3.2.34 \(\int \frac {e+f x}{((1-\sqrt {3}) \sqrt [3]{a}+\sqrt [3]{b} x) \sqrt {-a-b x^3}} \, dx\) [134]

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

[Out]

-1/3*(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))),2*I-I*3^(1/
2))*(b^(1/3)*e-a^(1/3)*f*(1+3^(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)*(1/2*6^(1/2)-1/2*2^(1/2))*3^(1/4)/a^(1/3)/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)-arctan(a^(1/6)*(a^(1/3)+b^(1/3)*x)*(-3+2*3^(1/2))^(1/2)/(-b*x^3-a)^(1/2)
)*(b^(1/3)*e-a^(1/3)*f*(1-3^(1/2)))/b^(2/3)/a^(1/2)/(-9+6*3^(1/2))^(1/2)

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Rubi [A]
time = 0.36, antiderivative size = 345, normalized size of antiderivative = 1.00, 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 = {2166, 225, 2165, 209} \begin {gather*} -\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 (\sqrt [3]{b} e-\left (1+\sqrt {3}\right ) \sqrt [3]{a} f\right ) F\left (\text {ArcSin}\left (\frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}\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 {\text {ArcTan}\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 ) \left (\sqrt [3]{b} e-\left (1-\sqrt {3}\right ) \sqrt [3]{a} f\right )}{\sqrt {3 \left (2 \sqrt {3}-3\right )} \sqrt {a} b^{2/3}} \end {gather*}

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 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 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 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 {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 ) \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 )}{\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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 11.02, size = 441, normalized size = 1.28 \begin {gather*} -\frac {4 \sqrt {\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (-\frac {i \sqrt [4]{3} f \left (\left ((-2-i)+\sqrt {3}\right ) \sqrt [3]{a}+\left ((1+2 i)-i \sqrt {3}\right ) \sqrt [3]{b} x\right ) \sqrt {i+\sqrt {3}-\frac {2 i \sqrt [3]{b} x}{\sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt {\frac {-2 i \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right )|\frac {1}{2} \left (1+i \sqrt {3}\right )\right )}{2 \sqrt {2}}+i \left (\sqrt [3]{b} e+\left (-1+\sqrt {3}\right ) \sqrt [3]{a} f\right ) \sqrt {\frac {-2 i \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\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 {-2 i \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\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*}

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))]*(((-1/2*I)*3^(1/4)*f*(((-2 - I) + Sqrt[3])*a^(1/3)
+ ((1 + 2*I) - I*Sqrt[3])*b^(1/3)*x)*Sqrt[I + Sqrt[3] - ((2*I)*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[((-2*
I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2])/Sqrt[2] + I*(b^(1/3)*e
+ (-1 + Sqrt[3])*a^(1/3)*f)*Sqrt[((-2*I)*a^(1/3) + (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[Sqr
t[((-2*I)*a^(1/3) + (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 \[\int \frac {f x +e}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )\right ) \sqrt {-b \,x^{3}-a}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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 = 0.00, 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)/(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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 44.15, size = 6561, normalized size = 19.02 \begin {gather*} \text {Too large to display} \end {gather*}

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]

