\(\int \frac {(a+b x)^{3/2} \sqrt {c+d x} (A+B x+C x^2)}{\sqrt {e+f x}} \, dx\) [89]

Optimal result

Integrand size = 38, antiderivative size = 1233 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx =\text {Too large to display} \] Output:

-2/945*(5*d*f*(7*b*c*f*(-9*A*b*d*f+C*a*c*f+3*C*a*d*e+5*C*b*c*e)-(a*c*f+a*d 
*e+5*b*c*e)*(4*a*C*d*f+b*(-9*B*d*f+6*C*c*f+8*C*d*e)))+(3*a*d*f-4*b*(c*f+d* 
e))*(7*d*f*(-9*A*b*d*f+C*a*c*f+3*C*a*d*e+5*C*b*c*e)-(2*a*d*f-b*c*f+6*b*d*e 
)*(4*a*C*d*f+b*(-9*B*d*f+6*C*c*f+8*C*d*e))/b))*(b*x+a)^(1/2)*(d*x+c)^(1/2) 
*(f*x+e)^(1/2)/b/d^3/f^4-2/315*(7*d*f*(-9*A*b*d*f+C*a*c*f+3*C*a*d*e+5*C*b* 
c*e)-(2*a*d*f-b*c*f+6*b*d*e)*(4*a*C*d*f+b*(-9*B*d*f+6*C*c*f+8*C*d*e))/b)*( 
b*x+a)^(3/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/d^2/f^3-2/63*(4*a*C*d*f+b*(-9*B 
*d*f+6*C*c*f+8*C*d*e))*(b*x+a)^(5/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^2/d/f^2 
+2/9*C*(b*x+a)^(5/2)*(d*x+c)^(3/2)*(f*x+e)^(1/2)/b/d/f+2/315*(a*d-b*c)^(1/ 
2)*(8*a^4*C*d^4*f^4+a^3*b*d^3*f^3*(-18*B*d*f-7*C*c*f+11*C*d*e)-3*a^2*b^2*d 
^2*f^2*(3*d*f*(-7*A*d*f-3*B*c*f+4*B*d*e)-C*(-3*c^2*f^2-5*c*d*e*f+9*d^2*e^2 
))-a*b^3*d*f*(2*C*(-16*c^3*f^3-18*c^2*d*e*f^2-33*c*d^2*e^2*f+92*d^3*e^3)+3 
*d*f*(7*A*d*f*(-7*c*f+13*d*e)-B*(-19*c^2*f^2-29*c*d*e*f+72*d^2*e^2)))+b^4* 
(C*(-16*c^4*f^4-16*c^3*d*e*f^3-21*c^2*d^2*e^2*f^2-40*c*d^3*e^3*f+128*d^4*e 
^4)+3*d*f*(7*A*d*f*(-2*c^2*f^2-3*c*d*e*f+8*d^2*e^2)-B*(-8*c^3*f^3-9*c^2*d* 
e*f^2-16*c*d^2*e^2*f+48*d^3*e^3))))*(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/d^(7/2)/f^5/(d*x+c)^(1/2)/(b*(f*x+e)/(-a*f+b*e))^(1/2)+2 
/315*(a*d-b*c)^(1/2)*(-a*f+b*e)*(-c*f+d*e)*(4*a^3*C*d^3*f^3+3*a^2*b*d^2*f^ 
2*(-3*B*d*f-C*c*f+3*C*d*e)-3*a*b^2*d*f*(3*d*f*(-21*A*d*f+3*B*c*f+16*B*d...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 34.51 (sec) , antiderivative size = 1470, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*x)^(3/2)*Sqrt[c + d*x]*(A + B*x + C*x^2))/Sqrt[e + f*x], 
x]
 

Output:

