3.103 \(\int \frac{1-\sqrt{3}+\sqrt [3]{\frac{b}{a}} x}{\sqrt{a+b x^3}} \, dx\)

Optimal. Leaf size=241 \[ \frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{a+b x^3}}{b \left (x \sqrt [3]{\frac{b}{a}}+\sqrt{3}+1\right )}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \left (x \sqrt [3]{\frac{b}{a}}+1\right ) \sqrt{\frac{x^2 \left (\frac{b}{a}\right )^{2/3}-x \sqrt [3]{\frac{b}{a}}+1}{\left (x \sqrt [3]{\frac{b}{a}}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{\frac{b}{a}} x-\sqrt{3}+1}{\sqrt [3]{\frac{b}{a}} x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{\frac{x \sqrt [3]{\frac{b}{a}}+1}{\left (x \sqrt [3]{\frac{b}{a}}+\sqrt{3}+1\right )^2}} \sqrt{a+b x^3}} \]

[Out]

(2*(b/a)^(2/3)*Sqrt[a + b*x^3])/(b*(1 + Sqrt[3] + (b/a)^(1/3)*x)) - (3^(1/4)*Sqr
t[2 - Sqrt[3]]*(1 + (b/a)^(1/3)*x)*Sqrt[(1 - (b/a)^(1/3)*x + (b/a)^(2/3)*x^2)/(1
+ Sqrt[3] + (b/a)^(1/3)*x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] + (b/a)^(1/3)*x)/(1
+ Sqrt[3] + (b/a)^(1/3)*x)], -7 - 4*Sqrt[3]])/((b/a)^(1/3)*Sqrt[(1 + (b/a)^(1/3
)*x)/(1 + Sqrt[3] + (b/a)^(1/3)*x)^2]*Sqrt[a + b*x^3])

_______________________________________________________________________________________

Rubi [A]  time = 0.154738, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.031 \[ \frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{a+b x^3}}{b \left (x \sqrt [3]{\frac{b}{a}}+\sqrt{3}+1\right )}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \left (x \sqrt [3]{\frac{b}{a}}+1\right ) \sqrt{\frac{x^2 \left (\frac{b}{a}\right )^{2/3}-x \sqrt [3]{\frac{b}{a}}+1}{\left (x \sqrt [3]{\frac{b}{a}}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{\frac{b}{a}} x-\sqrt{3}+1}{\sqrt [3]{\frac{b}{a}} x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{\frac{x \sqrt [3]{\frac{b}{a}}+1}{\left (x \sqrt [3]{\frac{b}{a}}+\sqrt{3}+1\right )^2}} \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - Sqrt[3] + (b/a)^(1/3)*x)/Sqrt[a + b*x^3],x]

[Out]

(2*(b/a)^(2/3)*Sqrt[a + b*x^3])/(b*(1 + Sqrt[3] + (b/a)^(1/3)*x)) - (3^(1/4)*Sqr
t[2 - Sqrt[3]]*(1 + (b/a)^(1/3)*x)*Sqrt[(1 - (b/a)^(1/3)*x + (b/a)^(2/3)*x^2)/(1
+ Sqrt[3] + (b/a)^(1/3)*x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] + (b/a)^(1/3)*x)/(1
+ Sqrt[3] + (b/a)^(1/3)*x)], -7 - 4*Sqrt[3]])/((b/a)^(1/3)*Sqrt[(1 + (b/a)^(1/3
)*x)/(1 + Sqrt[3] + (b/a)^(1/3)*x)^2]*Sqrt[a + b*x^3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.7379, size = 444, normalized size = 1.84 \[ - \frac{\sqrt [4]{3} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{b^{\frac{2}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} + \frac{2 \sqrt [3]{\frac{b}{a}} \sqrt{a + b x^{3}}}{b^{\frac{2}{3}} \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \left (- \sqrt{3} + 1\right ) \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (- \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} + \sqrt [3]{b}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 b^{\frac{2}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3**(1/4)*a**(1/3)*(b/a)**(1/3)*sqrt((a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*
x**2)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)**2)*sqrt(-sqrt(3) + 2)*(a**(1/3) + b
**(1/3)*x)*elliptic_e(asin((-a**(1/3)*(-1 + sqrt(3)) + b**(1/3)*x)/(a**(1/3)*(1
+ sqrt(3)) + b**(1/3)*x)), -7 - 4*sqrt(3))/(b**(2/3)*sqrt(a**(1/3)*(a**(1/3) + b
**(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)**2)*sqrt(a + b*x**3)) + 2*(b/a)
**(1/3)*sqrt(a + b*x**3)/(b**(2/3)*(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)) + 2*3*
*(3/4)*sqrt((a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(a**(1/3)*(1 + sqrt
(3)) + b**(1/3)*x)**2)*(-sqrt(3) + 1)*sqrt(sqrt(3) + 2)*(a**(1/3) + b**(1/3)*x)*
(-a**(1/3)*(b/a)**(1/3) + b**(1/3))*elliptic_f(asin((-a**(1/3)*(-1 + sqrt(3)) +
b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)), -7 - 4*sqrt(3))/(3*b**(2/3)*
sqrt(a**(1/3)*(a**(1/3) + b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)**2)*
sqrt(a + b*x**3))

