\(\int \frac {\sqrt {c+d x} (A+B x+C x^2)}{(a+b x)^{7/2} \sqrt {e+f x}} \, dx\) [94]

Optimal result

Integrand size = 38, antiderivative size = 1034 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2} \sqrt {e+f x}} \, dx=\frac {2 \left (4 a^3 C d f-b^3 (5 B c e-2 A d e-4 A c f)+a b^2 (10 c C e+3 B d e+B c f-6 A d f)-a^2 b (8 C d e+6 c C f-B d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}-\frac {2 \left (8 a^4 C d^2 f^2-a^3 b d f (23 C d e+13 c C f-2 B d f)-b^4 \left (2 A d^2 e^2-c d e (5 B e-3 A f)-c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a^2 b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d e f+3 c^2 f^2\right )\right )-a b^3 \left (d^2 e (3 B e-7 A f)+2 c^2 f (5 C e-B f)+c d \left (40 C e^2-13 f (B e-A f)\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{15 b^2 (b c-a d)^2 (b e-a f)^3 \sqrt {a+b x}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac {2 \sqrt {d} \left (8 a^4 C d^2 f^2-a^3 b d f (23 C d e+13 c C f-2 B d f)-b^4 \left (2 A d^2 e^2-c d e (5 B e-3 A f)-c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a^2 b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d e f+3 c^2 f^2\right )\right )-a b^3 \left (d^2 e (3 B e-7 A f)+2 c^2 f (5 C e-B f)+c d \left (40 C e^2-13 f (B e-A f)\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 (-b c+a d)^{3/2} (b e-a f)^3 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \sqrt {d} (d e-c f) \left (4 a^3 C d f-b^3 (5 B c e-2 A d e-4 A c f)+a b^2 (10 c C e+3 B d e+B c f-6 A d f)-a^2 b (8 C d e+6 c C f-B d f)\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 (-b c+a d)^{3/2} (b e-a f)^2 \sqrt {c+d x} \sqrt {e+f x}} \] Output:

2/15*(4*a^3*C*d*f-b^3*(-4*A*c*f-2*A*d*e+5*B*c*e)+a*b^2*(-6*A*d*f+B*c*f+3*B 
*d*e+10*C*c*e)-a^2*b*(-B*d*f+6*C*c*f+8*C*d*e))*(d*x+c)^(1/2)*(f*x+e)^(1/2) 
/b^2/(-a*d+b*c)/(-a*f+b*e)^2/(b*x+a)^(3/2)-2/15*(8*a^4*C*d^2*f^2-a^3*b*d*f 
*(-2*B*d*f+13*C*c*f+23*C*d*e)-b^4*(2*A*d^2*e^2-c*d*e*(-3*A*f+5*B*e)-c^2*(8 
*A*f^2-10*B*e*f+15*C*e^2))-a^2*b^2*(d*f*(-3*A*d*f+2*B*c*f+7*B*d*e)-C*(3*c^ 
2*f^2+37*c*d*e*f+23*d^2*e^2))-a*b^3*(d^2*e*(-7*A*f+3*B*e)+2*c^2*f*(-B*f+5* 
C*e)+c*d*(40*C*e^2-13*f*(-A*f+B*e))))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^2/(-a* 
d+b*c)^2/(-a*f+b*e)^3/(b*x+a)^(1/2)-2/5*(A*b^2-a*(B*b-C*a))*(d*x+c)^(3/2)* 
(f*x+e)^(1/2)/b/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^(5/2)+2/15*d^(1/2)*(8*a^4*C* 
d^2*f^2-a^3*b*d*f*(-2*B*d*f+13*C*c*f+23*C*d*e)-b^4*(2*A*d^2*e^2-c*d*e*(-3* 
A*f+5*B*e)-c^2*(8*A*f^2-10*B*e*f+15*C*e^2))-a^2*b^2*(d*f*(-3*A*d*f+2*B*c*f 
+7*B*d*e)-C*(3*c^2*f^2+37*c*d*e*f+23*d^2*e^2))-a*b^3*(d^2*e*(-7*A*f+3*B*e) 
+2*c^2*f*(-B*f+5*C*e)+c*d*(40*C*e^2-13*f*(-A*f+B*e))))*(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^3/(a*d-b*c)^(3/2)/(-a*f+b*e)^3/(d*x+c)^ 
(1/2)/(b*(f*x+e)/(-a*f+b*e))^(1/2)+2/15*d^(1/2)*(-c*f+d*e)*(4*a^3*C*d*f-b^ 
3*(-4*A*c*f-2*A*d*e+5*B*c*e)+a*b^2*(-6*A*d*f+B*c*f+3*B*d*e+10*C*c*e)-a^2*b 
*(-B*d*f+6*C*c*f+8*C*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^3/(a*d-b*c)^(3/2)/(-a*f+b*e)^2/(d*x+c)^(1/2)/(f*x...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 33.58 (sec) , antiderivative size = 1449, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2} \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(7/2)*Sqrt[e + f*x] 
),x]
 

