Integrand size = 38, antiderivative size = 524 \[ \int \frac {\sqrt {a+b x} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {2 \left (5 B d f-\frac {2 a C d f}{b}-4 C (d e+c f)\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{15 d^2 f^2}+\frac {2 C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{5 b d f}-\frac {2 \sqrt {-b c+a d} \left (3 d f (3 b c C e+a C d e+a c C f-5 A b d f)-\frac {(a d f-2 b (d e+c f)) (5 b B d f-2 a C d f-4 b C (d e+c f))}{b}\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b d^{5/2} f^3 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {-b c+a d} (b e-a f) \left (a C d f (d e-c f)+b C \left (8 d^2 e^2+3 c d e f+4 c^2 f^2\right )+5 b d f (3 A d f-B (2 d e+c f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^2 d^{5/2} f^3 \sqrt {c+d x} \sqrt {e+f x}} \] Output:
2/15*(5*B*d*f-2*a*C*d*f/b-4*C*(c*f+d*e))*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(f*x+ e)^(1/2)/d^2/f^2+2/5*C*(b*x+a)^(3/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/d/f-2/1 5*(a*d-b*c)^(1/2)*(3*d*f*(-5*A*b*d*f+C*a*c*f+C*a*d*e+3*C*b*c*e)-(a*d*f-2*b *(c*f+d*e))*(5*b*B*d*f-2*a*C*d*f-4*b*C*(c*f+d*e))/b)*(b*(d*x+c)/(-a*d+b*c) )^(1/2)*(f*x+e)^(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/d^(5/2)/f^3/(d*x+c)^(1/2)/(b*(f*x+e)/(-a* f+b*e))^(1/2)-2/15*(a*d-b*c)^(1/2)*(-a*f+b*e)*(a*C*d*f*(-c*f+d*e)+b*C*(4*c ^2*f^2+3*c*d*e*f+8*d^2*e^2)+5*b*d*f*(3*A*d*f-B*(c*f+2*d*e)))*(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^(5/2)/f^3/(d*x +c)^(1/2)/(f*x+e)^(1/2)
Result contains complex when optimal does not.
Time = 25.86 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {a+b x} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {2 \left (b^2 \sqrt {-a+\frac {b c}{d}} \left (2 a^2 C d^2 f^2+a b d f (-5 B d f+3 C (d e+c f))-b^2 \left (C \left (8 d^2 e^2+7 c d e f+8 c^2 f^2\right )+5 d f (3 A d f-2 B (d e+c f))\right )\right ) (c+d x) (e+f x)-b^2 \sqrt {-a+\frac {b c}{d}} d f (a+b x) (c+d x) (e+f x) (5 b B d f+a C d f+b C (-4 d e-4 c f+3 d f x))+i (b c-a d) f \left (2 a^2 C d^2 f^2+a b d f (-5 B d f+3 C (d e+c f))-b^2 \left (C \left (8 d^2 e^2+7 c d e f+8 c^2 f^2\right )+5 d f (3 A d f-2 B (d e+c f))\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )+i b (b c-a d) f \left (a C d f (-d e+c f)+b C \left (4 d^2 e^2+3 c d e f+8 c^2 f^2\right )+5 b d f (3 A d f-B (d e+2 c f))\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )}{15 b^3 \sqrt {-a+\frac {b c}{d}} d^3 f^3 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \] Input:
Integrate[(Sqrt[a + b*x]*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]), x]
Output:
(-2*(b^2*Sqrt[-a + (b*c)/d]*(2*a^2*C*d^2*f^2 + a*b*d*f*(-5*B*d*f + 3*C*(d* e + c*f)) - b^2*(C*(8*d^2*e^2 + 7*c*d*e*f + 8*c^2*f^2) + 5*d*f*(3*A*d*f - 2*B*(d*e + c*f))))*(c + d*x)*(e + f*x) - b^2*Sqrt[-a + (b*c)/d]*d*f*(a + b *x)*(c + d*x)*(e + f*x)*(5*b*B*d*f + a*C*d*f + b*C*(-4*d*e - 4*c*f + 3*d*f *x)) + I*(b*c - a*d)*f*(2*a^2*C*d^2*f^2 + a*b*d*f*(-5*B*d*f + 3*C*(d*e + c *f)) - b^2*(C*(8*d^2*e^2 + 7*c*d*e*f + 8*c^2*f^2) + 5*d*f*(3*A*d*f - 2*B*( d*e + c*f))))*(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*(a*C*d*f*(-(d*e) + c*f) + b*C*(4*d^2*e^2 + 3*c*d*e*f + 8*c^2*f^2) + 5*b*d*f*(3*A*d*f - B*( d*e + 2*c*f)))*(a + b*x)^(3/2)*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)]))/(15*b^3*Sqrt[-a + (b*c)/d]*d^3*f^ 3*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])
Time = 0.99 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2118, 27, 171, 27, 176, 124, 123, 131, 131, 130}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx\) |
| \(\Big \downarrow \) 2118 |
| \(\displaystyle \frac {2 \int -\frac {b \sqrt {a+b x} (3 b c C e+a C d e+a c C f-5 A b d f-(5 b B d f-2 a C d f-4 b C (d e+c f)) x)}{2 \sqrt {c+d x} \sqrt {e+f x}}dx}{5 b^2 d f}+\frac {2 C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{5 b d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{5 b d f}-\frac {\int \frac {\sqrt {a+b x} (3 b c C e+a C d e+a c C f-5 A b d f-(5 b B d f-2 a C d f-4 b C (d e+c f)) x)}{\sqrt {c+d x} \sqrt {e+f x}}dx}{5 b d f}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {2 C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{5 b d f}-\frac {\frac {2 \int \frac {3 a d f (3 b c C e+a C d e+a c C f-5 A b d f)+(b c e+a d e+a c f) (5 b B d f-2 a C d f-4 b C (d e+c f))+(3 b d f (3 b c C e+a C d e+a c C f-5 A b d f)-(a d f-2 b (d e+c f)) (5 b B d f-2 a C d f-4 b C (d e+c f))) x}{2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 