\(\int \frac {x \sqrt {c+d x^3}}{4 c+d x^3} \, dx\) [440]

Optimal result

Integrand size = 24, antiderivative size = 659 \[ \int \frac {x \sqrt {c+d x^3}}{4 c+d x^3} \, dx=\frac {2 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{2^{2/3} \sqrt {3} d^{2/3}}-\frac {\sqrt [6]{c} \arctan \left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{2^{2/3} \sqrt {3} d^{2/3}}+\frac {\sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{2^{2/3} d^{2/3}}-\frac {\sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3\ 2^{2/3} d^{2/3}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {2 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:

2*(d*x^3+c)^(1/2)/d^(2/3)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)+1/6*c^(1/6)*arct 
an(3^(1/2)*c^(1/6)*(c^(1/3)+2^(1/3)*d^(1/3)*x)/(d*x^3+c)^(1/2))*2^(1/3)*3^ 
(1/2)/d^(2/3)-1/6*c^(1/6)*arctan(1/3*(d*x^3+c)^(1/2)*3^(1/2)/c^(1/2))*2^(1 
/3)*3^(1/2)/d^(2/3)+1/2*c^(1/6)*arctanh(c^(1/6)*(c^(1/3)-2^(1/3)*d^(1/3)*x 
)/(d*x^3+c)^(1/2))*2^(1/3)/d^(2/3)-1/6*c^(1/6)*arctanh((d*x^3+c)^(1/2)/c^( 
1/2))*2^(1/3)/d^(2/3)-3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*c^(1/3)*(c^(1/3)+d 
^(1/3)*x)*((c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/((1+3^(1/2))*c^(1/3)+d^ 
(1/3)*x)^2)^(1/2)*EllipticE(((1-3^(1/2))*c^(1/3)+d^(1/3)*x)/((1+3^(1/2))*c 
^(1/3)+d^(1/3)*x),I*3^(1/2)+2*I)/d^(2/3)/(c^(1/3)*(c^(1/3)+d^(1/3)*x)/((1+ 
3^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2)/(d*x^3+c)^(1/2)+2/3*2^(1/2)*c^(1/3)*( 
c^(1/3)+d^(1/3)*x)*((c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/((1+3^(1/2))*c 
^(1/3)+d^(1/3)*x)^2)^(1/2)*EllipticF(((1-3^(1/2))*c^(1/3)+d^(1/3)*x)/((1+3 
^(1/2))*c^(1/3)+d^(1/3)*x),I*3^(1/2)+2*I)*3^(3/4)/d^(2/3)/(c^(1/3)*(c^(1/3 
)+d^(1/3)*x)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2)/(d*x^3+c)^(1/2)
 

Mathematica [C] (verified)

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

Time = 8.78 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.10 \[ \int \frac {x \sqrt {c+d x^3}}{4 c+d x^3} \, dx=\frac {x^2 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )}{8 \sqrt {c+d x^3}} \] Input:

Integrate[(x*Sqrt[c + d*x^3])/(4*c + d*x^3),x]
 

Output:

(x^2*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, -1/2, 1, 5/3, -((d*x^3)/c), -1/4*(d 
*x^3)/c])/(8*Sqrt[c + d*x^3])
 

Rubi [A] (warning: unable to verify)

Time = 1.15 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {984, 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.

\(\displaystyle \int \frac {x \sqrt {c+d x^3}}{4 c+d x^3} \, dx\)

\(\Big \downarrow \) 984

\(\displaystyle \int \frac {x}{\sqrt {d x^3+c}}dx-3 c \int \frac {x}{\sqrt {d x^3+c} \left (d x^3+4 c\right )}dx\)

\(\Big \downarrow \) 832

\(\displaystyle -\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c} \int \frac {1}{\sqrt {d x^3+c}}dx}{\sqrt [3]{d}}+\frac {\int \frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt {d x^3+c}}dx}{\sqrt [3]{d}}-3 c \int \frac {x}{\sqrt {d x^3+c} \left (d x^3+4 c\right )}dx\)

\(\Big \downarrow \) 759

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

\(\Big \downarrow \) 986

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

\(\Big \downarrow \) 2416

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

Input:

Int[(x*Sqrt[c + d*x^3])/(4*c + d*x^3),x]
 

Output:

-3*c*(-1/3*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + 2^(1/3)*d^(1/3)*x))/Sqrt[c + 
 d*x^3]]/(2^(2/3)*Sqrt[3]*c^(5/6)*d^(2/3)) + ArcTan[Sqrt[c + d*x^3]/(Sqrt[ 
3]*Sqrt[c])]/(3*2^(2/3)*Sqrt[3]*c^(5/6)*d^(2/3)) - ArcTanh[(c^(1/6)*(c^(1/ 
3) - 2^(1/3)*d^(1/3)*x))/Sqrt[c + d*x^3]]/(3*2^(2/3)*c^(5/6)*d^(2/3)) + Ar 
cTanh[Sqrt[c + d*x^3]/Sqrt[c]]/(9*2^(2/3)*c^(5/6)*d^(2/3))) + ((2*Sqrt[c + 
 d*x^3])/(d^(1/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (3^(1/4)*Sqrt[2 - 
 Sqrt[3]]*c^(1/3)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x 
+ d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticE[ArcSin[((1 
 - Sqrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 
- 4*Sqrt[3]])/(d^(1/3)*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3]) 
*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x^3]))/d^(1/3) - (2*(1 - Sqrt[3])*Sqrt 
[2 + Sqrt[3]]*c^(1/3)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3 
)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticF[ArcSin 
[((1 - Sqrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], 
 -7 - 4*Sqrt[3]])/(3^(1/4)*d^(2/3)*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/(( 
1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*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 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 [C] (warning: unable to verify)

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

Time = 1.02 (sec) , antiderivative size = 848, normalized size of antiderivative = 1.29

method	result	size
default	\(\text {Expression too large to display}\)	\(848\)
elliptic	\(\text {Expression too large to display}\)	\(848\)

Input:

int(x*(d*x^3+c)^(1/2)/(d*x^3+4*c),x,method=_RETURNVERBOSE)
 

Output:

-2/3*I*3^(1/2)/d*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d 
*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/( 
-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d* 
(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^( 
1/2)/(d*x^3+c)^(1/2)*((-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3 
))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^ 
2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/ 
2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))+1/d*(-c*d^2)^(1 
/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d 
^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3 
/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)))+1/3*I/d^3*2^( 
1/2)*sum(1/_alpha*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1 
/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*( 
-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/ 
2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*( 
I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/2)*(-c*d^2)^(2/3)+2*_alpha^2*d^2- 
(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d 
*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^ 
(1/2),1/6/d*(2*I*(-c*d^2)^(1/3)*_alpha^2*3^(1/2)*d-I*(-c*d^2)^(2/3)*_alpha 
*3^(1/2)+I*3^(1/2)*c*d-3*(-c*d^2)^(2/3)*_alpha-3*c*d)/c,(I*3^(1/2)/d*(-...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2202 vs. \(2 (470) = 940\).

Time = 1.07 (sec) , antiderivative size = 2202, normalized size of antiderivative = 3.34 \[ \int \frac {x \sqrt {c+d x^3}}{4 c+d x^3} \, dx=\text {Too large to display} \] Input:

integrate(x*(d*x^3+c)^(1/2)/(d*x^3+4*c),x, algorithm="fricas")
 

Output:

-1/12*((1/432)^(1/6)*(sqrt(-3)*d - d)*(-c/d^4)^(1/6)*log(1/2*(36*(1/432)^( 
5/6)*(d^6*x^9 - 66*c*d^5*x^6 - 72*c^2*d^4*x^3 - 32*c^3*d^3 + sqrt(-3)*(d^6 
*x^9 - 66*c*d^5*x^6 - 72*c^2*d^4*x^3 - 32*c^3*d^3))*(-c/d^4)^(5/6) + 24*sq 
rt(1/3)*(c*d^4*x^7 - c^2*d^3*x^4 - 2*c^3*d^2*x)*sqrt(-c/d^4) + (2*c*d^2*x^ 
7 - 32*c^2*d*x^4 - 16*c^3*x + 18*(1/2)^(2/3)*(sqrt(-3)*c*d^4*x^5 - c*d^4*x 
^5)*(-c/d^4)^(2/3) - (1/2)^(1/3)*(5*c*d^3*x^6 - 20*c^2*d^2*x^3 - 16*c^3*d 
+ sqrt(-3)*(5*c*d^3*x^6 - 20*c^2*d^2*x^3 - 16*c^3*d))*(-c/d^4)^(1/3))*sqrt 
(d*x^3 + c) - 6*(1/432)^(1/6)*(c*d^3*x^8 - 7*c^2*d^2*x^5 - 8*c^3*d*x^2 - s 
qrt(-3)*(c*d^3*x^8 - 7*c^2*d^2*x^5 - 8*c^3*d*x^2))*(-c/d^4)^(1/6))/(d^3*x^ 
9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3)) - (1/432)^(1/6)*(sqrt(-3)*d - d 
)*(-c/d^4)^(1/6)*log(-1/2*(36*(1/432)^(5/6)*(d^6*x^9 - 66*c*d^5*x^6 - 72*c 
^2*d^4*x^3 - 32*c^3*d^3 + sqrt(-3)*(d^6*x^9 - 66*c*d^5*x^6 - 72*c^2*d^4*x^ 
3 - 32*c^3*d^3))*(-c/d^4)^(5/6) + 24*sqrt(1/3)*(c*d^4*x^7 - c^2*d^3*x^4 - 
2*c^3*d^2*x)*sqrt(-c/d^4) - (2*c*d^2*x^7 - 32*c^2*d*x^4 - 16*c^3*x + 18*(1 
/2)^(2/3)*(sqrt(-3)*c*d^4*x^5 - c*d^4*x^5)*(-c/d^4)^(2/3) - (1/2)^(1/3)*(5 
*c*d^3*x^6 - 20*c^2*d^2*x^3 - 16*c^3*d + sqrt(-3)*(5*c*d^3*x^6 - 20*c^2*d^ 
2*x^3 - 16*c^3*d))*(-c/d^4)^(1/3))*sqrt(d*x^3 + c) - 6*(1/432)^(1/6)*(c*d^ 
3*x^8 - 7*c^2*d^2*x^5 - 8*c^3*d*x^2 - sqrt(-3)*(c*d^3*x^8 - 7*c^2*d^2*x^5 
- 8*c^3*d*x^2))*(-c/d^4)^(1/6))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 6 
4*c^3)) - (1/432)^(1/6)*(sqrt(-3)*d + d)*(-c/d^4)^(1/6)*log(1/2*(36*(1/...
 

Sympy [F]

\[ \int \frac {x \sqrt {c+d x^3}}{4 c+d x^3} \, dx=\int \frac {x \sqrt {c + d x^{3}}}{4 c + d x^{3}}\, dx \] Input:

integrate(x*(d*x**3+c)**(1/2)/(d*x**3+4*c),x)
 

Output:

Integral(x*sqrt(c + d*x**3)/(4*c + d*x**3), x)
 

Maxima [F]

\[ \int \frac {x \sqrt {c+d x^3}}{4 c+d x^3} \, dx=\int { \frac {\sqrt {d x^{3} + c} x}{d x^{3} + 4 \, c} \,d x } \] Input:

integrate(x*(d*x^3+c)^(1/2)/(d*x^3+4*c),x, algorithm="maxima")
 

Output:

integrate(sqrt(d*x^3 + c)*x/(d*x^3 + 4*c), x)
 

Giac [F]

\[ \int \frac {x \sqrt {c+d x^3}}{4 c+d x^3} \, dx=\int { \frac {\sqrt {d x^{3} + c} x}{d x^{3} + 4 \, c} \,d x } \] Input:

integrate(x*(d*x^3+c)^(1/2)/(d*x^3+4*c),x, algorithm="giac")
 

Output:

integrate(sqrt(d*x^3 + c)*x/(d*x^3 + 4*c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \sqrt {c+d x^3}}{4 c+d x^3} \, dx=\int \frac {x\,\sqrt {d\,x^3+c}}{d\,x^3+4\,c} \,d x \] Input:

int((x*(c + d*x^3)^(1/2))/(4*c + d*x^3),x)
 

Output:

int((x*(c + d*x^3)^(1/2))/(4*c + d*x^3), x)
 

Reduce [F]

\[ \int \frac {x \sqrt {c+d x^3}}{4 c+d x^3} \, dx=\int \frac {\sqrt {d \,x^{3}+c}\, x}{d \,x^{3}+4 c}d x \] Input:

int(x*(d*x^3+c)^(1/2)/(d*x^3+4*c),x)
 

Output:

int((sqrt(c + d*x**3)*x)/(4*c + d*x**3),x)