\(\int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx\) [99]

Optimal result

Integrand size = 38, antiderivative size = 642 \[ \int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac {2 \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)-b^3 (3 B c e-2 A (d e+c f))+a^2 b (B d f-4 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d)^2 (b e-a f)^2 \sqrt {a+b x}}-\frac {2 \sqrt {d} \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)-b^3 (3 B c e-2 A (d e+c f))+a^2 b (B d f-4 C (d e+c f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^2 (-b c+a d)^{3/2} (b e-a f)^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \left (a^2 C d (d e-c f)-b^2 \left (3 c^2 C e-3 B c d e+2 A d^2 e+A c d f\right )+a b \left (3 \left (c^2 C+A d^2\right ) f-B d (d e+2 c f)\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^2 \sqrt {d} (-b c+a d)^{3/2} (b e-a f) \sqrt {c+d x} \sqrt {e+f x}} \] Output:

-2/3*(A*b^2-a*(B*b-C*a))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(-a*f+b* 
e)/(b*x+a)^(3/2)+2/3*(2*a^3*C*d*f+a*b^2*(-4*A*d*f+B*c*f+B*d*e+6*C*c*e)-b^3 
*(3*B*c*e-2*A*(c*f+d*e))+a^2*b*(B*d*f-4*C*(c*f+d*e)))*(d*x+c)^(1/2)*(f*x+e 
)^(1/2)/b/(-a*d+b*c)^2/(-a*f+b*e)^2/(b*x+a)^(1/2)-2/3*d^(1/2)*(2*a^3*C*d*f 
+a*b^2*(-4*A*d*f+B*c*f+B*d*e+6*C*c*e)-b^3*(3*B*c*e-2*A*(c*f+d*e))+a^2*b*(B 
*d*f-4*C*(c*f+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 
^2/(a*d-b*c)^(3/2)/(-a*f+b*e)^2/(d*x+c)^(1/2)/(b*(f*x+e)/(-a*f+b*e))^(1/2) 
-2/3*(a^2*C*d*(-c*f+d*e)-b^2*(A*c*d*f+2*A*d^2*e-3*B*c*d*e+3*C*c^2*e)+a*b*( 
3*(A*d^2+C*c^2)*f-B*d*(2*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^2/d^(1/2)/(a*d-b*c)^(3/2)/(-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 = 27.50 (sec) , antiderivative size = 699, normalized size of antiderivative = 1.09 \[ \int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {2 \left (b^2 \sqrt {-a+\frac {b c}{d}} (c+d x) (e+f x) \left (\left (A b^2+a (-b B+a C)\right ) (b c-a d) (b e-a f)+\left (-2 a^3 C d f-a b^2 (6 c C e+B d e+B c f-4 A d f)+b^3 (3 B c e-2 A (d e+c f))+a^2 b (-B d f+4 C (d e+c f))\right ) (a+b x)\right )+(a+b x) \left (b^2 \sqrt {-a+\frac {b c}{d}} \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)+b^3 (-3 B c e+2 A (d e+c f))+a^2 b (B d f-4 C (d e+c f))\right ) (c+d x) (e+f x)+i (b c-a d) f \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)+b^3 (-3 B c e+2 A (d e+c f))+a^2 b (B d f-4 C (d e+c f))\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )-i b (b c-a d) \left (a^2 C f (d e-c f)+b^2 \left (3 c C e^2+A d e f+c f (-3 B e+2 A f)\right )+a b \left (-3 C d e^2+f (2 B d e+B c f-3 A d f)\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )\right )}{3 b^3 \sqrt {-a+\frac {b c}{d}} (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \] Input:

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

Output:

(-2*(b^2*Sqrt[-a + (b*c)/d]*(c + d*x)*(e + f*x)*((A*b^2 + a*(-(b*B) + a*C) 
)*(b*c - a*d)*(b*e - a*f) + (-2*a^3*C*d*f - a*b^2*(6*c*C*e + B*d*e + B*c*f 
 - 4*A*d*f) + b^3*(3*B*c*e - 2*A*(d*e + c*f)) + a^2*b*(-(B*d*f) + 4*C*(d*e 
 + c*f)))*(a + b*x)) + (a + b*x)*(b^2*Sqrt[-a + (b*c)/d]*(2*a^3*C*d*f + a* 
b^2*(6*c*C*e + B*d*e + B*c*f - 4*A*d*f) + b^3*(-3*B*c*e + 2*A*(d*e + c*f)) 
 + a^2*b*(B*d*f - 4*C*(d*e + c*f)))*(c + d*x)*(e + f*x) + I*(b*c - a*d)*f* 
(2*a^3*C*d*f + a*b^2*(6*c*C*e + B*d*e + B*c*f - 4*A*d*f) + b^3*(-3*B*c*e + 
 2*A*(d*e + c*f)) + a^2*b*(B*d*f - 4*C*(d*e + c*f)))*(a + b*x)^(3/2)*Sqrt[ 
(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I 
*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f 
)] - I*b*(b*c - a*d)*(a^2*C*f*(d*e - c*f) + b^2*(3*c*C*e^2 + A*d*e*f + c*f 
*(-3*B*e + 2*A*f)) + a*b*(-3*C*d*e^2 + f*(2*B*d*e + B*c*f - 3*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))]*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^3*Sqrt[-a + (b*c)/d]*(b*c - a*d)^2*(b*e - a*f 
)^2*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x])
 

Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2117, 27, 169, 27, 176, 124, 123, 131, 131, 130}

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

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

\(\Big \downarrow \) 2117

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 123

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 130

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

Input:

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

Output:

(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b*(b*c - a*d)* 
(b*e - a*f)*(a + b*x)^(3/2)) + ((2*(2*a^3*C*d*f + a*b^2*(6*c*C*e + B*d*e + 
 B*c*f - 4*A*d*f) - b^3*(3*B*c*e - 2*A*(d*e + c*f)) + a^2*b*(B*d*f - 4*C*( 
d*e + c*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*Sqrt[a 
+ b*x]) - ((2*Sqrt[d]*Sqrt[-(b*c) + a*d]*(2*a^3*C*d*f + a*b^2*(6*c*C*e + B 
*d*e + B*c*f - 4*A*d*f) - b^3*(3*B*c*e - 2*A*(d*e + c*f)) + a^2*b*(B*d*f - 
 4*C*(d*e + c*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*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[- 
(b*c) + a*d]*(b*e - a*f)*(a^2*C*d*(d*e - c*f) - b^2*(3*c^2*C*e - 3*B*c*d*e 
 + 2*A*d^2*e + A*c*d*f) + a*b*(3*(c^2*C + A*d^2)*f - B*d*(d*e + 2*c*f)))*S 
qrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[A 
rcSin[(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[e + f*x]))/((b*c - a*d)*(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 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. \(1248\) vs. \(2(588)=1176\).

Time = 21.47 (sec) , antiderivative size = 1249, normalized size of antiderivative = 1.95

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1249\)
default	\(\text {Expression too large to display}\)	\(13626\)

Input:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2344 vs. \(2 (587) = 1174\).

