3.1.27 \(\int \frac {(a+b x) (A+C x^2)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [27]

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

[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 = 0.88, 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 = {1615, 1629, 164, 115, 114, 122, 121} \begin {gather*} \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 (\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 ) \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 (\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 ) \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} \end {gather*}

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 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 1615

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 1629

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]

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] Result contains complex when optimal does not.
time = 25.93, size = 686, normalized size = 1.12 \begin {gather*} -\frac {2 \left (-d^2 \sqrt {-c+\frac {d e}{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 ) (e+f x) (g+h x)+C d^2 \sqrt {-c+\frac {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 \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 ) (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)}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )-i d h \left (5 a d f h \left (3 A d f^2 h+c C f (-f g+e h)+C d e (f g+2 e h)\right )-b \left (15 A d^2 e f^2 h^2+C \left (4 c^2 f^2 h (-f g+e h)+c d f \left (-4 f^2 g^2+e f g h+3 e^2 h^2\right )+d^2 e \left (4 f^2 g^2+3 e f g h+8 e^2 h^2\right )\right )\right )\right ) (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)}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )\right )}{15 d^4 \sqrt {-c+\frac {d e}{f}} f^3 h^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(5674\) vs. \(2(557)=1114\).
time = 0.11, size = 5675, normalized size = 9.29

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 C b x \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}}{5 d f h}+\frac {2 \left (a C -\frac {2 C b \left (2 h f c +2 d e h +2 g f d \right )}{5 d f h}\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}}{3 d f h}+\frac {2 \left (A a -\frac {2 C b c e g}{5 d f h}-\frac {2 \left (a C -\frac {2 C b \left (2 h f c +2 d e h +2 g f d \right )}{5 d f h}\right ) \left (\frac {1}{2} c e h +\frac {1}{2} c f g +\frac {1}{2} d e g \right )}{3 d f h}\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 )}{\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 {2 \left (A b -\frac {2 C b \left (\frac {3}{2} c e h +\frac {3}{2} c f g +\frac {3}{2} d e g \right )}{5 d f h}-\frac {2 \left (a C -\frac {2 C b \left (2 h f c +2 d e h +2 g f d \right )}{5 d f h}\right ) \left (h f c +d e h +g f d \right )}{3 d f h}\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 )}{\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}}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\)	\(824\)
default	\(\text {Expression too large to display}\)	\(5675\)

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,method=_RETURNVERBOSE)

[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((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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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.30, size = 1145, normalized size = 1.87 \begin {gather*} \frac {2 \, {\left (3 \, {\left (3 \, C b d^{3} f^{3} h^{3} x - 4 \, C b d^{3} f^{3} g h^{2} - 4 \, C b d^{3} f^{2} h^{3} e - {\left (4 \, C b c d^{2} - 5 \, C a d^{3}\right )} f^{3} h^{3}\right )} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g} - {\left (8 \, C b d^{3} f^{3} g^{3} + 8 \, C b d^{3} h^{3} e^{3} + {\left (3 \, C b c d^{2} - 10 \, C a d^{3}\right )} f^{3} g^{2} h + {\left (3 \, C b c^{2} d - 5 \, C a c d^{2} + 15 \, A b d^{3}\right )} f^{3} g h^{2} + {\left (8 \, C b c^{3} - 10 \, C a c^{2} d + 15 \, A b c d^{2} - 45 \, A a d^{3}\right )} f^{3} h^{3} + {\left (3 \, C b d^{3} f g h^{2} + {\left (3 \, C b c d^{2} - 10 \, C a d^{3}\right )} f h^{3}\right )} e^{2} + {\left (3 \, C b d^{3} f^{2} g^{2} h + {\left (3 \, C b c d^{2} - 5 \, C a d^{3}\right )} f^{2} g h^{2} + {\left (3 \, C b c^{2} d - 5 \, C a c d^{2} + 15 \, A b d^{3}\right )} f^{2} h^{3}\right )} e\right )} \sqrt {d f h} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - c d f^{2} g h + c^{2} f^{2} h^{2} + d^{2} h^{2} e^{2} - {\left (d^{2} f g h + c d f h^{2}\right )} e\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, c d^{2} f^{3} g^{2} h - 3 \, c^{2} d f^{3} g h^{2} + 2 \, c^{3} f^{3} h^{3} + 2 \, d^{3} h^{3} e^{3} - 3 \, {\left (d^{3} f g h^{2} + c d^{2} f h^{3}\right )} e^{2} - 3 \, {\left (d^{3} f^{2} g^{2} h - 4 \, c d^{2} f^{2} g h^{2} + c^{2} d f^{2} h^{3}\right )} e\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + c f h + d h e}{3 \, d f h}\right ) - 3 \, {\left (8 \, C b d^{3} f^{3} g^{2} h + 8 \, C b d^{3} f h^{3} e^{2} + {\left (7 \, C b c d^{2} - 10 \, C a d^{3}\right )} f^{3} g h^{2} + {\left (8 \, C b c^{2} d - 10 \, C a c d^{2} + 15 \, A b d^{3}\right )} f^{3} h^{3} + {\left (7 \, C b d^{3} f^{2} g h^{2} + {\left (7 \, C b c d^{2} - 10 \, C a d^{3}\right )} f^{2} h^{3}\right )} e\right )} \sqrt {d f h} {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - c d f^{2} g h + c^{2} f^{2} h^{2} + d^{2} h^{2} e^{2} - {\left (d^{2} f g h + c d f h^{2}\right )} e\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, c d^{2} f^{3} g^{2} h - 3 \, c^{2} d f^{3} g h^{2} + 2 \, c^{3} f^{3} h^{3} + 2 \, d^{3} h^{3} e^{3} - 3 \, {\left (d^{3} f g h^{2} + c d^{2} f h^{3}\right )} e^{2} - 3 \, {\left (d^{3} f^{2} g^{2} h - 4 \, c d^{2} f^{2} g h^{2} + c^{2} d f^{2} h^{3}\right )} e\right )}}{27 \, d^{3} f^{3} h^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - c d f^{2} g h + c^{2} f^{2} h^{2} + d^{2} h^{2} e^{2} - {\left (d^{2} f g h + c d f h^{2}\right )} e\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, c d^{2} f^{3} g^{2} h - 3 \, c^{2} d f^{3} g h^{2} + 2 \, c^{3} f^{3} h^{3} + 2 \, d^{3} h^{3} e^{3} - 3 \, {\left (d^{3} f g h^{2} + c d^{2} f h^{3}\right )} e^{2} - 3 \, {\left (d^{3} f^{2} g^{2} h - 4 \, c d^{2} f^{2} g h^{2} + c^{2} d f^{2} h^{3}\right )} e\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + c f h + d h e}{3 \, d f h}\right )\right )\right )}}{45 \, d^{4} f^{4} h^{4}} \end {gather*}