(-2*(-(b^2*Sqrt[-a + (b*c)/d]*(8*a^4*C*d^4*f^4 + a^3*b*d^3*f^3*(11*C*d*e - 
 7*c*C*f - 18*B*d*f) + 3*a^2*b^2*d^2*f^2*(3*d*f*(-4*B*d*e + 3*B*c*f + 7*A* 
d*f) + C*(9*d^2*e^2 - 5*c*d*e*f - 3*c^2*f^2)) + a*b^3*d*f*(C*(-184*d^3*e^3 
 + 66*c*d^2*e^2*f + 36*c^2*d*e*f^2 + 32*c^3*f^3) - 3*d*f*(7*A*d*f*(13*d*e 
- 7*c*f) + B*(-72*d^2*e^2 + 29*c*d*e*f + 19*c^2*f^2))) + b^4*(C*(128*d^4*e 
^4 - 40*c*d^3*e^3*f - 21*c^2*d^2*e^2*f^2 - 16*c^3*d*e*f^3 - 16*c^4*f^4) + 
3*d*f*(7*A*d*f*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2) + B*(-48*d^3*e^3 + 16*c 
*d^2*e^2*f + 9*c^2*d*e*f^2 + 8*c^3*f^3))))*(c + d*x)*(e + f*x)) + b^2*Sqrt 
[-a + (b*c)/d]*d*f*(a + b*x)*(c + d*x)*(e + f*x)*(4*a^3*C*d^3*f^3 - 3*a^2* 
b*d^2*f^2*(3*B*d*f + C*(-2*d*e + c*f + d*f*x)) - a*b^2*d*f*(9*d*f*(14*A*d* 
f + B*(-11*d*e + 3*c*f + 8*d*f*x)) + C*(-15*c^2*f^2 + c*d*f*(-19*e + 11*f* 
x) + d^2*(84*e^2 - 61*e*f*x + 50*f^2*x^2))) + b^3*(C*(-8*c^3*f^3 + 3*c^2*d 
*f^2*(-3*e + 2*f*x) + c*d^2*f*(-12*e^2 + 7*e*f*x - 5*f^2*x^2) + d^3*(64*e^ 
3 - 48*e^2*f*x + 40*e*f^2*x^2 - 35*f^3*x^3)) - 3*d*f*(7*A*d*f*(-4*d*e + c* 
f + 3*d*f*x) + B*(-4*c^2*f^2 + c*d*f*(-5*e + 3*f*x) + 3*d^2*(8*e^2 - 6*e*f 
*x + 5*f^2*x^2))))) - I*(b*c - a*d)*f*(8*a^4*C*d^4*f^4 + a^3*b*d^3*f^3*(11 
*C*d*e - 7*c*C*f - 18*B*d*f) + 3*a^2*b^2*d^2*f^2*(3*d*f*(-4*B*d*e + 3*B*c* 
f + 7*A*d*f) + C*(9*d^2*e^2 - 5*c*d*e*f - 3*c^2*f^2)) + a*b^3*d*f*(C*(-184 
*d^3*e^3 + 66*c*d^2*e^2*f + 36*c^2*d*e*f^2 + 32*c^3*f^3) - 3*d*f*(7*A*d*f* 
(13*d*e - 7*c*f) + B*(-72*d^2*e^2 + 29*c*d*e*f + 19*c^2*f^2))) + b^4*(C...
 

Rubi [A] (verified)

Time = 3.13 (sec) , antiderivative size = 1268, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2118, 27, 171, 27, 171, 27, 171, 27, 176, 124, 123, 131, 131, 130}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2118

