\(\int \frac {(A+B x) (d+e x)^{5/2}}{(b x-c x^2)^{5/2}} \, dx\) [162]

Optimal result

Integrand size = 29, antiderivative size = 502 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x-c x^2\right )^{5/2}} \, dx=\frac {2 (c d+b e) \left (8 A c^2 d+b^2 B e+b c (4 B d+7 A e)\right ) x^2 \sqrt {d+e x}}{3 b^3 c \left (b x-c x^2\right )^{3/2}}-\frac {2 (3 b B d+6 A c d+5 A b e) x (d+e x)^{3/2}}{3 b^2 \left (b x-c x^2\right )^{3/2}}-\frac {2 A (d+e x)^{5/2}}{3 b \left (b x-c x^2\right )^{3/2}}+\frac {2 \left (16 A c^3 d^2-2 b^3 B e^2+b^2 c e (3 B d+A e)+8 b c^2 d (B d+2 A e)\right ) x \sqrt {d+e x}}{3 b^4 c \sqrt {b x-c x^2}}-\frac {2 \left (16 A c^3 d^2-2 b^3 B e^2+b^2 c e (3 B d+A e)+8 b c^2 d (B d+2 A e)\right ) \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{3 b^{7/2} c^{3/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x-c x^2}}+\frac {2 d (c d+b e) \left (16 A c^2 d-b^2 B e+8 b c (B d+A e)\right ) \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),-\frac {b e}{c d}\right )}{3 b^{7/2} c^{3/2} \sqrt {d+e x} \sqrt {b x-c x^2}} \] Output:

2/3*(b*e+c*d)*(8*A*c^2*d+b^2*B*e+b*c*(7*A*e+4*B*d))*x^2*(e*x+d)^(1/2)/b^3/ 
c/(-c*x^2+b*x)^(3/2)-2/3*(5*A*b*e+6*A*c*d+3*B*b*d)*x*(e*x+d)^(3/2)/b^2/(-c 
*x^2+b*x)^(3/2)-2/3*A*(e*x+d)^(5/2)/b/(-c*x^2+b*x)^(3/2)+2/3*(16*A*c^3*d^2 
-2*b^3*B*e^2+b^2*c*e*(A*e+3*B*d)+8*b*c^2*d*(2*A*e+B*d))*x*(e*x+d)^(1/2)/b^ 
4/c/(-c*x^2+b*x)^(1/2)-2/3*(16*A*c^3*d^2-2*b^3*B*e^2+b^2*c*e*(A*e+3*B*d)+8 
*b*c^2*d*(2*A*e+B*d))*x^(1/2)*(1-c*x/b)^(1/2)*(e*x+d)^(1/2)*EllipticE(c^(1 
/2)*x^(1/2)/b^(1/2),(-b*e/c/d)^(1/2))/b^(7/2)/c^(3/2)/(1+e*x/d)^(1/2)/(-c* 
x^2+b*x)^(1/2)+2/3*d*(b*e+c*d)*(16*A*c^2*d-b^2*B*e+8*b*c*(A*e+B*d))*x^(1/2 
)*(1-c*x/b)^(1/2)*(1+e*x/d)^(1/2)*EllipticF(c^(1/2)*x^(1/2)/b^(1/2),(-b*e/ 
c/d)^(1/2))/b^(7/2)/c^(3/2)/(e*x+d)^(1/2)/(-c*x^2+b*x)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 27.51 (sec) , antiderivative size = 468, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x-c x^2\right )^{5/2}} \, dx=\frac {2 \left (\frac {c (d+e x) \left (b (b B+A c) (c d+b e)^2 x^2+(c d+b e) \left (8 A c^2 d-2 b^2 B e+b c (5 B d+A e)\right ) x^2 (b-c x)-A b c d^2 (b-c x)^2-c d (3 b B d+8 A c d+7 A b e) x (b-c x)^2\right )}{b-c x}+\frac {x \left (\sqrt {-\frac {b}{c}} \left (16 A c^3 d^2-2 b^3 B e^2+b^2 c e (3 B d+A e)+8 b c^2 d (B d+2 A e)\right ) (b-c x) (d+e x)+i b e \left (16 A c^3 d^2-2 b^3 B e^2+b^2 c e (3 B d+A e)+8 b c^2 d (B d+2 A e)\right ) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right )|-\frac {c d}{b e}\right )-i b e (c d+b e) \left (8 A c^2 d-2 b^2 B e+b c (4 B d+A e)\right ) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right ),-\frac {c d}{b e}\right )\right )}{\sqrt {-\frac {b}{c}}}\right )}{3 b^4 c^2 x \sqrt {x (b-c x)} \sqrt {d+e x}} \] Input:

Integrate[((A + B*x)*(d + e*x)^(5/2))/(b*x - c*x^2)^(5/2),x]
 

Output:

(2*((c*(d + e*x)*(b*(b*B + A*c)*(c*d + b*e)^2*x^2 + (c*d + b*e)*(8*A*c^2*d 
 - 2*b^2*B*e + b*c*(5*B*d + A*e))*x^2*(b - c*x) - A*b*c*d^2*(b - c*x)^2 - 
c*d*(3*b*B*d + 8*A*c*d + 7*A*b*e)*x*(b - c*x)^2))/(b - c*x) + (x*(Sqrt[-(b 
/c)]*(16*A*c^3*d^2 - 2*b^3*B*e^2 + b^2*c*e*(3*B*d + A*e) + 8*b*c^2*d*(B*d 
+ 2*A*e))*(b - c*x)*(d + e*x) + I*b*e*(16*A*c^3*d^2 - 2*b^3*B*e^2 + b^2*c* 
e*(3*B*d + A*e) + 8*b*c^2*d*(B*d + 2*A*e))*Sqrt[1 - b/(c*x)]*Sqrt[1 + d/(e 
*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[-(b/c)]/Sqrt[x]], -((c*d)/(b*e))] - 
I*b*e*(c*d + b*e)*(8*A*c^2*d - 2*b^2*B*e + b*c*(4*B*d + A*e))*Sqrt[1 - b/( 
c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[-(b/c)]/Sqrt[x]], 
 -((c*d)/(b*e))]))/Sqrt[-(b/c)]))/(3*b^4*c^2*x*Sqrt[x*(b - c*x)]*Sqrt[d + 
e*x])
 

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 467, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {1233, 27, 1234, 27, 1269, 1169, 122, 120, 127, 126}

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.

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

\(\Big \downarrow \) 1233

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1234

\(\displaystyle \frac {-\frac {2 \int -\frac {e \left (b d \left (B e b^2+c (4 B d+7 A e) b+8 A c^2 d\right )-\left (-2 B e^2 b^3+c e (3 B d+A e) b^2+8 c^2 d (B d+2 A e) b+16 A c^3 d^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {b x-c x^2}}dx}{b^2}-\frac {2 \sqrt {d+e x} \left (b d \left (b c (7 A e+4 B d)+8 A c^2 d+b^2 B e\right )-x \left (b^2 c e (A e+3 B d)+8 b c^2 d (2 A e+B d)+16 A c^3 d^2-2 b^3 B e^2\right )\right )}{b^2 \sqrt {b x-c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^{3/2} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{3 b^2 c \left (b x-c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {e \int \frac {b d \left (B e b^2+c (4 B d+7 A e) b+8 A c^2 d\right )-\left (-2 B e^2 b^3+c e (3 B d+A e) b^2+8 c^2 d (B d+2 A e) b+16 A c^3 d^2\right ) x}{\sqrt {d+e x} \sqrt {b x-c x^2}}dx}{b^2}-\frac {2 \sqrt {d+e x} \left (b d \left (b c (7 A e+4 B d)+8 A c^2 d+b^2 B e\right )-x \left (b^2 c e (A e+3 B d)+8 b c^2 d (2 A e+B d)+16 A c^3 d^2-2 b^3 B e^2\right )\right )}{b^2 \sqrt {b x-c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^{3/2} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{3 b^2 c \left (b x-c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {e \left (\frac {d (b e+c d) \left (8 b c (A e+B d)+16 A c^2 d+b^2 (-B) e\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x-c x^2}}dx}{e}-\frac {\left (b^2 c e (A e+3 B d)+8 b c^2 d (2 A e+B d)+16 A c^3 d^2-2 b^3 B e^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x-c x^2}}dx}{e}\right )}{b^2}-\frac {2 \sqrt {d+e x} \left (b d \left (b c (7 A e+4 B d)+8 A c^2 d+b^2 B e\right )-x \left (b^2 c e (A e+3 B d)+8 b c^2 d (2 A e+B d)+16 A c^3 d^2-2 b^3 B e^2\right )\right )}{b^2 \sqrt {b x-c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^{3/2} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{3 b^2 c \left (b x-c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1169

