\(\int \frac {x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx\) [146]

Optimal result

Integrand size = 25, antiderivative size = 202 \[ \int \frac {x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=\frac {2 \text {arctanh}\left (\frac {(c-2 d x)^2}{3 \sqrt {c} \sqrt {c^3-8 d^3 x^3}}\right )}{9 \sqrt {c} d^2}-\frac {\sqrt {2+\sqrt {3}} (c-2 d x) \sqrt {\frac {c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c-2 d x}{\left (1+\sqrt {3}\right ) c-2 d x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d^2 \sqrt {\frac {c (c-2 d x)}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} \sqrt {c^3-8 d^3 x^3}} \] Output:

2/9*arctanh(1/3*(-2*d*x+c)^2/c^(1/2)/(-8*d^3*x^3+c^3)^(1/2))/c^(1/2)/d^2-1 
/9*(1/2*6^(1/2)+1/2*2^(1/2))*(-2*d*x+c)*((4*d^2*x^2+2*c*d*x+c^2)/((1+3^(1/ 
2))*c-2*d*x)^2)^(1/2)*EllipticF(((1-3^(1/2))*c-2*d*x)/((1+3^(1/2))*c-2*d*x 
),I*3^(1/2)+2*I)*3^(3/4)/d^2/(c*(-2*d*x+c)/((1+3^(1/2))*c-2*d*x)^2)^(1/2)/ 
(-8*d^3*x^3+c^3)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.54 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.46 \[ \int \frac {x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=\frac {\sqrt {\frac {c-2 d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (\left (-2+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} c+2 d x\right ) \sqrt {\frac {\sqrt [3]{-1} \left (c+2 \sqrt [3]{-1} d x\right )}{\left (1+\sqrt [3]{-1}\right ) c}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}\right ),\sqrt [3]{-1}\right )+\frac {2 \sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) c \sqrt {\frac {c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt {\frac {c^2+2 c d x+4 d^2 x^2}{c^2}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{3 i+\sqrt {3}},\arcsin \left (\sqrt {\frac {c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}\right ),\sqrt [3]{-1}\right )}{\sqrt {3}}\right )}{\left (-2+\sqrt [3]{-1}\right ) d^2 \sqrt {\frac {c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt {c^3-8 d^3 x^3}} \] Input:

Integrate[x/((c + d*x)*Sqrt[c^3 - 8*d^3*x^3]),x]
 

Output:

(Sqrt[(c - 2*d*x)/((1 + (-1)^(1/3))*c)]*((-2 + (-1)^(1/3))*((-1)^(1/3)*c + 
 2*d*x)*Sqrt[((-1)^(1/3)*(c + 2*(-1)^(1/3)*d*x))/((1 + (-1)^(1/3))*c)]*Ell 
ipticF[ArcSin[Sqrt[(c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]], (-1)^(1/ 
3)] + (2*(-1)^(1/3)*(1 + (-1)^(1/3))*c*Sqrt[(c - 2*(-1)^(2/3)*d*x)/((1 + ( 
-1)^(1/3))*c)]*Sqrt[(c^2 + 2*c*d*x + 4*d^2*x^2)/c^2]*EllipticPi[(2*Sqrt[3] 
)/(3*I + Sqrt[3]), ArcSin[Sqrt[(c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c) 
]], (-1)^(1/3)])/Sqrt[3]))/((-2 + (-1)^(1/3))*d^2*Sqrt[(c - 2*(-1)^(2/3)*d 
*x)/((1 + (-1)^(1/3))*c)]*Sqrt[c^3 - 8*d^3*x^3])
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2564, 759, 2563, 219}

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.

\(\displaystyle \int \frac {x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx\)

\(\Big \downarrow \) 2564

\(\displaystyle \frac {\int \frac {1}{\sqrt {c^3-8 d^3 x^3}}dx}{3 d}-\frac {\int \frac {c-2 d x}{(c+d x) \sqrt {c^3-8 d^3 x^3}}dx}{3 d}\)

\(\Big \downarrow \) 759

\(\displaystyle -\frac {\int \frac {c-2 d x}{(c+d x) \sqrt {c^3-8 d^3 x^3}}dx}{3 d}-\frac {\sqrt {2+\sqrt {3}} (c-2 d x) \sqrt {\frac {c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c-2 d x}{\left (1+\sqrt {3}\right ) c-2 d x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d^2 \sqrt {\frac {c (c-2 d x)}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} \sqrt {c^3-8 d^3 x^3}}\)