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

\(\displaystyle \frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}-\frac {\frac {2 \int \frac {\sqrt {a+b x} \sqrt {c+d x} (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))+(7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) x)}{2 \sqrt {e+f x}}dx}{7 d f}+\frac {2 (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x} (4 a C d f+b (-9 B d f+6 c C f+8 C d e))}{7 d f}}{9 b d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}-\frac {\frac {\int \frac {\sqrt {a+b x} \sqrt {c+d x} (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))+(7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) x)}{\sqrt {e+f x}}dx}{7 d f}+\frac {2 (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x} (4 a C d f+b (-9 B d f+6 c C f+8 C d e))}{7 d f}}{9 b d f}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}-\frac {\frac {\frac {2 \int \frac {\sqrt {c+d x} \left (5 a d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))-(b c e+3 a d e+a c f) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+\left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right ) x\right )}{2 \sqrt {a+b x} \sqrt {e+f x}}dx}{5 d f}+\frac {2 \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x} (7 b d f (a c C f+3 a C d e-9 A b d f+5 b c C e)-(-3 a d f+4 b c f+6 b d e) (4 a C d f+b (-9 B d f+6 c C f+8 C d e)))}{5 d f}}{7 d f}+\frac {2 (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x} (4 a C d f+b (-9 B d f+6 c C f+8 C d e))}{7 d f}}{9 b d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}-\frac {\frac {\frac {\int \frac {\sqrt {c+d x} \left (5 a d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))-(b c e+3 a d e+a c f) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+\left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right ) x\right )}{\sqrt {a+b x} \sqrt {e+f x}}dx}{5 d f}+\frac {2 \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x} (7 b d f (a c C f+3 a C d e-9 A b d f+5 b c C e)-(-3 a d f+4 b c f+6 b d e) (4 a C d f+b (-9 B d f+6 c C f+8 C d e)))}{5 d f}}{7 d f}+\frac {2 (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x} (4 a C d f+b (-9 B d f+6 c C f+8 C d e))}{7 d f}}{9 b d f}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}-\frac {\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (a+b x)^{3/2} \sqrt {e+f x} (c+d x)^{3/2}}{7 d f}+\frac {\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) \sqrt {a+b x} \sqrt {e+f x} (c+d x)^{3/2}}{5 d f}+\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right )}{3 b f}+\frac {2 \int \frac {3 b c f (5 a d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))-(b c e+3 a d e+a c f) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))))-(b c e+a d e+a c f) \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right )-3 \left (\left (C \left (128 d^4 e^4-40 c d^3 f e^3-21 c^2 d^2 f^2 e^2-16 c^3 d f^3 e-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d f e-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 f e^2-9 c^2 d f^2 e-8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (184 d^3 e^3-66 c d^2 f e^2-36 c^2 d f^2 e-32 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d f e-19 c^2 f^2\right )\right )\right ) b^3-3 a^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d f e-3 c^2 f^2\right )\right ) b^2+a^3 d^3 f^3 (11 C d e-7 c C f-18 B d f) b+8 a^4 C d^4 f^4\right ) x}{2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 b f}}{5 d f}}{7 d f}}{9 b d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}-\frac {\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (a+b x)^{3/2} \sqrt {e+f x} (c+d x)^{3/2}}{7 d f}+\frac {\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) \sqrt {a+b x} \sqrt {e+f x} (c+d x)^{3/2}}{5 d f}+\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right )}{3 b f}+\frac {\int \frac {3 b c f (5 a d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))-(b c e+3 a d e+a c f) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))))-(b c e+a d e+a c f) \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right )-3 \left (\left (C \left (128 d^4 e^4-40 c d^3 f e^3-21 c^2 d^2 f^2 e^2-16 c^3 d f^3 e-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d f e-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 f e^2-9 c^2 d f^2 e-8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (184 d^3 e^3-66 c d^2 f e^2-36 c^2 d f^2 e-32 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d f e-19 c^2 f^2\right )\right )\right ) b^3-3 a^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d f e-3 c^2 f^2\right )\right ) b^2+a^3 d^3 f^3 (11 C d e-7 c C f-18 B d f) b+8 a^4 C d^4 f^4\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 b f}}{5 d f}}{7 d f}}{9 b d f}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}-\frac {\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (a+b x)^{3/2} \sqrt {e+f x} (c+d x)^{3/2}}{7 d f}+\frac {\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) \sqrt {a+b x} \sqrt {e+f x} (c+d x)^{3/2}}{5 d f}+\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right )}{3 b f}+\frac {-\frac {3 (b e-a f) (d e-c f) \left (-\left (\left (C \left (128 d^3 e^3+24 c d^2 f e^2+15 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (7 A d f (8 d e+c f)-4 B \left (12 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) b^3\right )-3 a d f \left (3 d f (16 B d e+3 B c f-21 A d f)-5 C \left (8 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) b^2+3 a^2 d^2 f^2 (3 C d e-c C f-3 B d f) b+4 a^3 C d^3 f^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}-\frac {3 \left (\left (C \left (128 d^4 e^4-40 c d^3 f e^3-21 c^2 d^2 f^2 e^2-16 c^3 d f^3 e-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d f e-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 f e^2-9 c^2 d f^2 e-8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (184 d^3 e^3-66 c d^2 f e^2-36 c^2 d f^2 e-32 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d f e-19 c^2 f^2\right )\right )\right ) b^3-3 a^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d f e-3 c^2 f^2\right )\right ) b^2+a^3 d^3 f^3 (11 C d e-7 c C f-18 B d f) b+8 a^4 C d^4 