\(\int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [20]

Optimal result

Integrand size = 60, antiderivative size = 680 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {b^2 (b B-2 a C) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac {b (b B-2 a C) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {(b B-2 a C) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) (b e-a f) \sqrt {e+f x} \sqrt {g+h x}}-\frac {\sqrt {-d e+c f} \left (4 a^3 C d f h+2 a b^2 B (d f g+d e h+c f h)-b^3 (B d e g-c (2 C e g-B f g-B e h))-a^2 b (3 B d f h+2 C (d f g+d e h+c f h))\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d)^2 \sqrt {f} (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {g+h x}} \]

[Out]

-b^2*(B*b-2*C*a)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(-a*h+b*g)/(b*x+a)+b*(B*b-2*C
*a)*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)*(c*f-d*e)^(1/2)
*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(-a*h+b*g)/(f*x+e)^(1/2)/(d*(h*x+g)/(-c*h+d*
g))^(1/2)-(4*a^3*C*d*f*h+2*a*b^2*B*(c*f*h+d*e*h+d*f*g)-b^3*(B*d*e*g-c*(-B*e*h-B*f*g+2*C*e*g))-a^2*b*(3*B*d*f*h
+2*C*(c*f*h+d*e*h+d*f*g)))*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)^2/(-a*f+b*e)/(-a*h+b*g)/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)-(B*b-2*C*a)*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)*(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)/(-a*f+b*e)/(f*x+e)^(1/2)/(h*x+g)^(1/2)

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.183, Rules used = {24, 1613, 1621, 175, 552, 551, 164, 115, 114, 122, 121} \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {\sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (4 a^3 C d f h-a^2 b (3 B d f h+2 C (c f h+d e h+d f g))+2 a b^2 B (c f h+d e h+d f g)-b^3 (B d e g-c (-B e h-B f g+2 C e g))\right ) \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {f} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)^2 (b e-a f) (b g-a h)}-\frac {\sqrt {f} (b B-2 a C) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {g+h x} (b c-a d) (b e-a f)}+\frac {b \sqrt {f} \sqrt {g+h x} (b B-2 a C) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)} \]

[In]

Int[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^3*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

-((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)*(a + b*x)
)) + (b*(b*B - 2*a*C)*Sqrt[f]*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSi
n[(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)*(b*e - a*f)*(b*g
- a*h)*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) - ((b*B - 2*a*C)*Sqrt[f]*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e
+ f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[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)*(b*e - a*f)*Sqrt[e + f*x]*Sqrt[g + h*x]) - (Sqrt[-(d*e) +
c*f]*(4*a^3*C*d*f*h + 2*a*b^2*B*(d*f*g + d*e*h + c*f*h) - b^3*(B*d*e*g - c*(2*C*e*g - B*f*g - B*e*h)) - a^2*b*
(3*B*d*f*h + 2*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*Ell
ipticPi[-((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)^2*Sqrt[f]*(b*e - a*f)*(b*g - a*h)*Sqrt[e + f*x]*Sqrt[g + h*x])

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

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 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 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]

Rule 1613

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] - Dist[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]*Sqrt[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 1621

