\(\int \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx\) [144]

Optimal result

Integrand size = 34, antiderivative size = 573 \[ \int \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=-\frac {2 (2 c d-b e) \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3}}{15 c e}+\frac {2 (d+e x) \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3}}{5 e}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^2 e^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (a+b x+c x^2\right )}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^2 e^2 (d+e x) \left (a+b x+c x^2\right )} \] Output:

-2/15*(-b*e+2*c*d)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2)/c/e+2/5*( 
e*x+d)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2)/e-2/15*2^(1/2)*(-4*a* 
c+b^2)^(1/2)*(c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d))*(-c*(c*x^2+b*x+a)/(-4*a*c+b 
^2))^(1/2)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2)*EllipticE(1/2*(1+ 
(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c* 
d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/c^2/e^2/(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b 
^2)^(1/2))*e))^(1/2)/(c*x^2+b*x+a)+2/15*2^(1/2)*(-4*a*c+b^2)^(1/2)*(-b*e+2 
*c*d)*(a*e^2-b*d*e+c*d^2)*(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/ 
2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c* 
e*x^3)^(1/2)*EllipticF(1/2*(1+(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2), 
(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/c^2/e^2/ 
(e*x+d)/(c*x^2+b*x+a)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 33.40 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.21 \[ \int \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=\frac {\sqrt {(d+e x) (a+x (b+c x))} \left (\frac {4 e^2 \left (-c^2 d^2-b^2 e^2+c e (b d+3 a e)\right )}{d+e x}+2 c e^2 (b e+c (d+3 e x))+\frac {i \sqrt {2} \sqrt {d+e x} \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {1+\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+\left (b^3 e^3-b^2 e^2 \left (2 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+b c e \left (-4 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+c \left (-c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (8 c d+3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{\sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} (a+x (b+c x))}\right )}{15 c^2 e^3} \] Input:

Integrate[Sqrt[a*d + (b*d + a*e)*x + (c*d + b*e)*x^2 + c*e*x^3],x]
 

Output:

(Sqrt[(d + e*x)*(a + x*(b + c*x))]*((4*e^2*(-(c^2*d^2) - b^2*e^2 + c*e*(b* 
d + 3*a*e)))/(d + e*x) + 2*c*e^2*(b*e + c*(d + 3*e*x)) + (I*Sqrt[2]*Sqrt[d 
 + e*x]*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 
- 4*a*c)*e^2])*(d + e*x))]*Sqrt[1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c* 
d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*c*d - b*e + Sqrt[(b^2 - 
 4*a*c)*e^2])*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*EllipticE[I*ArcSinh[ 
(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e 
^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - 
 b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (b^3*e^3 - b^2*e^2*(2*c*d + Sqrt[(b^2 
- 4*a*c)*e^2]) + b*c*e*(-4*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*(-(c*d^2 
*Sqrt[(b^2 - 4*a*c)*e^2]) + a*e^2*(8*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2])))*El 
lipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sq 
rt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a 
*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(Sqrt[(c*d^2 + e*(-( 
b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a + x*(b + c*x)))) 
)/(15*c^2*e^3)
 

Rubi [F]

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.

\(\displaystyle \int \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3} \, dx\)

\(\Big \downarrow \) 2481

\(\displaystyle \int \sqrt {\frac {(2 c d-b e) \left (-c e (b d-9 a e)-2 b^2 e^2+c^2 d^2\right )}{27 c^2 e^2}+\frac {\left (3 c e (a e+b d)-(b e+c d)^2\right ) \left (\frac {b e+c d}{3 c e}+x\right )}{3 c e}+c e \left (\frac {b e+c d}{3 c e}+x\right )^3}d\left (\frac {b e+c d}{3 c e}+x\right )\)

Input:

Int[Sqrt[a*d + (b*d + a*e)*x + (c*d + b*e)*x^2 + c*e*x^3],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2481

Int[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] 
, c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c* 
d + 27*a*d^2)/(27*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, c/(3 
*d) + x]] /; FreeQ[p, x] && PolyQ[Px, x, 3]
 

Maple [A] (verified)

Time = 3.32 (sec) , antiderivative size = 854, normalized size of antiderivative = 1.49

method	result	size
default	\(\frac {2 x \sqrt {c e \,x^{3}+x^{2} e b +c d \,x^{2}+a e x +b d x +a d}}{5}+\frac {2 \left (\frac {e b}{5}+\frac {c d}{5}\right ) \sqrt {c e \,x^{3}+x^{2} e b +c d \,x^{2}+a e x +b d x +a d}}{3 c e}+\frac {2 \left (\frac {3 a d}{5}-\frac {2 \left (\frac {e b}{5}+\frac {c d}{5}\right ) \left (\frac {a e}{2}+\frac {b d}{2}\right )}{3 c e}\right ) \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c e \,x^{3}+x^{2} e b +c d \,x^{2}+a e x +b d x +a d}}+\frac {2 \left (\frac {2 a e}{5}+\frac {2 b d}{5}-\frac {2 \left (\frac {e b}{5}+\frac {c d}{5}\right ) \left (e b +c d \right )}{3 c e}\right ) \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{\sqrt {c e \,x^{3}+x^{2} e b +c d \,x^{2}+a e x +b d x +a d}}\)	\(854\)
elliptic	\(\frac {2 x \sqrt {c e \,x^{3}+x^{2} e b +c d \,x^{2}+a e x +b d x +a d}}{5}+\frac {2 \left (\frac {e b}{5}+\frac {c d}{5}\right ) \sqrt {c e \,x^{3}+x^{2} e b +c d \,x^{2}+a e x +b d x +a d}}{3 c e}+\frac {2 \left (\frac {3 a d}{5}-\frac {2 \left (\frac {e b}{5}+\frac {c d}{5}\right ) \left (\frac {a e}{2}+\frac {b d}{2}\right )}{3 c e}\right ) \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c e \,x^{3}+x^{2} e b +c d \,x^{2}+a e x +b d x +a d}}+\frac {2 \left (\frac {2 a e}{5}+\frac {2 b d}{5}-\frac {2 \left (\frac {e b}{5}+\frac {c d}{5}\right ) \left (e b +c d \right )}{3 c e}\right ) \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{\sqrt {c e \,x^{3}+x^{2} e b +c d \,x^{2}+a e x +b d x +a d}}\)	\(854\)
risch	\(\text {Expression too large to display}\)	\(1666\)

