3.27 \(\int \frac {(a+b x) (A+C x^2)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\)

Optimal. Leaf size=611 \[ \frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (5 a d f h \left (3 A d f h^2+C (c h (f g-e h)+d g (e h+2 f g))\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right ) \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {h (d e-c f)}{f (d g-c h)}\right )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right ) \left (10 a C d f h (c f h+d e h+d f g)-b \left (15 A d^2 f^2 h^2+C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {4 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (a d f h-2 b (c f h+d e h+d f g))}{15 d^2 f^2 h^2}+\frac {2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h} \]

[Out]

4/15*C*(a*d*f*h-2*b*(c*f*h+d*e*h+d*f*g))*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/d^2/f^2/h^2+2/5*C*(b*x+a)*(
d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/d/f/h-2/15*(10*a*C*d*f*h*(c*f*h+d*e*h+d*f*g)-b*(15*A*d^2*f^2*h^2+C*(8
*c^2*f^2*h^2+7*c*d*f*h*(e*h+f*g)+d^2*(8*e^2*h^2+7*e*f*g*h+8*f^2*g^2))))*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))*(c*f-d*e)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(h*x+g)^(1/2)/d^3/f^
(5/2)/h^3/(f*x+e)^(1/2)/(d*(h*x+g)/(-c*h+d*g))^(1/2)+2/15*(5*a*d*f*h*(3*A*d*f*h^2+C*(c*h*(-e*h+f*g)+d*g*(e*h+2
*f*g)))-b*(15*A*d^2*f^2*g*h^2+C*(4*c^2*f*h^2*(-e*h+f*g)+c*d*h*(-4*e^2*h^2+e*f*g*h+3*f^2*g^2)+d^2*g*(4*e^2*h^2+
3*e*f*g*h+8*f^2*g^2))))*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))*(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)/d^3/f^(5/2)/h^3/(f*x+e)^(1/2)/(h*x+g)^(
1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.36, antiderivative size = 608, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1601, 1615, 158, 114, 113, 121, 120} \[ \frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right ) \left (5 a d f h \left (3 A d f h^2+c C h (f g-e h)+C d g (e h+2 f g)\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {g+h x}}+\frac {2 \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right ) \left (-10 a C d f h (c f h+d e h+d f g)+15 A b d^2 f^2 h^2+b C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {4 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (a d f h-2 b (c f h+d e h+d f g))}{15 d^2 f^2 h^2}+\frac {2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*C*(a*d*f*h - 2*b*(d*f*g + d*e*h + c*f*h))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(15*d^2*f^2*h^2) + (2*
C*(a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(5*d*f*h) + (2*Sqrt[-(d*e) + c*f]*(15*A*b*d^2*f^2*h^2 -
10*a*C*d*f*h*(d*f*g + d*e*h + c*f*h) + b*C*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) + d^2*(8*f^2*g^2 + 7*e*f*g*
h + 8*e^2*h^2)))*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))])/(15*d^3*f^(5/2)*h^3*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c
*h)]) + (2*Sqrt[-(d*e) + c*f]*(5*a*d*f*h*(3*A*d*f*h^2 + c*C*h*(f*g - e*h) + C*d*g*(2*f*g + e*h)) - b*(15*A*d^2
*f^2*g*h^2 + C*(4*c^2*f*h^2*(f*g - e*h) + c*d*h*(3*f^2*g^2 + e*f*g*h - 4*e^2*h^2) + d^2*g*(8*f^2*g^2 + 3*e*f*g
*h + 4*e^2*h^2))))*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))])/(15*d^3*f^(5/2)*h^3*Sqrt[e + f*x]*Sqrt[g
+ h*x])

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*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))])/b, 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 114

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 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*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))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &
& SimplerQ[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 121

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 158

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 1601

Int[(((a_.) + (b_.)*(x_))^(m_.)*((A_.) + (C_.)*(x_)^2))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqr
t[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[(2*C*(a + b*x)^m*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(d*f*h*(
2*m + 3)), x] + Dist[1/(d*f*h*(2*m + 3)), Int[((a + b*x)^(m - 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*
Simp[a*A*d*f*h*(2*m + 3) - C*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*m) + (A*b*d*f*h*(2*m + 3) - C*(2*a*(d*f*g
+ d*e*h + c*f*h) + b*(2*m + 1)*(d*e*g + c*f*g + c*e*h)))*x + 2*C*(a*d*f*h*m - b*(m + 1)*(d*f*g + d*e*h + c*f*h
))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, C}, x] && IntegerQ[2*m] && GtQ[m, 0]

Rule 1615

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[
{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[(k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*(e + f*x)^
(p + 1))/(d*f*b^(q - 1)*(m + n + p + q + 1)), x] + Dist[1/(d*f*b^q*(m + n + p + q + 1)), Int[(a + b*x)^m*(c +
d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a +
b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*
(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; F
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[2*m, 2*n, 2*p]