f^4\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}}dx}{f}}{3 b f}}{5 d f}}{7 d f}}{9 b d f}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}-\frac {\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (a+b x)^{3/2} \sqrt {e+f x} (c+d x)^{3/2}}{7 d f}+\frac {\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) \sqrt {a+b x} \sqrt {e+f x} (c+d x)^{3/2}}{5 d f}+\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right )}{3 b f}+\frac {-\frac {3 (b e-a f) (d e-c f) \left (-\left (\left (C \left (128 d^3 e^3+24 c d^2 f e^2+15 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (7 A d f (8 d e+c f)-4 B \left (12 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) b^3\right )-3 a d f \left (3 d f (16 B d e+3 B c f-21 A d f)-5 C \left (8 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) b^2+3 a^2 d^2 f^2 (3 C d e-c C f-3 B d f) b+4 a^3 C d^3 f^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}-\frac {3 \left (\left (C \left (128 d^4 e^4-40 c d^3 f e^3-21 c^2 d^2 f^2 e^2-16 c^3 d f^3 e-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d f e-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 f e^2-9 c^2 d f^2 e-8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (184 d^3 e^3-66 c d^2 f e^2-36 c^2 d f^2 e-32 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d f e-19 c^2 f^2\right )\right )\right ) b^3-3 a^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d f e-3 c^2 f^2\right )\right ) b^2+a^3 d^3 f^3 (11 C d e-7 c C f-18 B d f) b+8 a^4 C d^4 f^4\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}}}}{3 b f}}{5 d f}}{7 d f}}{9 b d f}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}-\frac {\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (a+b x)^{3/2} \sqrt {e+f x} (c+d x)^{3/2}}{7 d f}+\frac {\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) \sqrt {a+b x} \sqrt {e+f x} (c+d x)^{3/2}}{5 d f}+\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right )}{3 b f}+\frac {-\frac {6 \sqrt {a d-b c} \left (\left (C \left (128 d^4 e^4-40 c d^3 f e^3-21 c^2 d^2 f^2 e^2-16 c^3 d f^3 e-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d f e-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 f e^2-9 c^2 d f^2 e-8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (184 d^3 e^3-66 c d^2 f e^2-36 c^2 d f^2 e-32 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d f e-19 c^2 f^2\right )\right )\right ) b^3-3 a^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d f e-3 c^2 f^2\right )\right ) b^2+a^3 d^3 f^3 (11 C d e-7 c C f-18 B d f) b+8 a^4 C d^4 f^4\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 {3 (b e-a f) (d e-c f) \left (-\left (\left (C \left (128 d^3 e^3+24 c d^2 f e^2+15 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (7 A d f (8 d e+c f)-4 B \left (12 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) b^3\right )-3 a d f \left (3 d f (16 B d e+3 B c f-21 A d f)-5 C \left (8 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) b^2+3 a^2 d^2 f^2 (3 C d e-c C f-3 B d f) b+4 a^3 C d^3 f^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}}{3 b f}}{5 d f}}{7 d f}}{9 b d f}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}-\frac {\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (a+b x)^{3/2} \sqrt {e+f x} (c+d x)^{3/2}}{7 d f}+\frac {\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) \sqrt {a+b x} \sqrt {e+f x} (c+d x)^{3/2}}{5 d f}+\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right )}{3 b f}+\frac {-\frac {6 \sqrt {a d-b c} \left (\left (C \left (128 d^4 e^4-40 c d^3 f e^3-21 c^2 d^2 f^2 e^2-16 c^3 d f^3 e-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d f e-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 f e^2-9 c^2 d f^2 e-8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (184 d^3 e^3-66 c d^2 f e^2-36 c^2 d f^2 e-32 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d f e-19 c^2 f^2\right )\right )\right ) b^3-3 a^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d f e-3 c^2 f^2\right )\right ) b^2+a^3 d^3 f^3 (11 C d e-7 c C f-18 B d f) b+8 a^4 C d^4 f^4\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 {3 (b e-a f) (d e-c f) \left (-\left (\left (C \left (128 d^3 e^3+24 c d^2 f e^2+15 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (7 A d f (8 d e+c f)-4 B \left (12 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) b^3\right )-3 a d f \left (3 d f (16 B d e+3 B c f-21 A d f)-5 C \left (8 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) b^2+3 a^2 d^2 f^2 (3 C d e-c C f-3 B d f) b+4 a^3 C d^3 f^3\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}}}{3 b f}}{5 d f}}{7 d f}}{9 b d f}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}-\frac {\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (a+b x)^{3/2} \sqrt {e+f x} (c+d x)^{3/2}}{7 d f}+\frac {\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) \sqrt {a+b x} \sqrt {e+f x} (c+d x)^{3/2}}{5 d f}+\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right )}{3 b f}+\frac {-\frac {6 \sqrt {a d-b c} \left (\left (C \left (128 d^4 e^4-40 c d^3 f e^3-21 c^2 d^2 f^2 e^2-16 c^3 d f^3 e-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d f e-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 f e^2-9 c^2 d f^2 e-8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (184 d^3 e^3-66 c d^2 f e^2-36 c^2 d f^2 e-32 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d f e-19 c^2 f^2\right )\right )\right ) b^3-3 a^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d f e-3 c^2 f^2\right )\right ) b^2+a^3 d^3 f^3 (11 C d e-7 c C f-18 B d f) b+8 a^4 C d^4 f^4\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 {3 (b e-a f) (d e-c f) \left (-\left (\left (C \left (128 d^3 e^3+24 c d^2 f e^2+15 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (7 A d f (8 d e+c f)-4 B \left (12 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) b^3\right )-3 a d f \left (3 d f (16 B d e+3 B c f-21 A d f)-5 C \left (8 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) b^2+3 a^2 d^2 f^2 (3 C d e-c C f-3 B d f) b+4 a^3 C d^3 f^3\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}}}{3 b f}}{5 d f}}{7 d f}}{9 b d f}\)

