3.344 \(\int \frac{x \sqrt{-a+b x^3}}{-2 (5+3 \sqrt{3}) a+b x^3} \, dx\)

Optimal. Leaf size=774 \[ -\frac{2 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),4 \sqrt{3}-7\right )}{\sqrt [4]{3} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}+\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}-\frac{2 \sqrt{b x^3-a}}{b^{2/3} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}+\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{b x^3-a}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{b x^3-a}}\right )}{\sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{b x^3-a}}\right )}{2 \sqrt{2} b^{2/3}}-\frac{\sqrt [6]{a} \tanh ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt{b x^3-a}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{\sqrt{2} \sqrt [4]{3} b^{2/3}} \]

[Out]

(-2*Sqrt[-a + b*x^3])/(b^(2/3)*((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)) + (3^(1/4)*a^(1/6)*ArcTan[(3^(1/4)*(1 - Sq
rt[3])*a^(1/6)*(a^(1/3) - b^(1/3)*x))/(Sqrt[2]*Sqrt[-a + b*x^3])])/(2*Sqrt[2]*b^(2/3)) + (3^(1/4)*a^(1/6)*ArcT
an[(3^(1/4)*a^(1/6)*((1 + Sqrt[3])*a^(1/3) + 2*b^(1/3)*x))/(Sqrt[2]*Sqrt[-a + b*x^3])])/(Sqrt[2]*b^(2/3)) + (3
^(3/4)*a^(1/6)*ArcTanh[(3^(1/4)*(1 + Sqrt[3])*a^(1/6)*(a^(1/3) - b^(1/3)*x))/(Sqrt[2]*Sqrt[-a + b*x^3])])/(2*S
qrt[2]*b^(2/3)) - (a^(1/6)*ArcTanh[((1 - Sqrt[3])*Sqrt[-a + b*x^3])/(Sqrt[2]*3^(3/4)*Sqrt[a])])/(Sqrt[2]*3^(1/
4)*b^(2/3)) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) - b^(1/3)*x)*Sqrt[(a^(2/3) + a^(1/3)*b^(1/3)*x + b^(
2/3)*x^2)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)/((1 - Sq
rt[3])*a^(1/3) - b^(1/3)*x)], -7 + 4*Sqrt[3]])/(b^(2/3)*Sqrt[-((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 - Sqrt[3])*
a^(1/3) - b^(1/3)*x)^2)]*Sqrt[-a + b*x^3]) - (2*Sqrt[2]*a^(1/3)*(a^(1/3) - b^(1/3)*x)*Sqrt[(a^(2/3) + a^(1/3)*
b^(1/3)*x + b^(2/3)*x^2)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - b^(1
/3)*x)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*b^(2/3)*Sqrt[-((a^(1/3)*(a^(1/3) - b^(1
/3)*x))/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)^2)]*Sqrt[-a + b*x^3])

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Rubi [A]  time = 0.280537, antiderivative size = 774, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {489, 304, 219, 1879, 488} \[ -\frac{2 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}+\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}-\frac{2 \sqrt{b x^3-a}}{b^{2/3} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}+\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{b x^3-a}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{b x^3-a}}\right )}{\sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{b x^3-a}}\right )}{2 \sqrt{2} b^{2/3}}-\frac{\sqrt [6]{a} \tanh ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt{b x^3-a}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{\sqrt{2} \sqrt [4]{3} b^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(x*Sqrt[-a + b*x^3])/(-2*(5 + 3*Sqrt[3])*a + b*x^3),x]

[Out]

(-2*Sqrt[-a + b*x^3])/(b^(2/3)*((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)) + (3^(1/4)*a^(1/6)*ArcTan[(3^(1/4)*(1 - Sq
rt[3])*a^(1/6)*(a^(1/3) - b^(1/3)*x))/(Sqrt[2]*Sqrt[-a + b*x^3])])/(2*Sqrt[2]*b^(2/3)) + (3^(1/4)*a^(1/6)*ArcT
an[(3^(1/4)*a^(1/6)*((1 + Sqrt[3])*a^(1/3) + 2*b^(1/3)*x))/(Sqrt[2]*Sqrt[-a + b*x^3])])/(Sqrt[2]*b^(2/3)) + (3
^(3/4)*a^(1/6)*ArcTanh[(3^(1/4)*(1 + Sqrt[3])*a^(1/6)*(a^(1/3) - b^(1/3)*x))/(Sqrt[2]*Sqrt[-a + b*x^3])])/(2*S
qrt[2]*b^(2/3)) - (a^(1/6)*ArcTanh[((1 - Sqrt[3])*Sqrt[-a + b*x^3])/(Sqrt[2]*3^(3/4)*Sqrt[a])])/(Sqrt[2]*3^(1/
4)*b^(2/3)) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) - b^(1/3)*x)*Sqrt[(a^(2/3) + a^(1/3)*b^(1/3)*x + b^(
2/3)*x^2)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)/((1 - Sq
rt[3])*a^(1/3) - b^(1/3)*x)], -7 + 4*Sqrt[3]])/(b^(2/3)*Sqrt[-((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 - Sqrt[3])*
a^(1/3) - b^(1/3)*x)^2)]*Sqrt[-a + b*x^3]) - (2*Sqrt[2]*a^(1/3)*(a^(1/3) - b^(1/3)*x)*Sqrt[(a^(2/3) + a^(1/3)*
b^(1/3)*x + b^(2/3)*x^2)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - b^(1
/3)*x)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*b^(2/3)*Sqrt[-((a^(1/3)*(a^(1/3) - b^(1
/3)*x))/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)^2)]*Sqrt[-a + b*x^3])