\(\Big \downarrow \) 2563

\(\displaystyle \frac {2 c \int \frac {1}{9-\frac {(c-2 d x)^4}{c \left (c^3-8 d^3 x^3\right )}}d\frac {(c-2 d x)^2}{c^2 \sqrt {c^3-8 d^3 x^3}}}{3 d^2}-\frac {\sqrt {2+\sqrt {3}} (c-2 d x) \sqrt {\frac {c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c-2 d x}{\left (1+\sqrt {3}\right ) c-2 d x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d^2 \sqrt {\frac {c (c-2 d x)}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} \sqrt {c^3-8 d^3 x^3}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 \text {arctanh}\left (\frac {(c-2 d x)^2}{3 \sqrt {c} \sqrt {c^3-8 d^3 x^3}}\right )}{9 \sqrt {c} d^2}-\frac {\sqrt {2+\sqrt {3}} (c-2 d x) \sqrt {\frac {c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c-2 d x}{\left (1+\sqrt {3}\right ) c-2 d x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d^2 \sqrt {\frac {c (c-2 d x)}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} \sqrt {c^3-8 d^3 x^3}}\)

Input:

Int[x/((c + d*x)*Sqrt[c^3 - 8*d^3*x^3]),x]
 

Output:

(2*ArcTanh[(c - 2*d*x)^2/(3*Sqrt[c]*Sqrt[c^3 - 8*d^3*x^3])])/(9*Sqrt[c]*d^ 
2) - (Sqrt[2 + Sqrt[3]]*(c - 2*d*x)*Sqrt[(c^2 + 2*c*d*x + 4*d^2*x^2)/((1 + 
 Sqrt[3])*c - 2*d*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*c - 2*d*x)/((1 + S 
qrt[3])*c - 2*d*x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*d^2*Sqrt[(c*(c - 2*d*x))/ 
((1 + Sqrt[3])*c - 2*d*x)^2]*Sqrt[c^3 - 8*d^3*x^3])
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 2563

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ 
Symbol] :> Simp[-2*(e/d)   Subst[Int[1/(9 - a*x^2), x], x, (1 + f*(x/e))^2/ 
Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] & 
& EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
 

rule 2564

Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x 
_Symbol] :> Simp[(2*d*e + c*f)/(3*c*d)   Int[1/Sqrt[a + b*x^3], x], x] + Si 
mp[(d*e - c*f)/(3*c*d)   Int[(c - 2*d*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x 
] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && (EqQ[b*c^3 - 4*a* 
d^3, 0] || EqQ[b*c^3 + 8*a*d^3, 0]) && NeQ[2*d*e + c*f, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 508 vs. \(2 (173 ) = 346\).

Time = 0.37 (sec) , antiderivative size = 509, normalized size of antiderivative = 2.52

method	result	size
default	\(\frac {2 \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}\right ) \sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}, \sqrt {\frac {\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\right )}{d \sqrt {-8 d^{3} x^{3}+c^{3}}}-\frac {4 \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}\right ) \sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}, \frac {2 \left (\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}\right ) d}{3 c}, \sqrt {\frac {\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\right )}{3 d \sqrt {-8 d^{3} x^{3}+c^{3}}}\)	\(509\)
elliptic	\(\frac {2 \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}\right ) \sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}, \sqrt {\frac {\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\right )}{d \sqrt {-8 d^{3} x^{3}+c^{3}}}-\frac {4 \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}\right ) \sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}, \frac {2 \left (\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}\right ) d}{3 c}, \sqrt {\frac {\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\right )}{3 d \sqrt {-8 d^{3} x^{3}+c^{3}}}\)	\(509\)

Input:

int(x/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/d*(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d)*((x-1/2*c/d)/(1/2*(-1/2-1/2*I*3 
^(1/2))*c/d-1/2*c/d))^(1/2)*((x-1/2*(-1/2+1/2*I*3^(1/2))*c/d)/(1/2*c/d-1/2 
*(-1/2+1/2*I*3^(1/2))*c/d))^(1/2)*((x-1/2*(-1/2-1/2*I*3^(1/2))*c/d)/(1/2*c 
/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d))^(1/2)/(-8*d^3*x^3+c^3)^(1/2)*EllipticF(( 
(x-1/2*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d))^(1/2),((1/2*c/d-1/2*(- 
1/2-1/2*I*3^(1/2))*c/d)/(1/2*c/d-1/2*(-1/2+1/2*I*3^(1/2))*c/d))^(1/2))-4/3 
/d*(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d)*((x-1/2*c/d)/(1/2*(-1/2-1/2*I*3^ 
(1/2))*c/d-1/2*c/d))^(1/2)*((x-1/2*(-1/2+1/2*I*3^(1/2))*c/d)/(1/2*c/d-1/2* 
(-1/2+1/2*I*3^(1/2))*c/d))^(1/2)*((x-1/2*(-1/2-1/2*I*3^(1/2))*c/d)/(1/2*c/ 
d-1/2*(-1/2-1/2*I*3^(1/2))*c/d))^(1/2)/(-8*d^3*x^3+c^3)^(1/2)*EllipticPi(( 
(x-1/2*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d))^(1/2),2/3*(1/2*c/d-1/2 
*(-1/2-1/2*I*3^(1/2))*c/d)/c*d,((1/2*c/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d)/(1/ 
2*c/d-1/2*(-1/2+1/2*I*3^(1/2))*c/d))^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.76 \[ \int \frac {x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=\left [\frac {\sqrt {c} d^{2} \log \left (\frac {8 \, d^{6} x^{6} - 240 \, c d^{5} x^{5} + 408 \, c^{2} d^{4} x^{4} + 88 \, c^{3} d^{3} x^{3} + 156 \, c^{4} d^{2} x^{2} + 12 \, c^{5} d x + 17 \, c^{6} + 3 \, {\left (8 \, d^{4} x^{4} - 52 \, c d^{3} x^{3} + 12 \, c^{2} d^{2} x^{2} - 4 \, c^{3} d x + 5 \, c^{4}\right )} \sqrt {-8 \, d^{3} x^{3} + c^{3}} \sqrt {c}}{d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6}}\right ) - 3 \, \sqrt {2} \sqrt {-d^{3}} c {\rm weierstrassPInverse}\left (0, \frac {c^{3}}{2 \, d^{3}}, x\right )}{18 \, c d^{4}}, \frac {2 \, \sqrt {-c} d^{2} \arctan \left (\frac {{\left (4 \, d^{3} x^{3} - 24 \, c d^{2} x^{2} - 6 \, c^{2} d x - 5 \, c^{3}\right )} \sqrt {-8 \, d^{3} x^{3} + c^{3}} \sqrt {-c}}{3 \, {\left (16 \, c d^{4} x^{4} - 8 \, c^{2} d^{3} x^{3} - 2 \, c^{4} d x + c^{5}\right )}}\right ) - 3 \, \sqrt {2} \sqrt {-d^{3}} c {\rm weierstrassPInverse}\left (0, \frac {c^{3}}{2 \, d^{3}}, x\right )}{18 \, c d^{4}}\right ] \] Input:

