3.1.56 \(\int \frac {e+f x}{(2^{2/3}+x) \sqrt {1+x^3}} \, dx\) [56]

Optimal. Leaf size=159 \[ \frac {2 \left (e-2^{2/3} f\right ) \tan ^{-1}\left (\frac {\sqrt {3} \left (1+\sqrt [3]{2} x\right )}{\sqrt {1+x^3}}\right )}{3 \sqrt {3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{2} e+f\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \]

[Out]

2/9*(e-2^(2/3)*f)*arctan((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)),I*3^(1/2)+2*I)*(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)

________________________________________________________________________________________

Rubi [A]
time = 0.16, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2164, 224, 2162, 209} \begin {gather*} \frac {2 \sqrt {2+\sqrt {3}} (x+1) \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 {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {3} \left (\sqrt [3]{2} x+1\right )}{\sqrt {x^3+1}}\right ) \left (e-2^{2/3} f\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)*ArcTan[(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 209

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

Rule 224

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] && PosQ[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 &=\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 {1}{3} \left (\sqrt [3]{2} e+f\right ) \int \frac {1}{\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 ) \tan ^{-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*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 20.31, size = 340, normalized size = 2.14 \begin {gather*} \frac {2 \sqrt [6]{2} \sqrt {\frac {i (1+x)}{3 i+\sqrt {3}}} \left (f \sqrt {-i+\sqrt {3}+2 i x} \left (-6-3 \sqrt [3]{2}-2 i \sqrt {3}+i \sqrt [3]{2} \sqrt {3}+\left (3 \sqrt [3]{2}+4 i \sqrt {3}+i \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])]*(f*Sqrt[-I + Sqrt[3] + (2*I)*x]*(-6 - 3*2^(1/3) - (2*I)*Sqrt[3] +
I*2^(1/3)*Sqrt[3] + (3*2^(1/3) + (4*I)*Sqrt[3] + I*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 + Sqrt[3] -
(2*I)*x]*Sqrt[1 + x^3])

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (126 ) = 252\).
time = 0.29, size = 264, normalized size = 1.66

method	result	size
default	\(\frac {2 f \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 \left (e -2^{\frac {2}{3}} f \right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{2^{\frac {2}{3}}-1}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}\, \left (2^{\frac {2}{3}}-1\right )}\)	\(264\)
elliptic	\(\frac {2 f \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 \left (e -2^{\frac {2}{3}} f \right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{2^{\frac {2}{3}}-1}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}\, \left (2^{\frac {2}{3}}-1\right )}\)	\(264\)

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))

________________________________________________________________________________________

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)

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.26, size = 949, normalized size = 5.97 \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 + 84
224*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 + 10
816*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 +
608*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)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e + f x}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 2^{\frac {2}{3}}\right )}\, 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 + f*x)/(sqrt((x + 1)*(x**2 - x + 1))*(x + 2**(2/3))), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not 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="giac")

[Out]

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

________________________________________________________________________________________

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)

________________________________________________________________________________________