\(\int \frac {\sqrt {c+d x} \sqrt {e+f x} (A+B x+C x^2)}{(a+b x)^{9/2}} \, dx\) [66]

Optimal result

Integrand size = 38, antiderivative size = 1716 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx=-\frac {2 \left (24 a^4 C d^2 f^2-a^3 b d f (61 C d e+43 c C f-4 B d f)-3 a b^3 \left (d^2 e (B e-3 A f)+2 c^2 f (7 C e-B f)+c d \left (28 C e^2-5 B e f+5 A f^2\right )\right )-b^4 \left (4 A d^2 e^2-c d e (7 B e-A f)-c^2 \left (35 C e^2-14 B e f+8 A f^2\right )\right )-3 a^2 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d e f+5 c^2 f^2\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{105 b^3 (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac {2 \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{105 b^3 (b c-a d)^3 (b e-a f)^3 \sqrt {a+b x}}+\frac {2 \left (6 a^3 C d f+a b^2 (14 c C e+3 B d e+3 B c f-8 A d f)-b^3 (7 B c e-4 A (d e+c f))+a^2 b (B d f-10 C (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{35 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{5/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}+\frac {2 \sqrt {d} \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right )\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 )}{105 b^4 (-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} (d e-c f) \left (24 a^4 C d^2 f^2-a^3 b d f (43 C d e+61 c C f-4 B d f)+b^4 \left (8 A d^2 e^2-c d e (14 B e+A f)+c^2 \left (35 C e^2+7 B e f-4 A f^2\right )\right )+3 a b^3 \left (d^2 e (2 B e-5 A f)-c^2 f (28 C e+B f)-c d \left (14 C e^2-5 B e f-3 A f^2\right )\right )-3 a^2 b^2 \left (d f (2 B d e+3 B c f-A d f)-C \left (5 d^2 e^2+37 c d e f+15 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 )}{105 b^4 (-b c+a d)^{5/2} (b e-a f)^2 \sqrt {c+d x} \sqrt {e+f x}} \]

[Out]

-2/7*(A*b^2-a*(B*b-C*a))*(d*x+c)^(3/2)*(f*x+e)^(3/2)/b/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^(7/2)+2/35*(6*a^3*C*d*f+a
*b^2*(-8*A*d*f+3*B*c*f+3*B*d*e+14*C*c*e)-b^3*(7*B*c*e-4*A*(c*f+d*e))+a^2*b*(B*d*f-10*C*(c*f+d*e)))*(f*x+e)^(3/
2)*(d*x+c)^(1/2)/b^2/(-a*d+b*c)/(-a*f+b*e)^2/(b*x+a)^(5/2)-2/105*(24*a^4*C*d^2*f^2-a^3*b*d*f*(-4*B*d*f+43*C*c*
f+61*C*d*e)-3*a*b^3*(d^2*e*(-3*A*f+B*e)+2*c^2*f*(-B*f+7*C*e)+c*d*(5*A*f^2-5*B*e*f+28*C*e^2))-b^4*(4*A*d^2*e^2-
c*d*e*(-A*f+7*B*e)-c^2*(8*A*f^2-14*B*e*f+35*C*e^2))-3*a^2*b^2*(d*f*(-A*d*f+2*B*c*f+3*B*d*e)-C*(5*c^2*f^2+37*c*
d*e*f+15*d^2*e^2)))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^3/(-a*d+b*c)^2/(-a*f+b*e)^2/(b*x+a)^(3/2)+2/105*(48*a^5*C*d^
3*f^3+8*a^4*b*d^2*f^2*(B*d*f-16*C*(c*f+d*e))-b^5*(8*A*d^3*e^3-c*d^2*e^2*(5*A*f+14*B*e)+c^2*d*e*(-5*A*f^2+14*B*
e*f+35*C*e^2)+c^3*f*(8*A*f^2-14*B*e*f+35*C*e^2))-a*b^4*(d^3*e^2*(-19*A*f+6*B*e)-6*c^3*f^2*(-B*f+7*C*e)-c^2*d*f
*(238*C*e^2-19*f*(-A*f+B*e))-c*d^2*e*(42*C*e^2-f*(20*A*f+19*B*e)))+a^3*b^2*d*f*(C*(103*c^2*f^2+344*c*d*e*f+103
*d^2*e^2)+d*f*(6*A*d*f-19*B*(c*f+d*e)))-3*a^2*b^3*(C*(5*c^3*f^3+94*c^2*d*e*f^2+94*c*d^2*e^2*f+5*d^3*e^3)+d*f*(
3*A*d*f*(c*f+d*e)-B*(3*c^2*f^2+16*c*d*e*f+3*d^2*e^2))))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^3/(-a*d+b*c)^3/(-a*f+b*e
)^3/(b*x+a)^(1/2)+2/105*(48*a^5*C*d^3*f^3+8*a^4*b*d^2*f^2*(B*d*f-16*C*(c*f+d*e))-b^5*(8*A*d^3*e^3-c*d^2*e^2*(5
*A*f+14*B*e)+c^2*d*e*(-5*A*f^2+14*B*e*f+35*C*e^2)+c^3*f*(8*A*f^2-14*B*e*f+35*C*e^2))-a*b^4*(d^3*e^2*(-19*A*f+6
*B*e)-6*c^3*f^2*(-B*f+7*C*e)-c^2*d*f*(238*C*e^2-19*f*(-A*f+B*e))-c*d^2*e*(42*C*e^2-f*(20*A*f+19*B*e)))+a^3*b^2
*d*f*(C*(103*c^2*f^2+344*c*d*e*f+103*d^2*e^2)+d*f*(6*A*d*f-19*B*(c*f+d*e)))-3*a^2*b^3*(C*(5*c^3*f^3+94*c^2*d*e
*f^2+94*c*d^2*e^2*f+5*d^3*e^3)+d*f*(3*A*d*f*(c*f+d*e)-B*(3*c^2*f^2+16*c*d*e*f+3*d^2*e^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))*d^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(f*x+e)^
(1/2)/b^4/(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)+2/105*(-c*f+d*e)*(24*a^4*C*d
^2*f^2-a^3*b*d*f*(-4*B*d*f+61*C*c*f+43*C*d*e)+b^4*(8*A*d^2*e^2-c*d*e*(A*f+14*B*e)+c^2*(-4*A*f^2+7*B*e*f+35*C*e
^2))+3*a*b^3*(d^2*e*(-5*A*f+2*B*e)-c^2*f*(B*f+28*C*e)-c*d*(-3*A*f^2-5*B*e*f+14*C*e^2))-3*a^2*b^2*(d*f*(-A*d*f+
3*B*c*f+2*B*d*e)-C*(15*c^2*f^2+37*c*d*e*f+5*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*(d*x+c)/(-a*d+b*c))^(1/2)*(b*(f*x+e)/(-a*f+b*e))^(1/2)/b^4/(a*d-b*c)^(5
/2)/(-a*f+b*e)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2)

