\(\int \frac {(e x)^{2/3}}{\sqrt {a+b x} (4 a+b x)} \, dx\) [150]

Optimal result

Integrand size = 26, antiderivative size = 911 \[ \int \frac {(e x)^{2/3}}{\sqrt {a+b x} (4 a+b x)} \, dx =\text {Too large to display} \] Output:

6*e*(b*x+a)^(1/2)/b^(5/3)/((1+3^(1/2))*a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3) 
)-2/3*2^(1/3)*a^(1/6)*e^(2/3)*arctan(1/3*(b*x+a)^(1/2)*3^(1/2)/a^(1/2))*3^ 
(1/2)/b^(5/3)+2/3*2^(1/3)*a^(1/6)*e^(2/3)*arctan(3^(1/2)*a^(1/6)*(a^(1/3)* 
e^(1/3)+2^(1/3)*b^(1/3)*(e*x)^(1/3))/e^(1/3)/(b*x+a)^(1/2))*3^(1/2)/b^(5/3 
)-2/3*2^(1/3)*a^(1/6)*e^(2/3)*arctanh((b*x+a)^(1/2)/a^(1/2))/b^(5/3)+2*2^( 
1/3)*a^(1/6)*e^(2/3)*arctanh(a^(1/6)*(a^(1/3)*e^(1/3)-2^(1/3)*b^(1/3)*(e*x 
)^(1/3))/e^(1/3)/(b*x+a)^(1/2))/b^(5/3)-3*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2) 
)*a^(1/3)*e^(1/3)*(a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3))*((a^(2/3)*e^(2/3)- 
a^(1/3)*b^(1/3)*e^(1/3)*(e*x)^(1/3)+b^(2/3)*(e*x)^(2/3))/((1+3^(1/2))*a^(1 
/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3))^2)^(1/2)*EllipticE(((1-3^(1/2))*a^(1/3)*e 
^(1/3)+b^(1/3)*(e*x)^(1/3))/((1+3^(1/2))*a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/ 
3)),I*3^(1/2)+2*I)/b^(5/3)/(b*x+a)^(1/2)/(a^(1/3)*e^(1/3)*(a^(1/3)*e^(1/3) 
+b^(1/3)*(e*x)^(1/3))/((1+3^(1/2))*a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3))^2) 
^(1/2)+2*2^(1/2)*3^(3/4)*a^(1/3)*e^(1/3)*(a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1 
/3))*((a^(2/3)*e^(2/3)-a^(1/3)*b^(1/3)*e^(1/3)*(e*x)^(1/3)+b^(2/3)*(e*x)^( 
2/3))/((1+3^(1/2))*a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3))^2)^(1/2)*EllipticF 
(((1-3^(1/2))*a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3))/((1+3^(1/2))*a^(1/3)*e^ 
(1/3)+b^(1/3)*(e*x)^(1/3)),I*3^(1/2)+2*I)/b^(5/3)/(b*x+a)^(1/2)/(a^(1/3)*e 
^(1/3)*(a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3))/((1+3^(1/2))*a^(1/3)*e^(1/3)+ 
b^(1/3)*(e*x)^(1/3))^2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 11.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.07 \[ \int \frac {(e x)^{2/3}}{\sqrt {a+b x} (4 a+b x)} \, dx=\frac {3 x (e x)^{2/3} \sqrt {\frac {a+b x}{a}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {b x}{a},-\frac {b x}{4 a}\right )}{20 a \sqrt {a+b x}} \] Input:

Integrate[(e*x)^(2/3)/(Sqrt[a + b*x]*(4*a + b*x)),x]
 

Output:

(3*x*(e*x)^(2/3)*Sqrt[(a + b*x)/a]*AppellF1[5/3, 1/2, 1, 8/3, -((b*x)/a), 
-1/4*(b*x)/a])/(20*a*Sqrt[a + b*x])
 

Rubi [A] (warning: unable to verify)

Time = 0.79 (sec) , antiderivative size = 954, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {148, 27, 983, 832, 759, 986, 2416}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(e x)^{2/3}}{\sqrt {a+b x} (4 a+b x)} \, dx\)

\(\Big \downarrow \) 148

\(\displaystyle \frac {3 \int \frac {e (e x)^{4/3}}{\sqrt {a+b x} (4 a e+b x e)}d\sqrt [3]{e x}}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle 3 \int \frac {(e x)^{4/3}}{\sqrt {a+b x} (4 a e+b x e)}d\sqrt [3]{e x}\)