\(\Big \downarrow \) 130

\(\displaystyle \frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}-\frac {\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (a+b x)^{3/2} \sqrt {e+f x} (c+d x)^{3/2}}{7 d f}+\frac {\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) \sqrt {a+b x} \sqrt {e+f x} (c+d x)^{3/2}}{5 d f}+\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right )}{3 b f}+\frac {-\frac {6 \sqrt {a d-b c} \left (\left (C \left (128 d^4 e^4-40 c d^3 f e^3-21 c^2 d^2 f^2 e^2-16 c^3 d f^3 e-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d f e-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 f e^2-9 c^2 d f^2 e-8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (184 d^3 e^3-66 c d^2 f e^2-36 c^2 d f^2 e-32 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d f e-19 c^2 f^2\right )\right )\right ) b^3-3 a^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d f e-3 c^2 f^2\right )\right ) b^2+a^3 d^3 f^3 (11 C d e-7 c C f-18 B d f) b+8 a^4 C d^4 f^4\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 {6 \sqrt {a d-b c} (b e-a f) (d e-c f) \left (-\left (\left (C \left (128 d^3 e^3+24 c d^2 f e^2+15 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (7 A d f (8 d e+c f)-4 B \left (12 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) b^3\right )-3 a d f \left (3 d f (16 B d e+3 B c f-21 A d f)-5 C \left (8 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) b^2+3 a^2 d^2 f^2 (3 C d e-c C f-3 B d f) b+4 a^3 C d^3 f^3\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}}}{3 b f}}{5 d f}}{7 d f}}{9 b d f}\)

