\(\int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [72]

Optimal result

Integrand size = 35, antiderivative size = 875 \[ \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}}+\frac {2 b d^2 \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {c+d x}}-\frac {4 d^2 (d f g+d e h-2 c f h) \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt {c+d x}}+\frac {4 d \sqrt {f} (d f g+d e h-2 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (-d e+c f)^{3/2} (d g-c h)^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 b d \sqrt {h} \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}-\frac {2 \sqrt {f} (2 d f g+d e h-3 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (-d e+c f)^{3/2} (d g-c h) \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 b^2 \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d)^3 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}} \]

[Out]

2/3*d^2*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)/(-c*f+d*e)/(-c*h+d*g)/(d*x+c)^(3/2)+2*b*d^2*(f*x+e)^(1/2)*(h*x+
g)^(1/2)/(-a*d+b*c)^2/(-c*f+d*e)/(-c*h+d*g)/(d*x+c)^(1/2)-4/3*d^2*(-2*c*f*h+d*e*h+d*f*g)*(f*x+e)^(1/2)*(h*x+g)
^(1/2)/(-a*d+b*c)/(-c*f+d*e)^2/(-c*h+d*g)^2/(d*x+c)^(1/2)+4/3*d*(-2*c*f*h+d*e*h+d*f*g)*EllipticE(f^(1/2)*(d*x+
c)^(1/2)/(c*f-d*e)^(1/2),((-c*f+d*e)*h/f/(-c*h+d*g))^(1/2))*f^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(h*x+g)^(1/2)
/(-a*d+b*c)/(c*f-d*e)^(3/2)/(-c*h+d*g)^2/(f*x+e)^(1/2)/(d*(h*x+g)/(-c*h+d*g))^(1/2)-2/3*(-3*c*f*h+d*e*h+2*d*f*
g)*EllipticF(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),((-c*f+d*e)*h/f/(-c*h+d*g))^(1/2))*f^(1/2)*(d*(f*x+e)/(-c*f
+d*e))^(1/2)*(d*(h*x+g)/(-c*h+d*g))^(1/2)/(-a*d+b*c)/(c*f-d*e)^(3/2)/(-c*h+d*g)/(f*x+e)^(1/2)/(h*x+g)^(1/2)-2*
b^2*EllipticPi(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),-b*(-c*f+d*e)/(-a*d+b*c)/f,((-c*f+d*e)*h/f/(-c*h+d*g))^(1
/2))*(c*f-d*e)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(d*(h*x+g)/(-c*h+d*g))^(1/2)/(-a*d+b*c)^3/f^(1/2)/(f*x+e)^(1
/2)/(h*x+g)^(1/2)-2*b*d*EllipticE(h^(1/2)*(f*x+e)^(1/2)/(e*h-f*g)^(1/2),(-d*(-e*h+f*g)/(-c*f+d*e)/h)^(1/2))*h^
(1/2)*(e*h-f*g)^(1/2)*(d*x+c)^(1/2)*(f*(h*x+g)/(-e*h+f*g))^(1/2)/(-a*d+b*c)^2/(-c*f+d*e)/(-c*h+d*g)/(-f*(d*x+c
)/(-c*f+d*e))^(1/2)/(h*x+g)^(1/2)

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 875, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.343, Rules used = {185, 106, 157, 164, 115, 114, 122, 121, 21, 175, 552, 551} \[ \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right ) b^2}{(b c-a d)^3 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 d \sqrt {h} \sqrt {e h-f g} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right ) b}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}+\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x} b}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {c+d x}}+\frac {4 d \sqrt {f} (d f g+d e h-2 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (c f-d e)^{3/2} (d g-c h)^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {f} (2 d f g+d e h-3 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (c f-d e)^{3/2} (d g-c h) \sqrt {e+f x} \sqrt {g+h x}}-\frac {4 d^2 (d f g+d e h-2 c f h) \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt {c+d x}}+\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}} \]

[In]