\(\displaystyle \frac {\frac {e \left (\frac {d \sqrt {x} \sqrt {b-c x} (b e+c d) \left (8 b c (A e+B d)+16 A c^2 d+b^2 (-B) e\right ) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}-\frac {\sqrt {x} \sqrt {b-c x} \left (b^2 c e (A e+3 B d)+8 b c^2 d (2 A e+B d)+16 A c^3 d^2-2 b^3 B e^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b-c x}}dx}{e \sqrt {b x-c x^2}}\right )}{b^2}-\frac {2 \sqrt {d+e x} \left (b d \left (b c (7 A e+4 B d)+8 A c^2 d+b^2 B e\right )-x \left (b^2 c e (A e+3 B d)+8 b c^2 d (2 A e+B d)+16 A c^3 d^2-2 b^3 B e^2\right )\right )}{b^2 \sqrt {b x-c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^{3/2} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{3 b^2 c \left (b x-c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 122

\(\displaystyle \frac {\frac {e \left (\frac {d \sqrt {x} \sqrt {b-c x} (b e+c d) \left (8 b c (A e+B d)+16 A c^2 d+b^2 (-B) e\right ) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}-\frac {\sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (b^2 c e (A e+3 B d)+8 b c^2 d (2 A e+B d)+16 A c^3 d^2-2 b^3 B e^2\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {1-\frac {c x}{b}}}dx}{e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{b^2}-\frac {2 \sqrt {d+e x} \left (b d \left (b c (7 A e+4 B d)+8 A c^2 d+b^2 B e\right )-x \left (b^2 c e (A e+3 B d)+8 b c^2 d (2 A e+B d)+16 A c^3 d^2-2 b^3 B e^2\right )\right )}{b^2 \sqrt {b x-c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^{3/2} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{3 b^2 c \left (b x-c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 120

\(\displaystyle \frac {\frac {e \left (\frac {d \sqrt {x} \sqrt {b-c x} (b e+c d) \left (8 b c (A e+B d)+16 A c^2 d+b^2 (-B) e\right ) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (b^2 c e (A e+3 B d)+8 b c^2 d (2 A e+B d)+16 A c^3 d^2-2 b^3 B e^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{b^2}-\frac {2 \sqrt {d+e x} \left (b d \left (b c (7 A e+4 B d)+8 A c^2 d+b^2 B e\right )-x \left (b^2 c e (A e+3 B d)+8 b c^2 d (2 A e+B d)+16 A c^3 d^2-2 b^3 B e^2\right )\right )}{b^2 \sqrt {b x-c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^{3/2} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{3 b^2 c \left (b x-c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {\frac {e \left (\frac {d \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1} (b e+c d) \left (8 b c (A e+B d)+16 A c^2 d+b^2 (-B) e\right ) \int \frac {1}{\sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (b^2 c e (A e+3 B d)+8 b c^2 d (2 A e+B d)+16 A c^3 d^2-2 b^3 B e^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{b^2}-\frac {2 \sqrt {d+e x} \left (b d \left (b c (7 A e+4 B d)+8 A c^2 d+b^2 B e\right )-x \left (b^2 c e (A e+3 B d)+8 b c^2 d (2 A e+B d)+16 A c^3 d^2-2 b^3 B e^2\right )\right )}{b^2 \sqrt {b x-c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^{3/2} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{3 b^2 c \left (b x-c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {\frac {e \left (\frac {2 \sqrt {b} d \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1} (b e+c d) \left (8 b c (A e+B d)+16 A c^2 d+b^2 (-B) e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (b^2 c e (A e+3 B d)+8 b c^2 d (2 A e+B d)+16 A c^3 d^2-2 b^3 B e^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{b^2}-\frac {2 \sqrt {d+e x} \left (b d \left (b c (7 A e+4 B d)+8 A c^2 d+b^2 B e\right )-x \left (b^2 c e (A e+3 B d)+8 b c^2 d (2 A e+B d)+16 A c^3 d^2-2 b^3 B e^2\right )\right )}{b^2 \sqrt {b x-c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^{3/2} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{3 b^2 c \left (b x-c x^2\right )^{3/2}}\)