Rubi [A] (verified)

Time = 5.04 (sec) , antiderivative size = 1716, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1628, 155, 157, 164, 115, 114, 122, 121} \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx=-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}+\frac {2 \left (6 C d f a^3+b (B d f-10 C (d e+c f)) a^2+b^2 (14 c C e+3 B d e+3 B c f-8 A d f) a-b^3 (7 B c e-4 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{35 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{5/2}}+\frac {2 \sqrt {d} \left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} 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 ) \sqrt {e+f x}}{105 b^4 (a d-b c)^{5/2} (b e-a f)^3 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{105 b^3 (b c-a d)^3 (b e-a f)^3 \sqrt {a+b x}}-\frac {2 \left (24 C d^2 f^2 a^4-b d f (61 C d e+43 c C f-4 B d f) a^3-3 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d f e+5 c^2 f^2\right )\right ) a^2-3 b^3 \left (2 f (7 C e-B f) c^2+d \left (28 C e^2-5 B f e+5 A f^2\right ) c+d^2 e (B e-3 A f)\right ) a-b^4 \left (-\left (\left (35 C e^2-14 B f e+8 A f^2\right ) c^2\right )-d e (7 B e-A f) c+4 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{105 b^3 (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac {2 \sqrt {d} (d e-c f) \left (24 C d^2 f^2 a^4-b d f (43 C d e+61 c C f-4 B d f) a^3-3 b^2 \left (d f (2 B d e+3 B c f-A d f)-C \left (5 d^2 e^2+37 c d f e+15 c^2 f^2\right )\right ) a^2+3 b^3 \left (-f (28 C e+B f) c^2-d \left (14 C e^2-5 B f e-3 A f^2\right ) c+d^2 e (2 B e-5 A f)\right ) a+b^4 \left (\left (35 C e^2+7 B f e-4 A f^2\right ) c^2-d e (14 B e+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 )}{105 b^4 (a d-b c)^{5/2} (b e-a f)^2 \sqrt {c+d x} \sqrt {e+f x}} \]

[In]

Int[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^(9/2),x]

[Out]

