3.1996 \(\int \frac{\sqrt{a+\frac{b}{x^3}}}{x^{10}} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3} \]

[Out]

(-2*a^2*(a + b/x^3)^(3/2))/(9*b^3) + (4*a*(a + b/x^3)^(5/2))/(15*b^3) - (2*(a + b/x^3)^(7/2))/(21*b^3)

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Rubi [A]  time = 0.0300644, antiderivative size = 59, 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} \[ -\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x^3]/x^10,x]

[Out]

(-2*a^2*(a + b/x^3)^(3/2))/(9*b^3) + (4*a*(a + b/x^3)^(5/2))/(15*b^3) - (2*(a + b/x^3)^(7/2))/(21*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{\sqrt{a+\frac{b}{x^3}}}{x^{10}} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int x^2 \sqrt{a+b x} \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2 \sqrt{a+b x}}{b^2}-\frac{2 a (a+b x)^{3/2}}{b^2}+\frac{(a+b x)^{5/2}}{b^2}\right ) \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0116995, size = 49, normalized size = 0.83 \[ -\frac{2 \sqrt{a+\frac{b}{x^3}} \left (a x^3+b\right ) \left (8 a^2 x^6-12 a b x^3+15 b^2\right )}{315 b^3 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x^3]/x^10,x]

[Out]

(-2*Sqrt[a + b/x^3]*(b + a*x^3)*(15*b^2 - 12*a*b*x^3 + 8*a^2*x^6))/(315*b^3*x^9)

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Maple [A]  time = 0.006, size = 50, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 8\,{a}^{2}{x}^{6}-12\,{x}^{3}ab+15\,{b}^{2} \right ) }{315\,{b}^{3}{x}^{9}}\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^3)^(1/2)/x^10,x)

[Out]

-2/315*(a*x^3+b)*(8*a^2*x^6-12*a*b*x^3+15*b^2)*((a*x^3+b)/x^3)^(1/2)/b^3/x^9

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Maxima [A]  time = 0.955981, size = 63, normalized size = 1.07 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}}}{21 \, b^{3}} + \frac{4 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a}{15 \, b^{3}} - \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{2}}{9 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(1/2)/x^10,x, algorithm="maxima")

[Out]

-2/21*(a + b/x^3)^(7/2)/b^3 + 4/15*(a + b/x^3)^(5/2)*a/b^3 - 2/9*(a + b/x^3)^(3/2)*a^2/b^3

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Fricas [A]  time = 1.4758, size = 120, normalized size = 2.03 \begin{align*} -\frac{2 \,{\left (8 \, a^{3} x^{9} - 4 \, a^{2} b x^{6} + 3 \, a b^{2} x^{3} + 15 \, b^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{315 \, b^{3} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(1/2)/x^10,x, algorithm="fricas")

[Out]

-2/315*(8*a^3*x^9 - 4*a^2*b*x^6 + 3*a*b^2*x^3 + 15*b^3)*sqrt((a*x^3 + b)/x^3)/(b^3*x^9)

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Sympy [B]  time = 3.13166, size = 913, normalized size = 15.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**3)**(1/2)/x**10,x)

[Out]

-16*a**(19/2)*b**(9/2)*x**18*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) +
945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) - 40*a**(17/2)*b**(11/2)*x**15*sqrt(a*x**3/b + 1)
/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**
10*x**(21/2)) - 30*a**(15/2)*b**(13/2)*x**12*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*
b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) - 40*a**(13/2)*b**(15/2)*x**9*sqr
t(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 3
15*a**(7/2)*b**10*x**(21/2)) - 100*a**(11/2)*b**(17/2)*x**6*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) +
945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) - 96*a**(9/2)*b**(
19/2)*x**3*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9
*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) - 30*a**(7/2)*b**(21/2)*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(
39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) + 16*a**10*
b**4*x**(39/2)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 31
5*a**(7/2)*b**10*x**(21/2)) + 48*a**9*b**5*x**(33/2)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33
/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) + 48*a**8*b**6*x**(27/2)/(315*a**(13/2)*b**7
*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) + 16*a
**7*b**7*x**(21/2)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2)
+ 315*a**(7/2)*b**10*x**(21/2))

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Giac [A]  time = 1.23477, size = 58, normalized size = 0.98 \begin{align*} -\frac{2 \,{\left (15 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} - 42 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a + 35 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{2}\right )}}{315 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(1/2)/x^10,x, algorithm="giac")

[Out]

-2/315*(15*(a + b/x^3)^(7/2) - 42*(a + b/x^3)^(5/2)*a + 35*(a + b/x^3)^(3/2)*a^2)/b^3