Integrand size = 38, antiderivative size = 1116 \[ \int \frac {A+B x+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac {2 \left (2 a^3 C d f+a b^2 (10 c C e+B d e+B c f-8 A d f)-b^3 (5 B c e-4 A (d e+c f))+3 a^2 b (B d f-2 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{15 b (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac {2 \left (2 a^4 C d^2 f^2+a^3 b d f (3 B d f-7 C (d e+c f))-b^4 \left (8 A d^2 e^2-c d e (10 B e-7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a b^3 \left (d^2 e (2 B e-23 A f)-2 c^2 f (5 C e-B f)-c d \left (10 C e^2-33 B e f+23 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-13 c d e f+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{15 b (b c-a d)^3 (b e-a f)^3 \sqrt {a+b x}}+\frac {2 \sqrt {d} \left (2 a^4 C d^2 f^2+a^3 b d f (3 B d f-7 C (d e+c f))-b^4 \left (8 A d^2 e^2-c d e (10 B e-7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a b^3 \left (d^2 e (2 B e-23 A f)-2 c^2 f (5 C e-B f)-c d \left (10 C e^2-33 B e f+23 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-13 c d e f+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right )\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^2 (-b c+a d)^{5/2} (b e-a f)^3 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \sqrt {d} \left (a^3 C d f (d e-c f)+b^3 \left (8 A d^2 e^2-c d e (10 B e-3 A f)+c^2 \left (15 C e^2-5 B e f+4 A f^2\right )\right )+a b^2 \left (d^2 e (2 B e-19 A f)-c^2 f (20 C e-B f)-c d \left (10 C e^2-27 B e f+11 A f^2\right )\right )-3 a^2 b \left (d f (2 B d e+3 B c f-5 A d f)-C \left (d^2 e^2+c d e f+3 c^2 f^2\right )\right )\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 (-b c+a d)^{5/2} (b e-a f)^2 \sqrt {c+d x} \sqrt {e+f x}} \]
-2/5*(A*b^2-a*(B*b-C*a))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(-a*f+b* e)/(b*x+a)^(5/2)+2/15*(2*a^3*C*d*f+a*b^2*(-8*A*d*f+B*c*f+B*d*e+10*C*c*e)-b ^3*(5*B*c*e-4*A*(c*f+d*e))+3*a^2*b*(B*d*f-2*C*(c*f+d*e)))*(d*x+c)^(1/2)*(f *x+e)^(1/2)/b/(-a*d+b*c)^2/(-a*f+b*e)^2/(b*x+a)^(3/2)+2/15*(2*a^4*C*d^2*f^ 2+a^3*b*d*f*(3*B*d*f-7*C*(c*f+d*e))-b^4*(8*A*d^2*e^2-c*d*e*(-7*A*f+10*B*e) +c^2*(8*A*f^2-10*B*e*f+15*C*e^2))-a*b^3*(d^2*e*(-23*A*f+2*B*e)-2*c^2*f*(-B *f+5*C*e)-c*d*(23*A*f^2-33*B*e*f+10*C*e^2))-a^2*b^2*(C*(3*c^2*f^2-13*c*d*e *f+3*d^2*e^2)+d*f*(23*A*d*f-7*B*(c*f+d*e))))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b /(-a*d+b*c)^3/(-a*f+b*e)^3/(b*x+a)^(1/2)+2/15*(2*a^4*C*d^2*f^2+a^3*b*d*f*( 3*B*d*f-7*C*(c*f+d*e))-b^4*(8*A*d^2*e^2-c*d*e*(-7*A*f+10*B*e)+c^2*(8*A*f^2 -10*B*e*f+15*C*e^2))-a*b^3*(d^2*e*(-23*A*f+2*B*e)-2*c^2*f*(-B*f+5*C*e)-c*d *(23*A*f^2-33*B*e*f+10*C*e^2))-a^2*b^2*(C*(3*c^2*f^2-13*c*d*e*f+3*d^2*e^2) +d*f*(23*A*d*f-7*B*(c*f+d*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))*d^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1 /2)*(f*x+e)^(1/2)/b^2/(a*d-b*c)^(5/2)/(-a*f+b*e)^3/(d*x+c)^(1/2)/(b*(f*x+e )/(-a*f+b*e))^(1/2)+2/15*(a^3*C*d*f*(-c*f+d*e)+b^3*(8*A*d^2*e^2-c*d*e*(-3* A*f+10*B*e)+c^2*(4*A*f^2-5*B*e*f+15*C*e^2))+a*b^2*(d^2*e*(-19*A*f+2*B*e)-c ^2*f*(-B*f+20*C*e)-c*d*(11*A*f^2-27*B*e*f+10*C*e^2))-3*a^2*b*(d*f*(-5*A*d* f+3*B*c*f+2*B*d*e)-C*(3*c^2*f^2+c*d*e*f+d^2*e^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))*d^(1/2)*(b*...