Int[1/((a + b*x)*(c + d*x)^(5/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(2*d^2*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*d)*(d*e - c*f)*(d*g - c*h)*(c + d*x)^(3/2)) + (2*b*d^2*Sqrt[e
+ f*x]*Sqrt[g + h*x])/((b*c - a*d)^2*(d*e - c*f)*(d*g - c*h)*Sqrt[c + d*x]) - (4*d^2*(d*f*g + d*e*h - 2*c*f*h)
*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*d)*(d*e - c*f)^2*(d*g - c*h)^2*Sqrt[c + d*x]) + (4*d*Sqrt[f]*(d*f*g
+ d*e*h - 2*c*f*h)*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt
[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(3*(b*c - a*d)*(-(d*e) + c*f)^(3/2)*(d*g - c*h)^2*Sqrt[e +
f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) - (2*b*d*Sqrt[h]*Sqrt[-(f*g) + e*h]*Sqrt[c + d*x]*Sqrt[(f*(g + h*x))/(f*
g - e*h)]*EllipticE[ArcSin[(Sqrt[h]*Sqrt[e + f*x])/Sqrt[-(f*g) + e*h]], -((d*(f*g - e*h))/((d*e - c*f)*h))])/(
(b*c - a*d)^2*(d*e - c*f)*(d*g - c*h)*Sqrt[-((f*(c + d*x))/(d*e - c*f))]*Sqrt[g + h*x]) - (2*Sqrt[f]*(2*d*f*g
+ d*e*h - 3*c*f*h)*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcSin[(Sqrt[f]*S
qrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(3*(b*c - a*d)*(-(d*e) + c*f)^(3/2)*(d*g
- c*h)*Sqrt[e + f*x]*Sqrt[g + h*x]) - (2*b^2*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h
*x))/(d*g - c*h)]*EllipticPi[-((b*(d*e - c*f))/((b*c - a*d)*f)), ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) +
c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/((b*c - a*d)^3*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 106

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(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] + Dist[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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegersQ[2*m, 2*n, 2*p]

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> 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] &&  !LtQ[-(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 115

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[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] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]

Rule 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> 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] && Si
mplerQ[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 122

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[b*((c
+ d*x)/(b*c - a*d))]/Sqrt[c + d*x], Int[1/(Sqrt[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] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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 164

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(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] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rule 175

