3.1995 \(\int \frac{\sqrt{a+\frac{b}{x^3}}}{x^7} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 a \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^2}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0210704, antiderivative size = 38, 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} \[ \frac{2 a \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^2}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x^3]/x^7,x]

[Out]

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

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

Mathematica [A]  time = 0.0104663, size = 38, normalized size = 1. \[ \frac{2 \sqrt{a+\frac{b}{x^3}} \left (a x^3+b\right ) \left (2 a x^3-3 b\right )}{45 b^2 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x^3]/x^7,x]

[Out]

(2*Sqrt[a + b/x^3]*(b + a*x^3)*(-3*b + 2*a*x^3))/(45*b^2*x^6)

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Maple [A]  time = 0.004, size = 39, normalized size = 1. \begin{align*}{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 2\,a{x}^{3}-3\,b \right ) }{45\,{b}^{2}{x}^{6}}\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^7,x)

[Out]

2/45*(a*x^3+b)*(2*a*x^3-3*b)*((a*x^3+b)/x^3)^(1/2)/b^2/x^6

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Maxima [A]  time = 0.956515, size = 41, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}}}{15 \, b^{2}} + \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a}{9 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(1/2)/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.46503, size = 92, normalized size = 2.42 \begin{align*} \frac{2 \,{\left (2 \, a^{2} x^{6} - a b x^{3} - 3 \, b^{2}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{45 \, b^{2} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(1/2)/x^7,x, algorithm="fricas")

[Out]

2/45*(2*a^2*x^6 - a*b*x^3 - 3*b^2)*sqrt((a*x^3 + b)/x^3)/(b^2*x^6)

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Sympy [B]  time = 1.87429, size = 313, normalized size = 8.24 \begin{align*} \frac{4 a^{\frac{11}{2}} b^{\frac{3}{2}} x^{9} \sqrt{\frac{a x^{3}}{b} + 1}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} + \frac{2 a^{\frac{9}{2}} b^{\frac{5}{2}} x^{6} \sqrt{\frac{a x^{3}}{b} + 1}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} - \frac{8 a^{\frac{7}{2}} b^{\frac{7}{2}} x^{3} \sqrt{\frac{a x^{3}}{b} + 1}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} - \frac{6 a^{\frac{5}{2}} b^{\frac{9}{2}} \sqrt{\frac{a x^{3}}{b} + 1}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} - \frac{4 a^{6} b x^{\frac{21}{2}}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} - \frac{4 a^{5} b^{2} x^{\frac{15}{2}}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**3)**(1/2)/x**7,x)

[Out]

4*a**(11/2)*b**(3/2)*x**9*sqrt(a*x**3/b + 1)/(45*a**(7/2)*b**3*x**(21/2) + 45*a**(5/2)*b**4*x**(15/2)) + 2*a**
(9/2)*b**(5/2)*x**6*sqrt(a*x**3/b + 1)/(45*a**(7/2)*b**3*x**(21/2) + 45*a**(5/2)*b**4*x**(15/2)) - 8*a**(7/2)*
b**(7/2)*x**3*sqrt(a*x**3/b + 1)/(45*a**(7/2)*b**3*x**(21/2) + 45*a**(5/2)*b**4*x**(15/2)) - 6*a**(5/2)*b**(9/
2)*sqrt(a*x**3/b + 1)/(45*a**(7/2)*b**3*x**(21/2) + 45*a**(5/2)*b**4*x**(15/2)) - 4*a**6*b*x**(21/2)/(45*a**(7
/2)*b**3*x**(21/2) + 45*a**(5/2)*b**4*x**(15/2)) - 4*a**5*b**2*x**(15/2)/(45*a**(7/2)*b**3*x**(21/2) + 45*a**(
5/2)*b**4*x**(15/2))

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Giac [A]  time = 1.19324, size = 39, normalized size = 1.03 \begin{align*} -\frac{2 \,{\left (3 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} - 5 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a\right )}}{45 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(1/2)/x^7,x, algorithm="giac")

[Out]

-2/45*(3*(a + b/x^3)^(5/2) - 5*(a + b/x^3)^(3/2)*a)/b^2