Result contains complex when optimal does not.
Time = 33.54 (sec) , antiderivative size = 1258, normalized size of antiderivative = 1.13 \[ \int \frac {A+B x+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {2 \left (b^2 \sqrt {-a+\frac {b c}{d}} \left (2 a^4 C d^2 f^2+a^3 b d f (3 B d f-7 C (d e+c f))-b^4 \left (8 A d^2 e^2+c d e (-10 B e+7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )+a b^3 \left (d^2 e (-2 B e+23 A f)-2 c^2 f (-5 C e+B f)+c d \left (10 C e^2-33 B e f+23 A f^2\right )\right )+a^2 b^2 \left (d f (7 B d e+7 B c f-23 A d f)+C \left (-3 d^2 e^2+13 c d e f-3 c^2 f^2\right )\right )\right ) (a+b x)^2 (c+d x) (e+f x)+b^2 \sqrt {-a+\frac {b c}{d}} (c+d x) (e+f x) \left (3 \left (A b^2+a (-b B+a C)\right ) (b c-a d)^2 (b e-a f)^2+(b c-a d) (b e-a f) \left (-2 a^3 C d f-a b^2 (10 c C e+B d e+B c f-8 A d f)+b^3 (5 B c e-4 A (d e+c f))+3 a^2 b (-B d f+2 C (d e+c f))\right ) (a+b x)+\left (-2 a^4 C d^2 f^2+a^3 b d f (-3 B d f+7 C (d e+c f))+a b^3 \left (d^2 e (2 B e-23 A f)+2 c^2 f (-5 C e+B f)+c d \left (-10 C e^2+33 B e f-23 A f^2\right )\right )+b^4 \left (8 A d^2 e^2+c d e (-10 B e+7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )+a^2 b^2 \left (C \left (3 d^2 e^2-13 c d e f+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right )\right ) (a+b x)^2\right )+i (b c-a d) f \left (2 a^4 C d^2 f^2+a^3 b d f (3 B d f-7 C (d e+c f))-b^4 \left (8 A d^2 e^2+c d e (-10 B e+7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )+a b^3 \left (d^2 e (-2 B e+23 A f)-2 c^2 f (-5 C e+B f)+c d \left (10 C e^2-33 B e f+23 A f^2\right )\right )+a^2 b^2 \left (d f (7 B d e+7 B c f-23 A d f)+C \left (-3 d^2 e^2+13 c d e f-3 c^2 f^2\right )\right )\right ) (a+b x)^{7/2} \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^3 C d f (-d e+c f)+a b^2 \left (d^2 e (B e-11 A f)+2 c^2 f (-5 C e+B f)+c d \left (-20 C