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

Optimal. Leaf size=316 \[ -\frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{6 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{3 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}+\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}} \]

[Out]

-((2 + Sqrt[3])*ArcTan[(3^(1/4)*(1 + Sqrt[3])*a^(1/6)*(a^(1/3) - b^(1/3)*x))/(Sqrt[2]*Sqrt[a - b*x^3])])/(6*Sq
rt[2]*3^(1/4)*a^(5/6)*b^(2/3)) - ((2 + Sqrt[3])*ArcTan[(3^(1/4)*a^(1/6)*((1 - Sqrt[3])*a^(1/3) + 2*b^(1/3)*x))
/(Sqrt[2]*Sqrt[a - b*x^3])])/(3*Sqrt[2]*3^(1/4)*a^(5/6)*b^(2/3)) + ((2 + Sqrt[3])*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*Sqrt[2]*3^(3/4)*a^(5/6)*b^(2/3)) + ((2 + Sqrt
[3])*ArcTanh[((1 + Sqrt[3])*Sqrt[a - b*x^3])/(Sqrt[2]*3^(3/4)*Sqrt[a])])/(3*Sqrt[2]*3^(3/4)*a^(5/6)*b^(2/3))

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Rubi [A]  time = 0.0544255, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {487} \[ -\frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{6 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{3 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}+\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}} \]

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[3])*ArcTan[(3^(1/4)*(1 + Sqrt[3])*a^(1/6)*(a^(1/3) - b^(1/3)*x))/(Sqrt[2]*Sqrt[a - b*x^3])])/(6*Sq
rt[2]*3^(1/4)*a^(5/6)*b^(2/3)) - ((2 + Sqrt[3])*ArcTan[(3^(1/4)*a^(1/6)*((1 - Sqrt[3])*a^(1/3) + 2*b^(1/3)*x))
/(Sqrt[2]*Sqrt[a - b*x^3])])/(3*Sqrt[2]*3^(1/4)*a^(5/6)*b^(2/3)) + ((2 + Sqrt[3])*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*Sqrt[2]*3^(3/4)*a^(5/6)*b^(2/3)) + ((2 + Sqrt
[3])*ArcTanh[((1 + Sqrt[3])*Sqrt[a - b*x^3])/(Sqrt[2]*3^(3/4)*Sqrt[a])])/(3*Sqrt[2]*3^(3/4)*a^(5/6)*b^(2/3))

Rule 487

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)*ArcTan[((1 - r)*Sqrt[a + b*x^3])/(Sqrt[2]*Rt[a, 2]*r^(3/2))])/(3*Sqrt
[2]*Rt[a, 2]*d*r^(3/2)), x] + (-Simp[(q*(2 - r)*ArcTan[(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)*ArcTanh[(Rt[a, 2]*Sqrt[r]*(1 + r - 2*q*x))/(Sqr
t[2]*Sqrt[a + b*x^3])])/(3*Sqrt[2]*Rt[a, 2]*d*Sqrt[r]), x] - Simp[(q*(2 - r)*ArcTanh[(Rt[a, 2]*(1 - r)*Sqrt[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] && PosQ[a]

Rubi steps

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

Mathematica [C]  time = 0.0728146, size = 83, normalized size = 0.26 \[ \frac{x^2 \sqrt{1-\frac{b x^3}{a}} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right )}{\left (20 a-12 \sqrt{3} a\right ) \sqrt{a-b x^3}} \]

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[1 - (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)])/((20*a - 12*Sqrt
[3]*a)*Sqrt[a - b*x^3])

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Maple [C]  time = 0.063, size = 509, normalized size = 1.6 \begin{align*}{\frac{-{\frac{i}{27}}\sqrt{2}}{a{b}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+6\,a\sqrt{3}-10\,a \right ) }{\frac{1}{{\it \_alpha}}\sqrt [3]{a{b}^{2}}\sqrt{{-{\frac{i}{2}}b \left ( 2\,x+{\frac{1}{b} \left ( i\sqrt{3}\sqrt [3]{a{b}^{2}}+\sqrt [3]{a{b}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{a{b}^{2}}}}}}\sqrt{{b \left ( x-{\frac{1}{b}\sqrt [3]{a{b}^{2}}} \right ) \left ( -3\,\sqrt [3]{a{b}^{2}}-i\sqrt{3}\sqrt [3]{a{b}^{2}} \right ) ^{-1}}}\sqrt{{{\frac{i}{2}}b \left ( 2\,x+{\frac{1}{b} \left ( -i\sqrt{3}\sqrt [3]{a{b}^{2}}+\sqrt [3]{a{b}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{a{b}^{2}}}}}} \left ( -3\,i\sqrt [3]{a{b}^{2}}{\it \_alpha}\,\sqrt{3}b+4\,{b}^{2}{{\it \_alpha}}^{2}\sqrt{3}+3\,i \left ( a{b}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}-2\,\sqrt{3}\sqrt [3]{a{b}^{2}}{\it \_alpha}\,b-6\,i\sqrt [3]{a{b}^{2}}{\it \_alpha}\,b+6\,{b}^{2}{{\it \_alpha}}^{2}-2\,\sqrt{3} \left ( a{b}^{2} \right ) ^{2/3}+6\,i \left ( a{b}^{2} \right ) ^{{\frac{2}{3}}}-3\,\sqrt [3]{a{b}^{2}}{\it \_alpha}\,b-3\, \left ( a{b}^{2} \right ) ^{2/3} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{-i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{a{b}^{2}}}}}}},{\frac{1}{6\,ab} \left ( -2\,i\sqrt{3}\sqrt [3]{a{b}^{2}}{{\it \_alpha}}^{2}b+i\sqrt{3} \left ( a{b}^{2} \right ) ^{{\frac{2}{3}}}{\it \_alpha}-4\,i\sqrt [3]{a{b}^{2}}{{\it \_alpha}}^{2}b-2\,\sqrt{3} \left ( a{b}^{2} \right ) ^{2/3}{\it \_alpha}+i\sqrt{3}ab+2\,i \left ( a{b}^{2} \right ) ^{{\frac{2}{3}}}{\it \_alpha}+2\,\sqrt{3}ab-3\, \left ( a{b}^{2} \right ) ^{2/3}{\it \_alpha}+2\,iab+3\,ab \right ) },\sqrt{{\frac{-i\sqrt{3}}{b}\sqrt [3]{a{b}^{2}} \left ( -{\frac{3}{2\,b}\sqrt [3]{a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{a{b}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{-b{x}^{3}+a}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*I/b^3/a*2^(1/2)*sum(1/_alpha*(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))*El
lipticPi(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=RootO
f(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{x}{{\left (b x^{3} + 2 \, a{\left (3 \, \sqrt{3} - 5\right )}\right )} \sqrt{-b x^{3} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x/(-10*a*sqrt(a - b*x**3) + 6*sqrt(3)*a*sqrt(a - b*x**3) + b*x**3*sqrt(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+2*a*(5-3*3^(1/2)))/(-b*x^3+a)^(1/2),x, algorithm="giac")

[Out]

sage0*x