\(\int \frac {x \sqrt {a+b x^3}}{2 (5-3 \sqrt {3}) a+b x^3} \, dx\) [519]

Optimal result

Integrand size = 33, antiderivative size = 738 \[ \int \frac {x \sqrt {a+b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\frac {2 \sqrt {a+b x^3}}{b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt [6]{a} \arctan \left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a+b x^3}}\right )}{\sqrt {2} b^{2/3}}-\frac {\sqrt [4]{3} \sqrt [6]{a} \arctan \left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a+b x^3}}\right )}{2 \sqrt {2} b^{2/3}}+\frac {3^{3/4} \sqrt [6]{a} \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a+b x^3}}\right )}{2 \sqrt {2} b^{2/3}}+\frac {\sqrt [6]{a} \text {arctanh}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt {a+b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{\sqrt {2} \sqrt [4]{3} b^{2/3}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \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}} E\left (\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 )}{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 {2 \sqrt {2} \sqrt [3]{a} \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}} \operatorname {EllipticF}\left (\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}} \] Output:

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

Mathematica [C] (verified)

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

Time = 10.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {a+b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\frac {x^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )}{\left (20-12 \sqrt {3}\right ) \sqrt {a+b x^3}} \] Input:

Integrate[(x*Sqrt[a + b*x^3])/(2*(5 - 3*Sqrt[3])*a + b*x^3),x]
 

Output:

(x^2*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, -1/2, 1, 5/3, -((b*x^3)/a), -((b*x^ 
3)/(10*a - 6*Sqrt[3]*a))])/((20 - 12*Sqrt[3])*Sqrt[a + b*x^3])
 

Rubi [A] (warning: unable to verify)

Time = 1.36 (sec) , antiderivative size = 806, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {984, 832, 759, 989, 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.

\(\displaystyle \int \frac {x \sqrt {a+b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx\)

\(\Big \downarrow \) 984

\(\displaystyle \int \frac {x}{\sqrt {b x^3+a}}dx-3 \left (3-2 \sqrt {3}\right ) a \int \frac {x}{\sqrt {b x^3+a} \left (b x^3+2 \left (5-3 \sqrt {3}\right ) a\right )}dx\)

\(\Big \downarrow \) 832

\(\displaystyle -\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}+\frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-3 \left (3-2 \sqrt {3}\right ) a \int \frac {x}{\sqrt {b x^3+a} \left (b x^3+2 \left (5-3 \sqrt {3}\right ) a\right )}dx\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-3 \left (3-2 \sqrt {3}\right ) a \int \frac {x}{\sqrt {b x^3+a} \left (b x^3+2 \left (5-3 \sqrt {3}\right ) a\right )}dx-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 989

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

\(\Big \downarrow \) 2416

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

Input:

Int[(x*Sqrt[a + b*x^3])/(2*(5 - 3*Sqrt[3])*a + b*x^3),x]
 

Output:

-3*(3 - 2*Sqrt[3])*a*(-1/3*((2 + Sqrt[3])*ArcTan[(3^(1/4)*a^(1/6)*((1 - Sq 
rt[3])*a^(1/3) - 2*b^(1/3)*x))/(Sqrt[2]*Sqrt[a + b*x^3])])/(Sqrt[2]*3^(1/4 
)*a^(5/6)*b^(2/3)) - ((2 + Sqrt[3])*ArcTan[(3^(1/4)*(1 + Sqrt[3])*a^(1/6)* 
(a^(1/3) + b^(1/3)*x))/(Sqrt[2]*Sqrt[a + b*x^3])])/(6*Sqrt[2]*3^(1/4)*a^(5 
/6)*b^(2/3)) + ((2 + Sqrt[3])*ArcTanh[(3^(1/4)*(1 - Sqrt[3])*a^(1/6)*(a^(1 
/3) + b^(1/3)*x))/(Sqrt[2]*Sqrt[a + b*x^3])])/(2*Sqrt[2]*3^(3/4)*a^(5/6)*b 
^(2/3)) + ((2 + Sqrt[3])*ArcTanh[((1 + Sqrt[3])*Sqrt[a + b*x^3])/(Sqrt[2]* 
3^(3/4)*Sqrt[a])])/(3*Sqrt[2]*3^(3/4)*a^(5/6)*b^(2/3))) + ((2*Sqrt[a + b*x 
^3])/(b^(1/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (3^(1/4)*Sqrt[2 - Sqr 
t[3]]*a^(1/3)*(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]*EllipticE[ArcSin[((1 - S 
qrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4* 
Sqrt[3]])/(b^(1/3)*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]))/b^(1/3) - (2*(1 - Sqrt[3])*Sqrt[2 + 
 Sqrt[3]]*a^(1/3)*(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])
 

Defintions of rubi rules used

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 984

Int[((x_)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol 
] :> Simp[b/d   Int[x*(a + b*x^n)^(p - 1), x], x] - Simp[(b*c - a*d)/d   In 
t[x*((a + b*x^n)^(p - 1)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d}, x] && 
NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, 1, 
 1, n, p, -1, x]
 

