\(\int \frac {\sqrt {c+d x} (A+B x+C x^2)}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx\) [93]

Optimal result

Integrand size = 38, antiderivative size = 596 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=-\frac {2 \left (4 a^2 C f+b^2 (3 B e-2 A f)-a b (6 C e+B f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b^2 (b e-a f)^2 \sqrt {a+b x}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac {2 \sqrt {d} \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)-b^3 \left (3 c C e^2+A d e f+c f (3 B e-2 A f)\right )+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt {-b c+a d} f (b e-a f)^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 (d e-c f) \left (4 a^2 C d f+b^2 (3 c C e+A d f)-a b (B d f+3 C (d e+c f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt {d} \sqrt {-b c+a d} f (b e-a f) \sqrt {c+d x} \sqrt {e+f x}} \] Output:

-2/3*(4*a^2*C*f+b^2*(-2*A*f+3*B*e)-a*b*(B*f+6*C*e))*(d*x+c)^(1/2)*(f*x+e)^ 
(1/2)/b^2/(-a*f+b*e)^2/(b*x+a)^(1/2)-2/3*(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)^(3/2)+2/3*d^(1/2)*(8*a^3*C* 
d*f^2-a^2*b*f*(2*B*d*f+7*C*c*f+13*C*d*e)-b^3*(3*c*C*e^2+A*d*e*f+c*f*(-2*A* 
f+3*B*e))+a*b^2*(3*C*e*(4*c*f+d*e)+f*(-A*d*f+B*c*f+4*B*d*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)^(1/2)/f/(-a*f+b*e)^2 
/(d*x+c)^(1/2)/(b*(f*x+e)/(-a*f+b*e))^(1/2)+2/3*(-c*f+d*e)*(4*a^2*C*d*f+b^ 
2*(A*d*f+3*C*c*e)-a*b*(B*d*f+3*C*(c*f+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/d^(1/2)/(a*d-b*c)^(1/2)/f/(-a*f+ 
b*e)/(d*x+c)^(1/2)/(f*x+e)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 28.27 (sec) , antiderivative size = 724, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=-\frac {2 \left (b^2 \sqrt {-a+\frac {b c}{d}} f (c+d x) (e+f x) \left (\left (A b^2+a (-b B+a C)\right ) (b c-a d) (b e-a f)+\left (-5 a^3 C d f+b^3 (3 B c e+A d e-2 A c f)-a b^2 (6 c C e+4 B d e+B c f-A d f)+a^2 b (7 C d e+4 c C f+2 B d f)\right ) (a+b x)\right )+(a+b x) \left (b^2 \sqrt {-a+\frac {b c}{d}} \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)-b^3 \left (3 c C e^2+A d e f+c f (3 B e-2 A f)\right )+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d f))\right ) (c+d x) (e+f x)+i (b c-a d) f \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)-b^3 \left (3 c C e^2+A d e f+c f (3 B e-2 A f)\right )+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d f))\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )+i b (b c-a d) f (d e-c f) \left (-4 a^2 C f+b^2 (-3 B e+2 A f)+a b (6 C e+B f)\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )\right )}{3 b^4 \sqrt {-a+\frac {b c}{d}} (b c-a d) f (b e-a f)^2 (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \] Input:

Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(5/2)*Sqrt[e + f*x] 
),x]
 

Output:

(-2*(b^2*Sqrt[-a + (b*c)/d]*f*(c + d*x)*(e + f*x)*((A*b^2 + a*(-(b*B) + a* 
C))*(b*c - a*d)*(b*e - a*f) + (-5*a^3*C*d*f + b^3*(3*B*c*e + A*d*e - 2*A*c 
*f) - a*b^2*(6*c*C*e + 4*B*d*e + B*c*f - A*d*f) + a^2*b*(7*C*d*e + 4*c*C*f 
 + 2*B*d*f))*(a + b*x)) + (a + b*x)*(b^2*Sqrt[-a + (b*c)/d]*(8*a^3*C*d*f^2 
 - a^2*b*f*(13*C*d*e + 7*c*C*f + 2*B*d*f) - b^3*(3*c*C*e^2 + A*d*e*f + c*f 
*(3*B*e - 2*A*f)) + a*b^2*(3*C*e*(d*e + 4*c*f) + f*(4*B*d*e + B*c*f - A*d* 
f)))*(c + d*x)*(e + f*x) + I*(b*c - a*d)*f*(8*a^3*C*d*f^2 - a^2*b*f*(13*C* 
d*e + 7*c*C*f + 2*B*d*f) - b^3*(3*c*C*e^2 + A*d*e*f + c*f*(3*B*e - 2*A*f)) 
 + a*b^2*(3*C*e*(d*e + 4*c*f) + f*(4*B*d*e + B*c*f - A*d*f)))*(a + b*x)^(3 
/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*El 
lipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c* 
f - a*d*f)] + I*b*(b*c - a*d)*f*(d*e - c*f)*(-4*a^2*C*f + b^2*(-3*B*e + 2* 
A*f) + a*b*(6*C*e + B*f))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x)) 
]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d] 
/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])))/(3*b^4*Sqrt[-a + (b*c 
)/d]*(b*c - a*d)*f*(b*e - a*f)^2*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f* 
x])
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 630, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2117, 27, 167, 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.

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

\(\Big \downarrow \) 2117

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 167

\(\displaystyle \frac {\frac {2 \int -\frac {4 C d f (d e+c f) a^3-b \left (B d f (d e+c f)+C \left (6 d^2 e^2+11 c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (6 C e f c^2+d \left (9 C e^2+2 B f e+A f^2\right ) c+d^2 e (3 B e-2 A f)\right ) a-b^3 c e (3 c C e+3 B d e-A d f)+d \left (8 C d f^2 a^3-b f (13 C d e+7 c C f+2 B d f) a^2+b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d f)) a-b^3 \left (3 c C e^2+A d f e+c f (3 B e-2 A f)\right )\right ) x}{2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{b (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} (b c-a d) \left (4 a^2 C f-a b (B f+6 C e)+b^2 (3 B e-2 A f)\right )}{b \sqrt {a+b x} (b e-a f)}}{3 b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {4 C d f (d e+c f) a^3-b \left (B d f (d e+c f)+C \left (6 d^2 e^2+11 c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (6 C e f c^2+d \left (9 C e^2+2 B f e+A f^2\right ) c+d^2 e (3 B e-2 A f)\right ) a-b^3 c e (3 c C e+3 B d e-A d f)+d \left (8 C d f^2 a^3-b f (13 C d e+7 c C f+2 B d f) a^2+b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d f)) a-b^3 \left (3 c C e^2+A d f e+c f (3 B e-2 A f)\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{b (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} (b c-a d) \left (4 a^2 C f-a b (B f+6 C e)+b^2 (3 B e-2 A f)\right )}{b \sqrt {a+b x} (b e-a f)}}{3 b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 123

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 130

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

Input:

Int[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(5/2)*Sqrt[e + f*x]),x]
 

Output:

(-2*(A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*Sqrt[e + f*x])/(3*b*(b*c - a*d 
)*(b*e - a*f)*(a + b*x)^(3/2)) + ((-2*(b*c - a*d)*(4*a^2*C*f + b^2*(3*B*e 
- 2*A*f) - a*b*(6*C*e + B*f))*Sqrt[c + d*x]*Sqrt[e + f*x])/(b*(b*e - a*f)* 
Sqrt[a + b*x]) - ((2*Sqrt[d]*Sqrt[-(b*c) + a*d]*(8*a^3*C*d*f^2 - a^2*b*f*( 
13*C*d*e + 7*c*C*f + 2*B*d*f) - b^3*(3*c*C*e^2 + A*d*e*f + c*f*(3*B*e - 2* 
A*f)) + a*b^2*(3*C*e*(d*e + 4*c*f) + f*(4*B*d*e + B*c*f - A*d*f)))*Sqrt[(b 
*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + 
b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b*f*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^2*C*d*f + b^2*(3*c*C*e + A*d*f) - a*b*(B*d*f + 3*C*(d*e 
+ c*f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*E 
llipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)* 
f)/(d*(b*e - a*f))])/(b*Sqrt[d]*f*Sqrt[c + d*x]*Sqrt[e + f*x]))/(b*(b*e - 
a*f)))/(3*b*(b*c - a*d)*(b*e - a*f))
 

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 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. \(1268\) vs. \(2(542)=1084\).