e^2+27 B e f-19 A f^2\right )\right )+b^3 \left (4 A d^2 e^2+c d e (-5 B e+3 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )+3 a^2 b \left (d f (-3 B d e-2 B c f+5 A d f)+C \left (3 d^2 e^2+c d e f+c^2 f^2\right )\right )\right ) (a+b x)^{7/2} \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}} (b c-a d)^3 (b e-a f)^3 (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \]
(-2*(b^2*Sqrt[-a + (b*c)/d]*(2*a^4*C*d^2*f^2 + a^3*b*d*f*(3*B*d*f - 7*C*(d *e + c*f)) - b^4*(8*A*d^2*e^2 + c*d*e*(-10*B*e + 7*A*f) + c^2*(15*C*e^2 - 10*B*e*f + 8*A*f^2)) + a*b^3*(d^2*e*(-2*B*e + 23*A*f) - 2*c^2*f*(-5*C*e + B*f) + c*d*(10*C*e^2 - 33*B*e*f + 23*A*f^2)) + a^2*b^2*(d*f*(7*B*d*e + 7*B *c*f - 23*A*d*f) + C*(-3*d^2*e^2 + 13*c*d*e*f - 3*c^2*f^2)))*(a + b*x)^2*( c + d*x)*(e + f*x) + b^2*Sqrt[-a + (b*c)/d]*(c + d*x)*(e + f*x)*(3*(A*b^2 + a*(-(b*B) + a*C))*(b*c - a*d)^2*(b*e - a*f)^2 + (b*c - a*d)*(b*e - a*f)* (-2*a^3*C*d*f - a*b^2*(10*c*C*e + B*d*e + B*c*f - 8*A*d*f) + b^3*(5*B*c*e - 4*A*(d*e + c*f)) + 3*a^2*b*(-(B*d*f) + 2*C*(d*e + c*f)))*(a + b*x) + (-2 *a^4*C*d^2*f^2 + a^3*b*d*f*(-3*B*d*f + 7*C*(d*e + c*f)) + a*b^3*(d^2*e*(2* B*e - 23*A*f) + 2*c^2*f*(-5*C*e + B*f) + c*d*(-10*C*e^2 + 33*B*e*f - 23*A* f^2)) + b^4*(8*A*d^2*e^2 + c*d*e*(-10*B*e + 7*A*f) + c^2*(15*C*e^2 - 10*B* e*f + 8*A*f^2)) + a^2*b^2*(C*(3*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) + d*f*(2 3*A*d*f - 7*B*(d*e + c*f))))*(a + b*x)^2) + I*(b*c - a*d)*f*(2*a^4*C*d^2*f ^2 + a^3*b*d*f*(3*B*d*f - 7*C*(d*e + c*f)) - b^4*(8*A*d^2*e^2 + c*d*e*(-10 *B*e + 7*A*f) + c^2*(15*C*e^2 - 10*B*e*f + 8*A*f^2)) + a*b^3*(d^2*e*(-2*B* e + 23*A*f) - 2*c^2*f*(-5*C*e + B*f) + c*d*(10*C*e^2 - 33*B*e*f + 23*A*f^2 )) + a^2*b^2*(d*f*(7*B*d*e + 7*B*c*f - 23*A*d*f) + C*(-3*d^2*e^2 + 13*c*d* e*f - 3*c^2*f^2)))*(a + b*x)^(7/2)*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]/Sqr...