(-2*(24*a^4*C*d^2*f^2 - a^3*b*d*f*(61*C*d*e + 43*c*C*f - 4*B*d*f) - 3*a*b^3*(d^2*e*(B*e - 3*A*f) + 2*c^2*f*(7*
C*e - B*f) + c*d*(28*C*e^2 - 5*B*e*f + 5*A*f^2)) - b^4*(4*A*d^2*e^2 - c*d*e*(7*B*e - A*f) - c^2*(35*C*e^2 - 14
*B*e*f + 8*A*f^2)) - 3*a^2*b^2*(d*f*(3*B*d*e + 2*B*c*f - A*d*f) - C*(15*d^2*e^2 + 37*c*d*e*f + 5*c^2*f^2)))*Sq
rt[c + d*x]*Sqrt[e + f*x])/(105*b^3*(b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)^(3/2)) + (2*(48*a^5*C*d^3*f^3 + 8*a^
4*b*d^2*f^2*(B*d*f - 16*C*(d*e + c*f)) - b^5*(8*A*d^3*e^3 - c*d^2*e^2*(14*B*e + 5*A*f) + c^2*d*e*(35*C*e^2 + 1
4*B*e*f - 5*A*f^2) + c^3*f*(35*C*e^2 - 14*B*e*f + 8*A*f^2)) - a*b^4*(d^3*e^2*(6*B*e - 19*A*f) - 6*c^3*f^2*(7*C
*e - B*f) - c^2*d*f*(238*C*e^2 - 19*f*(B*e - A*f)) - c*d^2*e*(42*C*e^2 - f*(19*B*e + 20*A*f))) + a^3*b^2*d*f*(
C*(103*d^2*e^2 + 344*c*d*e*f + 103*c^2*f^2) + d*f*(6*A*d*f - 19*B*(d*e + c*f))) - 3*a^2*b^3*(C*(5*d^3*e^3 + 94
*c*d^2*e^2*f + 94*c^2*d*e*f^2 + 5*c^3*f^3) + d*f*(3*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 16*c*d*e*f + 3*c^2*f^2)
)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(105*b^3*(b*c - a*d)^3*(b*e - a*f)^3*Sqrt[a + b*x]) + (2*(6*a^3*C*d*f + a*b^2
*(14*c*C*e + 3*B*d*e + 3*B*c*f - 8*A*d*f) - b^3*(7*B*c*e - 4*A*(d*e + c*f)) + a^2*b*(B*d*f - 10*C*(d*e + c*f))
)*Sqrt[c + d*x]*(e + f*x)^(3/2))/(35*b^2*(b*c - a*d)*(b*e - a*f)^2*(a + b*x)^(5/2)) - (2*(A*b^2 - a*(b*B - a*C
))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(7*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(7/2)) + (2*Sqrt[d]*(48*a^5*C*d^3*f
^3 + 8*a^4*b*d^2*f^2*(B*d*f - 16*C*(d*e + c*f)) - b^5*(8*A*d^3*e^3 - c*d^2*e^2*(14*B*e + 5*A*f) + c^2*d*e*(35*
C*e^2 + 14*B*e*f - 5*A*f^2) + c^3*f*(35*C*e^2 - 14*B*e*f + 8*A*f^2)) - a*b^4*(d^3*e^2*(6*B*e - 19*A*f) - 6*c^3
*f^2*(7*C*e - B*f) - c^2*d*f*(238*C*e^2 - 19*f*(B*e - A*f)) - c*d^2*e*(42*C*e^2 - f*(19*B*e + 20*A*f))) + a^3*
b^2*d*f*(C*(103*d^2*e^2 + 344*c*d*e*f + 103*c^2*f^2) + d*f*(6*A*d*f - 19*B*(d*e + c*f))) - 3*a^2*b^3*(C*(5*d^3
*e^3 + 94*c*d^2*e^2*f + 94*c^2*d*e*f^2 + 5*c^3*f^3) + d*f*(3*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 16*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))])/(105*b^4*(-(b*c) + a*d)^(5/2)*(b*e - a*f)^3*Sqrt[c + d*x]*Sqrt[(b
*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[d]*(d*e - c*f)*(24*a^4*C*d^2*f^2 - a^3*b*d*f*(43*C*d*e + 61*c*C*f - 4*B*d*
f) + b^4*(8*A*d^2*e^2 - c*d*e*(14*B*e + A*f) + c^2*(35*C*e^2 + 7*B*e*f - 4*A*f^2)) + 3*a*b^3*(d^2*e*(2*B*e - 5
*A*f) - c^2*f*(28*C*e + B*f) - c*d*(14*C*e^2 - 5*B*e*f - 3*A*f^2)) - 3*a^2*b^2*(d*f*(2*B*d*e + 3*B*c*f - A*d*f
) - C*(5*d^2*e^2 + 37*c*d*e*f + 15*c^2*f^2)))*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))])/(105*b^4*(-(b*
c) + a*d)^(5/2)*(b*e - a*f)^2*Sqrt[c + d*x]*Sqrt[e + f*x])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> 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] &&  !LtQ[-(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])

Rule 115

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[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] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]

Rule 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> 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] && Si
mplerQ[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])

