3.1920 \(\int \frac{1}{\sqrt{a+\frac{b}{x^2}} x^7} \, dx\)

Optimal. Leaf size=57 \[ -\frac{a^2 \sqrt{a+\frac{b}{x^2}}}{b^3}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{5 b^3}+\frac{2 a \left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0289804, antiderivative size = 57, 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} \[ -\frac{a^2 \sqrt{a+\frac{b}{x^2}}}{b^3}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{5 b^3}+\frac{2 a \left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^2]*x^7),x]

[Out]

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

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

Mathematica [A]  time = 0.0180034, size = 42, normalized size = 0.74 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (8 a^2 x^4-4 a b x^2+3 b^2\right )}{15 b^3 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x^2]*x^7),x]

[Out]

-(Sqrt[a + b/x^2]*(3*b^2 - 4*a*b*x^2 + 8*a^2*x^4))/(15*b^3*x^4)

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Maple [A]  time = 0.005, size = 50, normalized size = 0.9 \begin{align*} -{\frac{ \left ( a{x}^{2}+b \right ) \left ( 8\,{a}^{2}{x}^{4}-4\,ab{x}^{2}+3\,{b}^{2} \right ) }{15\,{b}^{3}{x}^{6}}{\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^7,x)

[Out]

-1/15*(a*x^2+b)*(8*a^2*x^4-4*a*b*x^2+3*b^2)/x^6/b^3/((a*x^2+b)/x^2)^(1/2)

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Maxima [A]  time = 1.01328, size = 63, normalized size = 1.11 \begin{align*} -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}}}{5 \, b^{3}} + \frac{2 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} a}{3 \, b^{3}} - \frac{\sqrt{a + \frac{b}{x^{2}}} a^{2}}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(1/2)/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.4728, size = 96, normalized size = 1.68 \begin{align*} -\frac{{\left (8 \, a^{2} x^{4} - 4 \, a b x^{2} + 3 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{15 \, b^{3} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(1/2)/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 2.56191, size = 750, normalized size = 13.16 \begin{align*} - \frac{8 a^{\frac{15}{2}} b^{\frac{9}{2}} x^{10} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} - \frac{20 a^{\frac{13}{2}} b^{\frac{11}{2}} x^{8} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} - \frac{15 a^{\frac{11}{2}} b^{\frac{13}{2}} x^{6} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} - \frac{5 a^{\frac{9}{2}} b^{\frac{15}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} - \frac{5 a^{\frac{7}{2}} b^{\frac{17}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} - \frac{3 a^{\frac{5}{2}} b^{\frac{19}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} + \frac{8 a^{8} b^{4} x^{11}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} + \frac{24 a^{7} b^{5} x^{9}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} + \frac{24 a^{6} b^{6} x^{7}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} + \frac{8 a^{5} b^{7} x^{5}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**(1/2)/x**7,x)

[Out]

-8*a**(15/2)*b**(9/2)*x**10*sqrt(a*x**2/b + 1)/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*
b**9*x**7 + 15*a**(5/2)*b**10*x**5) - 20*a**(13/2)*b**(11/2)*x**8*sqrt(a*x**2/b + 1)/(15*a**(11/2)*b**7*x**11
+ 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x**7 + 15*a**(5/2)*b**10*x**5) - 15*a**(11/2)*b**(13/2)*x**6*sqrt(a
*x**2/b + 1)/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x**7 + 15*a**(5/2)*b**10*x**5
) - 5*a**(9/2)*b**(15/2)*x**4*sqrt(a*x**2/b + 1)/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2
)*b**9*x**7 + 15*a**(5/2)*b**10*x**5) - 5*a**(7/2)*b**(17/2)*x**2*sqrt(a*x**2/b + 1)/(15*a**(11/2)*b**7*x**11
+ 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x**7 + 15*a**(5/2)*b**10*x**5) - 3*a**(5/2)*b**(19/2)*sqrt(a*x**2/b
+ 1)/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x**7 + 15*a**(5/2)*b**10*x**5) + 8*a
**8*b**4*x**11/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x**7 + 15*a**(5/2)*b**10*x*
*5) + 24*a**7*b**5*x**9/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x**7 + 15*a**(5/2)
*b**10*x**5) + 24*a**6*b**6*x**7/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x**7 + 15
*a**(5/2)*b**10*x**5) + 8*a**5*b**7*x**5/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x
**7 + 15*a**(5/2)*b**10*x**5)

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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^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(1/2)/x^7,x, algorithm="giac")

[Out]

integrate(1/(sqrt(a + b/x^2)*x^7), x)