3.2022 \(\int \frac{1}{\sqrt{a+\frac{b}{x^3}} x^{10}} \, dx\)

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

[Out]

(-2*a^2*Sqrt[a + b/x^3])/(3*b^3) + (4*a*(a + b/x^3)^(3/2))/(9*b^3) - (2*(a + b/x^3)^(5/2))/(15*b^3)

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Rubi [A]  time = 0.030736, 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} \[ -\frac{2 a^2 \sqrt{a+\frac{b}{x^3}}}{3 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^3]*x^10),x]

[Out]

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

Mathematica [A]  time = 0.0173782, size = 42, normalized size = 0.71 \[ -\frac{2 \sqrt{a+\frac{b}{x^3}} \left (8 a^2 x^6-4 a b x^3+3 b^2\right )}{45 b^3 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x^3]*x^10),x]

[Out]

(-2*Sqrt[a + b/x^3]*(3*b^2 - 4*a*b*x^3 + 8*a^2*x^6))/(45*b^3*x^6)

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Maple [A]  time = 0.004, size = 50, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 8\,{a}^{2}{x}^{6}-4\,{x}^{3}ab+3\,{b}^{2} \right ) }{45\,{b}^{3}{x}^{9}}{\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^10/(a+b/x^3)^(1/2),x)

[Out]

-2/45*(a*x^3+b)*(8*a^2*x^6-4*a*b*x^3+3*b^2)/x^9/b^3/((a*x^3+b)/x^3)^(1/2)

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Maxima [A]  time = 0.952835, size = 63, normalized size = 1.07 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}}}{15 \, b^{3}} + \frac{4 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a}{9 \, b^{3}} - \frac{2 \, \sqrt{a + \frac{b}{x^{3}}} a^{2}}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.46216, size = 96, normalized size = 1.63 \begin{align*} -\frac{2 \,{\left (8 \, a^{2} x^{6} - 4 \, a b x^{3} + 3 \, b^{2}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{45 \, b^{3} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45*(8*a^2*x^6 - 4*a*b*x^3 + 3*b^2)*sqrt((a*x^3 + b)/x^3)/(b^3*x^6)

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Sympy [B]  time = 3.41594, size = 824, normalized size = 13.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16*a**(15/2)*b**(9/2)*x**15*sqrt(a*x**3/b + 1)/(45*a**(11/2)*b**7*x**(33/2) + 135*a**(9/2)*b**8*x**(27/2) + 1
35*a**(7/2)*b**9*x**(21/2) + 45*a**(5/2)*b**10*x**(15/2)) - 40*a**(13/2)*b**(11/2)*x**12*sqrt(a*x**3/b + 1)/(4
5*a**(11/2)*b**7*x**(33/2) + 135*a**(9/2)*b**8*x**(27/2) + 135*a**(7/2)*b**9*x**(21/2) + 45*a**(5/2)*b**10*x**
(15/2)) - 30*a**(11/2)*b**(13/2)*x**9*sqrt(a*x**3/b + 1)/(45*a**(11/2)*b**7*x**(33/2) + 135*a**(9/2)*b**8*x**(
27/2) + 135*a**(7/2)*b**9*x**(21/2) + 45*a**(5/2)*b**10*x**(15/2)) - 10*a**(9/2)*b**(15/2)*x**6*sqrt(a*x**3/b
+ 1)/(45*a**(11/2)*b**7*x**(33/2) + 135*a**(9/2)*b**8*x**(27/2) + 135*a**(7/2)*b**9*x**(21/2) + 45*a**(5/2)*b*
*10*x**(15/2)) - 10*a**(7/2)*b**(17/2)*x**3*sqrt(a*x**3/b + 1)/(45*a**(11/2)*b**7*x**(33/2) + 135*a**(9/2)*b**
8*x**(27/2) + 135*a**(7/2)*b**9*x**(21/2) + 45*a**(5/2)*b**10*x**(15/2)) - 6*a**(5/2)*b**(19/2)*sqrt(a*x**3/b
+ 1)/(45*a**(11/2)*b**7*x**(33/2) + 135*a**(9/2)*b**8*x**(27/2) + 135*a**(7/2)*b**9*x**(21/2) + 45*a**(5/2)*b*
*10*x**(15/2)) + 16*a**8*b**4*x**(33/2)/(45*a**(11/2)*b**7*x**(33/2) + 135*a**(9/2)*b**8*x**(27/2) + 135*a**(7
/2)*b**9*x**(21/2) + 45*a**(5/2)*b**10*x**(15/2)) + 48*a**7*b**5*x**(27/2)/(45*a**(11/2)*b**7*x**(33/2) + 135*
a**(9/2)*b**8*x**(27/2) + 135*a**(7/2)*b**9*x**(21/2) + 45*a**(5/2)*b**10*x**(15/2)) + 48*a**6*b**6*x**(21/2)/
(45*a**(11/2)*b**7*x**(33/2) + 135*a**(9/2)*b**8*x**(27/2) + 135*a**(7/2)*b**9*x**(21/2) + 45*a**(5/2)*b**10*x
**(15/2)) + 16*a**5*b**7*x**(15/2)/(45*a**(11/2)*b**7*x**(33/2) + 135*a**(9/2)*b**8*x**(27/2) + 135*a**(7/2)*b
**9*x**(21/2) + 45*a**(5/2)*b**10*x**(15/2))

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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^{10}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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