Rule 122

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[b*((c
+ d*x)/(b*c - a*d))]/Sqrt[c + d*x], Int[1/(Sqrt[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] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rule 155

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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]

Rule 164

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(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] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rule 1628

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] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}-\frac {2 \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (-\frac {3 a^2 C (d e+c f)-a b (7 c C e+3 B d e+3 B c f-7 A d f)+b^2 (7 B c e-4 A (d e+c f))}{2 b}+\frac {1}{2} \left (-7 b c C e+7 a C d e+7 a c C f+A b d f-a B d f-\frac {6 a^2 C d f}{b}\right ) x\right )}{(a+b x)^{7/2}} \, dx}{7 (b c-a d) (b e-a f)} \\ & = \frac {2 \left (6 a^3 C d f+a b^2 (14 c C e+3 B d e+3 B c f-8 A d f)-b^3 (7 B c e-4 A (d e+c f))+a^2 b (B d f-10 C (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{35 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{5/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}-\frac {4 \int \frac {\sqrt {e+f x} \left (\frac {6 a^3 C d f (d e+3 c f)+b^3 \left (4 A d^2 e^2-c d e (7 B e-A f)-c^2 \left (35 C e^2-14 B e f+8 A f^2\right )\right )+a b^2 \left (d^2 e (3 B e-8 A f)+6 c^2 f (7 C e-B f)+c d \left (49 C e^2-8 B e f+11 A f^2\right )\right )+a^2 b \left (B d f (d e+3 c f)-5 C \left (2 d^2 e^2+11 c d e f+3 c^2 f^2\right )\right )}{4 b}+\frac {d \left (24 a^3 C d f^2-b^3 \left (35 c C e^2-A d e f-c f (7 B e-4 A f)\right )+a^2 b f (4 B d f-5 C (11 d e+5 c f))+a b^2 (7 C e (5 d e+8 c f)-f (8 B d e+3 B c f-3 A d f))\right ) x}{4 b}\right )}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx}{35 b (b c-a d) (b e-a f)^2} \\ & = -\frac {2 \left (24 a^4 C d^2 f^2-a^3 b d f (61 C d e+43 c C f-4 B d f)-3 a b^3 \left (d^2 e (B e-3 A f)+2 c^2 f (7 C e-B f)+c d \left (28 C e^2-5 B e f+5 A f^2\right )\right )-b^4 \left (4 A d^2 e^2-c d e (7 B e-A f)-c^2 \left (35 C e^2-14 B e f+8 A f^2\right )\right )-3 a^2 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d e f+5 c^2 f^2\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{105 b^3 (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac {2 \left (6 a^3 C d f+a b^2 (14 c C e+3 B d e+3 B c f-8 A d f)-b^3 (7 B c e-4 A (d e+c f))+a^2 b (B d f-10 C (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{35 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{5/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}-\frac {8 \int \frac {-\frac {24 a^4 C d^2 f^2 (d e+c f)+3 a b^3 \left (d^3 e^2 (2 B e-5 A f)-2 c^3 f^2 (7 C e-B f)-2 c d^2 e \left (7 C e^2-2 B e f-3 A f^2\right )-c^2 d f \left (56 C e^2-4 B e f+5 A f^2\right )\right )+b^4 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )+a^3 b d f \left (4 B d f (d e+c f)-C \left (43 d^2 e^2+122 c d e f+43 c^2 f^2\right )\right )+3 a^2 b^2 \left (C \left (5 d^3 e^3+52 c d^2 e^2 f+52 c^2 d e f^2+5 c^3 f^3\right )+d f \left (A d f (d e+c f)-2 B \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right )\right )}{8 b}-\frac {d f \left (48 a^4 C d^2 f^2+8 a^3 b d f (B d f-13 C (d e+c f))+3 a b^3 \left (d^2 e (B e-2 A f)-c^2 f (42 C e-B f)-2 c d \left (21 C e^2-5 B e f+A f^2\right )\right )+b^4 \left (4 A d^2 e^2-c d e (7 B e+2 A f)+c^2 \left (70 C e^2-7 B e f+4 A f^2\right )\right )+3 a^2 b^2 \left (C \left (20 d^2 e^2+74 c d e f+20 c^2 f^2\right )+d f (2 A d f-5 B (d e+c f))\right )\right ) x}{8 b}}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{105 b^2 (b c-a d)^2 (b e-a f)^2} \\ & = -\frac {2 \left (24 a^4 C d^2 f^2-a^3 b d f (61 C d e+43 c C f-4 B d f)-3 a b^3 \left (d^2 e (B e-3 A f)+2 c^2 f (7 C e-B f)+c d \left (28 C e^2-5 B e f+5 A f^2\right )\right )-b^4 \left (4 A d^2 e^2-c d e (7 B e-A f)-c^2 \left (35 C e^2-14 B e f+8 A f^2\right )\right )-3 a^2 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d e f+5 c^2 f^2\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{105 b^3 (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac {2 \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{105 b^3 (b c-a d)^3 (b e-a f)^3 \sqrt {a+b x}}+\frac {2 \left (6 a^3 C d f+a b^2 (14 c C e+3 B d e+3 B c f-8 A d f)-b^3 (7 B c e-4 A (d e+c f))+a^2 b (B d f-10 