[1/12*(4*sqrt(3)*a^(2/3)*sqrt(-b)*b*e*weierstrassPInverse(0, -4*a/b, x) + a*b*sqrt(-(2*sqrt(3)*a^(2/3)*b^(2/3)
*f^2 + 2*(sqrt(3)*b*f*e + 3*b*f*e)*a^(1/3) + (2*sqrt(3)*b*e^2 + 3*b*e^2)*b^(1/3))/a)*log(-(8*a^2*b^9*f^6*x^24
- 14720*a^3*b^8*f^6*x^21 + 538112*a^4*b^7*f^6*x^18 - 468992*a^5*b^6*f^6*x^15 + 4032512*a^6*b^5*f^6*x^12 + 1712
1280*a^7*b^4*f^6*x^9 + 24805376*a^8*b^3*f^6*x^6 + 8716288*a^9*b^2*f^6*x^3 + 229376*a^10*b*f^6 + 32*(72*a^2*b^8
*f^6*x^22 - 6768*a^3*b^7*f^6*x^19 + 36936*a^4*b^6*f^6*x^16 + 43776*a^5*b^5*f^6*x^13 + 350208*a^6*b^4*f^6*x^10
+ 787968*a^7*b^3*f^6*x^7 + 474624*a^8*b^2*f^6*x^4 + 36864*a^9*b*f^6*x - 9*(b^10*x^22 - 94*a*b^9*x^19 + 513*a^2
*b^8*x^16 + 608*a^3*b^7*x^13 + 4864*a^4*b^6*x^10 + 10944*a^5*b^5*x^7 + 6592*a^6*b^4*x^4 + 512*a^7*b^3*x)*e^6 +
180*(a*b^9*f^3*x^22 - 94*a^2*b^8*f^3*x^19 + 513*a^3*b^7*f^3*x^16 + 608*a^4*b^6*f^3*x^13 + 4864*a^5*b^5*f^3*x^
10 + 10944*a^6*b^4*f^3*x^7 + 6592*a^7*b^3*f^3*x^4 + 512*a^8*b^2*f^3*x)*e^3 - sqrt(3)*(40*a^2*b^8*f^6*x^22 - 40
40*a^3*b^7*f^6*x^19 + 17040*a^4*b^6*f^6*x^16 - 39424*a^5*b^5*f^6*x^13 - 229504*a^6*b^4*f^6*x^10 - 430080*a^7*b
^3*f^6*x^7 - 281600*a^8*b^2*f^6*x^4 - 20480*a^9*b*f^6*x - (5*b^10*x^22 - 505*a*b^9*x^19 + 2130*a^2*b^8*x^16 -
4928*a^3*b^7*x^13 - 28688*a^4*b^6*x^10 - 53760*a^5*b^5*x^7 - 35200*a^6*b^4*x^4 - 2560*a^7*b^3*x)*e^6 + 20*(5*a
*b^9*f^3*x^22 - 505*a^2*b^8*f^3*x^19 + 2130*a^3*b^7*f^3*x^16 - 4928*a^4*b^6*f^3*x^13 - 28688*a^5*b^5*f^3*x^10
- 53760*a^6*b^4*f^3*x^7 - 35200*a^7*b^3*f^3*x^4 - 2560*a^8*b^2*f^3*x)*e^3))*a^(2/3)*b^(1/3) - 8*(24*a^2*b^8*f^
6*x^23 - 8616*a^3*b^7*f^6*x^20 + 106560*a^4*b^6*f^6*x^17 - 153600*a^5*b^5*f^6*x^14 - 890880*a^6*b^4*f^6*x^11 -
2760192*a^7*b^3*f^6*x^8 - 2629632*a^8*b^2*f^6*x^5 - 491520*a^9*b*f^6*x^2 - 3*(b^10*x^23 - 359*a*b^9*x^20 + 44
40*a^2*b^8*x^17 - 6400*a^3*b^7*x^14 - 37120*a^4*b^6*x^11 - 115008*a^5*b^5*x^8 - 109568*a^6*b^4*x^5 - 20480*a^7
*b^3*x^2)*e^6 + 60*(a*b^9*f^3*x^23 - 359*a^2*b^8*f^3*x^20 + 4440*a^3*b^7*f^3*x^17 - 6400*a^4*b^6*f^3*x^14 - 37
120*a^5*b^5*f^3*x^11 - 115008*a^6*b^4*f^3*x^8 - 109568*a^7*b^3*f^3*x^5 - 20480*a^8*b^2*f^3*x^2)*e^3 - 2*sqrt(3
)*(8*a^2*b^8*f^6*x^23 - 2392*a^3*b^7*f^6*x^20 + 34080*a^4*b^6*f^6*x^17 + 12160*a^5*b^5*f^6*x^14 + 213760*a^6*b
^4*f^6*x^11 + 840192*a^7*b^3*f^6*x^8 + 745472*a^8*b^2*f^6*x^5 + 143360*a^9*b*f^6*x^2 - (b^10*x^23 - 299*a*b^9*
x^20 + 4260*a^2*b^8*x^17 + 1520*a^3*b^7*x^14 + 26720*a^4*b^6*x^11 + 105024*a^5*b^5*x^8 + 93184*a^6*b^4*x^5 + 1
7920*a^7*b^3*x^2)*e^6 + 20*(a*b^9*f^3*x^23 - 299*a^2*b^8*f^3*x^20 + 4260*a^3*b^7*f^3*x^17 + 1520*a^4*b^6*f^3*x