Rule 489

Int[((x_)*Sqrt[(a_) + (b_.)*(x_)^3])/((c_) + (d_.)*(x_)^3), x_Symbol] :> Dist[b/d, Int[x/Sqrt[a + b*x^3], x],
x] - Dist[(b*c - a*d)/d, Int[x/((c + d*x^3)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{c, d, a, b}, x] && NeQ[b*c - a*
d, 0] && (EqQ[b*c - 4*a*d, 0] || EqQ[b*c + 8*a*d, 0] || EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0])

Rule 304

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, -Dist[(S
qrt[2]*s)/(Sqrt[2 - Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a
+ b*x^3], x], x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 219

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((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]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - S
qrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 1879

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 + Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 + Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 - Sqrt[3])*s + r*x)), x
] + Simp[(3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[-((s
*(s + r*x))/((1 - Sqrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rule 488

Int[(x_)/(Sqrt[(a_) + (b_.)*(x_)^3]*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[b/a, 3], r = Simplify[(b
*c - 10*a*d)/(6*a*d)]}, Simp[(q*(2 - r)*ArcTanh[((1 - r)*Sqrt[a + b*x^3])/(Sqrt[2]*Rt[-a, 2]*r^(3/2))])/(3*Sqr
t[2]*Rt[-a, 2]*d*r^(3/2)), x] + (-Simp[(q*(2 - r)*ArcTanh[(Rt[-a, 2]*Sqrt[r]*(1 + r)*(1 + q*x))/(Sqrt[2]*Sqrt[
a + b*x^3])])/(2*Sqrt[2]*Rt[-a, 2]*d*r^(3/2)), x] - Simp[(q*(2 - r)*ArcTan[(Rt[-a, 2]*Sqrt[r]*(1 + r - 2*q*x))
/(Sqrt[2]*Sqrt[a + b*x^3])])/(3*Sqrt[2]*Rt[-a, 2]*d*Sqrt[r]), x] - Simp[(q*(2 - r)*ArcTan[(Rt[-a, 2]*(1 - r)*S
qrt[r]*(1 + q*x))/(Sqrt[2]*Sqrt[a + b*x^3])])/(6*Sqrt[2]*Rt[-a, 2]*d*Sqrt[r]), x])] /; FreeQ[{a, b, c, d}, x]
&& NeQ[b*c - a*d, 0] && EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0] && NegQ[a]

Rubi steps

\begin{align*} \int \frac{x \sqrt{-a+b x^3}}{-2 \left (5+3 \sqrt{3}\right ) a+b x^3} \, dx &=\left (3 \left (3+2 \sqrt{3}\right ) a\right ) \int \frac{x}{\sqrt{-a+b x^3} \left (-2 \left (5+3 \sqrt{3}\right ) a+b x^3\right )} \, dx+\int \frac{x}{\sqrt{-a+b x^3}} \, dx\\ &=\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a+b x^3}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a+b x^3}}\right )}{\sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a+b x^3}}\right )}{2 \sqrt{2} b^{2/3}}-\frac{\sqrt [6]{a} \tanh ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt{-a+b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{\sqrt{2} \sqrt [4]{3} b^{2/3}}-\frac{\int \frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt{-a+b x^3}} \, dx}{\sqrt [3]{b}}+\frac{\left (\sqrt{2 \left (2+\sqrt{3}\right )} \sqrt [3]{a}\right ) \int \frac{1}{\sqrt{-a+b x^3}} \, dx}{\sqrt [3]{b}}\\ &=-\frac{2 \sqrt{-a+b x^3}}{b^{2/3} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}+\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a+b x^3}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a+b x^3}}\right )}{\sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a+b x^3}}\right )}{2 \sqrt{2} b^{2/3}}-\frac{\sqrt [6]{a} \tanh ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt{-a+b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{\sqrt{2} \sqrt [4]{3} b^{2/3}}+\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{-a+b x^3}}-\frac{2 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{-a+b x^3}}\\ \end{align*}