C (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{35 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{5/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}+\frac {16 \int \frac {-\frac {d f \left (24 a^5 C d^2 f^2 (d e+c f)-3 a^2 b^3 \left (d^3 e^2 (B e-3 A f)+c^3 f^2 (48 C e+B f)+3 c^2 d f \left (34 C e^2-4 B e f-A f^2\right )+12 c d^2 e \left (4 C e^2-B e f+A f^2\right )\right )-b^5 c e \left (4 A d^2 e^2-c d e (7 B e+2 A f)+c^2 \left (70 C e^2-7 B e f+4 A f^2\right )\right )-a b^4 \left (4 A d^3 e^3-c d^2 e^2 (4 B e+13 A f)-c^3 f \left (161 C e^2+4 B e f-4 A f^2\right )-c^2 d e \left (161 C e^2-58 B e f+13 A f^2\right )\right )+a^4 b d f \left (4 B d f (d e+c f)-C \left (61 d^2 e^2+134 c d e f+61 c^2 f^2\right )\right )+a^3 b^2 \left (5 C \left (9 d^3 e^3+46 c d^2 e^2 f+46 c^2 d e f^2+9 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (9 d^2 e^2+20 c d e f+9 c^2 f^2\right )\right )\right )\right )}{16 b}-\frac {d f \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right )\right )\right ) x}{16 b}}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{105 b^2 (b c-a d)^3 (b e-a f)^3} \\ & = -\frac {2 \left (24 a^4 C d^2 f^2-a^3 b d f (61 C d e+43 c C f-4 B d f)-3 a b^3 \left (d^2 e (B e-3 A f)+2 c^2 f (7 C e-B f)+c d \left (28 C e^2-5 B e f+5 A f^2\right )\right )-b^4 \left (4 A d^2 e^2-c d e (7 B e-A f)-c^2 \left (35 C e^2-14 B e f+8 A f^2\right )\right )-3 a^2 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d e f+5 c^2 f^2\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{105 b^3 (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac {2 \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{105 b^3 (b c-a d)^3 (b e-a f)^3 \sqrt {a+b x}}+\frac {2 \left (6 a^3 C d f+a b^2 (14 c C e+3 B d e+3 B c f-8 A d f)-b^3 (7 B c e-4 A (d e+c f))+a^2 b (B d f-10 C (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{35 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{5/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}-\frac {\left (d (d e-c f) \left (24 a^4 C d^2 f^2-a^3 b d f (43 C d e+61 c C f-4 B d f)+b^4 \left (8 A d^2 e^2-c d e (14 B e+A f)+c^2 \left (35 C e^2+7 B e f-4 A f^2\right )\right )+3 a b^3 \left (d^2 e (2 B e-5 A f)-c^2 f (28 C e+B f)-c d \left (14 C e^2-5 B e f-3 A f^2\right )\right )-3 a^2 b^2 \left (d f (2 B d e+3 B c f-A d f)-C \left (5 d^2 e^2+37 c d e f+15 c^2 f^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{105 b^3 (b c-a d)^3 (b e-a f)^2}-\frac {\left (d \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right )\right )\right )\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{105 b^3 (b c-a d)^3 (b e-a f)^3} \\ & = -\frac {2 \left (24 a^4 C d^2 f^2-a^3 b d f (61 C d e+43 c C f-4 B d f)-3 a b^3 \left (d^2 e (B e-3 A f)+2 c^2 f (7 C e-B f)+c d \left (28 C e^2-5 B e f+5 A f^2\right )\right )-b^4 \left (4 A d^2 e^2-c d e (7 B e-A f)-c^2 \left (35 C e^2-14 B e f+8 A f^2\right )\right )-3 a^2 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d e f+5 c^2 f^2\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{105 b^3 (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac {2 \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{105 b^3 (b c-a d)^3 (b e-a f)^3 \sqrt {a+b x}}+\frac {2 \left (6 a^3 C d f+a b^2 (14 c C e+3 B d e+3 B c f-8 A d f)-b^3 (7 B c e-4 A (d e+c f))+a^2 b (B d f-10 C (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{35 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{5/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}-\frac {\left (d (d e-c f) \left (24 a^4 C d^2 f^2-a^3 b d f (43 C d e+61 c C f-4 B d f)+b^4 \left (8 A d^2 e^2-c d e (14 B e+A f)+c^2 \left (35 C e^2+7 B e f-4 A f^2\right )\right )+3 a b^3 \left (d^2 e (2 B e-5 A f)-c^2 f (28 C e+B f)-c d \left (14 C e^2-5 B e f-3 A f^2\right )\right )-3 a^2 b^2 \left (d f (2 B d e+3 B c f-A d f)-C \left (5 d^2 e^2+37 c d e f+15 c^2 f^2\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}}\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}{105 b^3 (b c-a d)^3 (b e-a f)^2 \sqrt {c+d x}}-\frac {\left (d \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x}\right ) \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}{105 b^3 (b c-a d)^3 (b e-a f)^3 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 36.94 (sec) , antiderivative size = 2437, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx=\text {Result too large to show} \]