Time = 2.41 (sec) , antiderivative size = 1176, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2117, 27, 169, 27, 169, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx\) |
| \(\Big \downarrow \) 2117 |
| \(\displaystyle -\frac {2 \int -\frac {C (d e+c f) a^2-b (5 c C e+B d e+B c f-5 A d f) a+b^2 (5 B c e-4 A (d e+c f))+b \left (\frac {2 C d f a^2}{b}-5 C d e a-5 c C f a+3 B d f a+5 b c C e-3 A b d f\right ) x}{2 b (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}dx}{5 (b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{5 b (a+b x)^{5/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {C (d e+c f) a^2-b (5 c C e+B d e+B c f-5 A d f) a+b^2 (5 B c e-4 A (d e+c f))+b \left (\frac {2 C d f a^2}{b}+3 B d f a-5 C (d e+c f) a+b (5 c C e-3 A d f)\right ) x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}dx}{5 b (b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{5 b (a+b x)^{5/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+3 a^2 b (B d f-2 C (c f+d e))+a b^2 (-8 A d f+B c f+B d e+10 c C e)-b^3 (5 B c e-4 A (c f+d e))\right )}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}-\frac {2 \int -\frac {C d f (d e+c f) a^3+3 b \left (C \left (d^2 e^2-c d f e+c^2 f^2\right )+d f (5 A d f-2 B (d e+c f))\right ) a^2+b^2 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-28 B f e+19 A f^2\right ) c+d^2 e (2 B e-19 A f)\right ) a+b^3 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )+d f \left (2 C d f a^3+3 b (B d f-2 C (d e+c f)) a^2+b^2 (10 c C e+B d e+B c f-8 A d f) a-b^3 (5 B c e-4 A (d e+c f))\right ) x}{2 (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 (b c-a d) (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{5 b (a+b x)^{5/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {C d f (d e+c f) a^3+3 b \left (C \left (d^2 e^2-c d f e+c^2 f^2\right )+d f (5 A d f-2 B (d e+c f))\right ) a^2+b^2 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-28 B f e+19 A f^2\right ) c+d^2 e (2 B e-19 A f)\right ) a+b^3 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )+d f \left (2 C d f a^3+3 b (B d f-2 C (d e+c f)) a^2+b^2 (10 c C e+B d e+B c f-8 A d f) a-b^3 (5 B c e-4 A (d e+c f))\right ) x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 (b c-a d) (b e-a f)}+\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+3 a^2 b (B d f-2 C (c f+d e))+a b^2 (-8 A d f+B c f+B d e+10 c C e)-b^3 (5 B c e-4 A (c f+d e))\right )}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{5 b (a+b x)^{5/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 C d f a^3+3 b (B d f-2 C (d e+c f)) a^2+b^2 (10 c C e+B d e+B c f-8 A d f) a-b^3 (5 B c e-4 A (d e+c f))\right )}{3 (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac {\frac {2 \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {2 \int \frac {d f \left (C d f (d e+c f) a^4-b \left (C \left (9 d^2 e^2+11 c d f e+9 c^2 f^2\right )+3 d f (5 A d f-3 B (d e+c f))\right ) a^3-b^2 \left (-f (26 C e-B f) c^2-d \left (26 C e^2-29 B f e+11 A f^2\right ) c+d^2 e (B e-11 A f)\right ) a^2-b^3 \left (\left (25 C e^2-4 B f e+4 A f^2\right ) c^2-d e (4 B e+9 A f) c+4 A d^2 e^2\right ) a+b^4 c e (5 B c e-4 A (d e+c f))+\left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3+b^2 \left (d f (7 B d e+7 B c f-23 A d f)-C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) x\right )}{2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 C d f a^3+3 b (B d f-2 C (d e+c f)) a^2+b^2 (10 c C e+B d e+B c f-8 A d f) a-b^3 (5 B c e-4 A (d e+c f))\right )}{3 (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac {\frac {2 \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d f \int \frac {C d f (d e+c f) a^4-b \left (C \left (9 d^2 e^2+11 c d f e+9 c^2 f^2\right )+3 d f (5 A d f-3 B (d e+c f))\right ) a^3-b^2 \left (-f (26 C e-B f) c^2-d \left (26 C e^2-29 B f e+11 A f^2\right ) c+d^2 e (B e-11 A f)\right ) a^2-b^3 \left (\left (25 C e^2-4 B f e+4 A f^2\right ) c^2-d e (4 B e+9 A f) c+4 A d^2 e^2\right ) a+b^4 c e (5 B c e-4 A (d e+c f))+\left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3+b^2 \left (d f (7 B d e+7 B c f-23 A d f)-C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 C d f a^3+3 b (B d f-2 C (d e+c f)) a^2+b^2 (10 c C e+B d e+B c f-8 A d f) a-b^3 (5 B c e-4 A (d e+c f))\right )}{3 (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac {\frac {2 \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d f \left (\frac {(b e-a f) \left (C d f (d e-c f) a^3-3 b \left (d f (2 B d e+3 B c f-5 A d f)-C \left (d^2 e^2+c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (-f (20 C e-B f) c^2-d \left (10 C e^2-27 B f e+11 A f^2\right ) c+d^2 e (2 B e-19 A f)\right ) a+b^3 \left (\left (15 C e^2-5 B f e+4 A f^2\right ) c^2-d e (10 B e-3 A f) c+8 A d^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {\left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3+b^2 \left (d f (7 B d e+7 B c f-23 A d f)-C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}}dx}{f}\right )}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 C d f a^3+3 b (B d f-2 C (d e+c f)) a^2+b^2 (10 c C e+B d e+B c f-8 A d f) a-b^3 (5 B c e-4 A (d e+c f))\right )}{3 (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac {\frac {2 \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d f \left (\frac {(b e-a f) \left (C d f (d e-c f) a^3-3 b \left (d f (2 B d e+3 B c f-5 A d f)-C \left (d^2 e^2+c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (-f (20 C e-B f) c^2-d \left (10 C e^2-27 B f e+11 A f^2\right ) c+d^2 e (2 B e-19 A f)\right ) a+b^3 \left (\left (15 C e^2-5 B f e+4 A f^2\right ) c^2-d e (10 B e-3 A f) c+8 A d^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {\left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3+b^2 \left (d f (7 B d e+7 B c f-23 A d f)-C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}dx}{f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}\right )}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 C d f a^3+3 b (B d f-2 C (d e+c f)) a^2+b^2 (10 c C e+B d e+B c f-8 A d f) a-b^3 (5 B c e-4 A (d e+c f))\right )}{3 (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac {\frac {2 \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d f \left (\frac {2 \sqrt {a d-b c} \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3+b^2 \left (d f (7 B d e+7 B c f-23 A d f)-C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {(b e-a f) \left (C d f (d e-c f) a^3-3 b \left (d f (2 B d e+3 B c f-5 A d f)-C \left (d^2 e^2+c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (-f (20 C e-B f) c^2-d \left (10 C e^2-27 B f e+11 A f^2\right ) c+d^2 e (2 B e-19 A f)\right ) a+b^3 \left (\left (15 C e^2-5 B f e+4 A f^2\right ) c^2-d e (10 B e-3 A f) c+8 A d^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}\right )}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 C d f a^3+3 b (B d f-2 C (d e+c f)) a^2+b^2 (10 c C e+B d e+B c f-8 A d f) a-b^3 (5 B c e-4 A (d e+c f))\right )}{3 (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac {\frac {2 \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d f \left (\frac {2 \sqrt {a d-b c} \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3+b^2 \left (d f (7 B d e+7 B c f-23 A d f)-C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\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 {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}}}+\frac {(b e-a f) \left (C d f (d e-c f) a^3-3 b \left (d f (2 B d e+3 B c f-5 A d f)-C \left (d^2 e^2+c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (-f (20 C e-B f) c^2-d \left (10 C e^2-27 B f e+11 A f^2\right ) c+d^2 e (2 B e-19 A f)\right ) a+b^3 \left (\left (15 C e^2-5 B f e+4 A