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

Optimal. Leaf size=322 \[ \frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{-a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{3 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{6 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}} \]

[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)*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])])/(6*Sqrt[2]*3^(1/4)*a^(5/6)*b^(2/3))

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Rubi [A]  time = 0.191645, antiderivative size = 322, 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 \[ \frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{-a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{3 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{6 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}} \]

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))/(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)*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])])/(6*Sqrt[2]*3^(1/4)*a^(5/6)*b^(2/3))

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Rubi in Sympy [A]  time = 34.7465, size = 73, normalized size = 0.23 \[ - \frac{x^{2} \sqrt{- a - b x^{3}} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},1,\frac{5}{3},- \frac{b x^{3}}{a},- \frac{b x^{3}}{2 a \left (- 3 \sqrt{3} + 5\right )} \right )}}{4 a^{2} \sqrt{1 + \frac{b x^{3}}{a}} \left (- 3 \sqrt{3} + 5\right )} \]

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*s
qrt(3) + 5)))/(4*a**2*sqrt(1 + b*x**3/a)*(-3*sqrt(3) + 5))

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Mathematica [C]  time = 0.688283, size = 252, normalized size = 0.78 \[ -\frac{10 \left (26-15 \sqrt{3}\right ) a x^2 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a-10 a}\right )}{\left (3 \sqrt{3}-5\right ) \sqrt{-a-b x^3} \left (2 \left (3 \sqrt{3}-5\right ) a-b x^3\right ) \left (3 b x^3 \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right )+\left (5-3 \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right )\right )+10 \left (3 \sqrt{3}-5\right ) a F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x/(Sqrt[-a - b*x^3]*(2*(5 - 3*Sqrt[3])*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)/(10*a - 6*Sqrt[3]*a))])))

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Maple [C]  time = 0.096, size = 541, normalized size = 1.7 \[{\frac{{\frac{i}{27}}\sqrt{2}}{a{b}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}-6\,a\sqrt{3}+10\,a \right ) }{\frac{1}{{\it \_alpha}}\sqrt [3]{-a{b}^{2}}\sqrt{{{\frac{i}{2}}b \left ( 2\,x+{\frac{1}{b} \left ( \sqrt [3]{-a{b}^{2}}-i\sqrt{3}\sqrt [3]{-a{b}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}\sqrt{{b \left ( x-{\frac{1}{b}\sqrt [3]{-a{b}^{2}}} \right ) \left ( -3\,\sqrt [3]{-a{b}^{2}}+i\sqrt{3}\sqrt [3]{-a{b}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}b \left ( 2\,x+{\frac{1}{b} \left ( \sqrt [3]{-a{b}^{2}}+i\sqrt{3}\sqrt [3]{-a{b}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}} \left ( 3\,i\sqrt [3]{-a{b}^{2}}{\it \_alpha}\,\sqrt{3}b+4\,{b}^{2}{{\it \_alpha}}^{2}\sqrt{3}-3\,i \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}-2\,\sqrt{3}\sqrt [3]{-a{b}^{2}}{\it \_alpha}\,b+6\,i\sqrt [3]{-a{b}^{2}}{\it \_alpha}\,b+6\,{b}^{2}{{\it \_alpha}}^{2}-2\,\sqrt{3} \left ( -a{b}^{2} \right ) ^{2/3}-6\,i \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}}-3\,\sqrt [3]{-a{b}^{2}}{\it \_alpha}\,b-3\, \left ( -a{b}^{2} \right ) ^{2/3} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}},-{\frac{1}{6\,ab} \left ( 2\,i\sqrt{3}\sqrt [3]{-a{b}^{2}}{{\it \_alpha}}^{2}b-i\sqrt{3} \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}}{\it \_alpha}+4\,i\sqrt [3]{-a{b}^{2}}{{\it \_alpha}}^{2}b-2\,\sqrt{3} \left ( -a{b}^{2} \right ) ^{2/3}{\it \_alpha}+i\sqrt{3}ab-2\,i \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}}{\it \_alpha}-2\,\sqrt{3}ab-3\, \left ( -a{b}^{2} \right ) ^{2/3}{\it \_alpha}+2\,iab-3\,ab \right ) },\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}} \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{-b{x}^{3}-a}}}}} \]

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

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]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{- a - b x^{3}} \left (- 6 \sqrt{3} a + 10 a + b x^{3}\right )}\, dx \]

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

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GIAC/XCAS [A]  time = 0.560302, 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