Input:

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

Output:

2/5*x*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/3*(1/5*e*b+1/5*c*d 
)/c/e*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2*(3/5*a*d-2/3*(1/5* 
e*b+1/5*c*d)/c/e*(1/2*a*e+1/2*b*d))*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x 
+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^( 
1/2))/c)/(-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x+1/2*(b+(-4*a*c+b^ 
2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c 
*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^ 
(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2*(-b+(-4*a* 
c+b^2)^(1/2))/c))^(1/2))+2*(2/5*a*e+2/5*b*d-2/3*(1/5*e*b+1/5*c*d)/c/e*(b*e 
+c*d))*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2 
)^(1/2))/c))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2*(-b+(-4*a* 
c+b^2)^(1/2))/c))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4 
*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2) 
*((-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)*EllipticE(((x+d/e)/(d/e-1/2*(b+(-4* 
a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2*( 
-b+(-4*a*c+b^2)^(1/2))/c))^(1/2))+1/2*(-b+(-4*a*c+b^2)^(1/2))/c*EllipticF( 
((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b 
^2)^(1/2))/c)/(-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/c))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 494, normalized size of antiderivative = 0.86 \[ \int \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=\frac {2 \, {\left ({\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 6 \, {\left (c^{3} d^{2} e - b c^{2} d e^{2} + {\left (b^{2} c - 3 \, a c^{2}\right )} e^{3}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (3 \, c^{3} e^{3} x + c^{3} d e^{2} + b c^{2} e^{3}\right )} \sqrt {c e x^{3} + {\left (c d + b e\right )} x^{2} + a d + {\left (b d + a e\right )} x}\right )}}{45 \, c^{3} e^{3}} \] Input:

integrate((a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2),x, algorithm="fric 
as")
 

Output:

2/45*((2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9* 
a*b*c)*e^3)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 
3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c 
^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c 
*e)) + 6*(c^3*d^2*e - b*c^2*d*e^2 + (b^2*c - 3*a*c^2)*e^3)*sqrt(c*e)*weier 
strassZeta(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2 
*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e 
^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)* 
e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e 
^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) + 
 3*(3*c^3*e^3*x + c^3*d*e^2 + b*c^2*e^3)*sqrt(c*e*x^3 + (c*d + b*e)*x^2 + 
a*d + (b*d + a*e)*x))/(c^3*e^3)
 

Sympy [F]

\[ \int \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=\int \sqrt {a d + c e x^{3} + x^{2} \left (b e + c d\right ) + x \left (a e + b d\right )}\, dx \] Input:

integrate((a*d+(a*e+b*d)*x+(b*e+c*d)*x**2+c*e*x**3)**(1/2),x)
 

Output:

Integral(sqrt(a*d + c*e*x**3 + x**2*(b*e + c*d) + x*(a*e + b*d)), x)
 

Maxima [F]

\[ \int \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=\int { \sqrt {c e x^{3} + {\left (c d + b e\right )} x^{2} + a d + {\left (b d + a e\right )} x} \,d x } \] Input:

integrate((a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2),x, algorithm="maxi 
ma")
 

Output:

integrate(sqrt(c*e*x^3 + (c*d + b*e)*x^2 + a*d + (b*d + a*e)*x), x)
 

Giac [F]

\[ \int \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=\int { \sqrt {c e x^{3} + {\left (c d + b e\right )} x^{2} + a d + {\left (b d + a e\right )} x} \,d x } \] Input:

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

Output:

integrate(sqrt(c*e*x^3 + (c*d + b*e)*x^2 + a*d + (b*d + a*e)*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=\int \sqrt {c\,e\,x^3+\left (b\,e+c\,d\right )\,x^2+\left (a\,e+b\,d\right )\,x+a\,d} \,d x \] Input:

int((a*d + x*(a*e + b*d) + x^2*(b*e + c*d) + c*e*x^3)^(1/2),x)
 

Output:

int((a*d + x*(a*e + b*d) + x^2*(b*e + c*d) + c*e*x^3)^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=\int \sqrt {a d +\left (a e +b d \right ) x +\left (b e +c d \right ) x^{2}+c e \,x^{3}}d x \] Input:

int((a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2),x)
 

Output:

int((a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2),x)