Time = 0.28 (sec) , antiderivative size = 2344, normalized size of antiderivative = 3.65 \[ \int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(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*(C*a^3*b^3 - A*a*b^5)*d 
^2)*e*f - (3*(C*a^3*b^3 - A*a*b^5)*c*d - (C*a^4*b^2 + 2*B*a^3*b^3 - 5*A*a^ 
2*b^4)*d^2)*f^2 + ((3*(2*C*a*b^5 - B*b^6)*c*d - (4*C*a^2*b^4 - B*a*b^5 - 2 
*A*b^6)*d^2)*e*f - ((4*C*a^2*b^4 - B*a*b^5 - 2*A*b^6)*c*d - (2*C*a^3*b^3 + 
 B*a^2*b^4 - 4*A*a*b^5)*d^2)*f^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x 
+ e) + ((9*C*a^2*b^4*c^2 - 3*(4*C*a^3*b^3 + B*a^2*b^4)*c*d + (5*C*a^4*b^2 
+ B*a^3*b^3 + 2*A*a^2*b^4)*d^2)*e^2 - (3*(4*C*a^3*b^3 + B*a^2*b^4)*c^2 - ( 
13*C*a^4*b^2 + 11*B*a^3*b^3 + A*a^2*b^4)*c*d + (5*C*a^5*b + 4*B*a^4*b^2 + 
5*A*a^3*b^3)*d^2)*e*f + ((5*C*a^4*b^2 + B*a^3*b^3 + 2*A*a^2*b^4)*c^2 - (5* 
C*a^5*b + 4*B*a^4*b^2 + 5*A*a^3*b^3)*c*d + (2*C*a^6 + B*a^5*b + 5*A*a^4*b^ 
2)*d^2)*f^2 + ((9*C*b^6*c^2 - 3*(4*C*a*b^5 + B*b^6)*c*d + (5*C*a^2*b^4 + B 
*a*b^5 + 2*A*b^6)*d^2)*e^2 - (3*(4*C*a*b^5 + B*b^6)*c^2 - (13*C*a^2*b^4 + 
11*B*a*b^5 + A*b^6)*c*d + (5*C*a^3*b^3 + 4*B*a^2*b^4 + 5*A*a*b^5)*d^2)*e*f 
 + ((5*C*a^2*b^4 + B*a*b^5 + 2*A*b^6)*c^2 - (5*C*a^3*b^3 + 4*B*a^2*b^4 + 5 
*A*a*b^5)*c*d + (2*C*a^4*b^2 + B*a^3*b^3 + 5*A*a^2*b^4)*d^2)*f^2)*x^2 + 2* 
((9*C*a*b^5*c^2 - 3*(4*C*a^2*b^4 + B*a*b^5)*c*d + (5*C*a^3*b^3 + B*a^2*b^4 
 + 2*A*a*b^5)*d^2)*e^2 - (3*(4*C*a^2*b^4 + B*a*b^5)*c^2 - (13*C*a^3*b^3 + 
11*B*a^2*b^4 + A*a*b^5)*c*d + (5*C*a^4*b^2 + 4*B*a^3*b^3 + 5*A*a^2*b^4)*d^ 
2)*e*f + ((5*C*a^3*b^3 + B*a^2*b^4 + 2*A*a*b^5)*c^2 - (5*C*a^4*b^2 + 4*B*a 
^3*b^3 + 5*A*a^2*b^4)*c*d + (2*C*a^5*b + B*a^4*b^2 + 5*A*a^3*b^3)*d^2)*...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*b + 2*int((sqrt(e + f*x)*sqrt 
(c + d*x)*sqrt(a + b*x)*x**2)/(a**4*c*d*e*f + a**4*c*d*f**2*x + a**4*d**2* 
e*f*x + a**4*d**2*f**2*x**2 - a**3*b*c**2*e*f - a**3*b*c**2*f**2*x - a**3* 
b*c*d*e**2 + a**3*b*c*d*e*f*x + 2*a**3*b*c*d*f**2*x**2 - a**3*b*d**2*e**2* 
x + 2*a**3*b*d**2*e*f*x**2 + 3*a**3*b*d**2*f**2*x**3 - 3*a**2*b**2*c**2*e* 
f*x - 3*a**2*b**2*c**2*f**2*x**2 - 3*a**2*b**2*c*d*e**2*x - 3*a**2*b**2*c* 
d*e*f*x**2 - 3*a**2*b**2*d**2*e**2*x**2 + 3*a**2*b**2*d**2*f**2*x**4 - 3*a 
*b**3*c**2*e*f*x**2 - 3*a*b**3*c**2*f**2*x**3 - 3*a*b**3*c*d*e**2*x**2 - 5 
*a*b**3*c*d*e*f*x**3 - 2*a*b**3*c*d*f**2*x**4 - 3*a*b**3*d**2*e**2*x**3 - 
2*a*b**3*d**2*e*f*x**4 + a*b**3*d**2*f**2*x**5 - b**4*c**2*e*f*x**3 - b**4 
*c**2*f**2*x**4 - b**4*c*d*e**2*x**3 - 2*b**4*c*d*e*f*x**4 - b**4*c*d*f**2 
*x**5 - b**4*d**2*e**2*x**4 - b**4*d**2*e*f*x**5),x)*a**4*c*d**2*f**2 + in 
t((sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*x**2)/(a**4*c*d*e*f + a**4*c* 
d*f**2*x + a**4*d**2*e*f*x + a**4*d**2*f**2*x**2 - a**3*b*c**2*e*f - a**3* 
b*c**2*f**2*x - a**3*b*c*d*e**2 + a**3*b*c*d*e*f*x + 2*a**3*b*c*d*f**2*x** 
2 - a**3*b*d**2*e**2*x + 2*a**3*b*d**2*e*f*x**2 + 3*a**3*b*d**2*f**2*x**3 
- 3*a**2*b**2*c**2*e*f*x - 3*a**2*b**2*c**2*f**2*x**2 - 3*a**2*b**2*c*d*e* 
*2*x - 3*a**2*b**2*c*d*e*f*x**2 - 3*a**2*b**2*d**2*e**2*x**2 + 3*a**2*b**2 
*d**2*f**2*x**4 - 3*a*b**3*c**2*e*f*x**2 - 3*a*b**3*c**2*f**2*x**3 - 3*a*b 
**3*c*d*e**2*x**2 - 5*a*b**3*c*d*e*f*x**3 - 2*a*b**3*c*d*f**2*x**4 - 3*...