\(\int \frac {e+f x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx\) [137]

Optimal result

Integrand size = 29, antiderivative size = 221 \[ \int \frac {e+f x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=-\frac {2 (d e-c f) \text {arctanh}\left (\frac {(c-2 d x)^2}{3 \sqrt {c} \sqrt {c^3-8 d^3 x^3}}\right )}{9 c^{3/2} d^2}-\frac {\sqrt {2+\sqrt {3}} (2 d e+c f) (c-2 d x) \sqrt {\frac {c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c-2 d x}{\left (1+\sqrt {3}\right ) c-2 d x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt {\frac {c (c-2 d x)}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} \sqrt {c^3-8 d^3 x^3}} \] Output:

-2/9*(-c*f+d*e)*arctanh(1/3*(-2*d*x+c)^2/c^(1/2)/(-8*d^3*x^3+c^3)^(1/2))/c 
^(3/2)/d^2-1/9*(1/2*6^(1/2)+1/2*2^(1/2))*(c*f+2*d*e)*(-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)/c/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 = 10.93 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.74 \[ \int \frac {e+f x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=-\frac {i \sqrt {\frac {c-2 d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (f \sqrt {\frac {\left (-i+\sqrt {3}\right ) c+2 \left (i+\sqrt {3}\right ) d x}{\left (-3 i+\sqrt {3}\right ) c}} \left (\left (-3 i+\sqrt {3}\right ) c-2 \left (3 i+\sqrt {3}\right ) d x\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} \sqrt {\frac {i c+i d x+\sqrt {3} d x}{3 i c-\sqrt {3} c}}\right ),\frac {1}{2} \left (1+i \sqrt {3}\right )\right )+4 \sqrt {2} (d e-c f) \sqrt {\frac {i c+i d x+\sqrt {3} d x}{3 i c-\sqrt {3} c}} \sqrt {\frac {c^2+2 c d x+4 d^2 x^2}{c^2}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{3 i+\sqrt {3}},\arcsin \left (\sqrt {2} \sqrt {\frac {i c+i d x+\sqrt {3} d x}{3 i c-\sqrt {3} c}}\right ),\frac {1}{2} \left (1+i \sqrt {3}\right )\right )\right )}{2 \left (-2+\sqrt [3]{-1}\right ) d^2 \sqrt {\frac {c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt {c^3-8 d^3 x^3}} \] Input:

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

Output:

((-1/2*I)*Sqrt[(c - 2*d*x)/((1 + (-1)^(1/3))*c)]*(f*Sqrt[((-I + Sqrt[3])*c 
 + 2*(I + Sqrt[3])*d*x)/((-3*I + Sqrt[3])*c)]*((-3*I + Sqrt[3])*c - 2*(3*I 
 + Sqrt[3])*d*x)*EllipticF[ArcSin[Sqrt[2]*Sqrt[(I*c + I*d*x + Sqrt[3]*d*x) 
/((3*I)*c - Sqrt[3]*c)]], (1 + I*Sqrt[3])/2] + 4*Sqrt[2]*(d*e - c*f)*Sqrt[ 
(I*c + I*d*x + Sqrt[3]*d*x)/((3*I)*c - Sqrt[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[2]*Sqrt 
[(I*c + I*d*x + Sqrt[3]*d*x)/((3*I)*c - Sqrt[3]*c)]], (1 + I*Sqrt[3])/2])) 
/((-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 = 0.90 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, 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.

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

\(\Big \downarrow \) 2564

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

\(\Big \downarrow \) 759

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

\(\Big \downarrow \) 2563

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(-2*(d*e - c*f)*ArcTanh[(c - 2*d*x)^2/(3*Sqrt[c]*Sqrt[c^3 - 8*d^3*x^3])])/ 
(9*c^(3/2)*d^2) - (Sqrt[2 + Sqrt[3]]*(2*d*e + c*f)*(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 + Sqrt[3])*c - 2*d*x)], -7 - 4*Sqrt[3]])/(3*3^(1/4 
)*c*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. 520 vs. \(2 (192 ) = 384\).

Time = 0.42 (sec) , antiderivative size = 521, normalized size of antiderivative = 2.36

method	result	size
default	\(\frac {2 f \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 (c f -d e \right ) \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}}\, c}\)	\(521\)
elliptic	\(\frac {2 f \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 (c f -d e \right ) \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}}\, c}\)	\(521\)

Input:

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

Output:

2*f/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*(c*f-d*e)/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)/ 
c*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 = 0.24 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.79 \[ \int \frac {e+f x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=\left [-\frac {3 \, \sqrt {2} \sqrt {-d^{3}} {\left (2 \, c d e + c^{2} f\right )} {\rm weierstrassPInverse}\left (0, \frac {c^{3}}{2 \, d^{3}}, x\right ) + {\left (d^{3} e - c d^{2} f\right )} \sqrt {c} \log \left (\frac {8 \, d^{6} x^{6} - 240 \, c d^{5} x^{5} + 408 \, c^{2} d^{4} x^{4} + 88 \, c^{3} d^{3} x^{3} + 156 \, c^{4} d^{2} x^{2} + 12 \, c^{5} d x + 17 \, c^{6} + 3 \, {\left (8 \, d^{4} x^{4} - 52 \, c d^{3} x^{3} + 12 \, c^{2} d^{2} x^{2} - 4 \, c^{3} d x + 5 \, c^{4}\right )} \sqrt {-8 \, d^{3} x^{3} + c^{3}} \sqrt {c}}{d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6}}\right )}{18 \, c^{2} d^{4}}, -\frac {3 \, \sqrt {2} \sqrt {-d^{3}} {\left (2 \, c d e + c^{2} f\right )} {\rm weierstrassPInverse}\left (0, \frac {c^{3}}{2 \, d^{3}}, x\right ) + 2 \, {\left (d^{3} e - c d^{2} f\right )} \sqrt {-c} \arctan \left (\frac {{\left (4 \, d^{3} x^{3} - 24 \, c d^{2} x^{2} - 6 \, c^{2} d x - 5 \, c^{3}\right )} \sqrt {-8 \, d^{3} x^{3} + c^{3}} \sqrt {-c}}{3 \, {\left (16 \, c d^{4} x^{4} - 8 \, c^{2} d^{3} x^{3} - 2 \, c^{4} d x + c^{5}\right )}}\right )}{18 \, c^{2} d^{4}}\right ] \] Input:

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

Output:

[-1/18*(3*sqrt(2)*sqrt(-d^3)*(2*c*d*e + c^2*f)*weierstrassPInverse(0, 1/2* 
c^3/d^3, x) + (d^3*e - c*d^2*f)*sqrt(c)*log((8*d^6*x^6 - 240*c*d^5*x^5 + 4 
08*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)))/(c^2*d^4), -1/18*(3*sqrt(2)*s 
qrt(-d^3)*(2*c*d*e + c^2*f)*weierstrassPInverse(0, 1/2*c^3/d^3, x) + 2*(d^ 
3*e - c*d^2*f)*sqrt(-c)*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)))/(c^2*d^4)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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