Output:

Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*((-2*(A*b^2 - a*b*B + a^2*C))/(5 
*b^2*(b*e - a*f)*(a + b*x)^3) - (2*(5*b^3*B*c*e - 10*a*b^2*c*C*e + A*b^3*d 
*e - 6*a*b^2*B*d*e + 11*a^2*b*C*d*e - 4*A*b^3*c*f - a*b^2*B*c*f + 6*a^2*b* 
c*C*f + 3*a*A*b^2*d*f + 2*a^2*b*B*d*f - 7*a^3*C*d*f))/(15*b^2*(b*c - a*d)* 
(b*e - a*f)^2*(a + b*x)^2) - (2*(15*b^4*c^2*C*e^2 + 5*b^4*B*c*d*e^2 - 40*a 
*b^3*c*C*d*e^2 - 2*A*b^4*d^2*e^2 - 3*a*b^3*B*d^2*e^2 + 23*a^2*b^2*C*d^2*e^ 
2 - 10*b^4*B*c^2*e*f - 10*a*b^3*c^2*C*e*f - 3*A*b^4*c*d*e*f + 13*a*b^3*B*c 
*d*e*f + 37*a^2*b^2*c*C*d*e*f + 7*a*A*b^3*d^2*e*f - 7*a^2*b^2*B*d^2*e*f - 
23*a^3*b*C*d^2*e*f + 8*A*b^4*c^2*f^2 + 2*a*b^3*B*c^2*f^2 + 3*a^2*b^2*c^2*C 
*f^2 - 13*a*A*b^3*c*d*f^2 - 2*a^2*b^2*B*c*d*f^2 - 13*a^3*b*c*C*d*f^2 + 3*a 
^2*A*b^2*d^2*f^2 + 2*a^3*b*B*d^2*f^2 + 8*a^4*C*d^2*f^2))/(15*b^2*(b*c - a* 
d)^2*(b*e - a*f)^3*(a + b*x))) + (2*(a + b*x)^(3/2)*(Sqrt[-a + (b*c)/d]*(8 
*a^4*C*d^2*f^2 + a^3*b*d*f*(-23*C*d*e - 13*c*C*f + 2*B*d*f) + b^4*(-2*A*d^ 
2*e^2 + c*d*e*(5*B*e - 3*A*f) + c^2*(15*C*e^2 - 10*B*e*f + 8*A*f^2)) + a^2 
*b^2*(d*f*(-7*B*d*e - 2*B*c*f + 3*A*d*f) + C*(23*d^2*e^2 + 37*c*d*e*f + 3* 
c^2*f^2)) + a*b^3*(d^2*e*(-3*B*e + 7*A*f) + 2*c^2*f*(-5*C*e + B*f) + c*d*( 
-40*C*e^2 + 13*f*(B*e - A*f))))*(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*(-8*a^4*C*d^2* 
f^2 + a^3*b*d*f*(23*C*d*e + 13*c*C*f - 2*B*d*f) + b^4*(2*A*d^2*e^2 + c*d*e 
*(-5*B*e + 3*A*f) + c^2*(-15*C*e^2 + 10*B*e*f - 8*A*f^2)) - a^2*b^2*(d*...
 

Rubi [A] (verified)

Time = 2.55 (sec) , antiderivative size = 1073, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2117, 27, 167, 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.

\(\displaystyle \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2} \sqrt {e+f x}} \, dx\)