integrate(x/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x, algorithm="fricas")
 

Output:

[1/18*(sqrt(c)*d^2*log((8*d^6*x^6 - 240*c*d^5*x^5 + 408*c^2*d^4*x^4 + 88*c 
^3*d^3*x^3 + 156*c^4*d^2*x^2 + 12*c^5*d*x + 17*c^6 + 3*(8*d^4*x^4 - 52*c*d 
^3*x^3 + 12*c^2*d^2*x^2 - 4*c^3*d*x + 5*c^4)*sqrt(-8*d^3*x^3 + c^3)*sqrt(c 
))/(d^6*x^6 + 6*c*d^5*x^5 + 15*c^2*d^4*x^4 + 20*c^3*d^3*x^3 + 15*c^4*d^2*x 
^2 + 6*c^5*d*x + c^6)) - 3*sqrt(2)*sqrt(-d^3)*c*weierstrassPInverse(0, 1/2 
*c^3/d^3, x))/(c*d^4), 1/18*(2*sqrt(-c)*d^2*arctan(1/3*(4*d^3*x^3 - 24*c*d 
^2*x^2 - 6*c^2*d*x - 5*c^3)*sqrt(-8*d^3*x^3 + c^3)*sqrt(-c)/(16*c*d^4*x^4 
- 8*c^2*d^3*x^3 - 2*c^4*d*x + c^5)) - 3*sqrt(2)*sqrt(-d^3)*c*weierstrassPI 
nverse(0, 1/2*c^3/d^3, x))/(c*d^4)]
 

Sympy [F]

\[ \int \frac {x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=\int \frac {x}{\sqrt {- \left (- c + 2 d x\right ) \left (c^{2} + 2 c d x + 4 d^{2} x^{2}\right )} \left (c + d x\right )}\, dx \] Input:

integrate(x/(d*x+c)/(-8*d**3*x**3+c**3)**(1/2),x)
 

Output:

Integral(x/(sqrt(-(-c + 2*d*x)*(c**2 + 2*c*d*x + 4*d**2*x**2))*(c + d*x)), 
 x)
 

Maxima [F]

\[ \int \frac {x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=\int { \frac {x}{\sqrt {-8 \, d^{3} x^{3} + c^{3}} {\left (d x + c\right )}} \,d x } \] Input:

integrate(x/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)), x)
 

Giac [F]

\[ \int \frac {x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=\int { \frac {x}{\sqrt {-8 \, d^{3} x^{3} + c^{3}} {\left (d x + c\right )}} \,d x } \] Input:

integrate(x/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x, algorithm="giac")
 

Output:

integrate(x/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=\int \frac {x}{\sqrt {c^3-8\,d^3\,x^3}\,\left (c+d\,x\right )} \,d x \] Input:

int(x/((c^3 - 8*d^3*x^3)^(1/2)*(c + d*x)),x)
 

Output:

int(x/((c^3 - 8*d^3*x^3)^(1/2)*(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=\int \frac {\sqrt {-8 d^{3} x^{3}+c^{3}}\, x}{-8 d^{4} x^{4}-8 c \,d^{3} x^{3}+c^{3} d x +c^{4}}d x \] Input:

int(x/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

int((sqrt(c**3 - 8*d**3*x**3)*x)/(c**4 + c**3*d*x - 8*c*d**3*x**3 - 8*d**4 
*x**4),x)