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

Rule 185

Int[(((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_))/(Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)])
, x_Symbol] :> Int[ExpandIntegrand[1/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), (a + b*x)^m*(c + d*x)^(n + 1
/2), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IntegerQ[m] && IntegerQ[n + 1/2]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d/c)*x^2]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d}{(b c-a d) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}}-\frac {b d}{(b c-a d)^2 (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}+\frac {b^2}{(b c-a d)^2 (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}\right ) \, dx \\ & = \frac {b^2 \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{(b c-a d)^2}-\frac {(b d) \int \frac {1}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{(b c-a d)^2}-\frac {d \int \frac {1}{(c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b c-a d} \\ & = \frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}}+\frac {2 b d^2 \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {c+d x}}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{\left (b c-a d-b x^2\right ) \sqrt {e-\frac {c f}{d}+\frac {f x^2}{d}} \sqrt {g-\frac {c h}{d}+\frac {h x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{(b c-a d)^2}+\frac {(2 b d) \int \frac {-\frac {1}{2} c f h-\frac {1}{2} d f h x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{(b c-a d)^2 (d e-c f) (d g-c h)}+\frac {(2 d) \int \frac {\frac {1}{2} (2 d f g+2 d e h-3 c f h)+\frac {1}{2} d f h x}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{3 (b c-a d) (d e-c f) (d g-c h)} \\ & = \frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}}+\frac {2 b d^2 \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {c+d x}}-\frac {4 d^2 (d f g+d e h-2 c f h) \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt {c+d x}}-\frac {(4 d) \int \frac {-\frac {1}{4} f h \left (d^2 e g-3 c^2 f h+c d (f g+e h)\right )-\frac {1}{2} d f h (d f g+d e h-2 c f h) x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2}-\frac {(b d f h) \int \frac {\sqrt {c+d x}}{\sqrt {e+f x} \sqrt {g+h x}} \, dx}{(b c-a d)^2 (d e-c f) (d g-c h)}-\frac {\left (2 b^2 \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \text {Subst}\left (\int \frac {1}{\left (b c-a d-b x^2\right ) \sqrt {1+\frac {f x^2}{d \left (e-\frac {c f}{d}\right )}} \sqrt {g-\frac {c h}{d}+\frac {h x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{(b c-a d)^2 \sqrt {e+f x}} \\ & = \frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}}+\frac {2 b d^2 \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {c+d x}}-\frac {4 d^2 (d f g+d e h-2 c f h) \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt {c+d x}}-\frac {(d f (2 d f g+d e h-3 c f h)) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{3 (b c-a d) (d e-c f)^2 (d g-c h)}+\frac {\left (2 d^2 f (d f g+d e h-2 c f h)\right ) \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2}-\frac {\left (2 b^2 \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}}\right ) \text {Subst}\left (\int \frac {1}{\left (b c-a d-b x^2\right ) \sqrt {1+\frac {f x^2}{d \left (e-\frac {c f}{d}\right )}} \sqrt {1+\frac {h x^2}{d \left (g-\frac {c h}{d}\right )}}} \, dx,x,\sqrt {c+d x}\right )}{(b c-a d)^2 \sqrt {e+f x} \sqrt {g+h x}}-\frac {\left (b d f h \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}}\right ) \int \frac {\sqrt {\frac {c f}{-d e+c f}+\frac {d f x}{-d e+c f}}}{\sqrt {e+f x} \sqrt {\frac {f g}{f g-e h}+\frac {f h x}{f g-e h}}} \, dx}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {\frac {f (c+d x)}{-d e+c f}} \sqrt {g+h x}} \\ & = \frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}}+\frac {2 b d^2 \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {c+d x}}-\frac {4 d^2 (d f g+d e h-2 c f h) \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt {c+d x}}-\frac {2 b d \sqrt {h} \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}-\frac {2 b^2 \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d)^3 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {\left (d f (2 d f g+d e h-3 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}} \, dx}{3 (b c-a d) (d e-c f)^2 (d g-c h) \sqrt {e+f x}}+\frac {\left (2 d^2 f (d f g+d e h-2 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x}\right ) \int \frac {\sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}} \, dx}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}} \\ & = \frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}}+\frac {2 b d^2 \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {c+d x}}-\frac {4 d^2 (d f g+d e h-2 c f h) \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt {c+d x}}+\frac {4 d \sqrt {f} (d f g+d e h-2 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (-d e+c f)^{3/2} (d g-c h)^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 b d \sqrt {h} \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}-\frac {2 b^2 \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d)^3 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {\left (d f (2 d f g+d e h-3 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}} \, dx}{3 (b c-a d) (d e-c f)^2 (d g-c h) \sqrt {e+f x} \sqrt {g+h x}} \\ & = \frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}}+\frac {2 b d^2 \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {c+d x}}-\frac {4 d^2 (d f g+d e h-2 c f h) \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt {c+d x}}+\frac {4 d \sqrt {f} (d f g+d e h-2 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (-d e+c f)^{3/2} (d g-c h)^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 b d \sqrt {h} \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}+\frac {2 \sqrt {f} (3 c f h-d (2 f g+e h)) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (-d e+c f)^{3/2} (d g-c h) \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 b^2 \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d)^3 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 35.51 (sec) , antiderivative size = 4180, normalized size of antiderivative = 4.78 \[ \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]

[In]