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]

2/45*(3*(3*C*b*d^3*f^3*h^3*x - 4*C*b*d^3*f^3*g*h^2 - 4*C*b*d^3*f^2*h^3*e - (4*C*b*c*d^2 - 5*C*a*d^3)*f^3*h^3)*
sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g) - (8*C*b*d^3*f^3*g^3 + 8*C*b*d^3*h^3*e^3 + (3*C*b*c*d^2 - 10*C*a*d^3
)*f^3*g^2*h + (3*C*b*c^2*d - 5*C*a*c*d^2 + 15*A*b*d^3)*f^3*g*h^2 + (8*C*b*c^3 - 10*C*a*c^2*d + 15*A*b*c*d^2 -
45*A*a*d^3)*f^3*h^3 + (3*C*b*d^3*f*g*h^2 + (3*C*b*c*d^2 - 10*C*a*d^3)*f*h^3)*e^2 + (3*C*b*d^3*f^2*g^2*h + (3*C
*b*c*d^2 - 5*C*a*d^3)*f^2*g*h^2 + (3*C*b*c^2*d - 5*C*a*c*d^2 + 15*A*b*d^3)*f^2*h^3)*e)*sqrt(d*f*h)*weierstrass
PInverse(4/3*(d^2*f^2*g^2 - c*d*f^2*g*h + c^2*f^2*h^2 + d^2*h^2*e^2 - (d^2*f*g*h + c*d*f*h^2)*e)/(d^2*f^2*h^2)
, -4/27*(2*d^3*f^3*g^3 - 3*c*d^2*f^3*g^2*h - 3*c^2*d*f^3*g*h^2 + 2*c^3*f^3*h^3 + 2*d^3*h^3*e^3 - 3*(d^3*f*g*h^
2 + c*d^2*f*h^3)*e^2 - 3*(d^3*f^2*g^2*h - 4*c*d^2*f^2*g*h^2 + c^2*d*f^2*h^3)*e)/(d^3*f^3*h^3), 1/3*(3*d*f*h*x
+ d*f*g + c*f*h + d*h*e)/(d*f*h)) - 3*(8*C*b*d^3*f^3*g^2*h + 8*C*b*d^3*f*h^3*e^2 + (7*C*b*c*d^2 - 10*C*a*d^3)*
f^3*g*h^2 + (8*C*b*c^2*d - 10*C*a*c*d^2 + 15*A*b*d^3)*f^3*h^3 + (7*C*b*d^3*f^2*g*h^2 + (7*C*b*c*d^2 - 10*C*a*d
^3)*f^2*h^3)*e)*sqrt(d*f*h)*weierstrassZeta(4/3*(d^2*f^2*g^2 - c*d*f^2*g*h + c^2*f^2*h^2 + d^2*h^2*e^2 - (d^2*
f*g*h + c*d*f*h^2)*e)/(d^2*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*c*d^2*f^3*g^2*h - 3*c^2*d*f^3*g*h^2 + 2*c^3*f^3*
h^3 + 2*d^3*h^3*e^3 - 3*(d^3*f*g*h^2 + c*d^2*f*h^3)*e^2 - 3*(d^3*f^2*g^2*h - 4*c*d^2*f^2*g*h^2 + c^2*d*f^2*h^3
)*e)/(d^3*f^3*h^3), weierstrassPInverse(4/3*(d^2*f^2*g^2 - c*d*f^2*g*h + c^2*f^2*h^2 + d^2*h^2*e^2 - (d^2*f*g*
h + c*d*f*h^2)*e)/(d^2*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*c*d^2*f^3*g^2*h - 3*c^2*d*f^3*g*h^2 + 2*c^3*f^3*h^3
+ 2*d^3*h^3*e^3 - 3*(d^3*f*g*h^2 + c*d^2*f*h^3)*e^2 - 3*(d^3*f^2*g^2*h - 4*c*d^2*f^2*g*h^2 + c^2*d*f^2*h^3)*e)
/(d^3*f^3*h^3), 1/3*(3*d*f*h*x + d*f*g + c*f*h + d*h*e)/(d*f*h))))/(d^4*f^4*h^4)

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

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]

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

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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((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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

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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