Input:

Int[((A + B*x)*(d + e*x)^(5/2))/(b*x - c*x^2)^(5/2),x]
 

Output:

(-2*(d + e*x)^(3/2)*(A*b*c*d - (2*A*c^2*d + b^2*B*e + b*c*(B*d + A*e))*x)) 
/(3*b^2*c*(b*x - c*x^2)^(3/2)) + ((-2*Sqrt[d + e*x]*(b*d*(8*A*c^2*d + b^2* 
B*e + b*c*(4*B*d + 7*A*e)) - (16*A*c^3*d^2 - 2*b^3*B*e^2 + b^2*c*e*(3*B*d 
+ A*e) + 8*b*c^2*d*(B*d + 2*A*e))*x))/(b^2*Sqrt[b*x - c*x^2]) + (e*((-2*Sq 
rt[b]*(16*A*c^3*d^2 - 2*b^3*B*e^2 + b^2*c*e*(3*B*d + A*e) + 8*b*c^2*d*(B*d 
 + 2*A*e))*Sqrt[x]*Sqrt[1 - (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[ 
c]*Sqrt[x])/Sqrt[b]], -((b*e)/(c*d))])/(Sqrt[c]*e*Sqrt[1 + (e*x)/d]*Sqrt[b 
*x - c*x^2]) + (2*Sqrt[b]*d*(c*d + b*e)*(16*A*c^2*d - b^2*B*e + 8*b*c*(B*d 
 + A*e))*Sqrt[x]*Sqrt[1 - (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqr 
t[c]*Sqrt[x])/Sqrt[b]], -((b*e)/(c*d))])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[b*x 
 - c*x^2])))/b^2)/(3*b^2*c)
 

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 120

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- 
b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt 
Q[e, 0] &&  !LtQ[-b/d, 0]
 

rule 122

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) 
)   Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b 
, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

rule 126

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* 
Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & 
& GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
 

rule 127

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x 
]))   Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free 
Q[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

rule 1169

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> 
 Simp[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b*x + c*x^2])   Int[(d + e*x)^m/(Sqrt[x]* 
Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && Eq 
Q[m^2, 1/4]
 

rule 1233

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) 
^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c 
*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( 
p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim 
p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f 
*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( 
m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* 
p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && 
GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | 
|  !ILtQ[m + 2*p + 3, 0])
 

rule 1234

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( 
(f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 
 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g 
*(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)* 
(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1 
] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(944\) vs. \(2(436)=872\).

Time = 4.43 (sec) , antiderivative size = 945, normalized size of antiderivative = 1.88