\(\Big \downarrow \) 2117

\(\displaystyle -\frac {2 \int -\frac {\sqrt {c+d x} \left (C (3 d e+c f) a^2-b (5 c C e+3 B d e+B c f-5 A d f) a+b^2 (5 B c e-2 A d e-4 A c f)+b \left (\frac {4 C d f a^2}{b}-5 C d e a-5 c C f a+B d f a+5 b c C e-A b d f\right ) x\right )}{2 b (a+b x)^{5/2} \sqrt {e+f x}}dx}{5 (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{5 b (a+b x)^{5/2} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {c+d x} \left (C (3 d e+c f) a^2-b (5 c C e+3 B d e+B c f-5 A d f) a+b^2 (5 B c e-2 A (d e+2 c f))+b \left (\frac {4 C d f a^2}{b}+B d f a-5 C (d e+c f) a+b (5 c C e-A d f)\right ) x\right )}{(a+b x)^{5/2} \sqrt {e+f x}}dx}{5 b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{5 b (a+b x)^{5/2} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {\frac {2 \int -\frac {4 C d f (d e+c f) a^3+b \left (B d f (d e+c f)-C \left (8 d^2 e^2+17 c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (2 f (5 C e-B f) c^2+d \left (25 C e^2-8 B f e+9 A f^2\right ) c+3 d^2 e (B e-2 A f)\right ) a+b^3 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )+d \left (8 C d f^2 a^3-b f (19 C d e+9 c C f-2 B d f) a^2+b^2 (5 C e (3 d e+4 c f)-f (6 B d e+B c f-3 A d f)) a-b^3 \left (15 c C e^2-A d f e-c f (5 B e-4 A f)\right )\right ) x}{2 (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 b (b e-a f)}+\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (4 a^3 C d f-a^2 b (-B d f+6 c C f+8 C d e)+a b^2 (-6 A d f+B c f+3 B d e+10 c C e)-b^3 (-4 A c f-2 A d e+5 B c e)\right )}{3 b (a+b x)^{3/2} (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{5 b (a+b x)^{5/2} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (4 a^3 C d f-a^2 b (-B d f+6 c C f+8 C d e)+a b^2 (-6 A d f+B c f+3 B d e+10 c C e)-b^3 (-4 A c f-2 A d e+5 B c e)\right )}{3 b (a+b x)^{3/2} (b e-a f)}-\frac {\int \frac {4 C d f (d e+c f) a^3+b \left (B d f (d e+c f)-C \left (8 d^2 e^2+17 c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (2 f (5 C e-B f) c^2+d \left (25 C e^2-8 B f e+9 A f^2\right ) c+3 d^2 e (B e-2 A f)\right ) a+b^3 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )+d \left (8 C d f^2 a^3-b f (19 C d e+9 c C f-2 B d f) a^2+b^2 (5 C e (3 d e+4 c f)-f (6 B d e+B c f-3 A d f)) a-b^3 \left (15 c C e^2-A d f e-c f (5 B e-4 A f)\right )\right ) x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{5 b (a+b x)^{5/2} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\frac {2 \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b e-a f) (a+b x)^{3/2}}-\frac {\frac {2 \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {2 \int \frac {d \left (4 C d f^2 (d e+c f) a^4+b f \left (B d f (d e+c f)-C \left (11 d^2 e^2+19 c d f e+6 c^2 f^2\right )\right ) a^3+b^2 \left (C e \left (15 d^2 e^2+29 c d f e+19 c^2 f^2\right )+f \left (3 A d f (3 d e-2 c f)-B \left (9 d^2 e^2+c d f e-c^2 f^2\right )\right )\right ) a^2-b^3 \left (4 f \left (5 C e^2+f (B e-A f)\right ) c^2+d e \left (30 C e^2-f (16 B e-9 A f)\right ) c+A d^2 e^2 f\right ) a+b^4 c e \left (15 c C e^2-A d f e-c f (5 B e-4 A f)\right )+f \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) x\right )}{2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b e-a f) (a+b x)^{3/2}}-\frac {\frac {2 \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d \int \frac {4 C d f^2 (d e+c f) a^4+b f \left (B d f (d e+c f)-C \left (11 d^2 e^2+19 c d f e+6 c^2 f^2\right )\right ) a^3+b^2 \left (C e \left (15 d^2 e^2+29 c d f e+19 c^2 f^2\right )+f \left (3 A d f (3 d e-2 c f)-B \left (9 d^2 e^2+c d f e-c^2 f^2\right )\right )\right ) a^2-b^3 \left (4 f \left (5 C e^2+f (B e-A f)\right ) c^2+d e \left (30 C e^2-f (16 B e-9 A f)\right ) c+A d^2 e^2 f\right ) a+b^4 c e \left (15 c C e^2-A d f e-c f (5 B e-4 A f)\right )+f \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {\frac {2 \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b e-a f) (a+b x)^{3/2}}-\frac {\frac {2 \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d \left ((b e-a f) (d e-c f) \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx+\left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}}dx\right )}{(b c-a d) (b e-a f)}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {\frac {2 \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b e-a f) (a+b x)^{3/2}}-\frac {\frac {2 \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d \left ((b e-a f) (d e-c f) \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx+\frac {\left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}dx}{\sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}\right )}{(b c-a d) (b e-a f)}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {\frac {2 \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b e-a f) (a+b x)^{3/2}}-\frac {\frac {2 \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d \left (\frac {2 \sqrt {a d-b c} \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+(b e-a f) (d e-c f) \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx\right )}{(b c-a d) (b e-a f)}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {\frac {2 \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b e-a f) (a+b x)^{3/2}}-\frac {\frac {2 \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d \left (\frac {2 \sqrt {a d-b c} \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {(b e-a f) (d e-c f) \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}}dx}{\sqrt {c+d x}}\right )}{(b c-a d) (b e-a f)}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {\frac {2 \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b e-a f) (a+b x)^{3/2}}-\frac {\frac {2 \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d \left (\frac {2 \sqrt {a d-b c} \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {(b e-a f) (d e-c f) \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}dx}{\sqrt {c+d x} \sqrt {e+f x}}\right )}{(b c-a d) (b e-a f)}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\)