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.)*((g_.) + (h_.)*(x_)
)^(q_.), x_Symbol] :> Dist[PolynomialRemainder[Px, a + b*x, x], Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h
*x)^q, x], x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q,
x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && PolyQ[Px, x] && EqQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {b^2 (b B-a C)+b^3 C x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b^2} \\ & = -\frac {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) (a+b x)}+\frac {\int \frac {b^2 \left (b^2 C (2 b c e g-a (d e g+c f g+c e h))-(b B-a C) \left (2 a^2 d f h+b^2 (d e g+c f g+c e h)-2 a b (d f g+d e h+c f h)\right )\right )+2 a b^3 (b B-2 a C) d f h x+b^4 (b B-2 a C) d f h x^2}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{2 b^2 (b c-a d) (b e-a f) (b g-a h)} \\ & = -\frac {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) (a+b x)}+\frac {\int \frac {a b^3 B d f h-2 a^2 b^2 C d f h+\left (b^4 B d f h-2 a b^3 C d f h\right ) x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{2 b^2 (b c-a d) (b e-a f) (b g-a h)}+\frac {\left (4 a^3 C d f h+2 a b^2 B (d f g+d e h+c f h)-b^3 (B d e g-c (2 C e g-B f g-B e h))-a^2 b (3 B d f h+2 C (d f g+d e h+c f h))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{2 (b c-a d) (b e-a f) (b g-a h)} \\ & = -\frac {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) (a+b x)}-\frac {((b B-2 a C) d f) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{2 (b c-a d) (b e-a f)}+\frac {(b (b B-2 a C) d f) \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {\left (4 a^3 C d f h+2 a b^2 B (d f g+d e h+c f h)-b^3 (B d e g-c (2 C e g-B f g-B e h))-a^2 b (3 B d f h+2 C (d f g+d e h+c f h))\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) (b e-a f) (b g-a h)} \\ & = -\frac {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) (a+b x)}-\frac {\left ((b B-2 a C) d f \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}{2 (b c-a d) (b e-a f) \sqrt {e+f x}}-\frac {\left (\left (4 a^3 C d f h+2 a b^2 B (d f g+d e h+c f h)-b^3 (B d e g-c (2 C e g-B f g-B e h))-a^2 b (3 B d f h+2 C (d f g+d e h+c f h))\right ) \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) (b e-a f) (b g-a h) \sqrt {e+f x}}+\frac {\left (b (b B-2 a C) d f \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}{2 (b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}} \\ & = -\frac {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) (a+b x)}+\frac {b (b B-2 a C) \sqrt {f} \sqrt {-d e+c f} \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 )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {\left ((b B-2 a C) d f \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}{2 (b c-a d) (b e-a f) \sqrt {e+f x} \sqrt {g+h x}}-\frac {\left (\left (4 a^3 C d f h+2 a b^2 B (d f g+d e h+c f h)-b^3 (B d e g-c (2 C e g-B f g-B e h))-a^2 b (3 B d f h+2 C (d f g+d e h+c f h))\right ) \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) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {g+h x}} \\ & = -\frac {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) (a+b x)}+\frac {b (b B-2 a C) \sqrt {f} \sqrt {-d e+c f} \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 )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {(b B-2 a C) \sqrt {f} \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}} 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 )}{(b c-a d) (b e-a f) \sqrt {e+f x} \sqrt {g+h x}}-\frac {\sqrt {-d e+c f} \left (4 a^3 C d f h+2 a b^2 B (d f g+d e h+c f h)-b^3 (B d e g-c (2 C e g-B f g-B e h))-a^2 b (3 B d f h+2 C (d f g+d e h+c f h))\right ) \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)^2 \sqrt {f} (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {g+h x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 34.66 (sec) , antiderivative size = 3419, normalized size of antiderivative = 5.03 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]

[In]

Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^3*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