method	result	size
elliptic	\(\frac {\sqrt {\left (-c x +b \right ) x \left (e x +d \right )}\, \left (\frac {2 \left (A \,b^{2} e^{2} c +2 A b \,c^{2} d e +A \,c^{3} d^{2}+b^{3} B \,e^{2}+2 B \,b^{2} c d e +B b \,c^{2} d^{2}\right ) \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}{3 b^{3} c^{3} \left (x -\frac {b}{c}\right )^{2}}-\frac {2 \left (-c e \,x^{2}-c d x \right ) \left (A \,b^{2} e^{2} c +9 A b \,c^{2} d e +8 A \,c^{3} d^{2}-2 b^{3} B \,e^{2}+3 B \,b^{2} c d e +5 B b \,c^{2} d^{2}\right )}{3 b^{4} c^{2} \sqrt {\left (x -\frac {b}{c}\right ) \left (-c e \,x^{2}-c d x \right )}}-\frac {2 A \,d^{2} \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}{3 b^{3} x^{2}}-\frac {2 \left (-c e \,x^{2}+b e x -c d x +b d \right ) d \left (7 A b e +8 A c d +3 B b d \right )}{3 b^{4} \sqrt {x \left (-c e \,x^{2}+b e x -c d x +b d \right )}}+\frac {2 \left (\frac {B \,e^{3}}{c^{2}}-\frac {\left (A \,b^{2} e^{2} c +2 A b \,c^{2} d e +A \,c^{3} d^{2}+b^{3} B \,e^{2}+2 B \,b^{2} c d e +B b \,c^{2} d^{2}\right ) e}{3 c^{2} b^{3}}+\frac {\left (A \,b^{2} e^{2} c +9 A b \,c^{2} d e +8 A \,c^{3} d^{2}-2 b^{3} B \,e^{2}+3 B \,b^{2} c d e +5 B b \,c^{2} d^{2}\right ) \left (b e +c d \right )}{3 c^{2} b^{4}}-\frac {d \left (A \,b^{2} e^{2} c +9 A b \,c^{2} d e +8 A \,c^{3} d^{2}-2 b^{3} B \,e^{2}+3 B \,b^{2} c d e +5 B b \,c^{2} d^{2}\right )}{3 c \,b^{4}}+\frac {d^{2} A c e}{3 b^{3}}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {b}{c}}{-\frac {d}{e}-\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )}{e \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}+\frac {2 \left (-\frac {\left (A \,b^{2} e^{2} c +9 A b \,c^{2} d e +8 A \,c^{3} d^{2}-2 b^{3} B \,e^{2}+3 B \,b^{2} c d e +5 B b \,c^{2} d^{2}\right ) e}{3 c \,b^{4}}-\frac {d e c \left (7 A b e +8 A c d +3 B b d \right )}{3 b^{4}}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {b}{c}}{-\frac {d}{e}-\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}-\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )+\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )}{c}\right )}{e \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, \sqrt {x \left (-c x +b \right )}}\)	\(945\)
default	\(\text {Expression too large to display}\)	\(2626\)

Input:

int((B*x+A)*(e*x+d)^(5/2)/(-c*x^2+b*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/(e*x+d)^(1/2)*((-c*x+b)*x*(e*x+d))^(1/2)/(x*(-c*x+b))^(1/2)*(2/3*(A*b^2* 
c*e^2+2*A*b*c^2*d*e+A*c^3*d^2+B*b^3*e^2+2*B*b^2*c*d*e+B*b*c^2*d^2)/b^3/c^3 
*(-c*e*x^3+b*e*x^2-c*d*x^2+b*d*x)^(1/2)/(x-b/c)^2-2/3*(-c*e*x^2-c*d*x)*(A* 
b^2*c*e^2+9*A*b*c^2*d*e+8*A*c^3*d^2-2*B*b^3*e^2+3*B*b^2*c*d*e+5*B*b*c^2*d^ 
2)/b^4/c^2/((x-b/c)*(-c*e*x^2-c*d*x))^(1/2)-2/3*A*d^2/b^3*(-c*e*x^3+b*e*x^ 
2-c*d*x^2+b*d*x)^(1/2)/x^2-2/3*(-c*e*x^2+b*e*x-c*d*x+b*d)*d/b^4*(7*A*b*e+8 
*A*c*d+3*B*b*d)/(x*(-c*e*x^2+b*e*x-c*d*x+b*d))^(1/2)+2*(B*e^3/c^2-1/3*(A*b 
^2*c*e^2+2*A*b*c^2*d*e+A*c^3*d^2+B*b^3*e^2+2*B*b^2*c*d*e+B*b*c^2*d^2)/c^2* 
e/b^3+1/3*(A*b^2*c*e^2+9*A*b*c^2*d*e+8*A*c^3*d^2-2*B*b^3*e^2+3*B*b^2*c*d*e 
+5*B*b*c^2*d^2)/c^2*(b*e+c*d)/b^4-1/3/c*d*(A*b^2*c*e^2+9*A*b*c^2*d*e+8*A*c 
^3*d^2-2*B*b^3*e^2+3*B*b^2*c*d*e+5*B*b*c^2*d^2)/b^4+1/3*d^2/b^3*A*c*e)*d/e 
*((x+d/e)/d*e)^(1/2)*((x-b/c)/(-d/e-b/c))^(1/2)*(-e*x/d)^(1/2)/(-c*e*x^3+b 
*e*x^2-c*d*x^2+b*d*x)^(1/2)*EllipticF(((x+d/e)/d*e)^(1/2),(-d/e/(-d/e-b/c) 
)^(1/2))+2*(-1/3*(A*b^2*c*e^2+9*A*b*c^2*d*e+8*A*c^3*d^2-2*B*b^3*e^2+3*B*b^ 
2*c*d*e+5*B*b*c^2*d^2)/c*e/b^4-1/3*d*e*c*(7*A*b*e+8*A*c*d+3*B*b*d)/b^4)*d/ 
e*((x+d/e)/d*e)^(1/2)*((x-b/c)/(-d/e-b/c))^(1/2)*(-e*x/d)^(1/2)/(-c*e*x^3+ 
b*e*x^2-c*d*x^2+b*d*x)^(1/2)*((-d/e-b/c)*EllipticE(((x+d/e)/d*e)^(1/2),(-d 
/e/(-d/e-b/c))^(1/2))+b/c*EllipticF(((x+d/e)/d*e)^(1/2),(-d/e/(-d/e-b/c))^ 
(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1041 vs. \(2 (436) = 872\).