Rubi steps

\begin {align*} \int \frac {(a+b x) \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=\frac {2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}+\frac {\int \frac {-2 b c C e g+5 a A d f h-a C (d e g+c f g+c e h)+(5 A b d f h-3 b C (d e g+c f g+c e h)-2 a C (d f g+d e h+c f h)) x+2 C (a d f h-2 b (d f g+d e h+c f h)) x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{5 d f h}\\ &=\frac {4 C (a d f h-2 b (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{15 d^2 f^2 h^2}+\frac {2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}+\frac {2 \int \frac {\frac {1}{2} d \left (5 a d f h (3 A d f h-C (d e g+c f g+c e h))+2 b C \left (2 d^2 e g (f g+e h)+2 c^2 f h (f g+e h)+c d \left (2 f^2 g^2+3 e f g h+2 e^2 h^2\right )\right )\right )+\frac {1}{2} d \left (15 A b d^2 f^2 h^2-10 a C d f h (d f g+d e h+c f h)+b C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right ) x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{15 d^3 f^2 h^2}\\ &=\frac {4 C (a d f h-2 b (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{15 d^2 f^2 h^2}+\frac {2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}+\frac {\left (15 A b d^2 f^2 h^2-10 a C d f h (d f g+d e h+c f h)+b C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right ) \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{15 d^2 f^2 h^3}+\frac {\left (5 a d f h \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{15 d^2 f^2 h^3}\\ &=\frac {4 C (a d f h-2 b (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{15 d^2 f^2 h^2}+\frac {2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}+\frac {\left (\left (5 a d f h \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \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}{15 d^2 f^2 h^3 \sqrt {e+f x}}+\frac {\left (\left (15 A b d^2 f^2 h^2-10 a C d f h (d f g+d e h+c f h)+b C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right ) \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}{15 d^2 f^2 h^3 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}\\ &=\frac {4 C (a d f h-2 b (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{15 d^2 f^2 h^2}+\frac {2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}+\frac {2 \sqrt {-d e+c f} \left (15 A b d^2 f^2 h^2-10 a C d f h (d f g+d e h+c f h)+b C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right ) \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 )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {\left (\left (5 a d f h \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \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}{15 d^2 f^2 h^3 \sqrt {e+f x} \sqrt {g+h x}}\\ &=\frac {4 C (a d f h-2 b (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{15 d^2 f^2 h^2}+\frac {2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}+\frac {2 \sqrt {-d e+c f} \left (15 A b d^2 f^2 h^2-10 a C d f h (d f g+d e h+c f h)+b C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right ) \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 )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {2 \sqrt {-d e+c f} \left (5 a d f h \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \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 )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {g+h x}}\\ \end {align*}

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Mathematica [C]  time = 9.38, size = 686, normalized size = 1.12 \[ -\frac {2 \left (-i d h (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \left (5 a d f h \left (3 A d f^2 h+c C f (e h-f g)+C d e (2 e h+f g)\right )-b \left (15 A d^2 e f^2 h^2+C \left (4 c^2 f^2 h (e h-f g)+c d f \left (3 e^2 h^2+e f g h-4 f^2 g^2\right )+d^2 e \left (8 e^2 h^2+3 e f g h+4 f^2 g^2\right )\right )\right )\right ) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {d e}{f}-c}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )+d^2 (e+f x) (g+h x) \left (-\sqrt {\frac {d e}{f}-c}\right ) \left (-10 a C d f h (c f h+d e h+d f g)+15 A b d^2 f^2 h^2+b C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )-i h (c+d x)^{3/2} (d e-c f) \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {d e}{f}-c}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right ) \left (-10 a C d f h (c f h+d e h+d f g)+15 A b d^2 f^2 h^2+b C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )+C d^2 f h (c+d x) (e+f x) (g+h x) \sqrt {\frac {d e}{f}-c} (-5 a d f h+4 b c f h+b d (4 e h+4 f g-3 f h x))\right )}{15 d^4 f^3 h^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \sqrt {\frac {d e}{f}-c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-(d^2*Sqrt[-c + (d*e)/f]*(15*A*b*d^2*f^2*h^2 - 10*a*C*d*f*h*(d*f*g + d*e*h + c*f*h) + b*C*(8*c^2*f^2*h^2
+ 7*c*d*f*h*(f*g + e*h) + d^2*(8*f^2*g^2 + 7*e*f*g*h + 8*e^2*h^2)))*(e + f*x)*(g + h*x)) + C*d^2*Sqrt[-c + (d*
e)/f]*f*h*(c + d*x)*(e + f*x)*(g + h*x)*(4*b*c*f*h - 5*a*d*f*h + b*d*(4*f*g + 4*e*h - 3*f*h*x)) - I*(d*e - c*f
)*h*(15*A*b*d^2*f^2*h^2 - 10*a*C*d*f*h*(d*f*g + d*e*h + c*f*h) + b*C*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) +
d^2*(8*f^2*g^2 + 7*e*f*g*h + 8*e^2*h^2)))*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))
/(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)] - I*d*
h*(5*a*d*f*h*(3*A*d*f^2*h + c*C*f*(-(f*g) + e*h) + C*d*e*(f*g + 2*e*h)) - b*(15*A*d^2*e*f^2*h^2 + C*(4*c^2*f^2
*h*(-(f*g) + e*h) + c*d*f*(-4*f^2*g^2 + e*f*g*h + 3*e^2*h^2) + d^2*e*(4*f^2*g^2 + 3*e*f*g*h + 8*e^2*h^2))))*(c
+ d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(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)]))/(15*d^4*Sqrt[-c + (d*e)/f]*f^3*h^3*Sqrt[c + d*x]
*Sqrt[e + f*x]*Sqrt[g + h*x])

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fricas [F]  time = 1.23, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C b x^{3} + C a x^{2} + A b x + A a\right )} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}{d f h x^{3} + c e g + {\left (d f g + {\left (d e + c f\right )} h\right )} x^{2} + {\left (c e h + {\left (d e + c f\right )} g\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C x^{2} + A\right )} {\left (b x + a\right )}}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.04, size = 5679, normalized size = 9.29 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C x^{2} + A\right )} {\left (b x + a\right )}}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((C*x^2 + A)*(b*x + a)/(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 \[ \int \frac {\left (C\,x^2+A\right )\,\left (a+b\,x\right )}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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