\(\int \frac {x \sqrt {-a+b x^3}}{-2 (5+3 \sqrt {3}) a+b x^3} \, dx\) [517]

Optimal result

Integrand size = 35, antiderivative size = 774 \[ \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} \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 [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 {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)+1/4*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)+1/2*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^(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) 
,2*I-I*3^(1/2))/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),2*I-I*3^(1/2))*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 = 10.10 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {-a+b x^3}}{-2 \left (5+3 \sqrt {3}\right ) a+b x^3} \, dx=-\frac {x^2 \sqrt {-a+b x^3} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {1}{2},1,\frac {5}{3},\frac {b x^3}{a},\frac {b x^3}{10 a+6 \sqrt {3} a}\right )}{4 \left (5+3 \sqrt {3}\right ) a \sqrt {1-\frac {b x^3}{a}}} \] Input:

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

Output:

-1/4*(x^2*Sqrt[-a + b*x^3]*AppellF1[2/3, -1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/ 
(10*a + 6*Sqrt[3]*a)])/((5 + 3*Sqrt[3])*a*Sqrt[1 - (b*x^3)/a])
 

Rubi [A] (warning: unable to verify)

Time = 1.42 (sec) , antiderivative size = 852, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {984, 25, 833, 760, 990, 2418}

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.

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

\(\Big \downarrow \) 984

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 833

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

\(\Big \downarrow \) 760

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

\(\Big \downarrow \) 990

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

\(\Big \downarrow \) 2418

\(\displaystyle -3 \left (3+2 \sqrt {3}\right ) a \left (-\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 {b x^3-a}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (2 \sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\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 ) \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 {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 (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\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 )}{\sqrt [3]{b} \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 {b x^3-a}}}{\sqrt [3]{b}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\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 {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*(-1/6*((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])])/(Sqrt[2]*3^(1/4) 
*a^(5/6)*b^(2/3)) - ((2 - Sqrt[3])*ArcTan[(3^(1/4)*a^(1/6)*((1 + Sqrt[3])* 
a^(1/3) + 2*b^(1/3)*x))/(Sqrt[2]*Sqrt[-a + b*x^3])])/(3*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 
+ 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]*EllipticE[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]])/(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*Sqrt[2 - Sqrt 
[3]]*(1 + 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[A 
rcSin[((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 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 760

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 
] && NegQ[a]
 

rule 833

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] && NegQ[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 990

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)*( 
ArcTanh[(1 - r)*(Sqrt[a + b*x^3]/(Sqrt[2]*Rt[-a, 2]*r^(3/2)))]/(3*Sqrt[2]*R 
t[-a, 2]*d*r^(3/2))), x] + (-Simp[q*(2 - r)*(ArcTanh[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)*(ArcTan[Rt[-a, 2]*Sqrt[r]*((1 + r - 2*q*x)/(Sqrt[2]*S 
qrt[a + b*x^3]))]/(3*Sqrt[2]*Rt[-a, 2]*d*Sqrt[r])), x] - Simp[q*(2 - r)*(Ar 
cTan[Rt[-a, 2]*(1 - r)*Sqrt[r]*((1 + q*x)/(Sqrt[2]*Sqrt[a + b*x^3]))]/(6*Sq 
rt[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] && NegQ[a]
 

rule 2418

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 - S 
qrt[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] && NegQ[a] && 
 EqQ[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 = 2.03 (sec) , antiderivative size = 926, normalized size of antiderivative = 1.20

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(926\)
default	\(\text {Expression too large to display}\)	\(944\)

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*(I*3^(1/2)*(a*b^2)^(1/3)+(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*(-I*3^(1/2)*(a*b^2)^(1/3 
)+(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*(a*b^2)^(2/3)*3^(1/2)-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*(a*b^ 
2)^(2/3)*3^(1/2)+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)^(1/3))^(1/2),1/6/b*(-2*I*(a*b^2)^(1/3)*_alp...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4963 vs. \(2 (541) = 1082\).

Time = 4.25 (sec) , antiderivative size = 4963, normalized size of antiderivative = 6.41 \[ \int \frac {x \sqrt {-a+b x^3}}{-2 \left (5+3 \sqrt {3}\right ) a+b x^3} \, dx=\text {Too large to display} \] Input:

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

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="maxim 
a")
                                                                                    
                                                                                    
 

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

Output:

int((x*(b*x^3 - a)^(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