Time = 0.12 (sec) , antiderivative size = 1041, normalized size of antiderivative = 2.07 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x-c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^(5/2)/(-c*x^2+b*x)^(5/2),x, algorithm="fricas")
 

Output:

-2/9*(((8*(B*b*c^5 + 2*A*c^6)*d^3 + (7*B*b^2*c^4 + 24*A*b*c^5)*d^2*e - 2*( 
B*b^3*c^3 - 3*A*b^2*c^4)*d*e^2 + (2*B*b^4*c^2 - A*b^3*c^3)*e^3)*x^4 - 2*(8 
*(B*b^2*c^4 + 2*A*b*c^5)*d^3 + (7*B*b^3*c^3 + 24*A*b^2*c^4)*d^2*e - 2*(B*b 
^4*c^2 - 3*A*b^3*c^3)*d*e^2 + (2*B*b^5*c - A*b^4*c^2)*e^3)*x^3 + (8*(B*b^3 
*c^3 + 2*A*b^2*c^4)*d^3 + (7*B*b^4*c^2 + 24*A*b^3*c^3)*d^2*e - 2*(B*b^5*c 
- 3*A*b^4*c^2)*d*e^2 + (2*B*b^6 - A*b^5*c)*e^3)*x^2)*sqrt(-c*e)*weierstras 
sPInverse(4/3*(c^2*d^2 + b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 + 
3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d - 
 b*e)/(c*e)) + 3*((8*(B*b*c^5 + 2*A*c^6)*d^2*e + (3*B*b^2*c^4 + 16*A*b*c^5 
)*d*e^2 - (2*B*b^3*c^3 - A*b^2*c^4)*e^3)*x^4 - 2*(8*(B*b^2*c^4 + 2*A*b*c^5 
)*d^2*e + (3*B*b^3*c^3 + 16*A*b^2*c^4)*d*e^2 - (2*B*b^4*c^2 - A*b^3*c^3)*e 
^3)*x^3 + (8*(B*b^3*c^3 + 2*A*b^2*c^4)*d^2*e + (3*B*b^4*c^2 + 16*A*b^3*c^3 
)*d*e^2 - (2*B*b^5*c - A*b^4*c^2)*e^3)*x^2)*sqrt(-c*e)*weierstrassZeta(4/3 
*(c^2*d^2 + b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 + 3*b*c^2*d^2*e 
 - 3*b^2*c*d*e^2 - 2*b^3*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^2 
+ b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c 
*d*e^2 - 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d - b*e)/(c*e))) + 3*(A*b^ 
3*c^3*d^2*e + (8*(B*b*c^5 + 2*A*c^6)*d^2*e + (3*B*b^2*c^4 + 16*A*b*c^5)*d* 
e^2 - (2*B*b^3*c^3 - A*b^2*c^4)*e^3)*x^3 - (12*(B*b^2*c^4 + 2*A*b*c^5)*d^2 
*e + 5*(B*b^3*c^3 + 5*A*b^2*c^4)*d*e^2 - (B*b^4*c^2 - 2*A*b^3*c^3)*e^3)...
 