f^2\right ) c^2-d e (10 B e-3 A f) c+8 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \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}}\right )}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 C d f a^3+3 b (B d f-2 C (d e+c f)) a^2+b^2 (10 c C e+B d e+B c f-8 A d f) a-b^3 (5 B c e-4 A (d e+c f))\right )}{3 (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac {\frac {2 \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d f \left (\frac {2 \sqrt {a d-b c} \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3+b^2 \left (d f (7 B d e+7 B c f-23 A d f)-C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\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 {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}}}+\frac {(b e-a f) \left (C d f (d e-c f) a^3-3 b \left (d f (2 B d e+3 B c f-5 A d f)-C \left (d^2 e^2+c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (-f (20 C e-B f) c^2-d \left (10 C e^2-27 B f e+11 A f^2\right ) c+d^2 e (2 B e-19 A f)\right ) a+b^3 \left (\left (15 C e^2-5 B f e+4 A f^2\right ) c^2-d e (10 B e-3 A f) c+8 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \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}}\right )}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 130 |
| \(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 C d f a^3+3 b (B d f-2 C (d e+c f)) a^2+b^2 (10 c C e+B d e+B c f-8 A d f) a-b^3 (5 B c e-4 A (d e+c f))\right )}{3 (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac {\frac {2 \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d f \left (\frac {2 \sqrt {a d-b c} \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3+b^2 \left (d f (7 B d e+7 B c f-23 A d f)-C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\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 {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}}}+\frac {2 \sqrt {a d-b c} (b e-a f) \left (C d f (d e-c f) a^3-3 b \left (d f (2 B d e+3 B c f-5 A d f)-C \left (d^2 e^2+c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (-f (20 C e-B f) c^2-d \left (10 C e^2-27 B f e+11 A f^2\right ) c+d^2 e (2 B e-19 A f)\right ) a+b^3 \left (\left (15 C e^2-5 B f e+4 A f^2\right ) c^2-d e (10 B e-3 A f) c+8 A d^2 e^2\right )\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 {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}}\right )}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\) |
(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(5*b*(b*c - a*d)* (b*e - a*f)*(a + b*x)^(5/2)) + ((2*(2*a^3*C*d*f + a*b^2*(10*c*C*e + B*d*e + B*c*f - 8*A*d*f) - b^3*(5*B*c*e - 4*A*(d*e + c*f)) + 3*a^2*b*(B*d*f - 2* C*(d*e + c*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(3/2)) + ((2*(2*a^4*C*d^2*f^2 + a^3*b*d*f*(3*B*d*f - 7*C*(d*e + c* f)) - b^4*(8*A*d^2*e^2 - c*d*e*(10*B*e - 7*A*f) + c^2*(15*C*e^2 - 10*B*e*f + 8*A*f^2)) - a*b^3*(d^2*e*(2*B*e - 23*A*f) - 2*c^2*f*(5*C*e - B*f) - c*d *(10*C*e^2 - 33*B*e*f + 23*A*f^2)) - a^2*b^2*(C*(3*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) + d*f*(23*A*d*f - 7*B*(d*e + c*f))))*Sqrt[c + d*x]*Sqrt[e + f*x ])/((b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x]) - (d*f*((2*Sqrt[-(b*c) + a*d]*( 2*a^4*C*d^2*f^2 + a^3*b*d*f*(3*B*d*f - 7*C*(d*e + c*f)) - b^4*(8*A*d^2*e^2 - c*d*e*(10*B*e - 7*A*f) + c^2*(15*C*e^2 - 10*B*e*f + 8*A*f^2)) - a*b^3*( d^2*e*(2*B*e - 23*A*f) - 2*c^2*f*(5*C*e - B*f) - c*d*(10*C*e^2 - 33*B*e*f + 23*A*f^2)) + a^2*b^2*(d*f*(7*B*d*e + 7*B*c*f - 23*A*d*f) - C*(3*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]* EllipticE[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[(b*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(a^3*C*d*f*(d*e - c*f) + b^3*( 8*A*d^2*e^2 - c*d*e*(10*B*e - 3*A*f) + c^2*(15*C*e^2 - 5*B*e*f + 4*A*f^2)) + a*b^2*(d^2*e*(2*B*e - 19*A*f) - c^2*f*(20*C*e - B*f) - c*d*(10*C*e^2...
