Integrand size = 59, antiderivative size = 505 \[ \int \frac {A+B x+C x^2}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx=\frac {2 C (a+b x) (c+d x) (e+f x)}{3 b d f \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}}+\frac {2 \sqrt {-b c+a d} \left (3 B d f-\frac {2 C (b d e+b c f+a d f)}{b}\right ) \sqrt {a+b x} \sqrt {\frac {b (c+d x)}{b c-a d}} (e+f x) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b d^{3/2} f^2 \sqrt {\frac {b (e+f x)}{b e-a f}} \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}}+\frac {2 \sqrt {-b c+a d} (a C f (d e-c f)-b (3 d f (B e-A f)-C e (2 d e+c f))) \sqrt {a+b x} \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^2 d^{3/2} f^2 \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \] Output:
2/3*C*(b*x+a)*(d*x+c)*(f*x+e)/b/d/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)+2/3*(a*d-b*c)^(1/2)*(3*B*d*f-2*C*(a*d*f+b* c*f+b*d*e)/b)*(b*x+a)^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(f*x+e)*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/d^(3/2)/f^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)+2/3*(a*d-b*c)^(1/2)*(a*C*f*(-c*f+d* e)-b*(3*d*f*(-A*f+B*e)-C*e*(c*f+2*d*e)))*(b*x+a)^(1/2)*(b*(d*x+c)/(-a*d+b* c))^(1/2)*(b*(f*x+e)/(-a*f+b*e))^(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/(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)
Result contains complex when optimal does not.
Time = 28.24 (sec) , antiderivative size = 416, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x+C x^2}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx=\frac {(a+b x) \left (2 b^2 C d f (c+d x) (e+f x)-\frac {2 b^2 (-3 b B d f+2 a C d f+2 b C (d e+c f)) (c+d x) (e+f x)}{a+b x}+2 i \sqrt {-a+\frac {b c}{d}} d f (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )+\frac {2 i b f \left (a C d (-d e+c f)+b \left (2 c^2 C f+3 A d^2 f+c d (C e-3 B f)\right )\right ) \sqrt {a+b x} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )}{\sqrt {-a+\frac {b c}{d}}}\right )}{3 b^3 d^2 f^2 \sqrt {(a+b x) (c+d x) (e+f x)}} \] Input:
Integrate[(A + B*x + C*x^2)/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:
((a + b*x)*(2*b^2*C*d*f*(c + d*x)*(e + f*x) - (2*b^2*(-3*b*B*d*f + 2*a*C*d *f + 2*b*C*(d*e + c*f))*(c + d*x)*(e + f*x))/(a + b*x) + (2*I)*Sqrt[-a + ( b*c)/d]*d*f*(3*b*B*d*f - 2*a*C*d*f - 2*b*C*(d*e + c*f))*Sqrt[a + b*x]*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)] + ((2*I)*b*f*(a*C*d*(-(d*e) + c*f) + b*(2*c^2*C*f + 3*A*d^2*f + c*d*(C *e - 3*B*f)))*Sqrt[a + b*x]*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(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)])/Sqrt[-a + (b*c)/d]))/(3*b^3*d^2*f^2*S qrt[(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.
