3.1.55 \(\int \frac {2+3 x}{(2^{2/3}+x) \sqrt {-1-x^3}} \, dx\) [55]

Optimal. Leaf size=169 \[ \frac {2 \left (2-3\ 2^{2/3}\right ) \tanh ^{-1}\left (\frac {\sqrt {3} \left (1+\sqrt [3]{2} x\right )}{\sqrt {-1-x^3}}\right )}{3 \sqrt {3}}+\frac {2 \left (3+2 \sqrt [3]{2}\right ) \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \]

[Out]

2/9*(2-3*2^(2/3))*arctanh((1+2^(1/3)*x)*3^(1/2)/(-x^3-1)^(1/2))*3^(1/2)+2/9*(3+2*2^(1/3))*(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 = 0.15, antiderivative size = 169, 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 \left (3+2 \sqrt [3]{2}\right ) \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} 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 \left (2-3\ 2^{2/3}\right ) \tanh ^{-1}\left (\frac {\sqrt {3} \left (\sqrt [3]{2} x+1\right )}{\sqrt {-x^3-1}}\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2 - 3*2^(2/3))*ArcTanh[(Sqrt[3]*(1 + 2^(1/3)*x))/Sqrt[-1 - x^3]])/(3*Sqrt[3]) + (2*(3 + 2*2^(1/3))*Sqrt[2
- Sqrt[3]]*(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 {2+3 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx &=\frac {1}{3} \left (-3+\sqrt [3]{2}\right ) \int \frac {2^{2/3}-2 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx+\frac {1}{3} \left (3+2 \sqrt [3]{2}\right ) \int \frac {1}{\sqrt {-1-x^3}} \, dx\\ &=\frac {2 \left (3+2 \sqrt [3]{2}\right ) \sqrt {2-\sqrt {3}} (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 (2-3\ 2^{2/3}\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 (2-3\ 2^{2/3}\right ) \tanh ^{-1}\left (\frac {\sqrt {3} \left (1+\sqrt [3]{2} x\right )}{\sqrt {-1-x^3}}\right )}{3 \sqrt {3}}+\frac {2 \left (3+2 \sqrt [3]{2}\right ) \sqrt {2-\sqrt {3}} (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 = 20.20, size = 338, normalized size = 2.00 \begin {gather*} \frac {2 \sqrt [6]{2} \sqrt {\frac {i (1+x)}{3 i+\sqrt {3}}} \left (3 \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 )-4 \sqrt {3} \left (-3+\sqrt [3]{2}\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[(2 + 3*x)/((2^(2/3) + x)*Sqrt[-1 - x^3]),x]

[Out]

(2*2^(1/6)*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(3*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])] - 4*Sqrt[3]*(-3 + 2^(1/3))*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]/(S
qrt[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])

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Maple [A]
time = 1.94, size = 253, normalized size = 1.50

