\(\int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [25]

Optimal result

Integrand size = 62, antiderivative size = 458 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 b (b B-2 a C) (d g-c h) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right )|\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{(b c-a d) \sqrt {b e-a f} \sqrt {d e-c f} (b g-a h) \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}+\frac {2 (b C g-b B h+a C h) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{\sqrt {b e-a f} \sqrt {d e-c f} (b g-a h) \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}} \] Output:

-2*b*(B*b-2*C*a)*(-c*h+d*g)*(f*x+e)^(1/2)*(-(-a*d+b*c)*(h*x+g)/(-c*h+d*g)/ 
(b*x+a))^(1/2)*EllipticE((-a*f+b*e)^(1/2)*(d*x+c)^(1/2)/(-c*f+d*e)^(1/2)/( 
b*x+a)^(1/2),((-c*f+d*e)*(-a*h+b*g)/(-a*f+b*e)/(-c*h+d*g))^(1/2))/(-a*d+b* 
c)/(-a*f+b*e)^(1/2)/(-c*f+d*e)^(1/2)/(-a*h+b*g)/(-(-a*d+b*c)*(f*x+e)/(-c*f 
+d*e)/(b*x+a))^(1/2)/(h*x+g)^(1/2)+2*(-B*b*h+C*a*h+C*b*g)*(f*x+e)^(1/2)*(- 
(-a*d+b*c)*(h*x+g)/(-c*h+d*g)/(b*x+a))^(1/2)*EllipticF((-a*f+b*e)^(1/2)*(d 
*x+c)^(1/2)/(-c*f+d*e)^(1/2)/(b*x+a)^(1/2),((-c*f+d*e)*(-a*h+b*g)/(-a*f+b* 
e)/(-c*h+d*g))^(1/2))/(-a*f+b*e)^(1/2)/(-c*f+d*e)^(1/2)/(-a*h+b*g)/(-(-a*d 
+b*c)*(f*x+e)/(-c*f+d*e)/(b*x+a))^(1/2)/(h*x+g)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 26.06 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.74 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 (b e-a f) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} (e+f x)^{3/2} (g+h x)^{3/2} \left (b (b B-2 a C) (d g-c h) E\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )+(b c C-b B d+a C d) (b g-a h) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )\right )}{(b c-a d) (f g-e h)^3 (a+b x)^{5/2} \sqrt {c+d x} \left (-\frac {(b e-a f) (b g-a h) (e+f x) (g+h x)}{(f g-e h)^2 (a+b x)^2}\right )^{3/2}} \] Input:

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

Output:

(2*(b*e - a*f)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*(e + 
f*x)^(3/2)*(g + h*x)^(3/2)*(b*(b*B - 2*a*C)*(d*g - c*h)*EllipticE[ArcSin[S 
qrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((b*c - a*d)*(f* 
g - e*h))/((b*e - a*f)*(d*g - c*h))] + (b*c*C - b*B*d + a*C*d)*(b*g - a*h) 
*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))] 
], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))]))/((b*c - a*d)*(f* 
g - e*h)^3*(a + b*x)^(5/2)*Sqrt[c + d*x]*(-(((b*e - a*f)*(b*g - a*h)*(e + 
f*x)*(g + h*x))/((f*g - e*h)^2*(a + b*x)^2)))^(3/2))
 

Rubi [A] (verified)

Time = 1.60 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.28, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2004, 2102, 2105, 27, 188, 194, 321, 327}

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.

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

\(\Big \downarrow \) 2004

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

\(\Big \downarrow \) 2102

\(\displaystyle \frac {\int \frac {2 (b B-2 a C) d f h x^2 b^2+C (b c e g-a (d e g+c f g+c e h)) b^2+(b B-2 a C) (a d f h+b (d f g+d e h+c f h)) x b-a (b B-a C) (a d f h-b (d f g+d e h+c f h))}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 2105

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {b (b B-2 a C) (d e-c f) (d g-c h) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx+(b e-a f) (b g-a h) (a C d-b B d+b c C) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {2 b d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{\sqrt {c+d x}}}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 188

\(\displaystyle \frac {b (b B-2 a C) (d e-c f) (d g-c h) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {2 \sqrt {g+h x} (b e-a f) (b g-a h) (a C d-b B d+b c C) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \int \frac {1}{\sqrt {\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1} \sqrt {1-\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}}d\frac {\sqrt {e+f x}}{\sqrt {a+b x}}}{\sqrt {c+d x} (f g-e h) \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac {2 b d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{\sqrt {c+d x}}}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 194

