3.2021 \(\int \frac{1}{\sqrt{a+\frac{b}{x^3}} x^7} \, dx\)

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

[Out]

(2*a*Sqrt[a + b/x^3])/(3*b^2) - (2*(a + b/x^3)^(3/2))/(9*b^2)

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Rubi [A]  time = 0.0204016, 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} \[ \frac{2 a \sqrt{a+\frac{b}{x^3}}}{3 b^2}-\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^3]*x^7),x]

[Out]

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

Mathematica [A]  time = 0.0152488, size = 31, normalized size = 0.82 \[ \frac{2 \sqrt{a+\frac{b}{x^3}} \left (2 a x^3-b\right )}{9 b^2 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x^3]*x^7),x]

[Out]

(2*Sqrt[a + b/x^3]*(-b + 2*a*x^3))/(9*b^2*x^3)

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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}-b \right ) }{9\,{b}^{2}{x}^{6}}{\frac{1}{\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(a+b/x^3)^(1/2),x)

[Out]

2/9*(a*x^3+b)*(2*a*x^3-b)/x^6/b^2/((a*x^3+b)/x^3)^(1/2)

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Maxima [A]  time = 0.952738, size = 41, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}}}{9 \, b^{2}} + \frac{2 \, \sqrt{a + \frac{b}{x^{3}}} a}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.50814, size = 69, normalized size = 1.82 \begin{align*} \frac{2 \,{\left (2 \, a x^{3} - b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{9 \, b^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 2.05548, size = 255, normalized size = 6.71 \begin{align*} \frac{4 a^{\frac{7}{2}} b^{\frac{3}{2}} x^{6} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{5}{2}} b^{3} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{4} x^{\frac{9}{2}}} + \frac{2 a^{\frac{5}{2}} b^{\frac{5}{2}} x^{3} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{5}{2}} b^{3} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{4} x^{\frac{9}{2}}} - \frac{2 a^{\frac{3}{2}} b^{\frac{7}{2}} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{5}{2}} b^{3} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{4} x^{\frac{9}{2}}} - \frac{4 a^{4} b x^{\frac{15}{2}}}{9 a^{\frac{5}{2}} b^{3} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{4} x^{\frac{9}{2}}} - \frac{4 a^{3} b^{2} x^{\frac{9}{2}}}{9 a^{\frac{5}{2}} b^{3} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{4} x^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + \frac{b}{x^{3}}} x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a + b/x^3)*x^7), x)