Sympy [F(-1)]

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

integrate((B*x+A)*(e*x+d)**(5/2)/(-c*x**2+b*x)**(5/2),x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate((B*x+A)*(e*x+d)^(5/2)/(-c*x^2+b*x)^(5/2),x, algorithm="maxima")
 

Output:

integrate((B*x + A)*(e*x + d)^(5/2)/(-c*x^2 + b*x)^(5/2), x)
 

Giac [F]

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

integrate((B*x+A)*(e*x+d)^(5/2)/(-c*x^2+b*x)^(5/2),x, algorithm="giac")
 

Output:

integrate((B*x + A)*(e*x + d)^(5/2)/(-c*x^2 + b*x)^(5/2), x)
 

Mupad [F(-1)]

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

int(((A + B*x)*(d + e*x)^(5/2))/(b*x - c*x^2)^(5/2),x)
 

Output:

int(((A + B*x)*(d + e*x)^(5/2))/(b*x - c*x^2)^(5/2), x)
 

Reduce [F]

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

int((B*x+A)*(e*x+d)^(5/2)/(-c*x^2+b*x)^(5/2),x)
 

Output:

(2*sqrt(d + e*x)*sqrt(b - c*x)*a*b*c*e**2*x - 2*sqrt(d + e*x)*sqrt(b - c*x 
)*a*c**2*d**2 - 4*sqrt(d + e*x)*sqrt(b - c*x)*b**3*e**2*x + 2*sqrt(d + e*x 
)*sqrt(b - c*x)*b**2*c*d*e*x + 6*sqrt(d + e*x)*sqrt(b - c*x)*b**2*c*e**2*x 
**2 + sqrt(x)*int((sqrt(d + e*x)*sqrt(b - c*x))/(sqrt(x)*b**4*d*e*x + sqrt 
(x)*b**4*e**2*x**2 - 3*sqrt(x)*b**3*c*d**2*x - 6*sqrt(x)*b**3*c*d*e*x**2 - 
 3*sqrt(x)*b**3*c*e**2*x**3 + 9*sqrt(x)*b**2*c**2*d**2*x**2 + 12*sqrt(x)*b 
**2*c**2*d*e*x**3 + 3*sqrt(x)*b**2*c**2*e**2*x**4 - 9*sqrt(x)*b*c**3*d**2* 
x**3 - 10*sqrt(x)*b*c**3*d*e*x**4 - sqrt(x)*b*c**3*e**2*x**5 + 3*sqrt(x)*c 
**4*d**2*x**4 + 3*sqrt(x)*c**4*d*e*x**5),x)*a*b**5*c*d*e**3*x + 4*sqrt(x)* 
int((sqrt(d + e*x)*sqrt(b - c*x))/(sqrt(x)*b**4*d*e*x + sqrt(x)*b**4*e**2* 
x**2 - 3*sqrt(x)*b**3*c*d**2*x - 6*sqrt(x)*b**3*c*d*e*x**2 - 3*sqrt(x)*b** 
3*c*e**2*x**3 + 9*sqrt(x)*b**2*c**2*d**2*x**2 + 12*sqrt(x)*b**2*c**2*d*e*x 
**3 + 3*sqrt(x)*b**2*c**2*e**2*x**4 - 9*sqrt(x)*b*c**3*d**2*x**3 - 10*sqrt 
(x)*b*c**3*d*e*x**4 - sqrt(x)*b*c**3*e**2*x**5 + 3*sqrt(x)*c**4*d**2*x**4 
+ 3*sqrt(x)*c**4*d*e*x**5),x)*a*b**4*c**2*d**2*e**2*x - 2*sqrt(x)*int((sqr 
t(d + e*x)*sqrt(b - c*x))/(sqrt(x)*b**4*d*e*x + sqrt(x)*b**4*e**2*x**2 - 3 
*sqrt(x)*b**3*c*d**2*x - 6*sqrt(x)*b**3*c*d*e*x**2 - 3*sqrt(x)*b**3*c*e**2 
*x**3 + 9*sqrt(x)*b**2*c**2*d**2*x**2 + 12*sqrt(x)*b**2*c**2*d*e*x**3 + 3* 
sqrt(x)*b**2*c**2*e**2*x**4 - 9*sqrt(x)*b*c**3*d**2*x**3 - 10*sqrt(x)*b*c* 
*3*d*e*x**4 - sqrt(x)*b*c**3*e**2*x**5 + 3*sqrt(x)*c**4*d**2*x**4 + 3*s...