3.21.29 \(\int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^8} \, dx\) [2029]

Optimal. Leaf size=270 \[ -\frac {2 \sqrt {a+\frac {b}{x^3}}}{11 b x^4}+\frac {16 a \sqrt {a+\frac {b}{x^3}}}{55 b^2 x}-\frac {32 \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{55 \sqrt [4]{3} b^{7/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \]

[Out]

-2/11*(a+b/x^3)^(1/2)/b/x^4+16/55*a*(a+b/x^3)^(1/2)/b^2/x-32/165*a^2*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+
a^(1/3)*(1-3^(1/2)))/(b^(1/3)/x+a^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)+1/2*2^(1/2))*((a^(2/3)+b^(2/3
)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)*3^(3/4)/b^(7/3)/(a+b/x^3)^(1/2)/(a^(1/3)*(a^
(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {342, 327, 224} \begin {gather*} -\frac {32 \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} F\left (\text {ArcSin}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{55 \sqrt [4]{3} b^{7/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {16 a \sqrt {a+\frac {b}{x^3}}}{55 b^2 x}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{11 b x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^3]*x^8),x]

[Out]

(-2*Sqrt[a + b/x^3])/(11*b*x^4) + (16*a*Sqrt[a + b/x^3])/(55*b^2*x) - (32*Sqrt[2 + Sqrt[3]]*a^2*(a^(1/3) + b^(
1/3)/x)*Sqrt[(a^(2/3) + b^(2/3)/x^2 - (a^(1/3)*b^(1/3))/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^2]*EllipticF[Ar
cSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]])/(55*3^(1/4)*b^
(7/3)*Sqrt[a + b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)/x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^2])

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt
[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sq
rt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)
], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
 c, n, m, p, x]

Rule 342

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^8} \, dx &=-\text {Subst}\left (\int \frac {x^6}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 \sqrt {a+\frac {b}{x^3}}}{11 b x^4}+\frac {(8 a) \text {Subst}\left (\int \frac {x^3}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{11 b}\\ &=-\frac {2 \sqrt {a+\frac {b}{x^3}}}{11 b x^4}+\frac {16 a \sqrt {a+\frac {b}{x^3}}}{55 b^2 x}-\frac {\left (16 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{55 b^2}\\ &=-\frac {2 \sqrt {a+\frac {b}{x^3}}}{11 b x^4}+\frac {16 a \sqrt {a+\frac {b}{x^3}}}{55 b^2 x}-\frac {32 \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{55 \sqrt [4]{3} b^{7/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.02, size = 51, normalized size = 0.19 \begin {gather*} -\frac {2 \sqrt {1+\frac {a x^3}{b}} \, _2F_1\left (-\frac {11}{6},\frac {1}{2};-\frac {5}{6};-\frac {a x^3}{b}\right )}{11 \sqrt {a+\frac {b}{x^3}} x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x^3]*x^8),x]

[Out]

(-2*Sqrt[1 + (a*x^3)/b]*Hypergeometric2F1[-11/6, 1/2, -5/6, -((a*x^3)/b)])/(11*Sqrt[a + b/x^3]*x^7)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2008 vs. \(2 (207 ) = 414\).
time = 0.05, size = 2009, normalized size = 7.44

method result size
risch \(\frac {2 \left (a \,x^{3}+b \right ) \left (8 a \,x^{3}-5 b \right )}{55 b^{2} x^{7} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}+\frac {32 a^{3} \left (\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {\frac {\left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) x}{\left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}\, \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )^{2} \sqrt {\frac {\left (-a^{2} b \right )^{\frac {1}{3}} \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}{a \left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}\, \sqrt {\frac {\left (-a^{2} b \right )^{\frac {1}{3}} \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}{a \left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) x}{\left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}, \sqrt {\frac {\left (\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}{\left (\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right ) \sqrt {x \left (a \,x^{3}+b \right )}}{55 b^{2} \left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {a x \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right ) \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}\, x^{2} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}\) \(742\)
default \(\text {Expression too large to display}\) \(2009\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(a+b/x^3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/55/((a*x^3+b)/x^3)^(1/2)/x^7*(a*x^3+b)/(-a^2*b)^(1/3)/b^2*(-32*I*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-
a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/
2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I
*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(
1/2)-3))^(1/2))*3^(1/2)*a^3*x^8+64*I*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/
2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-
2*a*x-(-a^2*b)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))
/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-a^2*b)^(1/3)
*3^(1/2)*a^2*x^7-32*I*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/
3)+2*a*x+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^
(1/3))/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-a^2*b)
^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-a^2*b)^(2/3)*3^(1/2)*a*x^6+
32*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(
1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(-1+I*3^(1/2
))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*
3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*a^3*x^8-64*(-a^2*b)^(1/3)*(-(I*3^(1/2)-3)*x*a/(-
1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x
+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))
^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))
/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*a^2*x^7+32*(-a^2*b)^(2/3)*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-a^2*
b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(
(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(
1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)
-3))^(1/2))*a*x^6+8*I*(a*x^4+b*x)^(1/2)*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b
)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*(-a^2*b)^(1/3)*3^(1/2)*a*x^3-24*a*(a*x^4+b*x)^
(1/2)*x^3*(-a^2*b)^(1/3)*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(
1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)-5*I*(-a^2*b)^(1/3)*3^(1/2)*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*
3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*(a*x^4+b*x
)^(1/2)*b+15*(a*x^4+b*x)^(1/2)*b*(-a^2*b)^(1/3)*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x
+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2))/(x*(a*x^3+b))^(1/2)/(I*3^(1/2)-3)/(1/
a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-
a^2*b)^(1/3)))^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(a+b/x^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(a + b/x^3)*x^8), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.07, size = 59, normalized size = 0.22 \begin {gather*} -\frac {2 \, {\left (16 \, a^{2} \sqrt {b} x^{4} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, \frac {1}{x}\right ) - {\left (8 \, a b x^{3} - 5 \, b^{2}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right )}}{55 \, b^{3} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(a+b/x^3)^(1/2),x, algorithm="fricas")

[Out]

-2/55*(16*a^2*sqrt(b)*x^4*weierstrassPInverse(0, -4*a/b, 1/x) - (8*a*b*x^3 - 5*b^2)*sqrt((a*x^3 + b)/x^3))/(b^
3*x^4)

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Sympy [A]
time = 0.76, size = 39, normalized size = 0.14 \begin {gather*} - \frac {\Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt {a} x^{7} \Gamma \left (\frac {10}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**8/(a+b/x**3)**(1/2),x)

[Out]

-gamma(7/3)*hyper((1/2, 7/3), (10/3,), b*exp_polar(I*pi)/(a*x**3))/(3*sqrt(a)*x**7*gamma(10/3))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(a+b/x^3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a + b/x^3)*x^8), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^8\,\sqrt {a+\frac {b}{x^3}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8*(a + b/x^3)^(1/2)),x)

[Out]

int(1/(x^8*(a + b/x^3)^(1/2)), x)

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