\(\Big \downarrow \) 983

\(\displaystyle 3 \left (\frac {\int \frac {\sqrt [3]{e x}}{\sqrt {a+b x}}d\sqrt [3]{e x}}{b}-\frac {4 a e \int \frac {\sqrt [3]{e x}}{\sqrt {a+b x} (4 a e+b x e)}d\sqrt [3]{e x}}{b}\right )\)

\(\Big \downarrow \) 832

\(\displaystyle 3 \left (\frac {\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}}{\sqrt {a+b x}}d\sqrt [3]{e x}}{\sqrt [3]{b}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e} \int \frac {1}{\sqrt {a+b x}}d\sqrt [3]{e x}}{\sqrt [3]{b}}}{b}-\frac {4 a e \int \frac {\sqrt [3]{e x}}{\sqrt {a+b x} (4 a e+b x e)}d\sqrt [3]{e x}}{b}\right )\)

\(\Big \downarrow \) 759

\(\displaystyle 3 \left (\frac {\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}}{\sqrt {a+b x}}d\sqrt [3]{e x}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \sqrt [3]{e} \left (\sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right ) \sqrt {\frac {a^{2/3} e^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{e} \sqrt [3]{e x}+b^{2/3} (e x)^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {a+b x} \sqrt {\frac {\sqrt [3]{a} \sqrt [3]{e} \left (\sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )^2}}}}{b}-\frac {4 a e \int \frac {\sqrt [3]{e x}}{\sqrt {a+b x} (4 a e+b x e)}d\sqrt [3]{e x}}{b}\right )\)

\(\Big \downarrow \) 986

\(\displaystyle 3 \left (\frac {\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}}{\sqrt {a+b x}}d\sqrt [3]{e x}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \sqrt [3]{e} \left (\sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right ) \sqrt {\frac {a^{2/3} e^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{e} \sqrt [3]{e x}+b^{2/3} (e x)^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {a+b x} \sqrt {\frac {\sqrt [3]{a} \sqrt [3]{e} \left (\sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )^2}}}}{b}-\frac {4 a e \left (\frac {\arctan \left (\frac {\sqrt {a+b x}}{\sqrt {3} \sqrt {a}}\right )}{3\ 2^{2/3} \sqrt {3} a^{5/6} b^{2/3} \sqrt [3]{e}}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{2} \sqrt [3]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{e} \sqrt {a+b x}}\right )}{3\ 2^{2/3} \sqrt {3} a^{5/6} b^{2/3} \sqrt [3]{e}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{9\ 2^{2/3} a^{5/6} b^{2/3} \sqrt [3]{e}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{a} \left (\sqrt [3]{a} \sqrt [3]{e}-\sqrt [3]{2} \sqrt [3]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{e} \sqrt {a+b x}}\right )}{3\ 2^{2/3} a^{5/6} b^{2/3} \sqrt [3]{e}}\right )}{b}\right )\)

\(\Big \downarrow \) 2416

\(\displaystyle 3 \left (\frac {\frac {\frac {2 e \sqrt {a+b x}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \sqrt [3]{e} \left (\sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right ) \sqrt {\frac {a^{2/3} e^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{e x} \sqrt [3]{e}+b^{2/3} (e x)^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {a+b x} \sqrt {\frac {\sqrt [3]{a} \sqrt [3]{e} \left (\sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )^2}}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \sqrt [3]{e} \left (\sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right ) \sqrt {\frac {a^{2/3} e^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{e x} \sqrt [3]{e}+b^{2/3} (e x)^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {a+b x} \sqrt {\frac {\sqrt [3]{a} \sqrt [3]{e} \left (\sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )^2}}}}{b}-\frac {4 a e \left (\frac {\arctan \left (\frac {\sqrt {a+b x}}{\sqrt {3} \sqrt {a}}\right )}{3\ 2^{2/3} \sqrt {3} a^{5/6} b^{2/3} \sqrt [3]{e}}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{2} \sqrt [3]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{e} \sqrt {a+b x}}\right )}{3\ 2^{2/3} \sqrt {3} a^{5/6} b^{2/3} \sqrt [3]{e}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{9\ 2^{2/3} a^{5/6} b^{2/3} \sqrt [3]{e}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{a} \left (\sqrt [3]{a} \sqrt [3]{e}-\sqrt [3]{2} \sqrt [3]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{e} \sqrt {a+b x}}\right )}{3\ 2^{2/3} a^{5/6} b^{2/3} \sqrt [3]{e}}\right )}{b}\right )\)