Integrate[1/((a + b*x)*(c + d*x)^(5/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]*((2*d^2)/(3*(b*c - a*d)*(-(d*e) + c*f)*(-(d*g) + c*h)*(c + d*x)^2) +
(2*d^2*(3*b*d^2*e*g - 5*b*c*d*f*g + 2*a*d^2*f*g - 5*b*c*d*e*h + 2*a*d^2*e*h + 7*b*c^2*f*h - 4*a*c*d*f*h))/(3*
(b*c - a*d)^2*(-(d*e) + c*f)^2*(-(d*g) + c*h)^2*(c + d*x))) + (2*(c + d*x)^(3/2)*(-3*b^2*c*d^2*e*Sqrt[-c + (d*
e)/f]*f*g*h + 3*a*b*d^3*e*Sqrt[-c + (d*e)/f]*f*g*h + 5*b^2*c^2*d*Sqrt[-c + (d*e)/f]*f^2*g*h - 7*a*b*c*d^2*Sqrt
[-c + (d*e)/f]*f^2*g*h + 2*a^2*d^3*Sqrt[-c + (d*e)/f]*f^2*g*h + 5*b^2*c^2*d*e*Sqrt[-c + (d*e)/f]*f*h^2 - 7*a*b
*c*d^2*e*Sqrt[-c + (d*e)/f]*f*h^2 + 2*a^2*d^3*e*Sqrt[-c + (d*e)/f]*f*h^2 - 7*b^2*c^3*Sqrt[-c + (d*e)/f]*f^2*h^
2 + 11*a*b*c^2*d*Sqrt[-c + (d*e)/f]*f^2*h^2 - 4*a^2*c*d^2*Sqrt[-c + (d*e)/f]*f^2*h^2 - (3*b^2*c*d^4*e^2*Sqrt[-
c + (d*e)/f]*g^2)/(c + d*x)^2 + (3*a*b*d^5*e^2*Sqrt[-c + (d*e)/f]*g^2)/(c + d*x)^2 + (8*b^2*c^2*d^3*e*Sqrt[-c
+ (d*e)/f]*f*g^2)/(c + d*x)^2 - (10*a*b*c*d^4*e*Sqrt[-c + (d*e)/f]*f*g^2)/(c + d*x)^2 + (2*a^2*d^5*e*Sqrt[-c +
(d*e)/f]*f*g^2)/(c + d*x)^2 - (5*b^2*c^3*d^2*Sqrt[-c + (d*e)/f]*f^2*g^2)/(c + d*x)^2 + (7*a*b*c^2*d^3*Sqrt[-c
+ (d*e)/f]*f^2*g^2)/(c + d*x)^2 - (2*a^2*c*d^4*Sqrt[-c + (d*e)/f]*f^2*g^2)/(c + d*x)^2 + (8*b^2*c^2*d^3*e^2*S
qrt[-c + (d*e)/f]*g*h)/(c + d*x)^2 - (10*a*b*c*d^4*e^2*Sqrt[-c + (d*e)/f]*g*h)/(c + d*x)^2 + (2*a^2*d^5*e^2*Sq
rt[-c + (d*e)/f]*g*h)/(c + d*x)^2 - (20*b^2*c^3*d^2*e*Sqrt[-c + (d*e)/f]*f*g*h)/(c + d*x)^2 + (28*a*b*c^2*d^3*
e*Sqrt[-c + (d*e)/f]*f*g*h)/(c + d*x)^2 - (8*a^2*c*d^4*e*Sqrt[-c + (d*e)/f]*f*g*h)/(c + d*x)^2 + (12*b^2*c^4*d
*Sqrt[-c + (d*e)/f]*f^2*g*h)/(c + d*x)^2 - (18*a*b*c^3*d^2*Sqrt[-c + (d*e)/f]*f^2*g*h)/(c + d*x)^2 + (6*a^2*c^
2*d^3*Sqrt[-c + (d*e)/f]*f^2*g*h)/(c + d*x)^2 - (5*b^2*c^3*d^2*e^2*Sqrt[-c + (d*e)/f]*h^2)/(c + d*x)^2 + (7*a*
b*c^2*d^3*e^2*Sqrt[-c + (d*e)/f]*h^2)/(c + d*x)^2 - (2*a^2*c*d^4*e^2*Sqrt[-c + (d*e)/f]*h^2)/(c + d*x)^2 + (12
*b^2*c^4*d*e*Sqrt[-c + (d*e)/f]*f*h^2)/(c + d*x)^2 - (18*a*b*c^3*d^2*e*Sqrt[-c + (d*e)/f]*f*h^2)/(c + d*x)^2 +
(6*a^2*c^2*d^3*e*Sqrt[-c + (d*e)/f]*f*h^2)/(c + d*x)^2 - (7*b^2*c^5*Sqrt[-c + (d*e)/f]*f^2*h^2)/(c + d*x)^2 +
(11*a*b*c^4*d*Sqrt[-c + (d*e)/f]*f^2*h^2)/(c + d*x)^2 - (4*a^2*c^3*d^2*Sqrt[-c + (d*e)/f]*f^2*h^2)/(c + d*x)^