Mathematica [C]  time = 0.0638605, size = 87, normalized size = 0.11 \[ -\frac{x^2 \sqrt{b x^3-a} F_1\left (\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )}{4 \left (5+3 \sqrt{3}\right ) a \sqrt{1-\frac{b x^3}{a}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x*Sqrt[-a + b*x^3])/(-2*(5 + 3*Sqrt[3])*a + b*x^3),x]

[Out]

-(x^2*Sqrt[-a + b*x^3]*AppellF1[2/3, -1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)])/(4*(5 + 3*Sqrt[3]
)*a*Sqrt[1 - (b*x^3)/a])

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Maple [C]  time = 0.068, size = 926, normalized size = 1.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^3-a)^(1/2)/(b*x^3-2*a*(5+3*3^(1/2))),x)

[Out]

2/3*I*3^(1/2)/b*(a*b^2)^(1/3)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3
))^(1/2)*((x-1/b*(a*b^2)^(1/3))/(-3/2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3)))^(1/2)*(I*(x+1/2/b*(a*b^2
)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2)/(b*x^3-a)^(1/2)*((-3/2/b*(a*b^2)^(1/3)-1
/2*I*3^(1/2)/b*(a*b^2)^(1/3))*EllipticE(1/3*3^(1/2)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*
3^(1/2)*b/(a*b^2)^(1/3))^(1/2),(-I*3^(1/2)/b*(a*b^2)^(1/3)/(-3/2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3)
))^(1/2))+1/b*(a*b^2)^(1/3)*EllipticF(1/3*3^(1/2)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^
(1/2)*b/(a*b^2)^(1/3))^(1/2),(-I*3^(1/2)/b*(a*b^2)^(1/3)/(-3/2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3)))
^(1/2)))-1/9*I/b^3*2^(1/2)*sum(1/_alpha*(2*3^(1/2)+3)*(a*b^2)^(1/3)*(-1/2*I*b*(2*x+1/b*(I*3^(1/2)*(a*b^2)^(1/3
)+(a*b^2)^(1/3)))/(a*b^2)^(1/3))^(1/2)*(b*(x-1/b*(a*b^2)^(1/3))/(-3*(a*b^2)^(1/3)-I*3^(1/2)*(a*b^2)^(1/3)))^(1
/2)*(1/2*I*b*(2*x+1/b*(-I*3^(1/2)*(a*b^2)^(1/3)+(a*b^2)^(1/3)))/(a*b^2)^(1/3))^(1/2)/(b*x^3-a)^(1/2)*(3*I*(a*b
^2)^(1/3)*_alpha*3^(1/2)*b+4*b^2*_alpha^2*3^(1/2)-3*I*(a*b^2)^(2/3)*3^(1/2)-2*3^(1/2)*(a*b^2)^(1/3)*_alpha*b-6
*I*(a*b^2)^(1/3)*_alpha*b-6*b^2*_alpha^2-2*3^(1/2)*(a*b^2)^(2/3)+6*I*(a*b^2)^(2/3)+3*(a*b^2)^(1/3)*_alpha*b+3*
(a*b^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b
^2)^(1/3))^(1/2),1/6/b*(-2*I*3^(1/2)*(a*b^2)^(1/3)*_alpha^2*b+I*3^(1/2)*(a*b^2)^(2/3)*_alpha+4*I*(a*b^2)^(1/3)
*_alpha^2*b+2*3^(1/2)*(a*b^2)^(2/3)*_alpha+I*3^(1/2)*a*b-2*I*(a*b^2)^(2/3)*_alpha-2*3^(1/2)*a*b-3*(a*b^2)^(2/3
)*_alpha-2*I*a*b+3*a*b)/a,(-I*3^(1/2)/b*(a*b^2)^(1/3)/(-3/2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3)))^(1
/2)),_alpha=RootOf(b*_Z^3-6*a*3^(1/2)-10*a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x^{3} - a} x}{b x^{3} - 2 \, a{\left (3 \, \sqrt{3} + 5\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3-a)^(1/2)/(b*x^3-2*a*(5+3*3^(1/2))),x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^3 - a)*x/(b*x^3 - 2*a*(3*sqrt(3) + 5)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3-a)^(1/2)/(b*x^3-2*a*(5+3*3^(1/2))),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sqrt{- a + b x^{3}}}{- 6 \sqrt{3} a - 10 a + b x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**3-a)**(1/2)/(b*x**3-2*a*(5+3*3**(1/2))),x)

[Out]

Integral(x*sqrt(-a + b*x**3)/(-6*sqrt(3)*a - 10*a + b*x**3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3-a)^(1/2)/(b*x^3-2*a*(5+3*3^(1/2))),x, algorithm="giac")

[Out]

sage0*x