[In]

Integrate[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^(9/2),x]

[Out]

Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*((-2*(A*b^2 - a*b*B + a^2*C))/(7*b^3*(a + b*x)^4) - (2*(7*b^3*B*c*e
- 14*a*b^2*c*C*e + A*b^3*d*e - 8*a*b^2*B*d*e + 15*a^2*b*C*d*e + A*b^3*c*f - 8*a*b^2*B*c*f + 15*a^2*b*c*C*f - 2
*a*A*b^2*d*f + 9*a^2*b*B*d*f - 16*a^3*C*d*f))/(35*b^3*(b*c - a*d)*(b*e - a*f)*(a + b*x)^3) - (2*(35*b^4*c^2*C*
e^2 + 7*b^4*B*c*d*e^2 - 84*a*b^3*c*C*d*e^2 - 4*A*b^4*d^2*e^2 - 3*a*b^3*B*d^2*e^2 + 45*a^2*b^2*C*d^2*e^2 + 7*b^
4*B*c^2*e*f - 84*a*b^3*c^2*C*e*f + 2*A*b^4*c*d*e*f - 30*a*b^3*B*c*d*e*f + 198*a^2*b^2*c*C*d*e*f + 6*a*A*b^3*d^
2*e*f + 15*a^2*b^2*B*d^2*e*f - 106*a^3*b*C*d^2*e*f - 4*A*b^4*c^2*f^2 - 3*a*b^3*B*c^2*f^2 + 45*a^2*b^2*c^2*C*f^
2 + 6*a*A*b^3*c*d*f^2 + 15*a^2*b^2*B*c*d*f^2 - 106*a^3*b*c*C*d*f^2 - 6*a^2*A*b^2*d^2*f^2 - 8*a^3*b*B*d^2*f^2 +
57*a^4*C*d^2*f^2))/(105*b^3*(b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)^2) - (2*(35*b^5*c^2*C*d*e^3 - 14*b^5*B*c*d^
2*e^3 - 42*a*b^4*c*C*d^2*e^3 + 8*A*b^5*d^3*e^3 + 6*a*b^4*B*d^3*e^3 + 15*a^2*b^3*C*d^3*e^3 + 35*b^5*c^3*C*e^2*f
+ 14*b^5*B*c^2*d*e^2*f - 238*a*b^4*c^2*C*d*e^2*f - 5*A*b^5*c*d^2*e^2*f + 19*a*b^4*B*c*d^2*e^2*f + 282*a^2*b^3
*c*C*d^2*e^2*f - 19*a*A*b^4*d^3*e^2*f - 9*a^2*b^3*B*d^3*e^2*f - 103*a^3*b^2*C*d^3*e^2*f - 14*b^5*B*c^3*e*f^2 -
42*a*b^4*c^3*C*e*f^2 - 5*A*b^5*c^2*d*e*f^2 + 19*a*b^4*B*c^2*d*e*f^2 + 282*a^2*b^3*c^2*C*d*e*f^2 + 20*a*A*b^4*
c*d^2*e*f^2 - 48*a^2*b^3*B*c*d^2*e*f^2 - 344*a^3*b^2*c*C*d^2*e*f^2 + 9*a^2*A*b^3*d^3*e*f^2 + 19*a^3*b^2*B*d^3*
e*f^2 + 128*a^4*b*C*d^3*e*f^2 + 8*A*b^5*c^3*f^3 + 6*a*b^4*B*c^3*f^3 + 15*a^2*b^3*c^3*C*f^3 - 19*a*A*b^4*c^2*d*
f^3 - 9*a^2*b^3*B*c^2*d*f^3 - 103*a^3*b^2*c^2*C*d*f^3 + 9*a^2*A*b^3*c*d^2*f^3 + 19*a^3*b^2*B*c*d^2*f^3 + 128*a
^4*b*c*C*d^2*f^3 - 6*a^3*A*b^2*d^3*f^3 - 8*a^4*b*B*d^3*f^3 - 48*a^5*C*d^3*f^3))/(105*b^3*(b*c - a*d)^3*(b*e -
a*f)^3*(a + b*x))) - (2*(a + b*x)^(3/2)*(-(Sqrt[-a + (b*c)/d]*(-48*a^5*C*d^3*f^3 + 8*a^4*b*d^2*f^2*(-(B*d*f) +
16*C*(d*e + c*f)) + b^5*(8*A*d^3*e^3 - c*d^2*e^2*(14*B*e + 5*A*f) + c^2*d*e*(35*C*e^2 + 14*B*e*f - 5*A*f^2) +