2 - (3*b^2*c*d^3*e*Sqrt[-c + (d*e)/f]*f*g^2)/(c + d*x) + (3*a*b*d^4*e*Sqrt[-c + (d*e)/f]*f*g^2)/(c + d*x) + (5
*b^2*c^2*d^2*Sqrt[-c + (d*e)/f]*f^2*g^2)/(c + d*x) - (7*a*b*c*d^3*Sqrt[-c + (d*e)/f]*f^2*g^2)/(c + d*x) + (2*a
^2*d^4*Sqrt[-c + (d*e)/f]*f^2*g^2)/(c + d*x) - (3*b^2*c*d^3*e^2*Sqrt[-c + (d*e)/f]*g*h)/(c + d*x) + (3*a*b*d^4
*e^2*Sqrt[-c + (d*e)/f]*g*h)/(c + d*x) + (16*b^2*c^2*d^2*e*Sqrt[-c + (d*e)/f]*f*g*h)/(c + d*x) - (20*a*b*c*d^3
*e*Sqrt[-c + (d*e)/f]*f*g*h)/(c + d*x) + (4*a^2*d^4*e*Sqrt[-c + (d*e)/f]*f*g*h)/(c + d*x) - (17*b^2*c^3*d*Sqrt
[-c + (d*e)/f]*f^2*g*h)/(c + d*x) + (25*a*b*c^2*d^2*Sqrt[-c + (d*e)/f]*f^2*g*h)/(c + d*x) - (8*a^2*c*d^3*Sqrt[
-c + (d*e)/f]*f^2*g*h)/(c + d*x) + (5*b^2*c^2*d^2*e^2*Sqrt[-c + (d*e)/f]*h^2)/(c + d*x) - (7*a*b*c*d^3*e^2*Sqr
t[-c + (d*e)/f]*h^2)/(c + d*x) + (2*a^2*d^4*e^2*Sqrt[-c + (d*e)/f]*h^2)/(c + d*x) - (17*b^2*c^3*d*e*Sqrt[-c +
(d*e)/f]*f*h^2)/(c + d*x) + (25*a*b*c^2*d^2*e*Sqrt[-c + (d*e)/f]*f*h^2)/(c + d*x) - (8*a^2*c*d^3*e*Sqrt[-c + (
d*e)/f]*f*h^2)/(c + d*x) + (14*b^2*c^4*Sqrt[-c + (d*e)/f]*f^2*h^2)/(c + d*x) - (22*a*b*c^3*d*Sqrt[-c + (d*e)/f
]*f^2*h^2)/(c + d*x) + (8*a^2*c^2*d^2*Sqrt[-c + (d*e)/f]*f^2*h^2)/(c + d*x) - (I*(-(b*c) + a*d)*(-(d*e) + c*f)
*h*(2*a*d*(d*f*g + d*e*h - 2*c*f*h) + b*(3*d^2*e*g + 7*c^2*f*h - 5*c*d*(f*g + e*h)))*Sqrt[1 - c/(c + d*x) + (d
*e)/(f*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (d*g)/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c +
d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)])/Sqrt[c + d*x] + (I*(-(d*e) + c*f)*(a^2*d^2*h*(d*f*g + 2*d*e*h - 3*c*
f*h) + b^2*(3*d^3*e*g^2 - 9*c^3*f*h^2 - 3*c*d^2*g*(f*g + 3*e*h) + 2*c^2*d*h*(5*f*g + 4*e*h)) + a*b*d*h*(3*d^2*
e*g + 9*c^2*f*h - c*d*(5*f*g + 7*e*h)))*Sqrt[1 - c/(c + d*x) + (d*e)/(f*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (d*
g)/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)])/Sqr
t[c + d*x] + ((3*I)*b^2*d^4*e^2*g^2*Sqrt[1 - c/(c + d*x) + (d*e)/(f*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (d*g)/(
h*(c + d*x))]*EllipticPi[-((b*c*f - a*d*f)/(b*d*e - b*c*f)), I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f
*g - c*f*h)/(d*e*h - c*f*h)])/Sqrt[c + d*x] - ((6*I)*b^2*c*d^3*e*f*g^2*Sqrt[1 - c/(c + d*x) + (d*e)/(f*(c + d*
