3.65 \(\int \frac{e+f x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{a-b x^3}} \, dx\)

Optimal. Leaf size=304 \[ -\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\sqrt [3]{a} f+\sqrt [3]{b} e\right ) F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt [3]{a} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}-\frac{2 \left (\sqrt [3]{b} e-2 \sqrt [3]{a} f\right ) \tanh ^{-1}\left (\frac{\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{a-b x^3}}\right )}{9 \sqrt{a} b^{2/3}} \]

[Out]

(-2*(b^(1/3)*e - 2*a^(1/3)*f)*ArcTanh[(a^(1/3) - b^(1/3)*x)^2/(3*a^(1/6)*Sqrt[a
- b*x^3])])/(9*Sqrt[a]*b^(2/3)) - (2*Sqrt[2 + Sqrt[3]]*(b^(1/3)*e + a^(1/3)*f)*(
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*3^(1/4)*a^(1/3)*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])

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Rubi [A]  time = 0.609196, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114 \[ -\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\sqrt [3]{a} f+\sqrt [3]{b} e\right ) F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt [3]{a} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}-\frac{2 \left (\sqrt [3]{b} e-2 \sqrt [3]{a} f\right ) \tanh ^{-1}\left (\frac{\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{a-b x^3}}\right )}{9 \sqrt{a} b^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(e + f*x)/((2*a^(1/3) + b^(1/3)*x)*Sqrt[a - b*x^3]),x]

[Out]

(-2*(b^(1/3)*e - 2*a^(1/3)*f)*ArcTanh[(a^(1/3) - b^(1/3)*x)^2/(3*a^(1/6)*Sqrt[a
- b*x^3])])/(9*Sqrt[a]*b^(2/3)) - (2*Sqrt[2 + Sqrt[3]]*(b^(1/3)*e + a^(1/3)*f)*(
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*3^(1/4)*a^(1/3)*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])

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Rubi in Sympy [A]  time = 176.072, size = 745, normalized size = 2.45 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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(sqrt(3) + 2)*(a**(1/3) - b**(1/3)*x)*(a**(1/3)*f
*(1 + sqrt(3)) + b**(1/3)*e)*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*a**(1/3)*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(3) + 3)*sqrt(a - b*x**3)) + 3**(3/4)*sqrt(a**(2/3)*(1 + b**(1/3)*x/a**(1/
3) + b**(2/3)*x**2/a**(2/3))/(a**(1/3)*(1 + sqrt(3)) - b**(1/3)*x)**2)*sqrt(3 +
2*sqrt(3))*sqrt(-sqrt(3) + 2)*(a**(1/3) - b**(1/3)*x)*(2*a**(1/3)*f - b**(1/3)*e
)*atanh(sqrt(-(a**(1/3)*(-1 + sqrt(3)) + b**(1/3)*x)**2/(a**(1/3)*(1 + sqrt(3))
- b**(1/3)*x)**2 + 1)/(sqrt(3 + 2*sqrt(3))*sqrt((a**(1/3)*(-1 + sqrt(3)) + b**(1
/3)*x)**2/(a**(1/3)*(1 + sqrt(3)) - b**(1/3)*x)**2 - 4*sqrt(3) + 7)))/(9*a**(1/3
)*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)) - 4*3**(1/4)*sqrt(a**(2/3)*(1 + b**(1/3)*x/a**(1/3)
+ b**(2/3)*x**2/a**(2/3))/(a**(1/3)*(1 + sqrt(3)) - b**(1/3)*x)**2)*sqrt(-sqrt(
3) + 2)*(a**(1/3) - b**(1/3)*x)*(2*a**(1/3)*f - b**(1/3)*e)*elliptic_pi(4*sqrt(3
) + 7, 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))/(a**(1/3)*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(-4*sqrt(3) + 7)*(-sqrt(3) + 3
)*(sqrt(3) + 3)*sqrt(a - b*x**3))

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Mathematica [C]  time = 2.462, size = 447, normalized size = 1.47 \[ \frac{2 \sqrt{\frac{\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (-i \sqrt{-\frac{i \left (2 \sqrt [3]{a}+\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x\right )}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \left (\sqrt [3]{b} e-2 \sqrt [3]{a} f\right ) \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\sqrt{-\frac{i \left (\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x+2 \sqrt [3]{a}\right )}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )-\frac{1}{2} i f \sqrt{\frac{\left (\sqrt{3}-i\right ) \sqrt [3]{a}+\left (\sqrt{3}+i\right ) \sqrt [3]{b} x}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} \left (\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}-\left (\sqrt{3}+3 i\right ) \sqrt [3]{b} x\right ) F\left (\sin ^{-1}\left (\sqrt{-\frac{i \left (\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x+2 \sqrt [3]{a}\right )}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )\right )}{\left (\sqrt [3]{-1}-2\right ) b^{2/3} \sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{a-b x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e + f*x)/((2*a^(1/3) + b^(1/3)*x)*Sqrt[a - b*x^3]),x]

[Out]

(2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((-I/2)*f*Sqrt[((-I +
Sqrt[3])*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*((-3*I +
Sqrt[3])*a^(1/3) - (3*I + Sqrt[3])*b^(1/3)*x)*EllipticF[ArcSin[Sqrt[((-I)*(2*a^
(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3]
)/2] - I*(b^(1/3)*e - 2*a^(1/3)*f)*Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/
3)*x))/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/
a^(2/3)]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[((-I)*(2*a^(1/3) +
(1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/(
(-2 + (-1)^(1/3))*b^(2/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3)
)*a^(1/3))]*Sqrt[a - b*x^3])

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Maple [F]  time = 0.101, size = 0, normalized size = 0. \[ \int{(fx+e) \left ( 2\,\sqrt [3]{a}+\sqrt [3]{b}x \right ) ^{-1}{\frac{1}{\sqrt{-b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f x + e}{\sqrt{-b x^{3} + a}{\left (b^{\frac{1}{3}} x + 2 \, a^{\frac{1}{3}}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)/(sqrt(-b*x^3 + a)*(b^(1/3)*x + 2*a^(1/3))),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e + f x}{\left (2 \sqrt [3]{a} + \sqrt [3]{b} x\right ) \sqrt{a - b x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((e + f*x)/((2*a**(1/3) + b**(1/3)*x)*sqrt(a - b*x**3)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)/(sqrt(-b*x^3 + a)*(b^(1/3)*x + 2*a^(1/3))),x, algorithm="giac")

[Out]

Exception raised: TypeError