^14 + 26720*a^5*b^5*f^3*x^11 + 105024*a^6*b^4*f^3*x^8 + 93184*a^7*b^3*f^3*x^5 + 17920*a^8*b^2*f^3*x^2)*e^3))*a
^(1/3)*b^(2/3) - (b^11*x^24 - 1840*a*b^10*x^21 + 67264*a^2*b^9*x^18 - 58624*a^3*b^8*x^15 + 504064*a^4*b^7*x^12
+ 2140160*a^5*b^6*x^9 + 3100672*a^6*b^5*x^6 + 1089536*a^7*b^4*x^3 + 28672*a^8*b^3)*e^6 + 20*(a*b^10*f^3*x^24
- 1840*a^2*b^9*f^3*x^21 + 67264*a^3*b^8*f^3*x^18 - 58624*a^4*b^7*f^3*x^15 + 504064*a^5*b^6*f^3*x^12 + 2140160*
a^6*b^5*f^3*x^9 + 3100672*a^7*b^4*f^3*x^6 + 1089536*a^8*b^3*f^3*x^3 + 28672*a^9*b^2*f^3)*e^3 + 32*sqrt(3)*(280
*a^3*b^8*f^6*x^21 - 9128*a^4*b^7*f^6*x^18 + 20352*a^5*b^6*f^6*x^15 + 54080*a^6*b^5*f^6*x^12 + 316160*a^7*b^4*f
^6*x^9 + 445440*a^8*b^3*f^6*x^6 + 157696*a^9*b^2*f^6*x^3 + 4096*a^10*b*f^6 - (35*a*b^10*x^21 - 1141*a^2*b^9*x^
18 + 2544*a^3*b^8*x^15 + 6760*a^4*b^7*x^12 + 39520*a^5*b^6*x^9 + 55680*a^6*b^5*x^6 + 19712*a^7*b^4*x^3 + 512*a
^8*b^3)*e^6 + 20*(35*a^2*b^9*f^3*x^21 - 1141*a^3*b^8*f^3*x^18 + 2544*a^4*b^7*f^3*x^15 + 6760*a^5*b^6*f^3*x^12
+ 39520*a^6*b^5*f^3*x^9 + 55680*a^7*b^4*f^3*x^6 + 19712*a^8*b^3*f^3*x^3 + 512*a^9*b^2*f^3)*e^3) - 4*((104*a^2*
b^7*f^5*x^21 - 16720*a^3*b^6*f^5*x^18 + 158208*a^4*b^5*f^5*x^15 + 41728*a^5*b^4*f^5*x^12 + 1086976*a^6*b^3*f^5
*x^9 + 2798592*a^7*b^2*f^5*x^6 + 1138688*a^8*b*f^5*x^3 + 32768*a^9*f^5 - 2*(b^9*x^22 - 764*a*b^8*x^19 + 16860*
a^2*b^7*x^16 - 19792*a^3*b^6*x^13 + 42368*a^4*b^5*x^10 + 104448*a^5*b^4*x^7 + 90880*a^6*b^3*x^4 + 7168*a^7*b^2
*x)*e^5 - 24*(32*a*b^8*f*x^20 - 1869*a^2*b^7*f*x^17 + 5862*a^3*b^6*f*x^14 - 7280*a^4*b^5*f*x^11 - 1824*a^5*b^4
*f*x^8 - 7872*a^6*b^3*f*x^5 - 1408*a^7*b^2*f*x^2)*e^4 + 8*(32*a*b^8*f^2*x^21 - 5425*a^2*b^7*f^2*x^18 + 47184*a
^3*b^6*f^2*x^15 - 37256*a^4*b^5*f^2*x^12 - 16064*a^5*b^4*f^2*x^9 + 960*a^6*b^3*f^2*x^6 - 13312*a^7*b^2*f^2*x^3
+ 512*a^8*b*f^2)*e^3 + 2*(a*b^8*f^3*x^22 + 424*a^2*b^7*f^3*x^19 - 2256*a^3*b^6*f^3*x^16 + 82592*a^4*b^5*f^3*x
^13 + 614336*a^5*b^4*f^3*x^10 + 2178048*a^6*b^3*f^3*x^7 + 1853440*a^7*b^2*f^3*x^4 + 145408*a^8*b*f^3*x)*e^2 -
24*(13*a^2*b^7*f^4*x^20 - 696*a^3*b^6*f^4*x^17 + 3480*a^4*b^5*f^4*x^14 + 14336*a^5*b^4*f^4*x^11 + 104640*a^6*b
^3*f^4*x^8 + 144384*a^7*b^2*f^4*x^5 + 30208*a^8*b*f^4*x^2)*e - sqrt(3)*(56*a^2*b^7*f^5*x^21 - 10000*a^3*b^6*f^
5*x^18 + 79872*a^4*b^5*f^5*x^15 - 155648*a^5*b^4*f^5*x^12 - 660992*a^6*b^3*f^5*x^9 - 1551360*a^7*b^2*f^5*x^6 -
679936*a^8*b*f^5*x^3 - 16384*a^9*f^5 - (b^9*x^22 - 896*a*b^8*x^19 + 18984*a^2*b^7*x^16 - 31168*a^3*b^6*x^13 -
21184*a^4*b^5*x^10 - 125952*a^5*b^4*x^7 - 1049...

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

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