\(\int \frac {x^4}{\sqrt {c+d x^3} (4 c+d x^3)} \, dx\) [450]

Optimal result

Integrand size = 26, antiderivative size = 667 \[ \int \frac {x^4}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx=\frac {2 \sqrt {c+d x^3}}{d^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {2 \sqrt [3]{2} \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{3 \sqrt {3} d^{5/3}}-\frac {2 \sqrt [3]{2} \sqrt [6]{c} \arctan \left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{3 \sqrt {3} d^{5/3}}+\frac {2 \sqrt [3]{2} \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{3 d^{5/3}}-\frac {2 \sqrt [3]{2} \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{9 d^{5/3}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {2 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:

2*(d*x^3+c)^(1/2)/d^(5/3)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)+2/9*2^(1/3)*c^(1 
/6)*arctan(3^(1/2)*c^(1/6)*(c^(1/3)+2^(1/3)*d^(1/3)*x)/(d*x^3+c)^(1/2))*3^ 
(1/2)/d^(5/3)-2/9*2^(1/3)*c^(1/6)*arctan(1/3*(d*x^3+c)^(1/2)*3^(1/2)/c^(1/ 
2))*3^(1/2)/d^(5/3)+2/3*2^(1/3)*c^(1/6)*arctanh(c^(1/6)*(c^(1/3)-2^(1/3)*d 
^(1/3)*x)/(d*x^3+c)^(1/2))/d^(5/3)-2/9*2^(1/3)*c^(1/6)*arctanh((d*x^3+c)^( 
1/2)/c^(1/2))/d^(5/3)-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^(5/3)/(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^(5/3)/(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 = 10.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.10 \[ \int \frac {x^4}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx=\frac {x^5 \sqrt {\frac {c+d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )}{20 c \sqrt {c+d x^3}} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 1.13 (sec) , antiderivative size = 701, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {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.

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

\(\Big \downarrow \) 983

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

\(\Big \downarrow \) 832

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

\(\Big \downarrow \) 759

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

\(\Big \downarrow \) 986

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

\(\Big \downarrow \) 2416

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

Input:

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

Output:

(-4*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)) + A 
rcTanh[Sqrt[c + d*x^3]/Sqrt[c]]/(9*2^(2/3)*c^(5/6)*d^(2/3))))/d + (((2*Sqr 
t[c + d*x^3])/(d^(1/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (3^(1/4)*Sqr 
t[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[ArcSi 
n[((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 + Sqr 
t[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[A 
rcSin[((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]))/d
 

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 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 [C] (warning: unable to verify)

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

Time = 1.29 (sec) , antiderivative size = 848, normalized size of antiderivative = 1.27

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

Input:

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

Output:

-2/3*I/d^2*3^(1/2)*(-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)))+4/9*I/d^4*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)*_alp 
ha*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. 2268 vs. \(2 (470) = 940\).

Time = 2.06 (sec) , antiderivative size = 2268, normalized size of antiderivative = 3.40 \[ \int \frac {x^4}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

1/18*(2*(4/27)^(1/6)*d^2*(-c/d^10)^(1/6)*log(32*(9*(4/27)^(5/6)*(d^11*x^9 
- 66*c*d^10*x^6 - 72*c^2*d^9*x^3 - 32*c^3*d^8)*(-c/d^10)^(5/6) - 96*sqrt(1 
/3)*(c*d^7*x^7 - c^2*d^6*x^4 - 2*c^3*d^5*x)*sqrt(-c/d^10) + 4*(9*4^(2/3)*c 
*d^8*x^5*(-c/d^10)^(2/3) + 2*c*d^2*x^7 - 32*c^2*d*x^4 - 16*c^3*x + 4^(1/3) 
*(5*c*d^5*x^6 - 20*c^2*d^4*x^3 - 16*c^3*d^3)*(-c/d^10)^(1/3))*sqrt(d*x^3 + 
 c) - 24*(4/27)^(1/6)*(c*d^4*x^8 - 7*c^2*d^3*x^5 - 8*c^3*d^2*x^2)*(-c/d^10 
)^(1/6))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3)) - 2*(4/27)^(1/6 
)*d^2*(-c/d^10)^(1/6)*log(-32*(9*(4/27)^(5/6)*(d^11*x^9 - 66*c*d^10*x^6 - 
72*c^2*d^9*x^3 - 32*c^3*d^8)*(-c/d^10)^(5/6) - 96*sqrt(1/3)*(c*d^7*x^7 - c 
^2*d^6*x^4 - 2*c^3*d^5*x)*sqrt(-c/d^10) - 4*(9*4^(2/3)*c*d^8*x^5*(-c/d^10) 
^(2/3) + 2*c*d^2*x^7 - 32*c^2*d*x^4 - 16*c^3*x + 4^(1/3)*(5*c*d^5*x^6 - 20 
*c^2*d^4*x^3 - 16*c^3*d^3)*(-c/d^10)^(1/3))*sqrt(d*x^3 + c) - 24*(4/27)^(1 
/6)*(c*d^4*x^8 - 7*c^2*d^3*x^5 - 8*c^3*d^2*x^2)*(-c/d^10)^(1/6))/(d^3*x^9 
+ 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3)) - (4/27)^(1/6)*(sqrt(-3)*d^2 - d^ 
2)*(-c/d^10)^(1/6)*log(32*(9*(4/27)^(5/6)*(d^11*x^9 - 66*c*d^10*x^6 - 72*c 
^2*d^9*x^3 - 32*c^3*d^8 + sqrt(-3)*(d^11*x^9 - 66*c*d^10*x^6 - 72*c^2*d^9* 
x^3 - 32*c^3*d^8))*(-c/d^10)^(5/6) + 192*sqrt(1/3)*(c*d^7*x^7 - c^2*d^6*x^ 
4 - 2*c^3*d^5*x)*sqrt(-c/d^10) + 4*(4*c*d^2*x^7 - 64*c^2*d*x^4 - 32*c^3*x 
+ 9*4^(2/3)*(sqrt(-3)*c*d^8*x^5 - c*d^8*x^5)*(-c/d^10)^(2/3) - 4^(1/3)*(5* 
c*d^5*x^6 - 20*c^2*d^4*x^3 - 16*c^3*d^3 + sqrt(-3)*(5*c*d^5*x^6 - 20*c^...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^4}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx=\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{2} x^{6}+5 c d \,x^{3}+4 c^{2}}d x \] Input:

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

Output:

int((sqrt(c + d*x**3)*x**4)/(4*c**2 + 5*c*d*x**3 + d**2*x**6),x)