c^3*f*(35*C*e^2 - 14*B*e*f + 8*A*f^2)) + a*b^4*(d^3*e^2*(6*B*e - 19*A*f) + 6*c^3*f^2*(-7*C*e + B*f) + c^2*d*f
*(-238*C*e^2 + 19*f*(B*e - A*f)) + c*d^2*e*(-42*C*e^2 + f*(19*B*e + 20*A*f))) - a^3*b^2*d*f*(C*(103*d^2*e^2 +
344*c*d*e*f + 103*c^2*f^2) + d*f*(6*A*d*f - 19*B*(d*e + c*f))) + 3*a^2*b^3*(C*(5*d^3*e^3 + 94*c*d^2*e^2*f + 94
*c^2*d*e*f^2 + 5*c^3*f^3) + d*f*(3*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 16*c*d*e*f + 3*c^2*f^2))))*(d + (b*c)/(a
+ b*x) - (a*d)/(a + b*x))*(f + (b*e)/(a + b*x) - (a*f)/(a + b*x))) + (I*(-(b*c) + a*d)*f*(-48*a^5*C*d^3*f^3 +
8*a^4*b*d^2*f^2*(-(B*d*f) + 16*C*(d*e + c*f)) + b^5*(8*A*d^3*e^3 - c*d^2*e^2*(14*B*e + 5*A*f) + c^2*d*e*(35*C
*e^2 + 14*B*e*f - 5*A*f^2) + c^3*f*(35*C*e^2 - 14*B*e*f + 8*A*f^2)) + a*b^4*(d^3*e^2*(6*B*e - 19*A*f) + 6*c^3*
f^2*(-7*C*e + B*f) + c^2*d*f*(-238*C*e^2 + 19*f*(B*e - A*f)) + c*d^2*e*(-42*C*e^2 + f*(19*B*e + 20*A*f))) - a^
3*b^2*d*f*(C*(103*d^2*e^2 + 344*c*d*e*f + 103*c^2*f^2) + d*f*(6*A*d*f - 19*B*(d*e + c*f))) + 3*a^2*b^3*(C*(5*d
^3*e^3 + 94*c*d^2*e^2*f + 94*c^2*d*e*f^2 + 5*c^3*f^3) + d*f*(3*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 16*c*d*e*f +
3*c^2*f^2))))*Sqrt[1 - a/(a + b*x) + (b*c)/(d*(a + b*x))]*Sqrt[1 - a/(a + b*x) + (b*e)/(f*(a + b*x))]*Ellipti
cE[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*x] - (I*b*(-(b*c)
+ a*d)*f*(d*e - c*f)*(-24*a^4*C*d^2*f^2 + a^3*b*d*f*(61*C*d*e + 43*c*C*f - 4*B*d*f) + b^4*(4*A*d^2*e^2 + c*d*
e*(-7*B*e + A*f) + c^2*(-35*C*e^2 + 14*B*e*f - 8*A*f^2)) + 3*a*b^3*(d^2*e*(B*e - 3*A*f) - 2*c^2*f*(-7*C*e + B*
f) + c*d*(28*C*e^2 - 5*B*e*f + 5*A*f^2)) - 3*a^2*b^2*(d*f*(-3*B*d*e - 2*B*c*f + A*d*f) + C*(15*d^2*e^2 + 37*c*
d*e*f + 5*c^2*f^2)))*Sqrt[1 - a/(a + b*x) + (b*c)/(d*(a + b*x))]*Sqrt[1 - a/(a + b*x) + (b*e)/(f*(a + b*x))]*E
llipticF[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*x]))/(105*b
^5*Sqrt[-a + (b*c)/d]*(b*c - a*d)^3*(b*e - a*f)^3*Sqrt[c + ((a + b*x)*(d - (a*d)/(a + b*x)))/b]*Sqrt[e + ((a +
b*x)*(f - (a*f)/(a + b*x)))/b])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3899\) vs. \(2(1646)=3292\).