Time = 8.33 (sec) , antiderivative size = 1269, normalized size of antiderivative = 2.13

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1269\)
default	\(\text {Expression too large to display}\)	\(14757\)

Input:

int((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(5/2)/(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)* 
(2/3*(A*b^2-B*a*b+C*a^2)/b^4/(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+2/3*(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)/b^3*(A*a*b^2*d*f-2*A*b^ 
3*c*f+A*b^3*d*e+2*B*a^2*b*d*f-B*a*b^2*c*f-4*B*a*b^2*d*e+3*B*b^3*c*e-5*C*a^ 
3*d*f+4*C*a^2*b*c*f+7*C*a^2*b*d*e-6*C*a*b^2*c*e)/(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*((B*b*d-2*C*a*d+C*b*c)/b^3+1/3*(A*b^2 
-B*a*b+C*a^2)/b^3*d*f/(a*f-b*e)-1/3/b^3*(a*d*f-b*c*f-b*d*e)*(A*a*b^2*d*f-2 
*A*b^3*c*f+A*b^3*d*e+2*B*a^2*b*d*f-B*a*b^2*c*f-4*B*a*b^2*d*e+3*B*b^3*c*e-5 
*C*a^3*d*f+4*C*a^2*b*c*f+7*C*a^2*b*d*e-6*C*a*b^2*c*e)/(a^2*d*f-a*b*c*f-a*b 
*d*e+b^2*c*e)/(a*f-b*e)-1/3*(b*c*f+b*d*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e 
)/b^3*(A*a*b^2*d*f-2*A*b^3*c*f+A*b^3*d*e+2*B*a^2*b*d*f-B*a*b^2*c*f-4*B*a*b 
^2*d*e+3*B*b^3*c*e-5*C*a^3*d*f+4*C*a^2*b*c*f+7*C*a^2*b*d*e-6*C*a*b^2*c*e)/ 
(a*f-b*e))*(c/d-a/b)*((x+c/d)/(c/d-a/b))^(1/2)*((x+e/f)/(-c/d+e/f))^(1/2)* 
((x+a/b)/(-c/d+a/b))^(1/2)/(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f* 
x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)*EllipticF(((x+c/d)/(c/d-a/b))^(1/2),((-c/d+ 
a/b)/(-c/d+e/f))^(1/2))+2*(C*d/b^2-1/3*d*f/b^2*(A*a*b^2*d*f-2*A*b^3*c*f+A* 
b^3*d*e+2*B*a^2*b*d*f-B*a*b^2*c*f-4*B*a*b^2*d*e+3*B*b^3*c*e-5*C*a^3*d*f+4* 
C*a^2*b*c*f+7*C*a^2*b*d*e-6*C*a*b^2*c*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e) 
/(a*f-b*e))*(c/d-a/b)*((x+c/d)/(c/d-a/b))^(1/2)*((x+e/f)/(-c/d+e/f))^(1...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2429 vs. \(2 (541) = 1082\).

Time = 0.31 (sec) , antiderivative size = 2429, normalized size of antiderivative = 4.08 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(5/2)/(f*x+e)^(1/2),x, algor 
ithm="fricas")
 

Output:

