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

Optimal. Leaf size=345 \[ \frac{\sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}+\frac{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{b x^3-a}}\right )}{\sqrt{3 \left (2 \sqrt{3}-3\right )} \sqrt{a} b^{2/3}} \]

[Out]

((b^(1/3)*e + (1 - Sqrt[3])*a^(1/3)*f)*ArcTan[(Sqrt[-3 + 2*Sqrt[3]]*a^(1/6)*(a^(1/3) - b^(1/3)*x))/Sqrt[-a + b
*x^3]])/(Sqrt[3*(-3 + 2*Sqrt[3])]*Sqrt[a]*b^(2/3)) + (Sqrt[2 - Sqrt[3]]*(b^(1/3)*e + (1 + Sqrt[3])*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
^(3/4)*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 = 0.491108, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.089, Rules used = {2141, 219, 2140, 203} \[ \frac{\sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}+\frac{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{b x^3-a}}\right )}{\sqrt{3 \left (2 \sqrt{3}-3\right )} \sqrt{a} b^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2141

Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> -Dist[(6*a*d^4*e - c*f*
(b*c^3 - 22*a*d^3))/(c*d*(b*c^3 - 28*a*d^3)), Int[1/Sqrt[a + b*x^3], x], x] + Dist[(d*e - c*f)/(c*d*(b*c^3 - 2
8*a*d^3)), Int[(c*(b*c^3 - 22*a*d^3) + 6*a*d^4*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{a, b, c, d, e,
f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && NeQ[6*a*d^4*e - c*f*(b*c^3 - 2
2*a*d^3), 0]

Rule 219

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3
])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - S
qrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 2140

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{k = Simplify[(d*e
+ 2*c*f)/(c*f)]}, Dist[((1 + k)*e)/d, Subst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + ((1 + k)*d*x)/c)/Sqrt[a +
b*x^3]], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6
, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3), 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{e+f x}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{-a+b x^3}} \, dx &=\frac{\left (\sqrt [3]{b} e+\left (1-\sqrt{3}\right ) \sqrt [3]{a} f\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a} \left (-22 a b+\left (1-\sqrt{3}\right )^3 a b\right )-6 a b^{4/3} x}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{-a+b x^3}} \, dx}{12 \sqrt{3} a^{4/3} b^{4/3}}--\frac{\left (-6 a b^{4/3} e-\left (1-\sqrt{3}\right ) \sqrt [3]{a} \left (-22 a b+\left (1-\sqrt{3}\right )^3 a b\right ) f\right ) \int \frac{1}{\sqrt{-a+b x^3}} \, dx}{\left (1-\sqrt{3}\right ) \sqrt [3]{a} \sqrt [3]{b} \left (-28 a b+\left (1-\sqrt{3}\right )^3 a b\right )}\\ &=\frac{\sqrt{2-\sqrt{3}} \left (\sqrt [3]{b} e+\left (1+\sqrt{3}\right ) \sqrt [3]{a} f\right ) \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}} 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^{3/4} \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{\left (\sqrt [3]{b} e+\left (1-\sqrt{3}\right ) \sqrt [3]{a} f\right ) \operatorname{Subst}\left (\int \frac{1}{1-\left (3-2 \sqrt{3}\right ) a x^2} \, dx,x,\frac{1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{-a+b x^3}}\right )}{\sqrt{3} b^{2/3}}\\ &=\frac{\left (\sqrt [3]{b} e+\left (1-\sqrt{3}\right ) \sqrt [3]{a} f\right ) \tan ^{-1}\left (\frac{\sqrt{-3+2 \sqrt{3}} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{-a+b x^3}}\right )}{\sqrt{3 \left (-3+2 \sqrt{3}\right )} \sqrt{a} b^{2/3}}+\frac{\sqrt{2-\sqrt{3}} \left (\sqrt [3]{b} e+\left (1+\sqrt{3}\right ) \sqrt [3]{a} f\right ) \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}} 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^{3/4} \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}}\\ \end{align*}

Mathematica [C]  time = 1.10096, size = 467, normalized size = 1.35 \[ -\frac{4 \sqrt{\frac{\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\frac{1}{2} f \left (i \left (-3+(2+i) \sqrt{3}\right ) \sqrt [3]{a}+\left (3-(2-i) \sqrt{3}\right ) \sqrt [3]{b} x\right ) \sqrt{\frac{\left (\sqrt{3}-i\right ) \sqrt [3]{a}+\left (\sqrt{3}+i\right ) \sqrt [3]{b} x}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{-\frac{i \left (\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x+2 \sqrt [3]{a}\right )}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )-i \sqrt{-\frac{i \left (2 \sqrt [3]{a}+\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x\right )}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \left (\sqrt [3]{b} e-\left (\sqrt{3}-1\right ) \sqrt [3]{a} f\right ) \Pi \left (\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left (\sqrt{-\frac{i \left (\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x+2 \sqrt [3]{a}\right )}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )\right )}{\left (3-(2-i) \sqrt{3}\right ) b^{2/3} \sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{b x^3-a}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-4*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((f*(I*(-3 + (2 + I)*Sqrt[3])*a^(1/3) + (3 - (2 - I
)*Sqrt[3])*b^(1/3)*x)*Sqrt[((-I + Sqrt[3])*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*Elli
pticF[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 - I*(b^(1/3)*e - (-1 + Sqrt[3])*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 + (1 +
2*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]))/((3 - (2 - I)*Sqrt[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 = 0.063, size = 0, normalized size = 0. \begin{align*} \int{(fx+e) \left ( -\sqrt [3]{b}x+\sqrt [3]{a} \left ( 1-\sqrt{3} \right ) \right ) ^{-1}{\frac{1}{\sqrt{b{x}^{3}-a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{b x^{3} - a}{\left (2 \,{\left (2 \, b f x^{4} + 2 \, b e x^{3} - 2 \, a f x - 2 \, a e - \sqrt{3}{\left (b f x^{4} + b e x^{3} + 2 \, a f x + 2 \, a e\right )}\right )} a^{\frac{2}{3}} +{\left (b f x^{5} + b e x^{4} + 8 \, a f x^{2} + 8 \, a e x - \sqrt{3}{\left (b f x^{5} + b e x^{4} - 4 \, a f x^{2} - 4 \, a e x\right )}\right )} a^{\frac{1}{3}} b^{\frac{1}{3}} +{\left (b f x^{6} + b e x^{5} - 10 \, a f x^{3} - 10 \, a e x^{2} - 6 \, \sqrt{3}{\left (a f x^{3} + a e x^{2}\right )}\right )} b^{\frac{2}{3}}\right )}}{b^{3} x^{9} - 21 \, a b^{2} x^{6} + 12 \, a^{2} b x^{3} + 8 \, a^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(b*x^3 - a)*(2*(2*b*f*x^4 + 2*b*e*x^3 - 2*a*f*x - 2*a*e - sqrt(3)*(b*f*x^4 + b*e*x^3 + 2*a*f*x +
2*a*e))*a^(2/3) + (b*f*x^5 + b*e*x^4 + 8*a*f*x^2 + 8*a*e*x - sqrt(3)*(b*f*x^5 + b*e*x^4 - 4*a*f*x^2 - 4*a*e*x
))*a^(1/3)*b^(1/3) + (b*f*x^6 + b*e*x^5 - 10*a*f*x^3 - 10*a*e*x^2 - 6*sqrt(3)*(a*f*x^3 + a*e*x^2))*b^(2/3))/(b
^3*x^9 - 21*a*b^2*x^6 + 12*a^2*b*x^3 + 8*a^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

sage0*x