448 a b^3 x^6-128 b^4 x^8}{35 a^5 x \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x^2)^(9/2)),x]

[Out]

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

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Maple [A]  time = 0.004, size = 61, normalized size = 0.6 \begin{align*} -{\frac{128\,{b}^{4}{x}^{8}+448\,{b}^{3}{x}^{6}a+560\,{b}^{2}{x}^{4}{a}^{2}+280\,b{x}^{2}{a}^{3}+35\,{a}^{4}}{35\,{a}^{5}x} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x^2+a)^(9/2),x)

[Out]

-1/35*(128*b^4*x^8+448*a*b^3*x^6+560*a^2*b^2*x^4+280*a^3*b*x^2+35*a^4)/x/(b*x^2+a)^(7/2)/a^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(1/x^2/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.38824, size = 221, normalized size = 2.21 \begin{align*} -\frac{{\left (128 \, b^{4} x^{8} + 448 \, a b^{3} x^{6} + 560 \, a^{2} b^{2} x^{4} + 280 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt{b x^{2} + a}}{35 \,{\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 3.85121, size = 400, normalized size = 4. \begin{align*} - \frac{35 a^{4} b^{\frac{33}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac{280 a^{3} b^{\frac{35}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac{560 a^{2} b^{\frac{37}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac{448 a b^{\frac{39}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac{128 b^{\frac{41}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x**2+a)**(9/2),x)

[Out]

-35*a**4*b**(33/2)*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*
b**19*x**6 + 35*a**5*b**20*x**8) - 280*a**3*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**1
7*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 560*a**2*b**(37/2)*x**4*sqrt(a/(b*x
**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**
8) - 448*a*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17*x**2 + 210*a**7*b**18*x**4 + 14
0*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 128*b**(41/2)*x**8*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b*
*17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8)

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Giac [A]  time = 2.09056, size = 122, normalized size = 1.22 \begin{align*} -\frac{{\left ({\left (x^{2}{\left (\frac{93 \, b^{4} x^{2}}{a^{5}} + \frac{308 \, b^{3}}{a^{4}}\right )} + \frac{350 \, b^{2}}{a^{3}}\right )} x^{2} + \frac{140 \, b}{a^{2}}\right )} x}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} + \frac{2 \, \sqrt{b}}{{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

-1/35*((x^2*(93*b^4*x^2/a^5 + 308*b^3/a^4) + 350*b^2/a^3)*x^2 + 140*b/a^2)*x/(b*x^2 + a)^(7/2) + 2*sqrt(b)/(((
sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)*a^4)