2/9*(3*(((5*C*a^2*b^4 - 2*B*a*b^5 - A*b^6)*c*d - 3*(2*C*a^3*b^3 - B*a^2*b^ 
4)*d^2)*e*f^2 - (3*(C*a^3*b^3 - A*a*b^5)*c*d - (4*C*a^4*b^2 - B*a^3*b^3 - 
2*A*a^2*b^4)*d^2)*f^3 + ((3*(2*C*a*b^5 - B*b^6)*c*d - (7*C*a^2*b^4 - 4*B*a 
*b^5 + A*b^6)*d^2)*e*f^2 - ((4*C*a^2*b^4 - B*a*b^5 - 2*A*b^6)*c*d - (5*C*a 
^3*b^3 - 2*B*a^2*b^4 - A*a*b^5)*d^2)*f^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)*s 
qrt(f*x + e) - (3*(C*a^2*b^4*c*d - C*a^3*b^3*d^2)*e^3 - (6*C*a^2*b^4*c^2 - 
 3*(5*C*a^3*b^3 - 2*B*a^2*b^4)*c*d + (8*C*a^4*b^2 - 5*B*a^3*b^3 - A*a^2*b^ 
4)*d^2)*e^2*f + (3*(2*C*a^3*b^3 + B*a^2*b^4)*c^2 - (25*C*a^4*b^2 - 4*B*a^3 
*b^3 - 2*A*a^2*b^4)*c*d + (17*C*a^5*b - 5*B*a^4*b^2 - 4*A*a^3*b^3)*d^2)*e* 
f^2 - ((2*C*a^4*b^2 + B*a^3*b^3 + 2*A*a^2*b^4)*c^2 - (11*C*a^5*b - 2*B*a^4 
*b^2 + 2*A*a^3*b^3)*c*d + (8*C*a^6 - 2*B*a^5*b - A*a^4*b^2)*d^2)*f^3 + (3* 
(C*b^6*c*d - C*a*b^5*d^2)*e^3 - (6*C*b^6*c^2 - 3*(5*C*a*b^5 - 2*B*b^6)*c*d 
 + (8*C*a^2*b^4 - 5*B*a*b^5 - A*b^6)*d^2)*e^2*f + (3*(2*C*a*b^5 + B*b^6)*c 
^2 - (25*C*a^2*b^4 - 4*B*a*b^5 - 2*A*b^6)*c*d + (17*C*a^3*b^3 - 5*B*a^2*b^ 
4 - 4*A*a*b^5)*d^2)*e*f^2 - ((2*C*a^2*b^4 + B*a*b^5 + 2*A*b^6)*c^2 - (11*C 
*a^3*b^3 - 2*B*a^2*b^4 + 2*A*a*b^5)*c*d + (8*C*a^4*b^2 - 2*B*a^3*b^3 - A*a 
^2*b^4)*d^2)*f^3)*x^2 + 2*(3*(C*a*b^5*c*d - C*a^2*b^4*d^2)*e^3 - (6*C*a*b^ 
5*c^2 - 3*(5*C*a^2*b^4 - 2*B*a*b^5)*c*d + (8*C*a^3*b^3 - 5*B*a^2*b^4 - A*a 
*b^5)*d^2)*e^2*f + (3*(2*C*a^2*b^4 + B*a*b^5)*c^2 - (25*C*a^3*b^3 - 4*B*a^ 
2*b^4 - 2*A*a*b^5)*c*d + (17*C*a^4*b^2 - 5*B*a^3*b^3 - 4*A*a^2*b^4)*d^2...
 

Sympy [F]

\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=\int \frac {\sqrt {c + d x} \left (A + B x + C x^{2}\right )}{\left (a + b x\right )^{\frac {5}{2}} \sqrt {e + f x}}\, dx \] Input:

integrate((d*x+c)**(1/2)*(C*x**2+B*x+A)/(b*x+a)**(5/2)/(f*x+e)**(1/2),x)
 

Output:

Integral(sqrt(c + d*x)*(A + B*x + C*x**2)/((a + b*x)**(5/2)*sqrt(e + f*x)) 
, x)
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c}}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {f x + e}} \,d x } \] Input:

integrate((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(5/2)/(f*x+e)^(1/2),x, algor 
ithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(5/2)*sqrt(f*x + e)), 
 x)
 

Giac [F]

\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c}}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {f x + e}} \,d x } \] Input:

integrate((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(5/2)/(f*x+e)^(1/2),x, algor 
ithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(5/2)*sqrt(f*x + e)), 
 x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=\int \frac {\sqrt {c+d\,x}\,\left (C\,x^2+B\,x+A\right )}{\sqrt {e+f\,x}\,{\left (a+b\,x\right )}^{5/2}} \,d x \] Input:

int(((c + d*x)^(1/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(a + b*x)^(5/2)), 
x)
 

Output:

int(((c + d*x)^(1/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(a + b*x)^(5/2)), 
 x)
 

Reduce [F]

\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=\int \frac {\sqrt {d x +c}\, \left (C \,x^{2}+B x +A \right )}{\left (b x +a \right )^{\frac {5}{2}} \sqrt {f x +e}}d x \] Input:

int((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(5/2)/(f*x+e)^(1/2),x)
 

Output:

int((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(5/2)/(f*x+e)^(1/2),x)