Input:

Int[(e*x)^(2/3)/(Sqrt[a + b*x]*(4*a + b*x)),x]
 

Output:

3*((-4*a*e*(ArcTan[Sqrt[a + b*x]/(Sqrt[3]*Sqrt[a])]/(3*2^(2/3)*Sqrt[3]*a^( 
5/6)*b^(2/3)*e^(1/3)) - ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3)*e^(1/3) + 2^(1/3) 
*b^(1/3)*(e*x)^(1/3)))/(e^(1/3)*Sqrt[a + b*x])]/(3*2^(2/3)*Sqrt[3]*a^(5/6) 
*b^(2/3)*e^(1/3)) + ArcTanh[Sqrt[a + b*x]/Sqrt[a]]/(9*2^(2/3)*a^(5/6)*b^(2 
/3)*e^(1/3)) - ArcTanh[(a^(1/6)*(a^(1/3)*e^(1/3) - 2^(1/3)*b^(1/3)*(e*x)^( 
1/3)))/(e^(1/3)*Sqrt[a + b*x])]/(3*2^(2/3)*a^(5/6)*b^(2/3)*e^(1/3))))/b + 
(((2*e*Sqrt[a + b*x])/(b^(1/3)*((1 + Sqrt[3])*a^(1/3)*e^(1/3) + b^(1/3)*(e 
*x)^(1/3))) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*e^(1/3)*(a^(1/3)*e^(1/3) 
+ b^(1/3)*(e*x)^(1/3))*Sqrt[(a^(2/3)*e^(2/3) - a^(1/3)*b^(1/3)*e^(1/3)*(e* 
x)^(1/3) + b^(2/3)*(e*x)^(2/3))/((1 + Sqrt[3])*a^(1/3)*e^(1/3) + b^(1/3)*( 
e*x)^(1/3))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3)*e^(1/3) + b^(1/3)*( 
e*x)^(1/3))/((1 + Sqrt[3])*a^(1/3)*e^(1/3) + b^(1/3)*(e*x)^(1/3))], -7 - 4 
*Sqrt[3]])/(b^(1/3)*Sqrt[a + b*x]*Sqrt[(a^(1/3)*e^(1/3)*(a^(1/3)*e^(1/3) + 
 b^(1/3)*(e*x)^(1/3)))/((1 + Sqrt[3])*a^(1/3)*e^(1/3) + b^(1/3)*(e*x)^(1/3 
))^2]))/b^(1/3) - (2*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*a^(1/3)*e^(1/3)*(a^(1 
/3)*e^(1/3) + b^(1/3)*(e*x)^(1/3))*Sqrt[(a^(2/3)*e^(2/3) - a^(1/3)*b^(1/3) 
*e^(1/3)*(e*x)^(1/3) + b^(2/3)*(e*x)^(2/3))/((1 + Sqrt[3])*a^(1/3)*e^(1/3) 
 + b^(1/3)*(e*x)^(1/3))^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3)*e^(1/3) 
 + b^(1/3)*(e*x)^(1/3))/((1 + Sqrt[3])*a^(1/3)*e^(1/3) + b^(1/3)*(e*x)^(1/ 
3))], -7 - 4*Sqrt[3]])/(3^(1/4)*b^(2/3)*Sqrt[a + b*x]*Sqrt[(a^(1/3)*e^(...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 148

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))^(p_.), 
x_] :> With[{k = Denominator[m]}, Simp[k/b   Subst[Int[x^(k*(m + 1) - 1)*(c 
 + d*(x^k/b))^n*(e + f*(x^k/b))^p, x], x, (b*x)^(1/k)], x]] /; FreeQ[{b, c, 
 d, e, f, n, p}, x] && FractionQ[m] && IntegerQ[p]
 

rule 759

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]*Sqrt[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 832