| \(\displaystyle \int \frac {A+B x+C x^2}{\sqrt {x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3}} \, dx\) |
| \(\Big \downarrow \) 2526 |
| \(\displaystyle \frac {\int -\frac {a C (d e+c f)+b (c C e-3 A d f)-(3 b B d f-2 a C d f-2 b C (d e+c f)) x}{\sqrt {b d f x^3+(b d e+b c f+a d f) x^2+(b c e+a d e+a c f) x+a c e}}dx}{3 b d f}+\frac {2 C \sqrt {x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3}}{3 b d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 C \sqrt {x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3}}{3 b d f}-\frac {\int \frac {b c C e+a C d e+a c C f-3 A b d f-(3 b B d f-2 a C d f-2 b C (d e+c f)) x}{\sqrt {b d f x^3+(b d e+b c f+a d f) x^2+(b c e+a d e+a c f) x+a c e}}dx}{3 b d f}\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle \frac {2 C \sqrt {x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3}}{3 b d f}-\frac {\int \frac {\frac {3 b d f (b c C e+a C d e+a c C f-3 A b d f)-(b d e+b c f+a d f) (-3 b B d f+2 a C d f+2 b C (d e+c f))}{3 b d f}+(-3 b B d f+2 a C d f+2 b C (d e+c f)) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{\sqrt {b d f \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3+\frac {\left (3 b d f (b c e+a d e+a c f)-(b d e+b c f+a d f)^2\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{3 b d f}+\frac {(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{27 b^2 d^2 f^2}}}d\left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{3 b d f}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 C \sqrt {x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3}}{3 b d f}-\frac {\int \frac {\frac {3 b d f (b c C e+a C d e+a c C f-3 A b d f)-(b d e+b c f+a d f) (-3 b B d f+2 a C d f+2 b C (d e+c f))}{3 b d f}+(-3 b B d f+2 a C d f+2 b C (d e+c f)) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{\sqrt {b d f \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3+\frac {1}{3} \left (3 (b c e+a d e+a c f)-\frac {(b d e+b c f+a d f)^2}{b d f}\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+\frac {(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{27 b^2 d^2 f^2}}}d\left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{3 b d f}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \sqrt {x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3}}{3 b d f}-\frac {\int \left (\frac {-\left (\left (C \left (2 d^2 e^2+c d f e+2 c^2 f^2\right )+3 d f (3 A d f-B (d e+c f))\right ) b^2\right )-a d f (C d e+c C f-3 B d f) b-2 a^2 C d^2 f^2}{3 b d f \sqrt {b d f \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3+\frac {1}{3} \left (3 (b c e+a d e+a c f)-\frac {(b d e+b c f+a d f)^2}{b d f}\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+\frac {(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{27 b^2 d^2 f^2}}}+\frac {(-3 b B d f+2 a C d f+2 b C (d e+c f)) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{\sqrt {b d f \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3+\frac {1}{3} \left (3 (b c e+a d e+a c f)-\frac {(b d e+b c f+a d f)^2}{b d f}\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+\frac {(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{27 b^2 d^2 f^2}}}\right )d\left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{3 b d f}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \sqrt {x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3}}{3 b d f}-\frac {\int \left (\frac {\sqrt {3} \left (-\left (\left (C \left (2 d^2 e^2+c d f e+2 c^2 f^2\right )+3 d f (3 A d f-B (d e+c f))\right ) b^2\right )-a d f (C d e+c C f-3 B d f) b-2 a^2 C d^2 f^2\right )}{b d f \sqrt {\frac {27 b^3 d^3 f^3 \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3-9 b d f \left (\left (d^2 e^2-c d f e+c^2 f^2\right ) b^2-a d f (d e+c f) b+a^2 d^2 f^2\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{b^2 d^2 f^2}}}+\frac {3 \sqrt {3} (-3 b B d f+2 a C d f+2 b C (d e+c f)) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{\sqrt {\frac {27 b^3 d^3 f^3 \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3-9 b d f \left (\left (d^2 e^2-c d f e+c^2 f^2\right ) b^2-a d f (d e+c f) b+a^2 d^2 f^2\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{b^2 d^2 f^2}}}\right )d\left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{3 b d f}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {2 C \sqrt {x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3}}{3 b d f}-\frac {\int \left (\frac {\sqrt {3} \left (-\left (\left (C \left (2 d^2 e^2+c d f e+2 c^2 f^2\right )+3 d f (3 A d f-B (d e+c f))\right ) b^2\right )-a d f (C d e+c C f-3 B d f) b-2 a^2 C d^2 f^2\right )}{b d f \sqrt {\frac {27 b^3 d^3 f^3 \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3-9 b d f \left (\left (d^2 e^2-c d f e+c^2 f^2\right ) b^2-a d f (d e+c f) b+a^2 d^2 f^2\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{b^2 d^2 f^2}}}+\frac {3 \sqrt {3} (-3 b B d f+2 a C d f+2 b C (d e+c f)) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{\sqrt {\frac {27 b^3 d^3 f^3 \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3-9 b d f \left (\left (d^2 e^2-c d f e+c^2 f^2\right ) b^2-a d f (d e+c f) b+a^2 d^2 f^2\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{b^2 d^2 f^2}}}\right )d\left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{3 b d f}\) |