d f}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+5 b B d f-4 b C (c f+d e))}{3 d f}}{5 b d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{5 b d f}-\frac {\frac {\int \frac {3 a d f (3 b c C e+a C d e+a c C f-5 A b d f)+(b c e+a d e+a c f) (5 b B d f-2 a C d f-4 b C (d e+c f))+(3 b d f (3 b c C e+a C d e+a c C f-5 A b d f)-(a d f-2 b (d e+c f)) (5 b B d f-2 a C d f-4 b C (d e+c f))) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 d f}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+5 b B d f-4 b C (c f+d e))}{3 d f}}{5 b d f}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {2 C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{5 b d f}-\frac {\frac {\frac {(b e-a f) \left (a C d f (d e-c f)+5 b d f (3 A d f-B (c f+2 d e))+b C \left (4 c^2 f^2+3 c d e f+8 d^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {(3 b d f (a c C f+a C d e-5 A b d f+3 b c C e)-(a d f-2 b (c f+d e)) (-2 a C d f+5 b B d f-4 b C (c f+d e))) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}}dx}{f}}{3 d f}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+5 b B d f-4 b C (c f+d e))}{3 d f}}{5 b d f}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {2 C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{5 b d f}-\frac {\frac {\frac {(b e-a f) \left (a C d f (d e-c f)+5 b d f (3 A d f-B (c f+2 d e))+b C \left (4 c^2 f^2+3 c d e f+8 d^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {\sqrt {e+f x} \sqrt {\frac {b (c+d x)}{b c-a d}} (3 b d f (a c C f+a C d e-5 A b d f+3 b c C e)-(a d f-2 b (c f+d e)) (-2 a C d f+5 b B d f-4 b C (c f+d e))) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}dx}{f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 d f}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+5 b B d f-4 b C (c f+d e))}{3 d f}}{5 b d f}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {2 C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{5 b d f}-\frac {\frac {\frac {(b e-a f) \left (a C d f (d e-c f)+5 b d f (3 A d f-B (c f+2 d e))+b C \left (4 c^2 f^2+3 c d e f+8 d^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} (3 b d f (a c C f+a C d e-5 A b d f+3 b c C e)-(a d f-2 b (c f+d e)) (-2 a C d f+5 b B d f-4 b C (c f+d e))) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 d f}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+5 b B d f-4 b C (c f+d e))}{3 d f}}{5 b d f}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {2 C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{5 b d f}-\frac {\frac {\frac {(b e-a f) \sqrt {\frac {b (c+d x)}{b c-a d}} \left (a C d f (d e-c f)+5 b d f (3 A d f-B (c f+2 d e))+b C \left (4 c^2 f^2+3 c d e f+8 d^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}}dx}{f \sqrt {c+d x}}+\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} (3 b d f (a c C f+a C d e-5 A b d f+3 b c C e)-(a d f-2 b (c f+d e)) (-2 a C d f+5 b B d f-4 b C (c f+d e))) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 d f}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+5 b B d f-4 b C (c f+d e))}{3 d f}}{5 b d f}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {2 C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{5 b d f}-\frac {\frac {\frac {(b e-a f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \left (a C d f (d e-c f)+5 b d f (3 A d f-B (c f+2 d e))+b C \left (4 c^2 f^2+3 c d e f+8 d^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}dx}{f \sqrt {c+d x} \sqrt {e+f x}}+\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} (3 b d f (a c C f+a C d e-5 A b d f+3 b c C e)-(a d f-2 b (c f+d e)) (-2 a C d f+5 b B d f-4 b C (c f+d e))) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 d f}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+5 b B d f-4 b C (c f+d e))}{3 d f}}{5 b d f}\) |
| \(\Big \downarrow \) 130 |
| \(\displaystyle \frac {2 C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{5 b d f}-\frac {\frac {\frac {2 \sqrt {a d-b c} (b e-a f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \left (a C d f (d e-c f)+5 b d f (3 A d f-B (c f+2 d e))+b C \left (4 c^2 f^2+3 c d e f+8 d^2 e^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {e+f x}}+\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} (3 b d f (a c C f+a C d e-5 A b d f+3 b c C e)-(a d f-2 b (c f+d e)) (-2 a C d f+5 b B d f-4 b C (c f+d e))) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 d f}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+5 b B d f-4 b C (c f+d e))}{3 d f}}{5 b d f}\) |