_______________________________________________________________________________________

Mathematica [C]  time = 0.468901, size = 243, normalized size = 1.01 \[ \frac{2 i \sqrt [3]{a} \sqrt{\frac{(-1)^{5/6} \left (\sqrt [3]{-b} x-\sqrt [3]{a}\right )}{\sqrt [3]{a}}} \sqrt{\frac{(-b)^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+1} \left (\left (\sqrt{3} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}}+\left (\sqrt{3}-3\right ) \sqrt [3]{-b}\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-3 \sqrt [6]{-1} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{3^{3/4} (-b)^{2/3} \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(1 - Sqrt[3] + (b/a)^(1/3)*x)/Sqrt[a + b*x^3],x]

[Out]

((2*I)*a^(1/3)*Sqrt[((-1)^(5/6)*(-a^(1/3) + (-b)^(1/3)*x))/a^(1/3)]*Sqrt[1 + ((-
b)^(1/3)*x)/a^(1/3) + ((-b)^(2/3)*x^2)/a^(2/3)]*(-3*(-1)^(1/6)*a^(1/3)*(b/a)^(1/
3)*EllipticE[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-b)^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^
(1/3)] + ((-3 + Sqrt[3])*(-b)^(1/3) + Sqrt[3]*a^(1/3)*(b/a)^(1/3))*EllipticF[Arc
Sin[Sqrt[-(-1)^(5/6) - (I*(-b)^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)]))/(3^(3/4
)*(-b)^(2/3)*Sqrt[a + b*x^3])

_______________________________________________________________________________________

Maple [B]  time = 0.048, size = 1004, normalized size = 4.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*I/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)*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))-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)*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))-2/3*I*(b/a
)^(1/3)*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))
)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (\frac{b}{a}\right )^{\frac{1}{3}} - \sqrt{3} + 1}{\sqrt{b x^{3} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x*(b/a)^(1/3) - sqrt(3) + 1)/sqrt(b*x^3 + a),x, algorithm="maxima")

[Out]

integrate((x*(b/a)^(1/3) - sqrt(3) + 1)/sqrt(b*x^3 + a), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x \left (\frac{b}{a}\right )^{\frac{1}{3}} - \sqrt{3} + 1}{\sqrt{b x^{3} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x*(b/a)^(1/3) - sqrt(3) + 1)/sqrt(b*x^3 + a),x, algorithm="fricas")

[Out]

integral((x*(b/a)^(1/3) - sqrt(3) + 1)/sqrt(b*x^3 + a), x)

_______________________________________________________________________________________

Sympy [A]  time = 1.38769, size = 0, normalized size = 0. \[ \mathrm{NaN} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

nan

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (\frac{b}{a}\right )^{\frac{1}{3}} - \sqrt{3} + 1}{\sqrt{b x^{3} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x*(b/a)^(1/3) - sqrt(3) + 1)/sqrt(b*x^3 + a),x, algorithm="giac")

[Out]

integrate((x*(b/a)^(1/3) - sqrt(3) + 1)/sqrt(b*x^3 + a), x)