\(\displaystyle \frac {\frac {2 \sqrt {g+h x} (b e-a f) (b g-a h) (a C d-b B d+b c C) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \int \frac {1}{\sqrt {\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1} \sqrt {1-\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}}d\frac {\sqrt {e+f x}}{\sqrt {a+b x}}}{\sqrt {c+d x} (f g-e h) \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac {2 b \sqrt {a+b x} (b B-2 a C) (d g-c h) \sqrt {-\frac {(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}}}{\sqrt {g+h x} \sqrt {\frac {(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}+\frac {2 b d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{\sqrt {c+d x}}}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {-\frac {2 b \sqrt {a+b x} (b B-2 a C) (d g-c h) \sqrt {-\frac {(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}}}{\sqrt {g+h x} \sqrt {\frac {(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}+\frac {2 \sqrt {g+h x} (b e-a f) \sqrt {b g-a h} (a C d-b B d+b c C) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {c+d x} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac {2 b d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{\sqrt {c+d x}}}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 327

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

Input:

Int[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^(5/2)*Sqrt[c + d*x]*S 
qrt[e + f*x]*Sqrt[g + h*x]),x]
 

Output:

(-2*b^2*(b*B - 2*a*C)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a 
*d)*(b*e - a*f)*(b*g - a*h)*Sqrt[a + b*x]) + ((2*b*(b*B - 2*a*C)*d*Sqrt[a 
+ b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/Sqrt[c + d*x] - (2*b*(b*B - 2*a*C)*Sqr 
t[d*g - c*h]*Sqrt[f*g - e*h]*Sqrt[a + b*x]*Sqrt[-(((d*e - c*f)*(g + h*x))/ 
((f*g - e*h)*(c + d*x)))]*EllipticE[ArcSin[(Sqrt[d*g - c*h]*Sqrt[e + f*x]) 
/(Sqrt[f*g - e*h]*Sqrt[c + d*x])], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)* 
(d*g - c*h))])/(Sqrt[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]*Sqrt 
[g + h*x]) + (2*(b*c*C - b*B*d + a*C*d)*(b*e - a*f)*Sqrt[b*g - a*h]*Sqrt[( 
(b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]*EllipticF[Ar 
cSin[(Sqrt[b*g - a*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h]*Sqrt[a + b*x])], -(( 
(b*c - a*d)*(f*g - e*h))/((d*e - c*f)*(b*g - a*h)))])/(Sqrt[f*g - e*h]*Sqr 
t[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))]))/((b* 
c - a*d)*(b*e - a*f)*(b*g - a*h))
 

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 188

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.) 
*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[2*Sqrt[g + h*x]*(Sqrt[(b*e - 
a*f)*((c + d*x)/((d*e - c*f)*(a + b*x)))]/((f*g - e*h)*Sqrt[c + d*x]*Sqrt[( 
-(b*e - a*f))*((g + h*x)/((f*g - e*h)*(a + b*x)))]))   Subst[Int[1/(Sqrt[1 
+ (b*c - a*d)*(x^2/(d*e - c*f))]*Sqrt[1 - (b*g - a*h)*(x^2/(f*g - e*h))]), 
x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 
x]
 

rule 194

Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(e_.) + (f_.) 
*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2*Sqrt[c + d*x]*(Sqrt[(-(b*e 
 - a*f))*((g + h*x)/((f*g - e*h)*(a + b*x)))]/((b*e - a*f)*Sqrt[g + h*x]*Sq 
rt[(b*e - a*f)*((c + d*x)/((d*e - c*f)*(a + b*x)))]))   Subst[Int[Sqrt[1 + 
(b*c - a*d)*(x^2/(d*e - c*f))]/Sqrt[1 - (b*g - a*h)*(x^2/(f*g - e*h))], x], 
 x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 2004

Int[(u_)*((d_) + (e_.)*(x_))^(q_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.) 
, x_Symbol] :> Int[u*(d + e*x)^(p + q)*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, b 
, c, d, e, q}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
 

rule 2102

Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_)))/(Sqrt[(c_.) + (d_.)*(x 
_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[( 
A*b^2 - a*b*B)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x] 
/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))), x] - Simp[1/(2*(m + 1)*(b* 
c - a*d)*(b*e - a*f)*(b*g - a*h))   Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*S 
qrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)*(d 
*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) - b*B*(a*(d* 
e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*((A*b - a*B)*(a*d*f*h*(m + 1) 
 - b*(m + 2)*(d*f*g + d*e*h + c*f*h)))*x + d*f*h*(2*m + 5)*(A*b^2 - a*b*B)* 
x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && IntegerQ[2*m 
] && LtQ[m, -1]
 