Input:
Int[(Sqrt[a + b*x]*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]),x]
Output:
(2*C*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x])/(5*b*d*f) - ((-2*(5*b*B* d*f - 2*a*C*d*f - 4*b*C*(d*e + c*f))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*d*f) + ((2*Sqrt[-(b*c) + a*d]*(3*b*d*f*(3*b*c*C*e + a*C*d*e + a*c *C*f - 5*A*b*d*f) - (a*d*f - 2*b*(d*e + c*f))*(5*b*B*d*f - 2*a*C*d*f - 4*b *C*(d*e + c*f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[A rcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b*Sqrt[d]*f*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) + ( 2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(a*C*d*f*(d*e - c*f) + b*C*(8*d^2*e^2 + 3 *c*d*e*f + 4*c^2*f^2) + 5*b*d*f*(3*A*d*f - B*(2*d*e + c*f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[ d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/( b*Sqrt[d]*f*Sqrt[c + d*x]*Sqrt[e + f*x]))/(3*d*f))/(5*b*d*f)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d *x]*Sqrt[b*((e + f*x)/(b*e - a*f))])) Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x /(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && Gt Q[b/(b*e - a*f), 0]) && !LtQ[-(b*c - a*d)/d, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ [b/(b*e - a*f), 0] && SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f *x] && (PosQ[-(b*c - a*d)/d] || NegQ[-(b*e - a*f)/f])
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[Sqrt[b*((c + d*x)/(b*c - a*d))]/Sqrt[c + d*x] Int[1/(Sq rt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e + f*x]), x ], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[(b*c - a*d)/b, 0] && Simpler Q[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + q + 1)) Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x]
Time = 4.46 (sec) , antiderivative size = 812, normalized size of antiderivative = 1.55
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (f x +e \right ) \left (b x +a \right ) \left (x d +c \right )}\, \left (\frac {2 C 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 d f}+\frac {2 \left (B b +C a -\frac {2 C \left (2 a d f +2 b c f +2 b d e \right )}{5 d f}\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 (A a -\frac {2 C a c e}{5 d f}-\frac {2 \left (B b +C a -\frac {2 C \left (2 a d f +2 b c f +2 b d e \right )}{5 d f}\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 {c}{d}-\frac {a}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {a}{b}}{-\frac {c}{d}+\frac {e}{f}}}\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 (A b +B a -\frac {2 C \left (\frac {3}{2} a c f +\frac {3}{2} a d e +\frac {3}{2} b c e \right )}{5 d f}-\frac {2 \left (B b +C a -\frac {2 C \left (2 a d f +2 b c f +2 b d e \right )}{5 d f}\right ) \left (a d f +b c f +b d e \right )}{3 b d f}\right ) \left (\frac {c}{d}-\frac {a}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \left (\left (-\frac {c}{d}+\frac {e}{f}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {a}{b}}{-\frac {c}{d}+\frac {e}{f}}}\right )-\frac {e \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {a}{b}}{-\frac {c}{d}+\frac {e}{f}}}\right )}{f}\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}}\right )}{\sqrt {b x +a}\, \sqrt {x d +c}\, \sqrt {f x +e}}\) | \(812\) |
| default | \(\text {Expression too large to display}\) | \(6143\) |
Input:
int((b*x+a)^(1/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x,method=_RETU RNVERBOSE)
Output:
((f*x+e)*(b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)* (2/5*C/d/f*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*(B*b+C*a-2/5*C/d/f*(2*a*d*f+2*b*c*f+2*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*(A*a-2/5*C/d/f*a*c*e-2/3*(B*b+C*a-2/5*C/d/f*(2*a*d*f+2*b*c*f+2*b*d*e) )/b/d/f*(1/2*a*c*f+1/2*a*d*e+1/2*b*c*e))*(c/d-a/b)*((x+c/d)/(c/d-a/b))^(1/ 2)*((x+e/f)/(-c/d+e/f))^(1/2)*((x+a/b)/(-c/d+a/b))^(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 +c/d)/(c/d-a/b))^(1/2),((-c/d+a/b)/(-c/d+e/f))^(1/2))+2*(A*b+B*a-2/5*C/d/f *(3/2*a*c*f+3/2*a*d*e+3/2*b*c*e)-2/3*(B*b+C*a-2/5*C/d/f*(2*a*d*f+2*b*c*f+2 *b*d*e))/b/d/f*(a*d*f+b*c*f+b*d*e))*(c/d-a/b)*((x+c/d)/(c/d-a/b))^(1/2)*(( x+e/f)/(-c/d+e/f))^(1/2)*((x+a/b)/(-c/d+a/b))^(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)*((-c/d+e/f)*Ellipt icE(((x+c/d)/(c/d-a/b))^(1/2),((-c/d+a/b)/(-c/d+e/f))^(1/2))-e/f*EllipticF (((x+c/d)/(c/d-a/b))^(1/2),((-c/d+a/b)/(-c/d+e/f))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1036 vs. \(2 (469) = 938\).