rule 989

Int[(x_)/(Sqrt[(a_) + (b_.)*(x_)^3]*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wi 
th[{q = Rt[b/a, 3], r = Simplify[(b*c - 10*a*d)/(6*a*d)]}, Simp[(-q)*(2 - r 
)*(ArcTan[(1 - r)*(Sqrt[a + b*x^3]/(Sqrt[2]*Rt[a, 2]*r^(3/2)))]/(3*Sqrt[2]* 
Rt[a, 2]*d*r^(3/2))), x] + (-Simp[q*(2 - r)*(ArcTan[Rt[a, 2]*Sqrt[r]*(1 + r 
)*((1 + q*x)/(Sqrt[2]*Sqrt[a + b*x^3]))]/(2*Sqrt[2]*Rt[a, 2]*d*r^(3/2))), x 
] - Simp[q*(2 - r)*(ArcTanh[Rt[a, 2]*Sqrt[r]*((1 + r - 2*q*x)/(Sqrt[2]*Sqrt 
[a + b*x^3]))]/(3*Sqrt[2]*Rt[a, 2]*d*Sqrt[r])), x] - Simp[q*(2 - r)*(ArcTan 
h[Rt[a, 2]*(1 - r)*Sqrt[r]*((1 + q*x)/(Sqrt[2]*Sqrt[a + b*x^3]))]/(6*Sqrt[2 
]*Rt[a, 2]*d*Sqrt[r])), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
 && EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0] && PosQ[a]
 

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 [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.62 (sec) , antiderivative size = 977, normalized size of antiderivative = 1.32

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(977\)
default	\(\text {Expression too large to display}\)	\(995\)

Input:

int(x*(b*x^3+a)^(1/2)/(2*(5-3*3^(1/2))*a+b*x^3),x,method=_RETURNVERBOSE)
 

Output:

-2/3*I*3^(1/2)/b*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b 
*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/( 
-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b* 
(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^( 
1/2)/(b*x^3+a)^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3 
))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^ 
2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/ 
2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+1/b*(-a*b^2)^(1 
/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b 
^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3 
/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)))+1/9*I/b^3*2^( 
1/2)*sum(1/_alpha*(-3+2*3^(1/2))*(-a*b^2)^(1/3)*(1/2*I*b*(2*x+1/b*((-a*b^2 
)^(1/3)-I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)*(b*(x-1/b*(-a*b^2 
)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))^(1/2)*(-1/2*I*b*(2* 
x+1/b*((-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)/(b* 
x^3+a)^(1/2)*(3*I*(-a*b^2)^(1/3)*_alpha*3^(1/2)*b+4*b^2*_alpha^2*3^(1/2)-3 
*I*3^(1/2)*(-a*b^2)^(2/3)-2*(-a*b^2)^(1/3)*_alpha*3^(1/2)*b+6*I*(-a*b^2)^( 
1/3)*_alpha*b+6*_alpha^2*b^2-2*3^(1/2)*(-a*b^2)^(2/3)-6*I*(-a*b^2)^(2/3)-3 
*(-a*b^2)^(1/3)*_alpha*b-3*(-a*b^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/ 
2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4931 vs. \(2 (519) = 1038\).

Time = 4.36 (sec) , antiderivative size = 4931, normalized size of antiderivative = 6.68 \[ \int \frac {x \sqrt {a+b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\text {Too large to display} \] Input:

integrate(x*(b*x^3+a)^(1/2)/(2*(5-3*3^(1/2))*a+b*x^3),x, algorithm="fricas 
")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {x \sqrt {a+b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\int \frac {x \sqrt {a + b x^{3}}}{- 6 \sqrt {3} a + 10 a + b x^{3}}\, dx \] Input:

integrate(x*(b*x**3+a)**(1/2)/(2*(5-3*3**(1/2))*a+b*x**3),x)
 

Output:

Integral(x*sqrt(a + b*x**3)/(-6*sqrt(3)*a + 10*a + b*x**3), x)
 

Maxima [F]

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

integrate(x*(b*x^3+a)^(1/2)/(2*(5-3*3^(1/2))*a+b*x^3),x, algorithm="maxima 
")
 

Output:

integrate(sqrt(b*x^3 + a)*x/(b*x^3 - 2*a*(3*sqrt(3) - 5)), x)
 

Giac [F]

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

integrate(x*(b*x^3+a)^(1/2)/(2*(5-3*3^(1/2))*a+b*x^3),x, algorithm="giac")
 

Output:

integrate(sqrt(b*x^3 + a)*x/(b*x^3 - 2*a*(3*sqrt(3) - 5)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \sqrt {a+b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\int \frac {x\,\sqrt {b\,x^3+a}}{b\,x^3-2\,a\,\left (3\,\sqrt {3}-5\right )} \,d x \] Input:

int((x*(a + b*x^3)^(1/2))/(b*x^3 - 2*a*(3*3^(1/2) - 5)),x)
 

Output:

int((x*(a + b*x^3)^(1/2))/(b*x^3 - 2*a*(3*3^(1/2) - 5)), x)
 

Reduce [F]

\[ \int \frac {x \sqrt {a+b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=-6 \sqrt {3}\, \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{-b^{2} x^{6}-20 a b \,x^{3}+8 a^{2}}d x \right ) a -\left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{4}}{-b^{2} x^{6}-20 a b \,x^{3}+8 a^{2}}d x \right ) b -10 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{-b^{2} x^{6}-20 a b \,x^{3}+8 a^{2}}d x \right ) a \] Input:

int(x*(b*x^3+a)^(1/2)/(2*(5-3*3^(1/2))*a+b*x^3),x)
 

Output:

 - 6*sqrt(3)*int((sqrt(a + b*x**3)*x)/(8*a**2 - 20*a*b*x**3 - b**2*x**6),x 
)*a - int((sqrt(a + b*x**3)*x**4)/(8*a**2 - 20*a*b*x**3 - b**2*x**6),x)*b 
- 10*int((sqrt(a + b*x**3)*x)/(8*a**2 - 20*a*b*x**3 - b**2*x**6),x)*a