rule 2105

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_. 
) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbo 
l] :> Simp[C*Sqrt[a + b*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(b*f*h*Sqrt[c + d*x 
])), x] + (Simp[1/(2*b*d*f*h)   Int[(1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e 
+ f*x]*Sqrt[g + h*x]))*Simp[2*A*b*d*f*h - C*(b*d*e*g + a*c*f*h) + (2*b*B*d* 
f*h - C*(a*d*f*h + b*(d*f*g + d*e*h + c*f*h)))*x, x], x], x] + Simp[C*(d*e 
- c*f)*((d*g - c*h)/(2*b*d*f*h))   Int[Sqrt[a + b*x]/((c + d*x)^(3/2)*Sqrt[ 
e + f*x]*Sqrt[g + h*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C} 
, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2248\) vs. \(2(420)=840\).

Time = 24.02 (sec) , antiderivative size = 2249, normalized size of antiderivative = 4.91

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2249\)
default	\(\text {Expression too large to display}\)	\(20623\)

Input:

int((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1 
/2)/(h*x+g)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

((h*x+g)*(d*x+c)*(b*x+a)*(f*x+e))^(1/2)/(h*x+g)^(1/2)/(d*x+c)^(1/2)/(b*x+a 
)^(1/2)/(f*x+e)^(1/2)*(2*(b*d*f*h*x^3+b*c*f*h*x^2+b*d*e*h*x^2+b*d*f*g*x^2+ 
b*c*e*h*x+b*c*f*g*x+b*d*e*g*x+b*c*e*g)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h- 
a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)*b*(B*b-2*C*a)/( 
(x+a/b)*(b*d*f*h*x^3+b*c*f*h*x^2+b*d*e*h*x^2+b*d*f*g*x^2+b*c*e*h*x+b*c*f*g 
*x+b*d*e*g*x+b*c*e*g))^(1/2)+2*(C+(a^2*d*f*h-a*b*c*f*h-a*b*d*e*h-a*b*d*f*g 
+b^2*c*e*h+b^2*c*f*g+b^2*d*e*g)*(B*b-2*C*a)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d 
*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)-(b*c*e*h+b 
*c*f*g+b*d*e*g)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h 
+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)*b*(B*b-2*C*a))*(e/f-g/h)*((c/d-e/f)*(x 
+g/h)/(-e/f+g/h)/(x+c/d))^(1/2)*(x+c/d)^2*((-c/d+g/h)*(x+a/b)/(-a/b+g/h)/( 
x+c/d))^(1/2)*((-c/d+g/h)*(x+e/f)/(-e/f+g/h)/(x+c/d))^(1/2)/(c/d-e/f)/(-c/ 
d+g/h)/(h*d*b*f*(x+g/h)*(x+c/d)*(x+a/b)*(x+e/f))^(1/2)*EllipticF(((c/d-e/f 
)*(x+g/h)/(-e/f+g/h)/(x+c/d))^(1/2),((-c/d+a/b)*(e/f-g/h)/(a/b-g/h)/(-c/d+ 
e/f))^(1/2))+2*(-b*(a*d*f*h-b*c*f*h-b*d*e*h-b*d*f*g)*(B*b-2*C*a)/(a^3*d*f* 
h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g- 
b^3*c*e*g)-(2*b*c*f*h+2*b*d*e*h+2*b*d*f*g)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d* 
e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)*b*(B*b-2*C* 
a))*(e/f-g/h)*((c/d-e/f)*(x+g/h)/(-e/f+g/h)/(x+c/d))^(1/2)*(x+c/d)^2*((-c/ 
d+g/h)*(x+a/b)/(-a/b+g/h)/(x+c/d))^(1/2)*((-c/d+g/h)*(x+e/f)/(-e/f+g/h)...
 

Fricas [F]

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

integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x 
+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")
 

Output:

integral((C*b*x - C*a + B*b)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqr 
t(h*x + g)/(b^2*d*f*h*x^5 + a^2*c*e*g + (b^2*d*f*g + (b^2*d*e + (b^2*c + 2 
*a*b*d)*f)*h)*x^4 + ((b^2*d*e + (b^2*c + 2*a*b*d)*f)*g + ((b^2*c + 2*a*b*d 
)*e + (2*a*b*c + a^2*d)*f)*h)*x^3 + (((b^2*c + 2*a*b*d)*e + (2*a*b*c + a^2 
*d)*f)*g + (a^2*c*f + (2*a*b*c + a^2*d)*e)*h)*x^2 + (a^2*c*e*h + (a^2*c*f 
+ (2*a*b*c + a^2*d)*e)*g)*x), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x 
+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x 
+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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