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

Optimal. Leaf size=320 \[ \frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{b x^3-a}}\right )}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{b x^3-a}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{b x^3-a}}\right )}{6 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{b x^3-a}}\right )}{3 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}} \]

[Out]

((2 + Sqrt[3])*ArcTan[(3^(1/4)*(1 - Sqrt[3])*a^(1/6)*(a^(1/3) - b^(1/3)*x))/(Sqr
t[2]*Sqrt[-a + b*x^3])])/(2*Sqrt[2]*3^(3/4)*a^(5/6)*b^(2/3)) - ((2 + Sqrt[3])*Ar
cTan[((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)) - ((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])])/(6*Sqrt[2]*3^(1/4)*a^(5/6)*b^(
2/3)) - ((2 + Sqrt[3])*ArcTanh[(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))

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Rubi [A]  time = 0.189017, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.028 \[ \frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{b x^3-a}}\right )}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{b x^3-a}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{b x^3-a}}\right )}{6 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{b x^3-a}}\right )}{3 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[x/((2*(5 - 3*Sqrt[3])*a - b*x^3)*Sqrt[-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))/(Sqr
t[2]*Sqrt[-a + b*x^3])])/(2*Sqrt[2]*3^(3/4)*a^(5/6)*b^(2/3)) - ((2 + Sqrt[3])*Ar
cTan[((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)) - ((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])])/(6*Sqrt[2]*3^(1/4)*a^(5/6)*b^(
2/3)) - ((2 + Sqrt[3])*ArcTanh[(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))

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Rubi in Sympy [A]  time = 35.9728, size = 68, normalized size = 0.21 \[ - \frac{x^{2} \sqrt{- a + b x^{3}} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},1,\frac{5}{3},\frac{b x^{3}}{a},\frac{b x^{3}}{2 a \left (- 3 \sqrt{3} + 5\right )} \right )}}{4 a^{2} \sqrt{1 - \frac{b x^{3}}{a}} \left (- 3 \sqrt{3} + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-b*x**3+2*a*(5-3*3**(1/2)))/(b*x**3-a)**(1/2),x)

[Out]

-x**2*sqrt(-a + b*x**3)*appellf1(2/3, 1/2, 1, 5/3, b*x**3/a, b*x**3/(2*a*(-3*sqr
t(3) + 5)))/(4*a**2*sqrt(1 - b*x**3/a)*(-3*sqrt(3) + 5))

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Mathematica [C]  time = 0.684906, size = 243, normalized size = 0.76 \[ -\frac{10 \left (26-15 \sqrt{3}\right ) a x^2 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right )}{\left (3 \sqrt{3}-5\right ) \sqrt{b x^3-a} \left (2 \left (3 \sqrt{3}-5\right ) a+b x^3\right ) \left (10 \left (3 \sqrt{3}-5\right ) a F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right )-3 b x^3 \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right )+\left (5-3 \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-10*(26 - 15*Sqrt[3])*a*x^2*AppellF1[2/3, 1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a
- 6*Sqrt[3]*a)])/((-5 + 3*Sqrt[3])*Sqrt[-a + b*x^3]*(2*(-5 + 3*Sqrt[3])*a + b*x
^3)*(10*(-5 + 3*Sqrt[3])*a*AppellF1[2/3, 1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a -
6*Sqrt[3]*a)] - 3*b*x^3*(AppellF1[5/3, 1/2, 2, 8/3, (b*x^3)/a, (b*x^3)/(10*a -
6*Sqrt[3]*a)] + (5 - 3*Sqrt[3])*AppellF1[5/3, 3/2, 1, 8/3, (b*x^3)/a, (b*x^3)/(1
0*a - 6*Sqrt[3]*a)])))

