3.1.58 \(\int \frac {e+f x}{(2^{2/3}-x) \sqrt {-1+x^3}} \, dx\) [58]

Optimal. Leaf size=178 \[ -\frac {2 \left (e+2^{2/3} f\right ) \tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {-1+x^3}}\right )}{3 \sqrt {3}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{2} e-f\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \]

[Out]

-2/9*(e+2^(2/3)*f)*arctanh((1-2^(1/3)*x)*3^(1/2)/(x^3-1)^(1/2))*3^(1/2)-2/9*(2^(1/3)*e-f)*(1-x)*EllipticF((1-x
+3^(1/2))/(1-x-3^(1/2)),2*I-I*3^(1/2))*(1/2*6^(1/2)-1/2*2^(1/2))*((x^2+x+1)/(1-x-3^(1/2))^2)^(1/2)*3^(3/4)/(x^
3-1)^(1/2)/((-1+x)/(1-x-3^(1/2))^2)^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2164, 225, 2162, 212} \begin {gather*} -\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \left (\sqrt [3]{2} e-f\right ) F\left (\text {ArcSin}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {2 \left (e+2^{2/3} f\right ) \tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {x^3-1}}\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(e + 2^(2/3)*f)*ArcTanh[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[-1 + x^3]])/(3*Sqrt[3]) - (2*Sqrt[2 - Sqrt[3]]*(2^(
1/3)*e - f)*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] -
x)], -7 + 4*Sqrt[3]])/(3*3^(1/4)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3])

Rule 212

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

Rule 225

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

Rule 2162

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[2*(e/d), Subst[Int[
1/(1 + 3*a*x^2), x], x, (1 + 2*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*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]

Rule 2164

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

Rubi steps

\begin {align*} \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx &=-\left (\frac {1}{3} \left (-\sqrt [3]{2} e+f\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx\right )+\frac {1}{6} \left (\sqrt [3]{2} e+2 f\right ) \int \frac {2^{2/3}+2 x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx\\ &=-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{2} e-f\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {1}{3} \left (2 \left (e+2^{2/3} f\right )\right ) \text {Subst}\left (\int \frac {1}{1-3 x^2} \, dx,x,\frac {1-\sqrt [3]{2} x}{\sqrt {-1+x^3}}\right )\\ &=-\frac {2 \left (e+2^{2/3} f\right ) \tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {-1+x^3}}\right )}{3 \sqrt {3}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{2} e-f\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 20.22, size = 338, normalized size = 1.90 \begin {gather*} \frac {2 \sqrt [6]{2} \sqrt {-\frac {i (-1+x)}{3 i+\sqrt {3}}} \left (-i f \sqrt {-i+\sqrt {3}-2 i x} \left (-6 i-3 i \sqrt [3]{2}+2 \sqrt {3}-\sqrt [3]{2} \sqrt {3}+\left (-3 i \sqrt [3]{2}+4 \sqrt {3}+\sqrt [3]{2} \sqrt {3}\right ) x\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+2 \sqrt {3} \left (\sqrt [3]{2} e+2 f\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {1+x+x^2} \Pi \left (\frac {2 \sqrt {3}}{i+2 i 2^{2/3}+\sqrt {3}};\sin ^{-1}\left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\sqrt {3} \left (i+2 i 2^{2/3}+\sqrt {3}\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {-1+x^3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*2^(1/6)*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*((-I)*f*Sqrt[-I + Sqrt[3] - (2*I)*x]*(-6*I - (3*I)*2^(1/3) +
2*Sqrt[3] - 2^(1/3)*Sqrt[3] + ((-3*I)*2^(1/3) + 4*Sqrt[3] + 2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt
[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + 2*Sqrt[3]*(2^(1/3)*e + 2*f)*Sqrt[I + Sqrt[3]
+ (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] +
(2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(I + (2*I)*2^(2/3) + Sqrt[3])*Sqrt[I + Sq
rt[3] + (2*I)*x]*Sqrt[-1 + x^3])

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Maple [A]
time = 0.30, size = 270, normalized size = 1.52

