Integrand size = 48, antiderivative size = 588 \[ \int \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\frac {2}{15} \left (\frac {a}{b}+\frac {c}{d}-\frac {2 e}{f}\right ) \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}+\frac {2 (e+f x) \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}}{5 f}-\frac {4 \sqrt {-b c+a d} \left (a^2 d^2 f^2-a b d f (d e+c f)+b^2 \left (d^2 e^2-c d e f+c^2 f^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^2 d^{3/2} f^2 \sqrt {a+b x} (c+d x) \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \sqrt {-b c+a d} (b e-a f) (d e-c f) (2 b d e-b c f-a d f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^2 d^{3/2} f^2 \sqrt {a+b x} (c+d x) (e+f x)} \] Output:
2/15*(a/b+c/d-2*e/f)*(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+ b*d*f*x^3)^(1/2)+2/5*(f*x+e)*(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d *e)*x^2+b*d*f*x^3)^(1/2)/f-4/15*(a*d-b*c)^(1/2)*(a^2*d^2*f^2-a*b*d*f*(c*f+ d*e)+b^2*(c^2*f^2-c*d*e*f+d^2*e^2))*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(a*c*e+(a *c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^(1/2)*EllipticE(d^( 1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^(1/2))/b^2/ d^(3/2)/f^2/(b*x+a)^(1/2)/(d*x+c)/(b*(f*x+e)/(-a*f+b*e))^(1/2)+2/15*(a*d-b *c)^(1/2)*(-a*f+b*e)*(-c*f+d*e)*(-a*d*f-b*c*f+2*b*d*e)*(b*(d*x+c)/(-a*d+b* c))^(1/2)*(b*(f*x+e)/(-a*f+b*e))^(1/2)*(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f +b*c*f+b*d*e)*x^2+b*d*f*x^3)^(1/2)*EllipticF(d^(1/2)*(b*x+a)^(1/2)/(a*d-b* c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^(1/2))/b^2/d^(3/2)/f^2/(b*x+a)^(1/2)/ (d*x+c)/(f*x+e)
Result contains complex when optimal does not.
Time = 24.66 (sec) , antiderivative size = 485, normalized size of antiderivative = 0.82 \[ \int \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\frac {2 \left (-2 b^2 \sqrt {-a+\frac {b c}{d}} \left (a^2 d^2 f^2-a b d f (d e+c f)+b^2 \left (d^2 e^2-c d e f+c^2 f^2\right )\right ) (c+d x) (e+f x)+b^2 \sqrt {-a+\frac {b c}{d}} d f (a+b x) (c+d x) (e+f x) (b c f+a d f+b d (e+3 f x))-2 i (b c-a d) f \left (a^2 d^2 f^2-a b d f (d e+c f)+b^2 \left (d^2 e^2-c d e f+c^2 f^2\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )+i b (b c-a d) f (d e-c f) (b d e-2 b c f+a d f) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )}{15 b^3 \sqrt {-a+\frac {b c}{d}} d^2 f^2 \sqrt {(a+b x) (c+d x) (e+f x)}} \] Input:
Integrate[Sqrt[a*c*e + (b*c*e + a*d*e + a*c*f)*x + (b*d*e + b*c*f + a*d*f) *x^2 + b*d*f*x^3],x]
Output:
(2*(-2*b^2*Sqrt[-a + (b*c)/d]*(a^2*d^2*f^2 - a*b*d*f*(d*e + c*f) + b^2*(d^ 2*e^2 - c*d*e*f + c^2*f^2))*(c + d*x)*(e + f*x) + b^2*Sqrt[-a + (b*c)/d]*d *f*(a + b*x)*(c + d*x)*(e + f*x)*(b*c*f + a*d*f + b*d*(e + 3*f*x)) - (2*I) *(b*c - a*d)*f*(a^2*d^2*f^2 - a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 - c*d*e*f + c^2*f^2))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x ]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] + I*b*(b*c - a*d)*f*(d*e - c*f)*(b*d* e - 2*b*c*f + a*d*f)*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqr t[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt [a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)]))/(15*b^3*Sqrt[-a + (b*c)/d]* d^2*f^2*Sqrt[(a + b*x)*(c + d*x)*(e + f*x)])
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.