Time = 5.38 (sec) , antiderivative size = 3900, normalized size of antiderivative = 2.27

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(3900\)
default	\(\text {Expression too large to display}\)	\(65231\)

[In]

int((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^(9/2),x,method=_RETURNVERBOSE)

[Out]

((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)*(-2/7*(A*b^2-B*a*b+C*a^2)/b^7*(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)^4+2/35*(2*A*a*b^2*d*f-A*b^3*c*
f-A*b^3*d*e-9*B*a^2*b*d*f+8*B*a*b^2*c*f+8*B*a*b^2*d*e-7*B*b^3*c*e+16*C*a^3*d*f-15*C*a^2*b*c*f-15*C*a^2*b*d*e+1
4*C*a*b^2*c*e)/b^6/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)*(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+2/105*(6*A*a^2*b^2*d^2*f^2-6*A*a*b^3*c*d*f^2-6*A*a*b^3*d^2*e*f+4*A*b^4*c^2*f^2-
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b^4*d^2*e^2+8*B*a^3*b*d^2*f^2-15*B*a^2*b^2*c*d*f^2-15*B*a^2*b^2*d^2*e*f+3*B*a*b^3*c^2*f^2+30*B*a*b^3*c*d*e*f+3
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*d*e^2*f+14*B*b^5*c*d^2*e^3+48*C*a^5*d^3*f^3-128*C*a^4*b*c*d^2*f^3-128*C*a^4*b*d^3*e*f^2+103*C*a^3*b^2*c^2*d*f
^3+344*C*a^3*b^2*c*d^2*e*f^2+103*C*a^3*b^2*d^3*e^2*f-15*C*a^2*b^3*c^3*f^3-282*C*a^2*b^3*c^2*d*e*f^2-282*C*a^2*
b^3*c*d^2*e^2*f-15*C*a^2*b^3*d^3*e^3+42*C*a*b^4*c^3*e*f^2+238*C*a*b^4*c^2*d*e^2*f+42*C*a*b^4*c*d^2*e^3-35*C*b^
5*c^3*e^2*f-35*C*b^5*c^2*d*e^3))*(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)/
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^2*b^3*d^3*e*f^2+19*A*a*b^4*c^2*d*f^3-20*A*a*b^4*c*d^2*e*f^2+19*A*a*b^4*d^3*e^2*f-8*A*b^5*c^3*f^3+5*A*b^5*c^2*
d*e*f^2+5*A*b^5*c*d^2*e^2*f-8*A*b^5*d^3*e^3+8*B*a^4*b*d^3*f^3-19*B*a^3*b^2*c*d^2*f^3-19*B*a^3*b^2*d^3*e*f^2+9*
B*a^2*b^3*c^2*d*f^3+48*B*a^2*b^3*c*d^2*e*f^2+9*B*a^2*b^3*d^3*e^2*f-6*B*a*b^4*c^3*f^3-19*B*a*b^4*c^2*d*e*f^2-19
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*f^3-128*C*a^4*b*c*d^2*f^3-128*C*a^4*b*d^3*e*f^2+103*C*a^3*b^2*c^2*d*f^3+344*C*a^3*b^2*c*d^2*e*f^2+103*C*a^3*b
^2*d^3*e^2*f-15*C*a^2*b^3*c^3*f^3-282*C*a^2*b^3*c^2*d*e*f^2-282*C*a^2*b^3*c*d^2*e^2*f-15*C*a^2*b^3*d^3*e^3+42*
C*a*b^4*c^3*e*f^2+238*C*a*b^4*c^2*d*e^2*f+42*C*a*b^4*c*d^2*e^3-35*C*b^5*c^3*e^2*f-35*C*b^5*c^2*d*e^3)/(a^2*d*f
-a*b*c*f-a*b*d*e+b^2*c*e)^3*(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))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 3.01 (sec) , antiderivative size = 9150, normalized size of antiderivative = 5.33 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx=\text {Too large to display} \]

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^(9/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx=\text {Timed out} \]

[In]

integrate((C*x**2+B*x+A)*(d*x+c)**(1/2)*(f*x+e)**(1/2)/(b*x+a)**(9/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c} \sqrt {f x + e}}{{\left (b x + a\right )}^{\frac {9}{2}}} \,d x } \]

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^(9/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a)^(9/2), x)

Giac [F]

\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c} \sqrt {f x + e}}{{\left (b x + a\right )}^{\frac {9}{2}}} \,d x } \]

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^(9/2),x, algorithm="giac")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a)^(9/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx=\int \frac {\sqrt {e+f\,x}\,\sqrt {c+d\,x}\,\left (C\,x^2+B\,x+A\right )}{{\left (a+b\,x\right )}^{9/2}} \,d x \]

[In]

int(((e + f*x)^(1/2)*(c + d*x)^(1/2)*(A + B*x + C*x^2))/(a + b*x)^(9/2),x)

[Out]

int(((e + f*x)^(1/2)*(c + d*x)^(1/2)*(A + B*x + C*x^2))/(a + b*x)^(9/2), x)