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

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Rubi [A]  time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 64, normalized size = 0.64 \[ \frac {-35 a^4-280 a^3 b x^2-560 a^2 b^2 x^4-448 a b^3 x^6-128 b^4 x^8}{35 a^5 x \left (a+b x^2\right )^{7/2}} \]

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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fricas [A]  time = 0.83, size = 103, normalized size = 1.03 \[ -\frac {{\left (128 \, b^{4} x^{8} + 448 \, a b^{3} x^{6} + 560 \, a^{2} b^{2} x^{4} + 280 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt {b x^{2} + a}}{35 \, {\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}} \]

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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giac [A]  time = 1.12, size = 90, normalized size = 0.90 \[ -\frac {{\left ({\left (x^{2} {\left (\frac {93 \, b^{4} x^{2}}{a^{5}} + \frac {308 \, b^{3}}{a^{4}}\right )} + \frac {350 \, b^{2}}{a^{3}}\right )} x^{2} + \frac {140 \, b}{a^{2}}\right )} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a^{4}} \]

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)

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maple [A]  time = 0.01, size = 61, normalized size = 0.61 \[ -\frac {128 b^{4} x^{8}+448 a \,b^{3} x^{6}+560 a^{2} b^{2} x^{4}+280 a^{3} b \,x^{2}+35 a^{4}}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{5} x} \]

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 [A]  time = 1.43, size = 82, normalized size = 0.82 \[ -\frac {128 \, b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} - \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} \]

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]

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

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mupad [B]  time = 4.71, size = 76, normalized size = 0.76 \[ -\frac {\frac {1}{a^4}+\frac {128\,b\,x^2}{35\,a^5}}{x\,\sqrt {b\,x^2+a}}-\frac {29\,b\,x}{35\,a^4\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {13\,b\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {b\,x}{7\,a^2\,{\left (b\,x^2+a\right )}^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x^2)^(9/2)),x)

[Out]

- (1/a^4 + (128*b*x^2)/(35*a^5))/(x*(a + b*x^2)^(1/2)) - (29*b*x)/(35*a^4*(a + b*x^2)^(3/2)) - (13*b*x)/(35*a^
3*(a + b*x^2)^(5/2)) - (b*x)/(7*a^2*(a + b*x^2)^(7/2))

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sympy [B]  time = 2.93, size = 400, normalized size = 4.00 \[ - \frac {35 a^{4} b^{\frac {33}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac {280 a^{3} b^{\frac {35}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac {560 a^{2} b^{\frac {37}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac {448 a b^{\frac {39}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac {128 b^{\frac {41}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} \]

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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