3.2.31 \(\int \frac {e+f x}{((1-\sqrt {3}) \sqrt [3]{a}+\sqrt [3]{b} x) \sqrt {a+b x^3}} \, dx\) [131]

Optimal. Leaf size=332 \[ -\frac {\left (\sqrt [3]{b} e-\left (1-\sqrt {3}\right ) \sqrt [3]{a} f\right ) \tanh ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {3 \left (-3+2 \sqrt {3}\right )} \sqrt {a} b^{2/3}}-\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{b} e-\left (1+\sqrt {3}\right ) \sqrt [3]{a} f\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]

[Out]

-arctanh(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)-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))),I*3^(1/2)+2*I)*(b^(1/3)*e-a^(1/3)*f*(1+3^(1/2)))*(1/2*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^(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)

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Rubi [A]
time = 0.39, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2166, 224, 2165, 212} \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 {\left (\sqrt [3]{b} e-\left (1-\sqrt {3}\right ) \sqrt [3]{a} f\right ) \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 )}{\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)*ArcTanh[(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 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 {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 ) \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 )}{\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 = 11.26, size = 438, normalized size = 1.32 \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 = 43.68, size = 6491, normalized size = 19.55 \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)*b^(3/2)*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 - 1
4720*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 + 1712128
0*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 + 7
87968*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 + 18
0*(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 - 4040*
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 - 492
8*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 - 5
3760*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 - 27
60192*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 + 4440*
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 - 37120
*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^2
0 + 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 + 1792
0*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) - 4*sqrt(b*x^3 + a)*((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 + 41
728*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 + 9
60*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 - 2
256*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 - 104960*a^6*b^3*x^4 - 8192
*a^7*b^2*x)*e^5 - 12*(37*a*b^8*f*x^20 - 2154*a^2*b^7*f*x^17 + 6900*a^3*b^6*f*x^14 - 6496*a^4*b^5*f*x^11 + 9600
*a^5*b^4*f*x^8 + 7296*a^6*b^3*f*x^5 + 1792*a^7*b^2*f*x^2)*e^4 + 4*(37*a*b^8*f^2*x^21 - 6260*a^2*b^7*f^2*x^18 +
54624*a^3*b^6*f^2*x^15 - 40816*a^4*b^5*f^2*x^12 - 2752*a^5*b^4*f^2*x^9 + 39936*a^6*b^3*f^2*x^6 + 1024*a^7*b^2
*f^2*x^3 + 1024*a^8*b*f^2)*e^3 - 2*(a*b^8*f^3*x^22 - 104*a^2*b^7*f^3*x^19 + 6240*a^3*b^6*f^3*x^16 + 37088*a^4*
b^5*f^3*x^13 + 360128*a^5*b^4*f^3*x^10 + 1256448*a^6*b^3*f^3*x^7 + 1070080*a^7*b^2*f^3*x^4 + 83968*a^8*b*f^3*x
)*e^2 - 24*(7*a^2*b^7*f^4*x^20 - 444*a^3*b^6*f^4*x^17 + 672*a^4*b^5*f^4*x^14 - 11200*a^5*b^4*f^4*x^11 - 58944*
a^6*b^3*f^4*x^8 - 83712*a^7*b^2*f^4*x^5 - 17408*a^8*b*f^4*x^2)*e))*a^(2/3)*b^(2/3) - 2*(2*a^2*b^8*f^5*x^22 - 2
320*a^3*b^7*f^5*x^19 + 46464*a^4*b^6*f^5*x^16 - 107840*a^5*b^5*f^5*x^13 - 296576*a^6*b^4*f^5*x^10 - 1173504*a^
7*b^3*f^5*x^7 - 993280*a^8*b^2*f^5*x^4 - 77824*...

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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]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {e+f\,x}{\sqrt {b\,x^3+a}\,\left (b^{1/3}\,x-a^{1/3}\,\left (\sqrt {3}-1\right )\right )} \,d x \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]

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

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