Time = 0.13 (sec) , antiderivative size = 1036, normalized size of antiderivative = 1.98 \[ \int \frac {\sqrt {a+b x} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^(1/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algor ithm="fricas")
Output:
2/45*(3*(3*C*b^3*d^3*f^3*x - 4*C*b^3*d^3*e*f^2 - (4*C*b^3*c*d^2 - (C*a*b^2 + 5*B*b^3)*d^3)*f^3)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e) - (8*C*b^3 *d^3*e^3 + (3*C*b^3*c*d^2 - (7*C*a*b^2 + 10*B*b^3)*d^3)*e^2*f + (3*C*b^3*c ^2*d - (2*C*a*b^2 + 5*B*b^3)*c*d^2 - (2*C*a^2*b - 10*B*a*b^2 - 15*A*b^3)*d ^3)*e*f^2 + (8*C*b^3*c^3 - (7*C*a*b^2 + 10*B*b^3)*c^2*d - (2*C*a^2*b - 10* B*a*b^2 - 15*A*b^3)*c*d^2 - (2*C*a^3 - 5*B*a^2*b + 30*A*a*b^2)*d^3)*f^3)*s qrt(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*(8*C*b^3 *d^3*e^2*f + (7*C*b^3*c*d^2 - (3*C*a*b^2 + 10*B*b^3)*d^3)*e*f^2 + (8*C*b^3 *c^2*d - (3*C*a*b^2 + 10*B*b^3)*c*d^2 - (2*C*a^2*b - 5*B*a*b^2 - 15*A*b^3) *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...
\[ \int \frac {\sqrt {a+b x} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {\sqrt {a + b x} \left (A + B x + C x^{2}\right )}{\sqrt {c + d x} \sqrt {e + f x}}\, dx \] Input:
integrate((b*x+a)**(1/2)*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)
Output:
Integral(sqrt(a + b*x)*(A + B*x + C*x**2)/(sqrt(c + d*x)*sqrt(e + f*x)), x )
\[ \int \frac {\sqrt {a+b x} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {b x + a}}{\sqrt {d x + c} \sqrt {f x + e}} \,d x } \] Input:
integrate((b*x+a)^(1/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algor ithm="maxima")
Output:
integrate((C*x^2 + B*x + A)*sqrt(b*x + a)/(sqrt(d*x + c)*sqrt(f*x + e)), x )
\[ \int \frac {\sqrt {a+b x} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {b x + a}}{\sqrt {d x + c} \sqrt {f x + e}} \,d x } \] Input:
integrate((b*x+a)^(1/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algor ithm="giac")
Output:
integrate((C*x^2 + B*x + A)*sqrt(b*x + a)/(sqrt(d*x + c)*sqrt(f*x + e)), x )
Timed out. \[ \int \frac {\sqrt {a+b x} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {\sqrt {a+b\,x}\,\left (C\,x^2+B\,x+A\right )}{\sqrt {e+f\,x}\,\sqrt {c+d\,x}} \,d x \] Input:
int(((a + b*x)^(1/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(c + d*x)^(1/2)), x)
Output:
int(((a + b*x)^(1/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {\sqrt {a+b x} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {too large to display} \] Input:
int((b*x+a)^(1/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x)
Output:
(20*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*a*b*d*f - 6*sqrt(e + f*x)*sq rt(c + d*x)*sqrt(a + b*x)*a*c**2*f - 6*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*a*c*d*e + 4*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*a*c*d*f*x - 6 *sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*b*c**2*e + 4*sqrt(e + f*x)*sqrt (c + d*x)*sqrt(a + b*x)*b*c**2*f*x + 4*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*b*c*d*e*x + 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 **3*c*d**3*f**3 - 20*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*b**2*d**3*f**3 + 5*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...