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Maple [C]  time = 0.097, size = 510, normalized size = 1.6 \[{\frac{-{\frac{i}{27}}\sqrt{2}}{a{b}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+6\,a\sqrt{3}-10\,a \right ) }{\frac{1}{{\it \_alpha}}\sqrt [3]{a{b}^{2}}\sqrt{{-{\frac{i}{2}}b \left ( 2\,x+{\frac{1}{b} \left ( i\sqrt{3}\sqrt [3]{a{b}^{2}}+\sqrt [3]{a{b}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{a{b}^{2}}}}}}\sqrt{{b \left ( x-{\frac{1}{b}\sqrt [3]{a{b}^{2}}} \right ) \left ( -3\,\sqrt [3]{a{b}^{2}}-i\sqrt{3}\sqrt [3]{a{b}^{2}} \right ) ^{-1}}}\sqrt{{{\frac{i}{2}}b \left ( 2\,x+{\frac{1}{b} \left ( -i\sqrt{3}\sqrt [3]{a{b}^{2}}+\sqrt [3]{a{b}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{a{b}^{2}}}}}} \left ( -3\,i\sqrt [3]{a{b}^{2}}{\it \_alpha}\,\sqrt{3}b+4\,{b}^{2}{{\it \_alpha}}^{2}\sqrt{3}+3\,i \left ( a{b}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}-2\,\sqrt [3]{a{b}^{2}}{\it \_alpha}\,\sqrt{3}b-6\,i\sqrt [3]{a{b}^{2}}{\it \_alpha}\,b+6\,{b}^{2}{{\it \_alpha}}^{2}-2\, \left ( a{b}^{2} \right ) ^{2/3}\sqrt{3}+6\,i \left ( a{b}^{2} \right ) ^{{\frac{2}{3}}}-3\,\sqrt [3]{a{b}^{2}}{\it \_alpha}\,b-3\, \left ( a{b}^{2} \right ) ^{2/3} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{-i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{a{b}^{2}}}}}}},{\frac{1}{6\,ab} \left ( -2\,i\sqrt [3]{a{b}^{2}}{{\it \_alpha}}^{2}\sqrt{3}b+i \left ( a{b}^{2} \right ) ^{{\frac{2}{3}}}{\it \_alpha}\,\sqrt{3}-4\,i\sqrt [3]{a{b}^{2}}{{\it \_alpha}}^{2}b-2\, \left ( a{b}^{2} \right ) ^{2/3}{\it \_alpha}\,\sqrt{3}+2\,i \left ( a{b}^{2} \right ) ^{{\frac{2}{3}}}{\it \_alpha}+i\sqrt{3}ab-3\, \left ( a{b}^{2} \right ) ^{2/3}{\it \_alpha}+2\,\sqrt{3}ab+2\,iab+3\,ab \right ) },\sqrt{{\frac{-i\sqrt{3}}{b}\sqrt [3]{a{b}^{2}} \left ( -{\frac{3}{2\,b}\sqrt [3]{a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{a{b}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{b{x}^{3}-a}}}}} \]

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*(a*b^2)^(1
/3)*_alpha*3^(1/2)*b-6*I*(a*b^2)^(1/3)*_alpha*b+6*b^2*_alpha^2-2*(a*b^2)^(2/3)*3
^(1/2)+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*(a*b^2)^(1/3)*_alpha^2*3^(1/2)*b+I*(a*b^2)^(2/3)*
_alpha*3^(1/2)-4*I*(a*b^2)^(1/3)*_alpha^2*b-2*(a*b^2)^(2/3)*_alpha*3^(1/2)+2*I*(
a*b^2)^(2/3)*_alpha+I*3^(1/2)*a*b-3*(a*b^2)^(2/3)*_alpha+2*3^(1/2)*a*b+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. \[ -\int \frac{x}{{\left (b x^{3} + 2 \, a{\left (3 \, \sqrt{3} - 5\right )}\right )} \sqrt{b x^{3} - a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/((b*x^3 + 2*a*(3*sqrt(3) - 5))*sqrt(b*x^3 - a)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/((b*x^3 + 2*a*(3*sqrt(3) - 5))*sqrt(b*x^3 - a)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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GIAC/XCAS [A]  time = 0.575117, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/((b*x^3 + 2*a*(3*sqrt(3) - 5))*sqrt(b*x^3 - a)),x, algorithm="giac")

[Out]

sage0*x