Input:
Int[(A + B*x + C*x^2)/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[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 , x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*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, x + c/(3*d)] /; NeQ[c , 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] }, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n])), x] + Simp [1/(n*Coeff[Qn, x, n]) Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x , m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm , x] && PolyQ[Qn, x] && NeQ[p, -1]
Time = 4.12 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.14
| method | result | size |
| elliptic | \(\frac {2 C \sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}{3 b d f}+\frac {2 \left (A -\frac {2 C \left (\frac {1}{2} a c f +\frac {1}{2} a d e +\frac {1}{2} b c e \right )}{3 b d f}\right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}+\frac {2 \left (B -\frac {2 C \left (a d f +b c f +b d e \right )}{3 b d f}\right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \left (\left (-\frac {e}{f}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{b}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}\) | \(575\) |
| default | \(\text {Expression too large to display}\) | \(1023\) |
Input:
int((C*x^2+B*x+A)/(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:
2/3*C/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*(A-2/3*C/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*(B-2/3*C/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*El lipticF(((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 = 834, normalized size of antiderivative = 1.65 \[ \int \frac {A+B x+C x^2}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx =\text {Too large to display} \] Input:
integrate((C*x^2+B*x+A)/(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:
2/9*(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)*C*b^2*d^2*f^2 + (2*C*b^2*d^2*e^2 + (C*b^2*c*d + (C*a*b - 3* B*b^2)*d^2)*e*f + (2*C*b^2*c^2 + (C*a*b - 3*B*b^2)*c*d + (2*C*a^2 - 3*B*a* b + 9*A*b^2)*d^2)*f^2)*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)) + 3*(2*C*b^2*d^2*e*f + (2*C*b^2*c*d + (2*C*a*b - 3*B*b^2)*d^2) *f^2)*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))))/(b^3* d^3*f^3)
\[ \int \frac {A+B x+C x^2}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx=\int \frac {A + B x + C x^{2}}{\sqrt {\left (a + b x\right ) \left (c + d x\right ) \left (e + f x\right )}}\, dx \] Input:
integrate((C*x**2+B*x+A)/(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((A + B*x + C*x**2)/sqrt((a + b*x)*(c + d*x)*(e + f*x)), x)
\[ \int \frac {A+B x+C x^2}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {b d f x^{3} + a c e + {\left (b d e + b c f + a d f\right )} x^{2} + {\left (b c e + a d e + a c f\right )} x}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/(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((C*x^2 + B*x + A)/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)
\[ \int \frac {A+B x+C x^2}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {b d f x^{3} + a c e + {\left (b d e + b c f + a d f\right )} x^{2} + {\left (b c e + a d e + a c f\right )} x}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/(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((C*x^2 + B*x + A)/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. \[ \int \frac {A+B x+C x^2}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx=\int \frac {C\,x^2+B\,x+A}{\sqrt {b\,d\,f\,x^3+\left (a\,d\,f+b\,c\,f+b\,d\,e\right )\,x^2+\left (a\,c\,f+a\,d\,e+b\,c\,e\right )\,x+a\,c\,e}} \,d x \] Input:
int((A + B*x + C*x^2)/(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((A + B*x + C*x^2)/(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)
\[ \int \frac {A+B x+C x^2}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx=\text {too large to display} \] Input:
int((C*x^2+B*x+A)/(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)*b + 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**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**2*c*d**2*f**2 - 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*d**2*f**2 + 4*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*c**2*d*f**2 + 4*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...