x))]*Sqrt[1 - c/(c + d*x) + (d*g)/(h*(c + d*x))]*EllipticPi[-((b*c*f - a*d*f)/(b*d*e - b*c*f)), I*ArcSinh[Sqrt
[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)])/Sqrt[c + d*x] + ((3*I)*b^2*c^2*d^2*f^2*g^2*Sq
rt[1 - c/(c + d*x) + (d*e)/(f*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (d*g)/(h*(c + d*x))]*EllipticPi[-((b*c*f - a*
d*f)/(b*d*e - b*c*f)), I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)])/Sqrt[c +
d*x] - ((6*I)*b^2*c*d^3*e^2*g*h*Sqrt[1 - c/(c + d*x) + (d*e)/(f*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (d*g)/(h*(
c + d*x))]*EllipticPi[-((b*c*f - a*d*f)/(b*d*e - b*c*f)), I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g
- c*f*h)/(d*e*h - c*f*h)])/Sqrt[c + d*x] + ((12*I)*b^2*c^2*d^2*e*f*g*h*Sqrt[1 - c/(c + d*x) + (d*e)/(f*(c + d*
x))]*Sqrt[1 - c/(c + d*x) + (d*g)/(h*(c + d*x))]*EllipticPi[-((b*c*f - a*d*f)/(b*d*e - b*c*f)), I*ArcSinh[Sqrt
[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)])/Sqrt[c + d*x] - ((6*I)*b^2*c^3*d*f^2*g*h*Sqrt
[1 - c/(c + d*x) + (d*e)/(f*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (d*g)/(h*(c + d*x))]*EllipticPi[-((b*c*f - a*d*
f)/(b*d*e - b*c*f)), I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)])/Sqrt[c + d
*x] + ((3*I)*b^2*c^2*d^2*e^2*h^2*Sqrt[1 - c/(c + d*x) + (d*e)/(f*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (d*g)/(h*(
c + d*x))]*EllipticPi[-((b*c*f - a*d*f)/(b*d*e - b*c*f)), I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g
- c*f*h)/(d*e*h - c*f*h)])/Sqrt[c + d*x] - ((6*I)*b^2*c^3*d*e*f*h^2*Sqrt[1 - c/(c + d*x) + (d*e)/(f*(c + d*x))
]*Sqrt[1 - c/(c + d*x) + (d*g)/(h*(c + d*x))]*EllipticPi[-((b*c*f - a*d*f)/(b*d*e - b*c*f)), I*ArcSinh[Sqrt[-c
+ (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)])/Sqrt[c + d*x] + ((3*I)*b^2*c^4*f^2*h^2*Sqrt[1 -
c/(c + d*x) + (d*e)/(f*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (d*g)/(h*(c + d*x))]*EllipticPi[-((b*c*f - a*d*f)/(b
*d*e - b*c*f)), I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)])/Sqrt[c + d*x]))
/(3*(b*c - a*d)*(-(b*c) + a*d)^2*Sqrt[-c + (d*e)/f]*(-(d*e) + c*f)^2*(-(d*g) + c*h)^2*Sqrt[e + ((c + d*x)*(f -
(c*f)/(c + d*x)))/d]*Sqrt[g + ((c + d*x)*(h - (c*h)/(c + d*x)))/d])

