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3.1921 \(\int \frac{1}{\sqrt{a+\frac{b}{x^2}} x^9} \, dx\)

Optimal. Leaf size=75 \[ -\frac{a^2 \left (a+\frac{b}{x^2}\right )^{3/2}}{b^4}+\frac{a^3 \sqrt{a+\frac{b}{x^2}}}{b^4}-\frac{\left (a+\frac{b}{x^2}\right )^{7/2}}{7 b^4}+\frac{3 a \left (a+\frac{b}{x^2}\right )^{5/2}}{5 b^4} \]

[Out]

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

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Rubi [A]  time = 0.0392734, antiderivative size = 75, 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} \[ -\frac{a^2 \left (a+\frac{b}{x^2}\right )^{3/2}}{b^4}+\frac{a^3 \sqrt{a+\frac{b}{x^2}}}{b^4}-\frac{\left (a+\frac{b}{x^2}\right )^{7/2}}{7 b^4}+\frac{3 a \left (a+\frac{b}{x^2}\right )^{5/2}}{5 b^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^2]*x^9),x]

[Out]

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

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

Mathematica [A]  time = 0.0161335, size = 53, normalized size = 0.71 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (-8 a^2 b x^4+16 a^3 x^6+6 a b^2 x^2-5 b^3\right )}{35 b^4 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x^2]*x^9),x]

[Out]

(Sqrt[a + b/x^2]*(-5*b^3 + 6*a*b^2*x^2 - 8*a^2*b*x^4 + 16*a^3*x^6))/(35*b^4*x^6)

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Maple [A]  time = 0.005, size = 61, normalized size = 0.8 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) \left ( 16\,{a}^{3}{x}^{6}-8\,{a}^{2}b{x}^{4}+6\,a{b}^{2}{x}^{2}-5\,{b}^{3} \right ) }{35\,{x}^{8}{b}^{4}}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35*(a*x^2+b)*(16*a^3*x^6-8*a^2*b*x^4+6*a*b^2*x^2-5*b^3)/x^8/b^4/((a*x^2+b)/x^2)^(1/2)

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Maxima [A]  time = 1.00915, size = 85, normalized size = 1.13 \begin{align*} -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{7}{2}}}{7 \, b^{4}} + \frac{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} a}{5 \, b^{4}} - \frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} a^{2}}{b^{4}} + \frac{\sqrt{a + \frac{b}{x^{2}}} a^{3}}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.5285, size = 117, normalized size = 1.56 \begin{align*} \frac{{\left (16 \, a^{3} x^{6} - 8 \, a^{2} b x^{4} + 6 \, a b^{2} x^{2} - 5 \, b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{35 \, b^{4} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35*(16*a^3*x^6 - 8*a^2*b*x^4 + 6*a*b^2*x^2 - 5*b^3)*sqrt((a*x^2 + b)/x^2)/(b^4*x^6)

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Sympy [B]  time = 3.92326, size = 1969, normalized size = 26.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16*a**(25/2)*b**(23/2)*x**18*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 210*a**(17/2)*b**16*x**17 + 525*a*
*(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**
(7/2)*b**21*x**7) + 88*a**(23/2)*b**(25/2)*x**16*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 210*a**(17/2)*
b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2)*b**19*x**11 + 210*a**(9/2)
*b**20*x**9 + 35*a**(7/2)*b**21*x**7) + 198*a**(21/2)*b**(27/2)*x**14*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x
**19 + 210*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2)*b**19
*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) + 231*a**(19/2)*b**(29/2)*x**12*sqrt(a*x**2/b + 1)/
(35*a**(19/2)*b**15*x**19 + 210*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13
+ 525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) + 140*a**(17/2)*b**(31/2)*x**1
0*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 210*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a
**(13/2)*b**18*x**13 + 525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) + 21*a**(
15/2)*b**(33/2)*x**8*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 210*a**(17/2)*b**16*x**17 + 525*a**(15/2)*
b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b*
*21*x**7) - 42*a**(13/2)*b**(35/2)*x**6*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 210*a**(17/2)*b**16*x**
17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x*
*9 + 35*a**(7/2)*b**21*x**7) - 47*a**(11/2)*b**(37/2)*x**4*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 210*
a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2)*b**19*x**11 + 21
0*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) - 24*a**(9/2)*b**(39/2)*x**2*sqrt(a*x**2/b + 1)/(35*a**(19/2)*
b**15*x**19 + 210*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2
)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) - 5*a**(7/2)*b**(41/2)*sqrt(a*x**2/b + 1)/(3
5*a**(19/2)*b**15*x**19 + 210*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 +
525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) - 16*a**13*b**11*x**19/(35*a**(1
9/2)*b**15*x**19 + 210*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**
(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) - 96*a**12*b**12*x**17/(35*a**(19/2)*b*
*15*x**19 + 210*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2)*
b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) - 240*a**11*b**13*x**15/(35*a**(19/2)*b**15*x*
*19 + 210*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2)*b**19*
x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) - 320*a**10*b**14*x**13/(35*a**(19/2)*b**15*x**19 +
210*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2)*b**19*x**11
+ 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) - 240*a**9*b**15*x**11/(35*a**(19/2)*b**15*x**19 + 210*a**
(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2)*b**19*x**11 + 210*a
**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) - 96*a**8*b**16*x**9/(35*a**(19/2)*b**15*x**19 + 210*a**(17/2)*b*
*16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b
**20*x**9 + 35*a**(7/2)*b**21*x**7) - 16*a**7*b**17*x**7/(35*a**(19/2)*b**15*x**19 + 210*a**(17/2)*b**16*x**17
 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9
 + 35*a**(7/2)*b**21*x**7)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + \frac{b}{x^{2}}} x^{9}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a + b/x^2)*x^9), x)

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