method	result	size
default	\(-\frac {2 f \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}+\frac {2 \left (-e -2^{\frac {2}{3}} f \right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-2^{\frac {2}{3}}+1}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}\, \left (-2^{\frac {2}{3}}+1\right )}\)	\(270\)
elliptic	\(-\frac {2 f \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}+\frac {2 \left (-e -2^{\frac {2}{3}} f \right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-2^{\frac {2}{3}}+1}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}\, \left (-2^{\frac {2}{3}}+1\right )}\)	\(270\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)/(2^(2/3)-x)/(x^3-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2*f*(-3/2-1/2*I*3^(1/2))*((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2
)*((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*EllipticF(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2
),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))+2*(-e-2^(2/3)*f)*(-3/2-1/2*I*3^(1/2))*((-1+x)/(-3/2-1/2*I*3
^(1/2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(
1/2)/(x^3-1)^(1/2)/(-2^(2/3)+1)*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(-2^(2/3)+1
),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((f*x + e)/(sqrt(x^3 - 1)*(x - 2^(2/3))), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.23, size = 947, normalized size = 5.32 \begin {gather*} \left [\frac {1}{18} \, \sqrt {3} \sqrt {2 \cdot 2^{\frac {2}{3}} f e + 2 \cdot 2^{\frac {1}{3}} f^{2} + e^{2}} \log \left (\frac {4 \, f^{3} x^{18} + 5760 \, f^{3} x^{15} + 69600 \, f^{3} x^{12} - 84224 \, f^{3} x^{9} - 41472 \, f^{3} x^{6} + 61440 \, f^{3} x^{3} - 8192 \, f^{3} + 4 \, \sqrt {3} {\left (252 \, f^{2} x^{14} + 5328 \, f^{2} x^{11} - 9216 \, f^{2} x^{5} + 4608 \, f^{2} x^{2} + {\left (17 \, x^{15} + 1058 \, x^{12} + 2528 \, x^{9} - 5408 \, x^{6} + 2560 \, x^{3} - 512\right )} e^{2} - 2 \, {\left (f x^{16} + 310 \, f x^{13} + 2332 \, f x^{10} - 2656 \, f x^{7} - 256 \, f x^{4} + 512 \, f x\right )} e + 2^{\frac {2}{3}} {\left (2 \, f^{2} x^{16} + 620 \, f^{2} x^{13} + 4664 \, f^{2} x^{10} - 5312 \, f^{2} x^{7} - 512 \, f^{2} x^{4} + 1024 \, f^{2} x + 9 \, {\left (7 \, x^{14} + 148 \, x^{11} - 256 \, x^{5} + 128 \, x^{2}\right )} e^{2} - {\left (17 \, f x^{15} + 1058 \, f x^{12} + 2528 \, f x^{9} - 5408 \, f x^{6} + 2560 \, f x^{3} - 512 \, f\right )} e\right )} + 2^{\frac {1}{3}} {\left (34 \, f^{2} x^{15} + 2116 \, f^{2} x^{12} + 5056 \, f^{2} x^{9} - 10816 \, f^{2} x^{6} + 5120 \, f^{2} x^{3} - 1024 \, f^{2} + {\left (x^{16} + 310 \, x^{13} + 2332 \, x^{10} - 2656 \, x^{7} - 256 \, x^{4} + 512 \, x\right )} e^{2} - 18 \, {\left (7 \, f x^{14} + 148 \, f x^{11} - 256 \, f x^{5} + 128 \, f x^{2}\right )} e\right )}\right )} \sqrt {x^{3} - 1} \sqrt {2 \cdot 2^{\frac {2}{3}} f e + 2 \cdot 2^{\frac {1}{3}} f^{2} + e^{2}} + {\left (x^{18} + 1440 \, x^{15} + 17400 \, x^{12} - 21056 \, x^{9} - 10368 \, x^{6} + 15360 \, x^{3} - 2048\right )} e^{3} + 24 \cdot 2^{\frac {2}{3}} {\left (4 \, f^{3} x^{17} + 484 \, f^{3} x^{14} + 1912 \, f^{3} x^{11} - 4576 \, f^{3} x^{8} + 2432 \, f^{3} x^{5} - 256 \, f^{3} x^{2} + {\left (x^{17} + 121 \, x^{14} + 478 \, x^{11} - 1144 \, x^{8} + 608 \, x^{5} - 64 \, x^{2}\right )} e^{3}\right )} + 48 \cdot 2^{\frac {1}{3}} {\left (20 \, f^{3} x^{16} + 704 \, f^{3} x^{13} + 332 \, f^{3} x^{10} - 2720 \, f^{3} x^{7} + 2176 \, f^{3} x^{4} - 512 \, f^{3} x + {\left (5 \, x^{16} + 176 \, x^{13} + 83 \, x^{10} - 680 \, x^{7} + 544 \, x^{4} - 128 \, x\right )} e^{3}\right )}}{x^{18} - 24 \, x^{15} + 240 \, x^{12} - 1280 \, x^{9} + 3840 \, x^{6} - 6144 \, x^{3} + 4096}\right ) + \frac {2}{3} \, {\left (2^{\frac {1}{3}} e - f\right )} {\rm weierstrassPInverse}\left (0, 4, x\right ), -\frac {1}{9} \, \sqrt {3} \sqrt {-2 \cdot 2^{\frac {2}{3}} f e - 2 \cdot 2^{\frac {1}{3}} f^{2} - e^{2}} \arctan \left (\frac {\sqrt {3} {\left (4 \, f^{2} x^{5} - 4 \, f^{2} x^{2} - {\left (5 \, x^{3} - 2\right )} e^{2} - 2 \, {\left (7 \, f x^{4} - 4 \, f x\right )} e + 2^{\frac {2}{3}} {\left (14 \, f^{2} x^{4} - 8 \, f^{2} x + {\left (x^{5} - x^{2}\right )} e^{2} + {\left (5 \, f x^{3} - 2 \, f\right )} e\right )} - 2^{\frac {1}{3}} {\left (10 \, f^{2} x^{3} - 4 \, f^{2} - {\left (7 \, x^{4} - 4 \, x\right )} e^{2} + 2 \, {\left (f x^{5} - f x^{2}\right )} e\right )}\right )} \sqrt {x^{3} - 1} \sqrt {-2 \cdot 2^{\frac {2}{3}} f e - 2 \cdot 2^{\frac {1}{3}} f^{2} - e^{2}}}{6 \, {\left (8 \, f^{3} x^{6} - 12 \, f^{3} x^{3} + 4 \, f^{3} + {\left (2 \, x^{6} - 3 \, x^{3} + 1\right )} e^{3}\right )}}\right ) + \frac {2}{3} \, {\left (2^{\frac {1}{3}} e - f\right )} {\rm weierstrassPInverse}\left (0, 4, x\right )\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/18*sqrt(3)*sqrt(2*2^(2/3)*f*e + 2*2^(1/3)*f^2 + e^2)*log((4*f^3*x^18 + 5760*f^3*x^15 + 69600*f^3*x^12 - 842
24*f^3*x^9 - 41472*f^3*x^6 + 61440*f^3*x^3 - 8192*f^3 + 4*sqrt(3)*(252*f^2*x^14 + 5328*f^2*x^11 - 9216*f^2*x^5
+ 4608*f^2*x^2 + (17*x^15 + 1058*x^12 + 2528*x^9 - 5408*x^6 + 2560*x^3 - 512)*e^2 - 2*(f*x^16 + 310*f*x^13 +
2332*f*x^10 - 2656*f*x^7 - 256*f*x^4 + 512*f*x)*e + 2^(2/3)*(2*f^2*x^16 + 620*f^2*x^13 + 4664*f^2*x^10 - 5312*
f^2*x^7 - 512*f^2*x^4 + 1024*f^2*x + 9*(7*x^14 + 148*x^11 - 256*x^5 + 128*x^2)*e^2 - (17*f*x^15 + 1058*f*x^12
+ 2528*f*x^9 - 5408*f*x^6 + 2560*f*x^3 - 512*f)*e) + 2^(1/3)*(34*f^2*x^15 + 2116*f^2*x^12 + 5056*f^2*x^9 - 108
16*f^2*x^6 + 5120*f^2*x^3 - 1024*f^2 + (x^16 + 310*x^13 + 2332*x^10 - 2656*x^7 - 256*x^4 + 512*x)*e^2 - 18*(7*
f*x^14 + 148*f*x^11 - 256*f*x^5 + 128*f*x^2)*e))*sqrt(x^3 - 1)*sqrt(2*2^(2/3)*f*e + 2*2^(1/3)*f^2 + e^2) + (x^
18 + 1440*x^15 + 17400*x^12 - 21056*x^9 - 10368*x^6 + 15360*x^3 - 2048)*e^3 + 24*2^(2/3)*(4*f^3*x^17 + 484*f^3
*x^14 + 1912*f^3*x^11 - 4576*f^3*x^8 + 2432*f^3*x^5 - 256*f^3*x^2 + (x^17 + 121*x^14 + 478*x^11 - 1144*x^8 + 6
08*x^5 - 64*x^2)*e^3) + 48*2^(1/3)*(20*f^3*x^16 + 704*f^3*x^13 + 332*f^3*x^10 - 2720*f^3*x^7 + 2176*f^3*x^4 -
512*f^3*x + (5*x^16 + 176*x^13 + 83*x^10 - 680*x^7 + 544*x^4 - 128*x)*e^3))/(x^18 - 24*x^15 + 240*x^12 - 1280*
x^9 + 3840*x^6 - 6144*x^3 + 4096)) + 2/3*(2^(1/3)*e - f)*weierstrassPInverse(0, 4, x), -1/9*sqrt(3)*sqrt(-2*2^
(2/3)*f*e - 2*2^(1/3)*f^2 - e^2)*arctan(1/6*sqrt(3)*(4*f^2*x^5 - 4*f^2*x^2 - (5*x^3 - 2)*e^2 - 2*(7*f*x^4 - 4*
f*x)*e + 2^(2/3)*(14*f^2*x^4 - 8*f^2*x + (x^5 - x^2)*e^2 + (5*f*x^3 - 2*f)*e) - 2^(1/3)*(10*f^2*x^3 - 4*f^2 -
(7*x^4 - 4*x)*e^2 + 2*(f*x^5 - f*x^2)*e))*sqrt(x^3 - 1)*sqrt(-2*2^(2/3)*f*e - 2*2^(1/3)*f^2 - e^2)/(8*f^3*x^6
- 12*f^3*x^3 + 4*f^3 + (2*x^6 - 3*x^3 + 1)*e^3)) + 2/3*(2^(1/3)*e - f)*weierstrassPInverse(0, 4, x)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {e}{x \sqrt {x^{3} - 1} - 2^{\frac {2}{3}} \sqrt {x^{3} - 1}}\, dx - \int \frac {f x}{x \sqrt {x^{3} - 1} - 2^{\frac {2}{3}} \sqrt {x^{3} - 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{1,[2]%%%} / %%%{%%{[2,0]:[1,0,0,-2]%%},[2]%%%} Error: Ba
d Argument

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {e+f\,x}{\sqrt {x^3-1}\,\left (x-2^{2/3}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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