Input:

Int[((a + b*x)^(3/2)*Sqrt[c + d*x]*(A + B*x + C*x^2))/Sqrt[e + f*x],x]
 

Output:

(2*C*(a + b*x)^(5/2)*(c + d*x)^(3/2)*Sqrt[e + f*x])/(9*b*d*f) - ((2*(4*a*C 
*d*f + b*(8*C*d*e + 6*c*C*f - 9*B*d*f))*(a + b*x)^(3/2)*(c + d*x)^(3/2)*Sq 
rt[e + f*x])/(7*d*f) + ((2*(7*b*d*f*(5*b*c*C*e + 3*a*C*d*e + a*c*C*f - 9*A 
*b*d*f) - (6*b*d*e + 4*b*c*f - 3*a*d*f)*(4*a*C*d*f + b*(8*C*d*e + 6*c*C*f 
- 9*B*d*f)))*Sqrt[a + b*x]*(c + d*x)^(3/2)*Sqrt[e + f*x])/(5*d*f) + ((2*(5 
*b*d*f*(7*a*d*f*(5*b*c*C*e + 3*a*C*d*e + a*c*C*f - 9*A*b*d*f) - (3*b*c*e + 
 3*a*d*e + a*c*f)*(4*a*C*d*f + b*(8*C*d*e + 6*c*C*f - 9*B*d*f))) + 2*((a*d 
*f)/2 - b*(2*d*e + c*f))*(7*b*d*f*(5*b*c*C*e + 3*a*C*d*e + a*c*C*f - 9*A*b 
*d*f) - (6*b*d*e + 4*b*c*f - 3*a*d*f)*(4*a*C*d*f + b*(8*C*d*e + 6*c*C*f - 
9*B*d*f))))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b*f) + ((-6*Sqrt 
[-(b*c) + a*d]*(8*a^4*C*d^4*f^4 + a^3*b*d^3*f^3*(11*C*d*e - 7*c*C*f - 18*B 
*d*f) - 3*a^2*b^2*d^2*f^2*(3*d*f*(4*B*d*e - 3*B*c*f - 7*A*d*f) - C*(9*d^2* 
e^2 - 5*c*d*e*f - 3*c^2*f^2)) - a*b^3*d*f*(C*(184*d^3*e^3 - 66*c*d^2*e^2*f 
 - 36*c^2*d*e*f^2 - 32*c^3*f^3) + 3*d*f*(7*A*d*f*(13*d*e - 7*c*f) - B*(72* 
d^2*e^2 - 29*c*d*e*f - 19*c^2*f^2))) + b^4*(C*(128*d^4*e^4 - 40*c*d^3*e^3* 
f - 21*c^2*d^2*e^2*f^2 - 16*c^3*d*e*f^3 - 16*c^4*f^4) + 3*d*f*(7*A*d*f*(8* 
d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2) - B*(48*d^3*e^3 - 16*c*d^2*e^2*f - 9*c^2* 
d*e*f^2 - 8*c^3*f^3))))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*Elli 
pticE[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 - ...
 

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 171

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

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 2118

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f 
_.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo 
n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 
1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + 
q + 1))   Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + 
n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q 
- 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + 
 c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m 
 + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] && PolyQ[Px, x]
 

Maple [A] (verified)

Time = 7.33 (sec) , antiderivative size = 2108, normalized size of antiderivative = 1.71

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2108\)
default	\(\text {Expression too large to display}\)	\(15963\)