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && PosQ[a]
 

rule 983

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^( 
n_)), x_Symbol] :> Simp[e^n/b   Int[(e*x)^(m - n)*(c + d*x^n)^q, x], x] - S 
imp[a*(e^n/b)   Int[(e*x)^(m - n)*((c + d*x^n)^q/(a + b*x^n)), x], x] /; Fr 
eeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, 
m, 2*n - 1] && IntBinomialQ[a, b, c, d, e, m, n, -1, q, x]
 

rule 986

Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> Wi 
th[{q = Rt[d/c, 3]}, Simp[q*(ArcTanh[Sqrt[c + d*x^3]/Rt[c, 2]]/(9*2^(2/3)*b 
*Rt[c, 2])), x] + (-Simp[q*(ArcTanh[Rt[c, 2]*((1 - 2^(1/3)*q*x)/Sqrt[c + d* 
x^3])]/(3*2^(2/3)*b*Rt[c, 2])), x] + Simp[q*(ArcTan[Sqrt[c + d*x^3]/(Sqrt[3 
]*Rt[c, 2])]/(3*2^(2/3)*Sqrt[3]*b*Rt[c, 2])), x] - Simp[q*(ArcTan[Sqrt[3]*R 
t[c, 2]*((1 + 2^(1/3)*q*x)/Sqrt[c + d*x^3])]/(3*2^(2/3)*Sqrt[3]*b*Rt[c, 2]) 
), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[4*b*c - a*d, 
0] && PosQ[c]
 

rule 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S 
imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt 
[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) 
*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

\[\int \frac {\left (e x \right )^{\frac {2}{3}}}{\sqrt {b x +a}\, \left (b x +4 a \right )}d x\]

Input:

int((e*x)^(2/3)/(b*x+a)^(1/2)/(b*x+4*a),x)
 

Output:

int((e*x)^(2/3)/(b*x+a)^(1/2)/(b*x+4*a),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(e x)^{2/3}}{\sqrt {a+b x} (4 a+b x)} \, dx=\text {Timed out} \] Input:

integrate((e*x)^(2/3)/(b*x+a)^(1/2)/(b*x+4*a),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {(e x)^{2/3}}{\sqrt {a+b x} (4 a+b x)} \, dx=\int \frac {\left (e x\right )^{\frac {2}{3}}}{\sqrt {a + b x} \left (4 a + b x\right )}\, dx \] Input:

integrate((e*x)**(2/3)/(b*x+a)**(1/2)/(b*x+4*a),x)
 

Output:

Integral((e*x)**(2/3)/(sqrt(a + b*x)*(4*a + b*x)), x)
 

Maxima [F]

\[ \int \frac {(e x)^{2/3}}{\sqrt {a+b x} (4 a+b x)} \, dx=\int { \frac {\left (e x\right )^{\frac {2}{3}}}{{\left (b x + 4 \, a\right )} \sqrt {b x + a}} \,d x } \] Input:

integrate((e*x)^(2/3)/(b*x+a)^(1/2)/(b*x+4*a),x, algorithm="maxima")
 

Output:

integrate((e*x)^(2/3)/((b*x + 4*a)*sqrt(b*x + a)), x)
 

Giac [F]

\[ \int \frac {(e x)^{2/3}}{\sqrt {a+b x} (4 a+b x)} \, dx=\int { \frac {\left (e x\right )^{\frac {2}{3}}}{{\left (b x + 4 \, a\right )} \sqrt {b x + a}} \,d x } \] Input:

integrate((e*x)^(2/3)/(b*x+a)^(1/2)/(b*x+4*a),x, algorithm="giac")
 

Output:

integrate((e*x)^(2/3)/((b*x + 4*a)*sqrt(b*x + a)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{2/3}}{\sqrt {a+b x} (4 a+b x)} \, dx=\int \frac {{\left (e\,x\right )}^{2/3}}{\left (4\,a+b\,x\right )\,\sqrt {a+b\,x}} \,d x \] Input:

int((e*x)^(2/3)/((4*a + b*x)*(a + b*x)^(1/2)),x)
 

Output:

int((e*x)^(2/3)/((4*a + b*x)*(a + b*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {(e x)^{2/3}}{\sqrt {a+b x} (4 a+b x)} \, dx=e^{\frac {2}{3}} \left (\int \frac {x^{\frac {2}{3}} \sqrt {b x +a}}{b^{2} x^{2}+5 a b x +4 a^{2}}d x \right ) \] Input:

int((e*x)^(2/3)/(b*x+a)^(1/2)/(b*x+4*a),x)
 

Output:

e**(2/3)*int((x**(2/3)*sqrt(a + b*x))/(4*a**2 + 5*a*b*x + b**2*x**2),x)