-((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)*(a + b*x)
)) - ((c + d*x)^(3/2)*(b^3*B*c*Sqrt[-c + (d*e)/f]*f*h - 2*a*b^2*c*C*Sqrt[-c + (d*e)/f]*f*h - a*b^2*B*d*Sqrt[-c
+ (d*e)/f]*f*h + 2*a^2*b*C*d*Sqrt[-c + (d*e)/f]*f*h + (b^3*B*c*d^2*e*Sqrt[-c + (d*e)/f]*g)/(c + d*x)^2 - (2*a
*b^2*c*C*d^2*e*Sqrt[-c + (d*e)/f]*g)/(c + d*x)^2 - (a*b^2*B*d^3*e*Sqrt[-c + (d*e)/f]*g)/(c + d*x)^2 + (2*a^2*b
*C*d^3*e*Sqrt[-c + (d*e)/f]*g)/(c + d*x)^2 - (b^3*B*c^2*d*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x)^2 + (2*a*b^2*c^2*C
*d*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x)^2 + (a*b^2*B*c*d^2*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x)^2 - (2*a^2*b*c*C*d^2
*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x)^2 - (b^3*B*c^2*d*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x)^2 + (2*a*b^2*c^2*C*d*e*S
qrt[-c + (d*e)/f]*h)/(c + d*x)^2 + (a*b^2*B*c*d^2*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x)^2 - (2*a^2*b*c*C*d^2*e*Sqr
t[-c + (d*e)/f]*h)/(c + d*x)^2 + (b^3*B*c^3*Sqrt[-c + (d*e)/f]*f*h)/(c + d*x)^2 - (2*a*b^2*c^3*C*Sqrt[-c + (d*
e)/f]*f*h)/(c + d*x)^2 - (a*b^2*B*c^2*d*Sqrt[-c + (d*e)/f]*f*h)/(c + d*x)^2 + (2*a^2*b*c^2*C*d*Sqrt[-c + (d*e)
/f]*f*h)/(c + d*x)^2 + (b^3*B*c*d*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x) - (2*a*b^2*c*C*d*Sqrt[-c + (d*e)/f]*f*g)/(
c + d*x) - (a*b^2*B*d^2*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x) + (2*a^2*b*C*d^2*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x) +
(b^3*B*c*d*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x) - (2*a*b^2*c*C*d*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x) - (a*b^2*B*d^
2*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x) + (2*a^2*b*C*d^2*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x) - (2*b^3*B*c^2*Sqrt[-c
+ (d*e)/f]*f*h)/(c + d*x) + (4*a*b^2*c^2*C*Sqrt[-c + (d*e)/f]*f*h)/(c + d*x) + (2*a*b^2*B*c*d*Sqrt[-c + (d*e)/
f]*f*h)/(c + d*x) - (4*a^2*b*c*C*d*Sqrt[-c + (d*e)/f]*f*h)/(c + d*x) + (I*b*(b*B - 2*a*C)*(-(b*c) + a*d)*(-(d*
e) + c*f)*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*(2*a*b*B*d
*f - 2*a^2*C*d*f + b^2*(2*c*C*e - B*d*e - B*c*f))*(-(b*g) + a*h)*Sqrt[1 - c/(c + d*x) + (d*e)/(f*(c + d*x))]*S
qrt[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)])/Sqrt[c + d*x] + ((2*I)*b^3*c*C*d*e*g*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] - (I*b^3*B*d^2*e*g*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] - (I*b^3*B*c*d*f
*g*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)])/Sqr
t[c + d*x] + ((2*I)*a*b^2*B*d^2*f*g*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] - ((2*I)*a^2*b*C*d^2*f*g*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] - (I*b^3*B*c*d*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))]*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] + ((2*
I)*a*b^2*B*d^2*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))]*Ell
ipticPi[-((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] - ((2*I)*a^2*b*C*d^2*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))]*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] + ((2*I)*a*b^2*B*c*d*f*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] - ((2*I)*a^2*b*c*
C*d*f*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)*a^2*b*B*d^2*f*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] + ((4*I)*a^3*C*d^2*f*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[Sqr
t[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)])/Sqrt[c + d*x]))/(d*(b*c - a*d)*(-(b*c) + a*d
)*Sqrt[-c + (d*e)/f]*(-(b*e) + a*f)*(-(b*g) + a*h)*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 = 3.93 (sec) , antiderivative size = 1211, normalized size of antiderivative = 1.78

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1211\)
default	\(\text {Expression too large to display}\)	\(13369\)

[In]

int((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^3/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,method=_RETURNVERBOS
E)

[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)*(b^2/(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-2*C*a)*(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)/(b*x+a)-a*d*f*h*(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)*(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))-d*f*h*b*(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)*(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*Ellip
ticF(((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2)))+(3*B*a^2*b*d*f*h-2*B*a*b^2*c*f*h-2*B*a*b^2*d*e*
h-2*B*a*b^2*d*f*g+B*b^3*c*e*h+B*b^3*c*f*g+B*b^3*d*e*g-4*C*a^3*d*f*h+2*C*a^2*b*c*f*h+2*C*a^2*b*d*e*h+2*C*a^2*b*
d*f*g-2*C*b^3*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*(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. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]

[In]

integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^3/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fr
icas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{3} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]

[In]

integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^3/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="ma
xima")

[Out]

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

Giac [F]

\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{3} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]

[In]

integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^3/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="gi
ac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Hanged} \]

[In]

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)^3*(c + d*x)^(1/2)),x)

[Out]

\text{Hanged}