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

Optimal. Leaf size=680 \[ -\frac {b^2 (b B-2 a C) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac {b (b B-2 a C) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {(b B-2 a C) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) (b e-a f) \sqrt {e+f x} \sqrt {g+h x}}-\frac {\sqrt {-d e+c f} \left (4 a^3 C d f h+2 a b^2 B (d f g+d e h+c f h)-b^3 (B d e g-c (2 C e g-B f g-B e h))-a^2 b (3 B d f h+2 C (d f g+d e h+c f h))\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d)^2 \sqrt {f} (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {g+h x}} \]

[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]
time = 1.12, antiderivative size = 680, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.183, Rules used = {24, 1613, 1621, 175, 552, 551, 164, 115, 114, 122, 121} \begin {gather*} -\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 ) \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\text {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}} F\left (\text {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 (\text {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)} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {\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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 34.26, size = 3409, normalized size = 5.01 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[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*f*Sqrt[-c + (d*g)/h]*h) + 2*a*b^2*c*C*f*Sqrt[-c + (d*g)/h]*h + a*b^2*B*d*f*Sq
rt[-c + (d*g)/h]*h - 2*a^2*b*C*d*f*Sqrt[-c + (d*g)/h]*h - (b^3*B*c*d^2*e*g*Sqrt[-c + (d*g)/h])/(c + d*x)^2 + (
2*a*b^2*c*C*d^2*e*g*Sqrt[-c + (d*g)/h])/(c + d*x)^2 + (a*b^2*B*d^3*e*g*Sqrt[-c + (d*g)/h])/(c + d*x)^2 - (2*a^
2*b*C*d^3*e*g*Sqrt[-c + (d*g)/h])/(c + d*x)^2 + (b^3*B*c^2*d*f*g*Sqrt[-c + (d*g)/h])/(c + d*x)^2 - (2*a*b^2*c^
2*C*d*f*g*Sqrt[-c + (d*g)/h])/(c + d*x)^2 - (a*b^2*B*c*d^2*f*g*Sqrt[-c + (d*g)/h])/(c + d*x)^2 + (2*a^2*b*c*C*
d^2*f*g*Sqrt[-c + (d*g)/h])/(c + d*x)^2 + (b^3*B*c^2*d*e*Sqrt[-c + (d*g)/h]*h)/(c + d*x)^2 - (2*a*b^2*c^2*C*d*
e*Sqrt[-c + (d*g)/h]*h)/(c + d*x)^2 - (a*b^2*B*c*d^2*e*Sqrt[-c + (d*g)/h]*h)/(c + d*x)^2 + (2*a^2*b*c*C*d^2*e*
Sqrt[-c + (d*g)/h]*h)/(c + d*x)^2 - (b^3*B*c^3*f*Sqrt[-c + (d*g)/h]*h)/(c + d*x)^2 + (2*a*b^2*c^3*C*f*Sqrt[-c
+ (d*g)/h]*h)/(c + d*x)^2 + (a*b^2*B*c^2*d*f*Sqrt[-c + (d*g)/h]*h)/(c + d*x)^2 - (2*a^2*b*c^2*C*d*f*Sqrt[-c +
(d*g)/h]*h)/(c + d*x)^2 - (b^3*B*c*d*f*g*Sqrt[-c + (d*g)/h])/(c + d*x) + (2*a*b^2*c*C*d*f*g*Sqrt[-c + (d*g)/h]
)/(c + d*x) + (a*b^2*B*d^2*f*g*Sqrt[-c + (d*g)/h])/(c + d*x) - (2*a^2*b*C*d^2*f*g*Sqrt[-c + (d*g)/h])/(c + d*x
) - (b^3*B*c*d*e*Sqrt[-c + (d*g)/h]*h)/(c + d*x) + (2*a*b^2*c*C*d*e*Sqrt[-c + (d*g)/h]*h)/(c + d*x) + (a*b^2*B
*d^2*e*Sqrt[-c + (d*g)/h]*h)/(c + d*x) - (2*a^2*b*C*d^2*e*Sqrt[-c + (d*g)/h]*h)/(c + d*x) + (2*b^3*B*c^2*f*Sqr
t[-c + (d*g)/h]*h)/(c + d*x) - (4*a*b^2*c^2*C*f*Sqrt[-c + (d*g)/h]*h)/(c + d*x) - (2*a*b^2*B*c*d*f*Sqrt[-c + (
d*g)/h]*h)/(c + d*x) + (4*a^2*b*c*C*d*f*Sqrt[-c + (d*g)/h]*h)/(c + d*x) + (I*b*(b*B - 2*a*C)*(b*c - a*d)*f*(-(
d*g) + c*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*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)])/Sqrt[c + d*x] - (I*d*(-(b*e) +
a*f)*(2*a*b*B*d*h - 2*a^2*C*d*h + b^2*(2*c*C*g - B*d*g - B*c*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*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f
*h)/(d*f*g - 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*h - a*d*h)/(b*d*g - b*c*h)), I*ArcSinh[Sqrt[-c + (d*g)/
h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - 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*h - a*d*h)/(b*d*g - b*c*h)),
I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - 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*h
- a*d*h)/(b*d*g - b*c*h)), I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - 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*h - a*d*h)/(b*d*g - b*c*h)), I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e
*h - c*f*h)/(d*f*g - 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*h - a*d*h)/(b*d*g - b*c*h)), I*ArcSinh[Sqrt[-
c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - 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*h - a*d*h)/(b*d*g
- b*c*h)), I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - 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*h - a*d*h)/(b*d*g - b*c*h)), I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*
g - 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*h - a*d*h)/(b*d*g - b*c*h)), I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt
[c + d*x]], (d*e*h - c*f*h)/(d*f*g - 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*h - a*d*h)/(b*d*g - b*c*h)),
I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - 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*h - a*d*h)/(b*d*g - b*c*h)), I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - 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*h - a*d*h)/(b*d*g - b*c*h)), I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]],
(d*e*h - c*f*h)/(d*f*g - c*f*h)])/Sqrt[c + d*x...

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(13368\) vs. \(2(623)=1246\).
time = 0.12, size = 13369, normalized size = 19.66

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {b^{2} \left (B b -2 a C \right ) \sqrt {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}}{\left (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 \right ) \left (b x +a \right )}-\frac {a d f h \left (B b -2 a C \right ) \left (-\frac {e}{f}+\frac {g}{h}\right ) \sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{\left (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 \right ) \sqrt {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}}-\frac {b d f h \left (B b -2 a C \right ) \left (-\frac {e}{f}+\frac {g}{h}\right ) \sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \left (\left (-\frac {g}{h}+\frac {c}{d}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )-\frac {c \EllipticF \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{d}\right )}{\left (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 \right ) \sqrt {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}}+\frac {\left (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 a^{3} C 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 \right ) \left (-\frac {e}{f}+\frac {g}{h}\right ) \sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {a}{b}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{\left (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 \right ) b \sqrt {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}\, \left (-\frac {g}{h}+\frac {a}{b}\right )}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\)	\(1211\)
default	\(\text {Expression too large to display}\)	\(13369\)

Verification of antiderivative is not currently implemented for this CAS.

[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]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 )}^3\,\sqrt {c+d\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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)

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