3.2.41 \(\int \frac {x}{((1-\sqrt {3}) \sqrt [3]{a}-\sqrt [3]{b} x) \sqrt {a-b x^3}} \, dx\) [141]

Optimal. Leaf size=286 \[ -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {a-b x^3}}\right )}{3^{3/4} \sqrt [6]{a} b^{2/3}}+\frac {2 \sqrt {\frac {7}{6}+\frac {2}{\sqrt {3}}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}} \]

[Out]

-1/3*arctanh(a^(1/6)*(a^(1/3)-b^(1/3)*x)*(-3+2*3^(1/2))^(1/2)/(-b*x^3+a)^(1/2))*2^(1/2)*3^(1/4)/a^(1/6)/b^(2/3
)+2/3*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1-3^(1/2)))/(-b^(1/3)*x+a^(1/3)*(1+3^(1/2))),I*3^(1/2
)+2*I)*(1/3*6^(1/2)+1/2*2^(1/2))*((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1+3^(1/2)))^2)^
(1/2)*3^(3/4)/b^(2/3)/(-b*x^3+a)^(1/2)/(a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)

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Rubi [A]
time = 0.31, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2166, 224, 2165, 212} \begin {gather*} \frac {2 \sqrt {\frac {7}{6}+\frac {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}} F\left (\text {ArcSin}\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 )}{\sqrt [4]{3} 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 {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {a-b x^3}}\right )}{3^{3/4} \sqrt [6]{a} b^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[2]*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*a^(1/6)*(a^(1/3) - b^(1/3)*x))/Sqrt[a - b*x^3]])/(3^(3/4)*a^(1/6)*b^(
2/3))) + (2*Sqrt[7/6 + 2/Sqrt[3]]*(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^(1/4)*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 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 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 2165

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 2166

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]