| \(\displaystyle \int \sqrt {x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3} \, dx\) |
| \(\Big \downarrow \) 2481 |
| \(\displaystyle \int \sqrt {\frac {(-2 a d f+b c f+b d e) (-a d f-b c f+2 b d e) (a d f-2 b c f+b d e)}{27 b^2 d^2 f^2}+b d f \left (\frac {a d f+b c f+b d e}{3 b d f}+x\right )^3+\frac {\left (3 b d f (a c f+a d e+b c e)-(a d f+b c f+b d e)^2\right ) \left (\frac {a d f+b c f+b d e}{3 b d f}+x\right )}{3 b d f}}d\left (\frac {a d f+b c f+b d e}{3 b d f}+x\right )\) |
Input:
Int[Sqrt[a*c*e + (b*c*e + a*d*e + a*c*f)*x + (b*d*e + b*c*f + a*d*f)*x^2 + b*d*f*x^3],x]
Output:
$Aborted
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]
Time = 0.59 (sec) , antiderivative size = 691, normalized size of antiderivative = 1.18
| method | result | size |
| default | \(\frac {2 x \sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}{5}+\frac {2 \left (\frac {1}{5} a d f +\frac {1}{5} b c f +\frac {1}{5} b d e \right ) \sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}{3 b d f}+\frac {2 \left (\frac {3 a c e}{5}-\frac {2 \left (\frac {1}{5} a d f +\frac {1}{5} b c f +\frac {1}{5} b d e \right ) \left (\frac {1}{2} a c f +\frac {1}{2} a d e +\frac {1}{2} b c e \right )}{3 b d f}\right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}+\frac {2 \left (\frac {2 a c f}{5}+\frac {2 a d e}{5}+\frac {2 b c e}{5}-\frac {2 \left (\frac {1}{5} a d f +\frac {1}{5} b c f +\frac {1}{5} b d e \right ) \left (a d f +b c f +b d e \right )}{3 b d f}\right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \left (\left (-\frac {e}{f}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{b}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}\) | \(691\) |
| elliptic | \(\frac {2 x \sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}{5}+\frac {2 \left (\frac {1}{5} a d f +\frac {1}{5} b c f +\frac {1}{5} b d e \right ) \sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}{3 b d f}+\frac {2 \left (\frac {3 a c e}{5}-\frac {2 \left (\frac {1}{5} a d f +\frac {1}{5} b c f +\frac {1}{5} b d e \right ) \left (\frac {1}{2} a c f +\frac {1}{2} a d e +\frac {1}{2} b c e \right )}{3 b d f}\right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}+\frac {2 \left (\frac {2 a c f}{5}+\frac {2 a d e}{5}+\frac {2 b c e}{5}-\frac {2 \left (\frac {1}{5} a d f +\frac {1}{5} b c f +\frac {1}{5} b d e \right ) \left (a d f +b c f +b d e \right )}{3 b d f}\right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \left (\left (-\frac {e}{f}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{b}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}\) | \(691\) |
Input:
int((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^(1/2), x,method=_RETURNVERBOSE)
Output:
2/5*x*(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c *e)^(1/2)+2/3*(1/5*a*d*f+1/5*b*c*f+1/5*b*d*e)/b/d/f*(b*d*f*x^3+a*d*f*x^2+b *c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)+2*(3/5*a*c*e-2/3*( 1/5*a*d*f+1/5*b*c*f+1/5*b*d*e)/b/d/f*(1/2*a*c*f+1/2*a*d*e+1/2*b*c*e))*(e/f -c/d)*((x+e/f)/(e/f-c/d))^(1/2)*((x+a/b)/(-e/f+a/b))^(1/2)*((x+c/d)/(-e/f+ c/d))^(1/2)/(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e *x+a*c*e)^(1/2)*EllipticF(((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f+a/b) )^(1/2))+2*(2/5*a*c*f+2/5*a*d*e+2/5*b*c*e-2/3*(1/5*a*d*f+1/5*b*c*f+1/5*b*d *e)/b/d/f*(a*d*f+b*c*f+b*d*e))*(e/f-c/d)*((x+e/f)/(e/f-c/d))^(1/2)*((x+a/b )/(-e/f+a/b))^(1/2)*((x+c/d)/(-e/f+c/d))^(1/2)/(b*d*f*x^3+a*d*f*x^2+b*c*f* x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)*((-e/f+a/b)*EllipticE(( (x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f+a/b))^(1/2))-a/b*EllipticF(((x+ e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f+a/b))^(1/2)))
Time = 0.15 (sec) , antiderivative size = 914, normalized size of antiderivative = 1.55 \[ \int \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx =\text {Too large to display} \] Input:
integrate((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^ (1/2),x, algorithm="fricas")
Output:
2/45*((2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4* a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^ 2 + 2*a^3*d^3)*f^3)*sqrt(b*d*f)*weierstrassPInverse(4/3*(b^2*d^2*e^2 - (b^ 2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^2*f^2), - 4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a *b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/ (b*d*f)) + 6*(b^3*d^3*e^2*f - (b^3*c*d^2 + a*b^2*d^3)*e*f^2 + (b^3*c^2*d - a*b^2*c*d^2 + a^2*b*d^3)*f^3)*sqrt(b*d*f)*weierstrassZeta(4/3*(b^2*d^2*e^ 2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^2* f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2* d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2* b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), weierstrassPInverse(4/3*(b^2*d^2* e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^ 2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^ 2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^ 2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f))) + 3*(3*b^3*d^3*f^3*x + b^3*d^3*e*f^2 + (b^3*c*d^2 + a*b ^2*d^3)*f^3)*sqrt(b*d*f*x^3 + a*c*e + (b*d*e + (b*c + a*d)*f)*x^2 + (a*c*f + (b*c + a*d)*e)*x))/(b^3*d^3*f^3)
\[ \int \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\int \sqrt {a c e + b d f x^{3} + x^{2} \left (a d f + b c f + b d e\right ) + x \left (a c f + a d e + b c e\right )}\, dx \] Input:
integrate((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x**2+b*d*f*x**3 )**(1/2),x)
Output:
Integral(sqrt(a*c*e + b*d*f*x**3 + x**2*(a*d*f + b*c*f + b*d*e) + x*(a*c*f + a*d*e + b*c*e)), x)
\[ \int \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\int { \sqrt {b d f x^{3} + a c e + {\left (b d e + b c f + a d f\right )} x^{2} + {\left (b c e + a d e + a c f\right )} x} \,d x } \] Input:
integrate((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^ (1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*d*f*x^3 + a*c*e + (b*d*e + b*c*f + a*d*f)*x^2 + (b*c*e + a*d*e + a*c*f)*x), x)
\[ \int \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\int { \sqrt {b d f x^{3} + a c e + {\left (b d e + b c f + a d f\right )} x^{2} + {\left (b c e + a d e + a c f\right )} x} \,d x } \] Input:
integrate((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^ (1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*d*f*x^3 + a*c*e + (b*d*e + b*c*f + a*d*f)*x^2 + (b*c*e + a*d*e + a*c*f)*x), x)
Timed out. \[ \int \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\int \sqrt {b\,d\,f\,x^3+\left (a\,d\,f+b\,c\,f+b\,d\,e\right )\,x^2+\left (a\,c\,f+a\,d\,e+b\,c\,e\right )\,x+a\,c\,e} \,d x \] Input:
int((x^2*(a*d*f + b*c*f + b*d*e) + x*(a*c*f + a*d*e + b*c*e) + a*c*e + b*d *f*x^3)^(1/2),x)
Output:
int((x^2*(a*d*f + b*c*f + b*d*e) + x*(a*c*f + a*d*e + b*c*e) + a*c*e + b*d *f*x^3)^(1/2), x)
\[ \int \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\text {too large to display} \] Input:
int((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^(1/2), x)
Output:
(2*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*a*c*f + 2*sqrt(e + f*x)*sqrt( c + d*x)*sqrt(a + b*x)*a*d*e + 2*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x) *a*d*f*x + 2*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*b*c*e + 2*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*b*c*f*x + 2*sqrt(e + f*x)*sqrt(c + d*x)*s qrt(a + b*x)*b*d*e*x + int((sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*x**2 )/(a**2*c*d*e*f + a**2*c*d*f**2*x + a**2*d**2*e*f*x + a**2*d**2*f**2*x**2 + a*b*c**2*e*f + a*b*c**2*f**2*x + a*b*c*d*e**2 + 3*a*b*c*d*e*f*x + 2*a*b* c*d*f**2*x**2 + a*b*d**2*e**2*x + 2*a*b*d**2*e*f*x**2 + a*b*d**2*f**2*x**3 + b**2*c**2*e*f*x + b**2*c**2*f**2*x**2 + b**2*c*d*e**2*x + 2*b**2*c*d*e* f*x**2 + b**2*c*d*f**2*x**3 + b**2*d**2*e**2*x**2 + b**2*d**2*e*f*x**3),x) *a**3*d**3*f**3 - 3*int((sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*x**2)/( a**2*c*d*e*f + a**2*c*d*f**2*x + a**2*d**2*e*f*x + a**2*d**2*f**2*x**2 + a *b*c**2*e*f + a*b*c**2*f**2*x + a*b*c*d*e**2 + 3*a*b*c*d*e*f*x + 2*a*b*c*d *f**2*x**2 + a*b*d**2*e**2*x + 2*a*b*d**2*e*f*x**2 + a*b*d**2*f**2*x**3 + b**2*c**2*e*f*x + b**2*c**2*f**2*x**2 + b**2*c*d*e**2*x + 2*b**2*c*d*e*f*x **2 + b**2*c*d*f**2*x**3 + b**2*d**2*e**2*x**2 + b**2*d**2*e*f*x**3),x)*a* b**2*c*d**2*e*f**2 + int((sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*x**2)/ (a**2*c*d*e*f + a**2*c*d*f**2*x + a**2*d**2*e*f*x + a**2*d**2*f**2*x**2 + a*b*c**2*e*f + a*b*c**2*f**2*x + a*b*c*d*e**2 + 3*a*b*c*d*e*f*x + 2*a*b*c* d*f**2*x**2 + a*b*d**2*e**2*x + 2*a*b*d**2*e*f*x**2 + a*b*d**2*f**2*x**...