method	result	size
default	\(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}-1}}-\frac {2 i \left (2-3 \,2^{\frac {2}{3}}\right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{2^{\frac {2}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (2^{\frac {2}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}\)	\(253\)
elliptic	\(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}-1}}-\frac {2 i \left (2-3 \,2^{\frac {2}{3}}\right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{2^{\frac {2}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (2^{\frac {2}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}\)	\(253\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((1+x)/(3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x-1/2+1/2*I*3^(1/2
))*3^(1/2))^(1/2)/(-x^3-1)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(3/2
+1/2*I*3^(1/2)))^(1/2))-2/3*I*(2-3*2^(2/3))*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((1+x)/(3/2+1/2*I*
3^(1/2)))^(1/2)*(-I*(x-1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^3-1)^(1/2)/(2^(2/3)+1/2+1/2*I*3^(1/2))*EllipticPi
(1/3*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(2^(2/3)+1/2+1/2*I*3^(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((2+3*x)/(2^(2/3)+x)/(-x^3-1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (130) = 260\).
time = 0.18, size = 415, normalized size = 2.46 \begin {gather*} \frac {1}{18} \, \sqrt {3} \sqrt {-12 \cdot 2^{\frac {2}{3}} + 18 \cdot 2^{\frac {1}{3}} + 4} \log \left (\frac {25 \, x^{18} - 36000 \, x^{15} + 435000 \, x^{12} + 526400 \, x^{9} - 259200 \, x^{6} - 384000 \, x^{3} + 2 \, \sqrt {3} {\left (6 \, x^{16} - 34 \, x^{15} + 1134 \, x^{14} - 1860 \, x^{13} + 2116 \, x^{12} - 23976 \, x^{11} + 13992 \, x^{10} - 5056 \, x^{9} + 15936 \, x^{7} - 10816 \, x^{6} + 41472 \, x^{5} - 1536 \, x^{4} - 5120 \, x^{3} + 20736 \, x^{2} + 3 \cdot 2^{\frac {2}{3}} {\left (3 \, x^{16} - 17 \, x^{15} + 42 \, x^{14} - 930 \, x^{13} + 1058 \, x^{12} - 888 \, x^{11} + 6996 \, x^{10} - 2528 \, x^{9} + 7968 \, x^{7} - 5408 \, x^{6} + 1536 \, x^{5} - 768 \, x^{4} - 2560 \, x^{3} + 768 \, x^{2} - 1536 \, x - 512\right )} + 2^{\frac {1}{3}} {\left (2 \, x^{16} - 153 \, x^{15} + 378 \, x^{14} - 620 \, x^{13} + 9522 \, x^{12} - 7992 \, x^{11} + 4664 \, x^{10} - 22752 \, x^{9} + 5312 \, x^{7} - 48672 \, x^{6} + 13824 \, x^{5} - 512 \, x^{4} - 23040 \, x^{3} + 6912 \, x^{2} - 1024 \, x - 4608\right )} - 3072 \, x - 1024\right )} \sqrt {-x^{3} - 1} \sqrt {-12 \cdot 2^{\frac {2}{3}} + 18 \cdot 2^{\frac {1}{3}} + 4} - 600 \cdot 2^{\frac {2}{3}} {\left (x^{17} - 121 \, x^{14} + 478 \, x^{11} + 1144 \, x^{8} + 608 \, x^{5} + 64 \, x^{2}\right )} + 1200 \cdot 2^{\frac {1}{3}} {\left (5 \, x^{16} - 176 \, x^{13} + 83 \, x^{10} + 680 \, x^{7} + 544 \, x^{4} + 128 \, x\right )} - 51200}{x^{18} + 24 \, x^{15} + 240 \, x^{12} + 1280 \, x^{9} + 3840 \, x^{6} + 6144 \, x^{3} + 4096}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*sqrt(3)*sqrt(-12*2^(2/3) + 18*2^(1/3) + 4)*log((25*x^18 - 36000*x^15 + 435000*x^12 + 526400*x^9 - 259200*
x^6 - 384000*x^3 + 2*sqrt(3)*(6*x^16 - 34*x^15 + 1134*x^14 - 1860*x^13 + 2116*x^12 - 23976*x^11 + 13992*x^10 -
5056*x^9 + 15936*x^7 - 10816*x^6 + 41472*x^5 - 1536*x^4 - 5120*x^3 + 20736*x^2 + 3*2^(2/3)*(3*x^16 - 17*x^15
+ 42*x^14 - 930*x^13 + 1058*x^12 - 888*x^11 + 6996*x^10 - 2528*x^9 + 7968*x^7 - 5408*x^6 + 1536*x^5 - 768*x^4
- 2560*x^3 + 768*x^2 - 1536*x - 512) + 2^(1/3)*(2*x^16 - 153*x^15 + 378*x^14 - 620*x^13 + 9522*x^12 - 7992*x^1
1 + 4664*x^10 - 22752*x^9 + 5312*x^7 - 48672*x^6 + 13824*x^5 - 512*x^4 - 23040*x^3 + 6912*x^2 - 1024*x - 4608)
- 3072*x - 1024)*sqrt(-x^3 - 1)*sqrt(-12*2^(2/3) + 18*2^(1/3) + 4) - 600*2^(2/3)*(x^17 - 121*x^14 + 478*x^11
+ 1144*x^8 + 608*x^5 + 64*x^2) + 1200*2^(1/3)*(5*x^16 - 176*x^13 + 83*x^10 + 680*x^7 + 544*x^4 + 128*x) - 5120
0)/(x^18 + 24*x^15 + 240*x^12 + 1280*x^9 + 3840*x^6 + 6144*x^3 + 4096))

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

[Out]

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

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

[Out]

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

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

[Out]

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

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