\(\Big \downarrow \) 130

\(\displaystyle \frac {\frac {2 \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b e-a f) (a+b x)^{3/2}}-\frac {\frac {2 \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d \left (\frac {2 \sqrt {a d-b c} \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2\right )-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \sqrt {a d-b c} (b e-a f) (d e-c f) \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {c+d x} \sqrt {e+f x}}\right )}{(b c-a d) (b e-a f)}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\)

Input:

Int[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(7/2)*Sqrt[e + f*x]),x]
 

Output:

(-2*(A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*Sqrt[e + f*x])/(5*b*(b*c - a*d 
)*(b*e - a*f)*(a + b*x)^(5/2)) + ((2*(4*a^3*C*d*f - b^3*(5*B*c*e - 2*A*d*e 
 - 4*A*c*f) + a*b^2*(10*c*C*e + 3*B*d*e + B*c*f - 6*A*d*f) - a^2*b*(8*C*d* 
e + 6*c*C*f - B*d*f))*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b*(b*e - a*f)*(a + b 
*x)^(3/2)) - ((2*(8*a^4*C*d^2*f^2 - a^3*b*d*f*(23*C*d*e + 13*c*C*f - 2*B*d 
*f) - b^4*(2*A*d^2*e^2 - c*d*e*(5*B*e - 3*A*f) - c^2*(15*C*e^2 - 10*B*e*f 
+ 8*A*f^2)) - a^2*b^2*(d*f*(7*B*d*e + 2*B*c*f - 3*A*d*f) - C*(23*d^2*e^2 + 
 37*c*d*e*f + 3*c^2*f^2)) - a*b^3*(d^2*e*(3*B*e - 7*A*f) + 2*c^2*f*(5*C*e 
- B*f) + c*d*(40*C*e^2 - 13*f*(B*e - A*f))))*Sqrt[c + d*x]*Sqrt[e + f*x])/ 
((b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x]) - (d*((2*Sqrt[-(b*c) + a*d]*(8*a^4 
*C*d^2*f^2 - a^3*b*d*f*(23*C*d*e + 13*c*C*f - 2*B*d*f) - b^4*(2*A*d^2*e^2 
- c*d*e*(5*B*e - 3*A*f) - c^2*(15*C*e^2 - 10*B*e*f + 8*A*f^2)) - a^2*b^2*( 
d*f*(7*B*d*e + 2*B*c*f - 3*A*d*f) - C*(23*d^2*e^2 + 37*c*d*e*f + 3*c^2*f^2 
)) - a*b^3*(d^2*e*(3*B*e - 7*A*f) + 2*c^2*f*(5*C*e - B*f) + c*d*(40*C*e^2 
- 13*f*(B*e - A*f))))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*Ellipt 
icE[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]*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) 
+ (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(d*e - c*f)*(4*a^3*C*d*f - b^3*(5*B*c* 
e - 2*A*d*e - 4*A*c*f) + a*b^2*(10*c*C*e + 3*B*d*e + B*c*f - 6*A*d*f) - a^ 
2*b*(8*C*d*e + 6*c*C*f - B*d*f))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 123

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])
 

rule 124

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]
 

rule 130

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])
 

rule 131

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]
 

rule 167

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*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[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] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
 

rule 169

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]
 

rule 176

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]
 

rule 2117

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2329\) vs. \(2(972)=1944\).