Rubi steps

\begin {align*} \int \frac {x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx &=\frac {\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}{6 \left (3+\sqrt {3}\right ) a b^{4/3}}-\frac {\left (2+\sqrt {3}\right ) \int \frac {1}{\sqrt {a-b x^3}} \, dx}{\left (3+\sqrt {3}\right ) \sqrt [3]{b}}\\ &=\frac {2 \sqrt {\frac {7}{6}+\frac {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}} 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 )}{\sqrt [4]{3} 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 (2 \sqrt [3]{a}\right ) \text {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 )}{\left (3+\sqrt {3}\right ) b^{2/3}}\\ &=-\frac {\sqrt {2} \tanh ^{-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 )}{3^{3/4} \sqrt [6]{a} b^{2/3}}+\frac {2 \sqrt {\frac {7}{6}+\frac {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}} 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 )}{\sqrt [4]{3} 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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.84, size = 454, normalized size = 1.59 \begin {gather*} -\frac {4 \sqrt {\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\frac {1}{2} \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 (-i+\sqrt {3}\right ) \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt {-\frac {i \left (2 \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right )|\frac {1}{2} \left (1+i \sqrt {3}\right )\right )+i \left (-1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt {-\frac {i \left (2 \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}} \sqrt {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+\frac {b^{2/3} x^2}{a^{2/3}}} \Pi \left (\frac {2 \sqrt {3}}{-3 i+(1+2 i) \sqrt {3}};\sin ^{-1}\left (\sqrt {-\frac {i \left (2 \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x\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 {a-b x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[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))]*(((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))]*Ellipt
icF[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*(-1 + Sqrt[3])*a^(1/3)*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]), Arc
Sin[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 = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )\right ) \sqrt {-b \,x^{3}+a}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x/(-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 = 0.00, 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(x/(-b^(1/3)*x+a^(1/3)*(1-3^(1/2)))/(-b*x^3+a)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.85, size = 1354, normalized size = 4.73 \begin {gather*} \left [\frac {\sqrt {2} a^{\frac {1}{3}} b^{\frac {4}{3}} \sqrt {\frac {\sqrt {3}}{a}} \log \left (\frac {b^{8} x^{24} + 1840 \, a b^{7} x^{21} + 67264 \, a^{2} b^{6} x^{18} + 58624 \, a^{3} b^{5} x^{15} + 504064 \, a^{4} b^{4} x^{12} - 2140160 \, a^{5} b^{3} x^{9} + 3100672 \, a^{6} b^{2} x^{6} - 1089536 \, a^{7} b x^{3} + 28672 \, a^{8} + 32 \, {\left (9 \, b^{7} x^{22} + 846 \, a b^{6} x^{19} + 4617 \, a^{2} b^{5} x^{16} - 5472 \, a^{3} b^{4} x^{13} + 43776 \, a^{4} b^{3} x^{10} - 98496 \, a^{5} b^{2} x^{7} + 59328 \, a^{6} b x^{4} - 4608 \, a^{7} x - \sqrt {3} {\left (5 \, b^{7} x^{22} + 505 \, a b^{6} x^{19} + 2130 \, a^{2} b^{5} x^{16} + 4928 \, a^{3} b^{4} x^{13} - 28688 \, a^{4} b^{3} x^{10} + 53760 \, a^{5} b^{2} x^{7} - 35200 \, a^{6} b x^{4} + 2560 \, a^{7} x\right )}\right )} a^{\frac {2}{3}} b^{\frac {1}{3}} + 8 \, {\left (3 \, b^{7} x^{23} + 1077 \, a b^{6} x^{20} + 13320 \, a^{2} b^{5} x^{17} + 19200 \, a^{3} b^{4} x^{14} - 111360 \, a^{4} b^{3} x^{11} + 345024 \, a^{5} b^{2} x^{8} - 328704 \, a^{6} b x^{5} + 61440 \, a^{7} x^{2} - 2 \, \sqrt {3} {\left (b^{7} x^{23} + 299 \, a b^{6} x^{20} + 4260 \, a^{2} b^{5} x^{17} - 1520 \, a^{3} b^{4} x^{14} + 26720 \, a^{4} b^{3} x^{11} - 105024 \, a^{5} b^{2} x^{8} + 93184 \, a^{6} b x^{5} - 17920 \, a^{7} x^{2}\right )}\right )} a^{\frac {1}{3}} b^{\frac {2}{3}} + 2 \, \sqrt {2} {\left ({\left (b^{7} x^{22} + 1160 \, a b^{6} x^{19} + 23232 \, a^{2} b^{5} x^{16} + 53920 \, a^{3} b^{4} x^{13} - 148288 \, a^{4} b^{3} x^{10} + 586752 \, a^{5} b^{2} x^{7} - 496640 \, a^{6} b x^{4} + 38912 \, a^{7} x - \sqrt {3} {\left (b^{7} x^{22} + 632 \, a b^{6} x^{19} + 14736 \, a^{2} b^{5} x^{16} + 8416 \, a^{3} b^{4} x^{13} + 105920 \, a^{4} b^{3} x^{10} - 334848 \, a^{5} b^{2} x^{7} + 286720 \, a^{6} b x^{4} - 22528 \, a^{7} x\right )}\right )} \sqrt {-b x^{3} + a} a^{\frac {2}{3}} b^{\frac {1}{3}} + 12 \, {\left (17 \, a b^{6} x^{20} + 1014 \, a^{2} b^{5} x^{17} + 2748 \, a^{3} b^{4} x^{14} + 9632 \, a^{4} b^{3} x^{11} - 36096 \, a^{5} b^{2} x^{8} + 53376 \, a^{6} b x^{5} - 11008 \, a^{7} x^{2} - 2 \, \sqrt {3} {\left (5 \, a b^{6} x^{20} + 285 \, a^{2} b^{5} x^{17} + 1038 \, a^{3} b^{4} x^{14} - 784 \, a^{4} b^{3} x^{11} + 11424 \, a^{5} b^{2} x^{8} - 15168 \, a^{6} b x^{5} + 3200 \, a^{7} x^{2}\right )}\right )} \sqrt {-b x^{3} + a} a^{\frac {1}{3}} b^{\frac {2}{3}} + 2 \, {\left (13 \, a b^{7} x^{21} + 2090 \, a^{2} b^{6} x^{18} + 19776 \, a^{3} b^{5} x^{15} - 5216 \, a^{4} b^{4} x^{12} + 135872 \, a^{5} b^{3} x^{9} - 349824 \, a^{6} b^{2} x^{6} + 142336 \, a^{7} b x^{3} - 4096 \, a^{8} - \sqrt {3} {\left (7 \, a b^{7} x^{21} + 1250 \, a^{2} b^{6} x^{18} + 9984 \, a^{3} b^{5} x^{15} + 19456 \, a^{4} b^{4} x^{12} - 82624 \, a^{5} b^{3} x^{9} + 193920 \, a^{6} b^{2} x^{6} - 84992 \, a^{7} b x^{3} + 2048 \, a^{8}\right )}\right )} \sqrt {-b x^{3} + a}\right )} \sqrt {\frac {\sqrt {3}}{a}} - 32 \, \sqrt {3} {\left (35 \, a b^{7} x^{21} + 1141 \, a^{2} b^{6} x^{18} + 2544 \, a^{3} b^{5} x^{15} - 6760 \, a^{4} b^{4} x^{12} + 39520 \, a^{5} b^{3} x^{9} - 55680 \, a^{6} b^{2} x^{6} + 19712 \, a^{7} b x^{3} - 512 \, a^{8}\right )}}{b^{8} x^{24} - 80 \, a b^{7} x^{21} + 2368 \, a^{2} b^{6} x^{18} - 30080 \, a^{3} b^{5} x^{15} + 121984 \, a^{4} b^{4} x^{12} + 240640 \, a^{5} b^{3} x^{9} + 151552 \, a^{6} b^{2} x^{6} + 40960 \, a^{7} b x^{3} + 4096 \, a^{8}}\right ) + 4 \, \sqrt {-b} b^{\frac {2}{3}} {\left (\sqrt {3} + 3\right )} {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right )}{12 \, b^{2}}, -\frac {\sqrt {2} a^{\frac {1}{3}} b^{\frac {4}{3}} \sqrt {-\frac {\sqrt {3}}{a}} \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {-b x^{3} + a} {\left (\sqrt {3} x - 3 \, x\right )} a^{\frac {2}{3}} b^{\frac {1}{3}} \sqrt {-\frac {\sqrt {3}}{a}} - \sqrt {2} \sqrt {-b x^{3} + a} {\left (\sqrt {3} x^{2} + 3 \, x^{2}\right )} a^{\frac {1}{3}} b^{\frac {2}{3}} \sqrt {-\frac {\sqrt {3}}{a}} - 4 \, \sqrt {3} \sqrt {2} \sqrt {-b x^{3} + a} a \sqrt {-\frac {\sqrt {3}}{a}}}{12 \, {\left (b x^{3} - a\right )}}\right ) - 2 \, \sqrt {-b} b^{\frac {2}{3}} {\left (\sqrt {3} + 3\right )} {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right )}{6 \, b^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(sqrt(2)*a^(1/3)*b^(4/3)*sqrt(sqrt(3)/a)*log((b^8*x^24 + 1840*a*b^7*x^21 + 67264*a^2*b^6*x^18 + 58624*a^
3*b^5*x^15 + 504064*a^4*b^4*x^12 - 2140160*a^5*b^3*x^9 + 3100672*a^6*b^2*x^6 - 1089536*a^7*b*x^3 + 28672*a^8 +
32*(9*b^7*x^22 + 846*a*b^6*x^19 + 4617*a^2*b^5*x^16 - 5472*a^3*b^4*x^13 + 43776*a^4*b^3*x^10 - 98496*a^5*b^2*
x^7 + 59328*a^6*b*x^4 - 4608*a^7*x - sqrt(3)*(5*b^7*x^22 + 505*a*b^6*x^19 + 2130*a^2*b^5*x^16 + 4928*a^3*b^4*x
^13 - 28688*a^4*b^3*x^10 + 53760*a^5*b^2*x^7 - 35200*a^6*b*x^4 + 2560*a^7*x))*a^(2/3)*b^(1/3) + 8*(3*b^7*x^23
+ 1077*a*b^6*x^20 + 13320*a^2*b^5*x^17 + 19200*a^3*b^4*x^14 - 111360*a^4*b^3*x^11 + 345024*a^5*b^2*x^8 - 32870
4*a^6*b*x^5 + 61440*a^7*x^2 - 2*sqrt(3)*(b^7*x^23 + 299*a*b^6*x^20 + 4260*a^2*b^5*x^17 - 1520*a^3*b^4*x^14 + 2
6720*a^4*b^3*x^11 - 105024*a^5*b^2*x^8 + 93184*a^6*b*x^5 - 17920*a^7*x^2))*a^(1/3)*b^(2/3) + 2*sqrt(2)*((b^7*x
^22 + 1160*a*b^6*x^19 + 23232*a^2*b^5*x^16 + 53920*a^3*b^4*x^13 - 148288*a^4*b^3*x^10 + 586752*a^5*b^2*x^7 - 4
96640*a^6*b*x^4 + 38912*a^7*x - sqrt(3)*(b^7*x^22 + 632*a*b^6*x^19 + 14736*a^2*b^5*x^16 + 8416*a^3*b^4*x^13 +
105920*a^4*b^3*x^10 - 334848*a^5*b^2*x^7 + 286720*a^6*b*x^4 - 22528*a^7*x))*sqrt(-b*x^3 + a)*a^(2/3)*b^(1/3) +
12*(17*a*b^6*x^20 + 1014*a^2*b^5*x^17 + 2748*a^3*b^4*x^14 + 9632*a^4*b^3*x^11 - 36096*a^5*b^2*x^8 + 53376*a^6
*b*x^5 - 11008*a^7*x^2 - 2*sqrt(3)*(5*a*b^6*x^20 + 285*a^2*b^5*x^17 + 1038*a^3*b^4*x^14 - 784*a^4*b^3*x^11 + 1
1424*a^5*b^2*x^8 - 15168*a^6*b*x^5 + 3200*a^7*x^2))*sqrt(-b*x^3 + a)*a^(1/3)*b^(2/3) + 2*(13*a*b^7*x^21 + 2090
*a^2*b^6*x^18 + 19776*a^3*b^5*x^15 - 5216*a^4*b^4*x^12 + 135872*a^5*b^3*x^9 - 349824*a^6*b^2*x^6 + 142336*a^7*
b*x^3 - 4096*a^8 - sqrt(3)*(7*a*b^7*x^21 + 1250*a^2*b^6*x^18 + 9984*a^3*b^5*x^15 + 19456*a^4*b^4*x^12 - 82624*
a^5*b^3*x^9 + 193920*a^6*b^2*x^6 - 84992*a^7*b*x^3 + 2048*a^8))*sqrt(-b*x^3 + a))*sqrt(sqrt(3)/a) - 32*sqrt(3)
*(35*a*b^7*x^21 + 1141*a^2*b^6*x^18 + 2544*a^3*b^5*x^15 - 6760*a^4*b^4*x^12 + 39520*a^5*b^3*x^9 - 55680*a^6*b^
2*x^6 + 19712*a^7*b*x^3 - 512*a^8))/(b^8*x^24 - 80*a*b^7*x^21 + 2368*a^2*b^6*x^18 - 30080*a^3*b^5*x^15 + 12198
4*a^4*b^4*x^12 + 240640*a^5*b^3*x^9 + 151552*a^6*b^2*x^6 + 40960*a^7*b*x^3 + 4096*a^8)) + 4*sqrt(-b)*b^(2/3)*(
sqrt(3) + 3)*weierstrassPInverse(0, 4*a/b, x))/b^2, -1/6*(sqrt(2)*a^(1/3)*b^(4/3)*sqrt(-sqrt(3)/a)*arctan(-1/1
2*(2*sqrt(2)*sqrt(-b*x^3 + a)*(sqrt(3)*x - 3*x)*a^(2/3)*b^(1/3)*sqrt(-sqrt(3)/a) - sqrt(2)*sqrt(-b*x^3 + a)*(s
qrt(3)*x^2 + 3*x^2)*a^(1/3)*b^(2/3)*sqrt(-sqrt(3)/a) - 4*sqrt(3)*sqrt(2)*sqrt(-b*x^3 + a)*a*sqrt(-sqrt(3)/a))/
(b*x^3 - a)) - 2*sqrt(-b)*b^(2/3)*(sqrt(3) + 3)*weierstrassPInverse(0, 4*a/b, x))/b^2]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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