Input:

int((b*x+a)^(3/2)*(d*x+c)^(1/2)*(C*x^2+B*x+A)/(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)* 
(2/9*C*b/f*x^3*(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b* 
c*e*x+a*c*e)^(1/2)+2/7*(b^2*B*d+2*C*a*b*d+C*b^2*c-2/9*C*b/f*(4*a*d*f+4*b*c 
*f+4*b*d*e))/b/d/f*x^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)+2/5*(b^2*A*d+2*a*b*B*d+B*b^2*c+a^2*C*d+2*C*a*b* 
c-2/9*C*b/f*(7/2*a*c*f+7/2*a*d*e+7/2*b*c*e)-2/7*(b^2*B*d+2*C*a*b*d+C*b^2*c 
-2/9*C*b/f*(4*a*d*f+4*b*c*f+4*b*d*e))/b/d/f*(3*a*d*f+3*b*c*f+3*b*d*e))/b/d 
/f*x*(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c* 
e)^(1/2)+2/3*(2*A*a*b*d+b^2*A*c+B*a^2*d+2*a*b*B*c+C*a^2*c-2/3*C*b/f*a*c*e- 
2/7*(b^2*B*d+2*C*a*b*d+C*b^2*c-2/9*C*b/f*(4*a*d*f+4*b*c*f+4*b*d*e))/b/d/f* 
(5/2*a*c*f+5/2*a*d*e+5/2*b*c*e)-2/5*(b^2*A*d+2*a*b*B*d+B*b^2*c+a^2*C*d+2*C 
*a*b*c-2/9*C*b/f*(7/2*a*c*f+7/2*a*d*e+7/2*b*c*e)-2/7*(b^2*B*d+2*C*a*b*d+C* 
b^2*c-2/9*C*b/f*(4*a*d*f+4*b*c*f+4*b*d*e))/b/d/f*(3*a*d*f+3*b*c*f+3*b*d*e) 
)/b/d/f*(2*a*d*f+2*b*c*f+2*b*d*e))/b/d/f*(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b* 
d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)+2*(a^2*A*c-2/5*(b^2*A*d+2*a*b 
*B*d+B*b^2*c+a^2*C*d+2*C*a*b*c-2/9*C*b/f*(7/2*a*c*f+7/2*a*d*e+7/2*b*c*e)-2 
/7*(b^2*B*d+2*C*a*b*d+C*b^2*c-2/9*C*b/f*(4*a*d*f+4*b*c*f+4*b*d*e))/b/d/f*( 
3*a*d*f+3*b*c*f+3*b*d*e))/b/d/f*a*c*e-2/3*(2*A*a*b*d+b^2*A*c+B*a^2*d+2*a*b 
*B*c+C*a^2*c-2/3*C*b/f*a*c*e-2/7*(b^2*B*d+2*C*a*b*d+C*b^2*c-2/9*C*b/f*(4*a 
*d*f+4*b*c*f+4*b*d*e))/b/d/f*(5/2*a*c*f+5/2*a*d*e+5/2*b*c*e)-2/5*(b^2*A...
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 1931, normalized size of antiderivative = 1.57 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(3/2)*(d*x+c)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^(1/2),x, algor 
ithm="fricas")
 

Output:

2/945*(3*(35*C*b^5*d^5*f^5*x^3 - 64*C*b^5*d^5*e^3*f^2 + 12*(C*b^5*c*d^4 + 
(7*C*a*b^4 + 6*B*b^5)*d^5)*e^2*f^3 + (9*C*b^5*c^2*d^3 - (19*C*a*b^4 + 15*B 
*b^5)*c*d^4 - 3*(2*C*a^2*b^3 + 33*B*a*b^4 + 28*A*b^5)*d^5)*e*f^4 + (8*C*b^ 
5*c^3*d^2 - 3*(5*C*a*b^4 + 4*B*b^5)*c^2*d^3 + 3*(C*a^2*b^3 + 9*B*a*b^4 + 7 
*A*b^5)*c*d^4 - (4*C*a^3*b^2 - 9*B*a^2*b^3 - 126*A*a*b^4)*d^5)*f^5 - 5*(8* 
C*b^5*d^5*e*f^4 - (C*b^5*c*d^4 + (10*C*a*b^4 + 9*B*b^5)*d^5)*f^5)*x^2 + (4 
8*C*b^5*d^5*e^2*f^3 - (7*C*b^5*c*d^4 + (61*C*a*b^4 + 54*B*b^5)*d^5)*e*f^4 
- (6*C*b^5*c^2*d^3 - (11*C*a*b^4 + 9*B*b^5)*c*d^4 - 3*(C*a^2*b^3 + 24*B*a* 
b^4 + 21*A*b^5)*d^5)*f^5)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e) - ( 
128*C*b^5*d^5*e^5 - 8*(13*C*b^5*c*d^4 + (31*C*a*b^4 + 18*B*b^5)*d^5)*e^4*f 
 - (25*C*b^5*c^2*d^3 - 2*(113*C*a*b^4 + 60*B*b^5)*c*d^4 - (95*C*a^2*b^3 + 
288*B*a*b^4 + 168*A*b^5)*d^5)*e^3*f^2 - (10*C*b^5*c^3*d^2 - 15*(3*C*a*b^4 
+ 2*B*b^5)*c^2*d^3 + 3*(37*C*a^2*b^3 + 91*B*a*b^4 + 49*A*b^5)*c*d^4 - (20* 
C*a^3*b^2 - 117*B*a^2*b^3 - 357*A*a*b^4)*d^5)*e^2*f^3 - (8*C*b^5*c^4*d - ( 
22*C*a*b^4 + 15*B*b^5)*c^3*d^2 + 3*(5*C*a^2*b^3 + 21*B*a*b^4 + 14*A*b^5)*c 
^2*d^3 + 7*(2*C*a^3*b^2 - 21*B*a^2*b^3 - 54*A*a*b^4)*c*d^4 - (7*C*a^4*b - 
27*B*a^3*b^2 + 168*A*a^2*b^3)*d^5)*e*f^4 - (16*C*b^5*c^5 - 8*(5*C*a*b^4 + 
3*B*b^5)*c^4*d + (22*C*a^2*b^3 + 69*B*a*b^4 + 42*A*b^5)*c^3*d^2 + (7*C*a^3 
*b^2 - 51*B*a^2*b^3 - 168*A*a*b^4)*c^2*d^3 + (11*C*a^4*b - 36*B*a^3*b^2 + 
357*A*a^2*b^3)*c*d^4 - (8*C*a^5 - 18*B*a^4*b + 63*A*a^3*b^2)*d^5)*f^5)*...
 

Sympy [F]

\[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x} \left (A + B x + C x^{2}\right )}{\sqrt {e + f x}}\, dx \] Input:

integrate((b*x+a)**(3/2)*(d*x+c)**(1/2)*(C*x**2+B*x+A)/(f*x+e)**(1/2),x)
 

Output:

Integral((a + b*x)**(3/2)*sqrt(c + d*x)*(A + B*x + C*x**2)/sqrt(e + f*x), 
x)
 

Maxima [F]

\[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c}}{\sqrt {f x + e}} \,d x } \] Input:

integrate((b*x+a)^(3/2)*(d*x+c)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^(1/2),x, algor 
ithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)*(b*x + a)^(3/2)*sqrt(d*x + c)/sqrt(f*x + e), x 
)
 

Giac [F]

\[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c}}{\sqrt {f x + e}} \,d x } \] Input:

integrate((b*x+a)^(3/2)*(d*x+c)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^(1/2),x, algor 
ithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)*(b*x + a)^(3/2)*sqrt(d*x + c)/sqrt(f*x + e), x 
)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}\,\left (C\,x^2+B\,x+A\right )}{\sqrt {e+f\,x}} \,d x \] Input:

int(((a + b*x)^(3/2)*(c + d*x)^(1/2)*(A + B*x + C*x^2))/(e + f*x)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

int(((a + b*x)^(3/2)*(c + d*x)^(1/2)*(A + B*x + C*x^2))/(e + f*x)^(1/2), x 
)
 

Reduce [F]

\[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\int \frac {\left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}\, \left (C \,x^{2}+B x +A \right )}{\sqrt {f x +e}}d x \] Input:

int((b*x+a)^(3/2)*(d*x+c)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^(1/2),x)
 

Output:

int((b*x+a)^(3/2)*(d*x+c)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^(1/2),x)