Time = 21.72 (sec) , antiderivative size = 2330, normalized size of antiderivative = 2.25

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2330\)
default	\(\text {Expression too large to display}\)	\(34865\)

Input:

int((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(7/2)/(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*(A*b^2-B*a*b+C*a^2)/b^5/(a*f-b*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+2/15*(3*A*a*b^2*d*f-4 
*A*b^3*c*f+A*b^3*d*e+2*B*a^2*b*d*f-B*a*b^2*c*f-6*B*a*b^2*d*e+5*B*b^3*c*e-7 
*C*a^3*d*f+6*C*a^2*b*c*f+11*C*a^2*b*d*e-10*C*a*b^2*c*e)/b^4/(a^2*d*f-a*b*c 
*f-a*b*d*e+b^2*c*e)/(a*f-b*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)^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)^2/b^3*(3*A*a^2*b^2*d^2*f^2-13*A 
*a*b^3*c*d*f^2+7*A*a*b^3*d^2*e*f+8*A*b^4*c^2*f^2-3*A*b^4*c*d*e*f-2*A*b^4*d 
^2*e^2+2*B*a^3*b*d^2*f^2-2*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+13*B*a*b^3*c*d*e*f-3*B*a*b^3*d^2*e^2-10*B*b^4*c^2*e*f+5*B*b^4*c*d 
*e^2+8*C*a^4*d^2*f^2-13*C*a^3*b*c*d*f^2-23*C*a^3*b*d^2*e*f+3*C*a^2*b^2*c^2 
*f^2+37*C*a^2*b^2*c*d*e*f+23*C*a^2*b^2*d^2*e^2-10*C*a*b^3*c^2*e*f-40*C*a*b 
^3*c*d*e^2+15*C*b^4*c^2*e^2)/(a*f-b*e)/((x+a/b)*(b*d*f*x^2+b*c*f*x+b*d*e*x 
+b*c*e))^(1/2)+2*(C*d/b^3+1/15*d*f*(3*A*a*b^2*d*f-4*A*b^3*c*f+A*b^3*d*e+2* 
B*a^2*b*d*f-B*a*b^2*c*f-6*B*a*b^2*d*e+5*B*b^3*c*e-7*C*a^3*d*f+6*C*a^2*b*c* 
f+11*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)/(a* 
f-b*e)-1/15/b^3*(a*d*f-b*c*f-b*d*e)*(3*A*a^2*b^2*d^2*f^2-13*A*a*b^3*c*d*f^ 
2+7*A*a*b^3*d^2*e*f+8*A*b^4*c^2*f^2-3*A*b^4*c*d*e*f-2*A*b^4*d^2*e^2+2*B*a^ 
3*b*d^2*f^2-2*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+1...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4867 vs. \(2 (971) = 1942\).

Time = 0.96 (sec) , antiderivative size = 4867, normalized size of antiderivative = 4.71 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2} \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(7/2)/(f*x+e)^(1/2),x, algor 
ithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2} \sqrt {e+f x}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(1/2)*(C*x**2+B*x+A)/(b*x+a)**(7/2)/(f*x+e)**(1/2),x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2} \sqrt {e+f x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c}}{{\left (b x + a\right )}^{\frac {7}{2}} \sqrt {f x + e}} \,d x } \] Input:

integrate((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(7/2)/(f*x+e)^(1/2),x, algor 
ithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(7/2)*sqrt(f*x + e)), 
 x)
 

Giac [F]

\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2} \sqrt {e+f x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c}}{{\left (b x + a\right )}^{\frac {7}{2}} \sqrt {f x + e}} \,d x } \] Input:

integrate((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(7/2)/(f*x+e)^(1/2),x, algor 
ithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(7/2)*sqrt(f*x + e)), 
 x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2} \sqrt {e+f x}} \, dx=\int \frac {\sqrt {c+d\,x}\,\left (C\,x^2+B\,x+A\right )}{\sqrt {e+f\,x}\,{\left (a+b\,x\right )}^{7/2}} \,d x \] Input:

int(((c + d*x)^(1/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(a + b*x)^(7/2)), 
x)
 

Output:

int(((c + d*x)^(1/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(a + b*x)^(7/2)), 
 x)
 

Reduce [F]

\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2} \sqrt {e+f x}} \, dx=\int \frac {\sqrt {d x +c}\, \left (C \,x^{2}+B x +A \right )}{\left (b x +a \right )^{\frac {7}{2}} \sqrt {f x +e}}d x \] Input:

int((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(7/2)/(f*x+e)^(1/2),x)
 

Output:

int((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(7/2)/(f*x+e)^(1/2),x)