3.1.78.3.1 Defintions of rubi rules used
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[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[b*R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Si mp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n* (e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f*R*(m + 1 ) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x] , x], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && PolyQ[Px, x] && LtQ[m, - 1] && IntegersQ[2*m, 2*n, 2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(2282\) vs. \(2(1054)=2108\).
Time = 6.00 (sec) , antiderivative size = 2283, normalized size of antiderivative = 2.05
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2283\) |
| default | \(\text {Expression too large to display}\) | \(32154\) |
((b*x+a)*(d*x+c)*(f*x+e))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)* (-2/5/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^4*(A*b^2-B*a*b+C*a^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)/(x+a/b )^3-2/15*(8*A*a*b^2*d*f-4*A*b^3*c*f-4*A*b^3*d*e-3*B*a^2*b*d*f-B*a*b^2*c*f- B*a*b^2*d*e+5*B*b^3*c*e-2*C*a^3*d*f+6*C*a^2*b*c*f+6*C*a^2*b*d*e-10*C*a*b^2 *c*e)/b^3/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)^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)/(x+a/b)^2-2/15*(b*d*f*x^ 2+b*c*f*x+b*d*e*x+b*c*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)^3/b^2*(23*A*a^2 *b^2*d^2*f^2-23*A*a*b^3*c*d*f^2-23*A*a*b^3*d^2*e*f+8*A*b^4*c^2*f^2+7*A*b^4 *c*d*e*f+8*A*b^4*d^2*e^2-3*B*a^3*b*d^2*f^2-7*B*a^2*b^2*c*d*f^2-7*B*a^2*b^2 *d^2*e*f+2*B*a*b^3*c^2*f^2+33*B*a*b^3*c*d*e*f+2*B*a*b^3*d^2*e^2-10*B*b^4*c ^2*e*f-10*B*b^4*c*d*e^2-2*C*a^4*d^2*f^2+7*C*a^3*b*c*d*f^2+7*C*a^3*b*d^2*e* f+3*C*a^2*b^2*c^2*f^2-13*C*a^2*b^2*c*d*e*f+3*C*a^2*b^2*d^2*e^2-10*C*a*b^3* c^2*e*f-10*C*a*b^3*c*d*e^2+15*C*b^4*c^2*e^2)/((x+a/b)*(b*d*f*x^2+b*c*f*x+b *d*e*x+b*c*e))^(1/2)+2*(-1/15*d*f*(8*A*a*b^2*d*f-4*A*b^3*c*f-4*A*b^3*d*e-3 *B*a^2*b*d*f-B*a*b^2*c*f-B*a*b^2*d*e+5*B*b^3*c*e-2*C*a^3*d*f+6*C*a^2*b*c*f +6*C*a^2*b*d*e-10*C*a*b^2*c*e)/b^2/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)^2+1/1 5/b^2*(a*d*f-b*c*f-b*d*e)*(23*A*a^2*b^2*d^2*f^2-23*A*a*b^3*c*d*f^2-23*A*a* b^3*d^2*e*f+8*A*b^4*c^2*f^2+7*A*b^4*c*d*e*f+8*A*b^4*d^2*e^2-3*B*a^3*b*d^2* f^2-7*B*a^2*b^2*c*d*f^2-7*B*a^2*b^2*d^2*e*f+2*B*a*b^3*c^2*f^2+33*B*a*b^...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.05 (sec) , antiderivative size = 5108, normalized size of antiderivative = 4.58 \[ \int \frac {A+B x+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B x+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (b x + a\right )}^{\frac {7}{2}} \sqrt {d x + c} \sqrt {f x + e}} \,d x } \]
\[ \int \frac {A+B x+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (b x + a\right )}^{\frac {7}{2}} \sqrt {d x + c} \sqrt {f x + e}} \,d x } \]
Timed out. \[ \int \frac {A+B x+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {C\,x^2+B\,x+A}{\sqrt {e+f\,x}\,{\left (a+b\,x\right )}^{7/2}\,\sqrt {c+d\,x}} \,d x \]