Maple [A] (verified)

Time = 2.77 (sec) , antiderivative size = 1335, normalized size of antiderivative = 1.53

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1335\)
default	\(\text {Expression too large to display}\)	\(16647\)

[In]

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

[Out]

((d*x+c)*(f*x+e)*(h*x+g))^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)*(-2/3/(c^2*f*h-c*d*e*h-c*d*f*g+d^2*e
*g)/(a*d-b*c)*(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)/(x+c/d)^2-2/3*(d*f
*h*x^2+d*e*h*x+d*f*g*x+d*e*g)/(c^2*f*h-c*d*e*h-c*d*f*g+d^2*e*g)^2*d*(4*a*c*d*f*h-2*a*d^2*e*h-2*a*d^2*f*g-7*b*c
^2*f*h+5*b*c*d*e*h+5*b*c*d*f*g-3*b*d^2*e*g)/(a*d-b*c)^2/((x+c/d)*(d*f*h*x^2+d*e*h*x+d*f*g*x+d*e*g))^(1/2)+2*(-
1/3*d*f*h/(c^2*f*h-c*d*e*h-c*d*f*g+d^2*e*g)/(a*d-b*c)+1/3*d*(c*f*h-d*e*h-d*f*g)*(4*a*c*d*f*h-2*a*d^2*e*h-2*a*d
^2*f*g-7*b*c^2*f*h+5*b*c*d*e*h+5*b*c*d*f*g-3*b*d^2*e*g)/(c^2*f*h-c*d*e*h-c*d*f*g+d^2*e*g)^2/(a*d-b*c)^2+1/3*(d
*e*h+d*f*g)/(c^2*f*h-c*d*e*h-c*d*f*g+d^2*e*g)^2*d*(4*a*c*d*f*h-2*a*d^2*e*h-2*a*d^2*f*g-7*b*c^2*f*h+5*b*c*d*e*h
+5*b*c*d*f*g-3*b*d^2*e*g)/(a*d-b*c)^2)*(g/h-e/f)*((x+g/h)/(g/h-e/f))^(1/2)*((x+c/d)/(-g/h+c/d))^(1/2)*((x+e/f)
/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)*EllipticF(((x
+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2))+2/3*f*h*d^2*(4*a*c*d*f*h-2*a*d^2*e*h-2*a*d^2*f*g-7*b*c^2
*f*h+5*b*c*d*e*h+5*b*c*d*f*g-3*b*d^2*e*g)/(c^2*f*h-c*d*e*h-c*d*f*g+d^2*e*g)^2/(a*d-b*c)^2*(g/h-e/f)*((x+g/h)/(
g/h-e/f))^(1/2)*((x+c/d)/(-g/h+c/d))^(1/2)*((x+e/f)/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2
+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)*((-g/h+c/d)*EllipticE(((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^
(1/2))-c/d*EllipticF(((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2)))+2/(a*d-b*c)^2*b*(g/h-e/f)*((x+g
/h)/(g/h-e/f))^(1/2)*((x+c/d)/(-g/h+c/d))^(1/2)*((x+e/f)/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*
g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)/(-g/h+a/b)*EllipticPi(((x+g/h)/(g/h-e/f))^(1/2),(-g/h+e/f)/(-g/h+a/
b),((-g/h+e/f)/(-g/h+c/d))^(1/2)))

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {1}{\left (a + b x\right ) \left (c + d x\right )^{\frac {5}{2}} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]

[In]

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

[Out]

Integral(1/((a + b*x)*(c + d*x)**(5/2)*sqrt(e + f*x)*sqrt(g + h*x)), x)

Maxima [F]

\[ \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {1}{{\left (b x + a\right )} {\left (d x + c\right )}^{\frac {5}{2}} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]

[In]

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

[Out]

integrate(1/((b*x + a)*(d*x + c)^(5/2)*sqrt(f*x + e)*sqrt(h*x + g)), x)

Giac [F]

\[ \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {1}{{\left (b x + a\right )} {\left (d x + c\right )}^{\frac {5}{2}} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]

[In]

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

[Out]

integrate(1/((b*x + a)*(d*x + c)^(5/2)*sqrt(f*x + e)*sqrt(h*x + g)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {1}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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