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

Optimal. Leaf size=410 \[ \frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right ),\frac{h (d e-c f)}{f (d g-c h)}\right ) \left (-3 a^2 C d f h^2+3 a b B d f h^2+b^2 (-(3 B d f g h-C (c h (f g-e h)+d g (e h+2 f g))))\right )}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 b^2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} (3 B d f h-2 C (c f h+d e h+d f g)) E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 b^2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h} \]

[Out]

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

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Rubi [A]  time = 0.58561, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 53, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.113, Rules used = {1615, 158, 114, 113, 121, 120} \[ \frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) \left (-3 a^2 C d f h^2+3 a b B d f h^2+b^2 (-(3 B d f g h-c C h (f g-e h)-C d g (e h+2 f g)))\right )}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 b^2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} (3 B d f h-2 C (c f h+d e h+d f g)) E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 b^2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 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 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 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 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)
])

Rubi steps

\begin{align*} \int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=\frac{2 b^2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h}+\frac{2 \int \frac{\frac{1}{2} d \left (3 a b B d f h-3 a^2 C d f h-b^2 C (d e g+c f g+c e h)\right )+\frac{1}{2} b^2 d (3 B d f h-2 C (d f g+d e h+c f h)) x}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{3 d^2 f h}\\ &=\frac{2 b^2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h}+\frac{\left (b^2 (3 B d f h-2 C (d f g+d e h+c f h))\right ) \int \frac{\sqrt{g+h x}}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{3 d f h^2}+\frac{1}{3} \left (3 a b B-3 a^2 C-\frac{b^2 (3 B d f g h-c C h (f g-e h)-C d g (2 f g+e h))}{d f h^2}\right ) \int \frac{1}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx\\ &=\frac{2 b^2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h}+\frac{\left (\left (3 a b B-3 a^2 C-\frac{b^2 (3 B d f g h-c C h (f g-e h)-C d g (2 f g+e h))}{d f h^2}\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}{3 \sqrt{e+f x}}+\frac{\left (b^2 (3 B d f h-2 C (d f g+d e h+c f h)) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x}\right ) \int \frac{\sqrt{\frac{d g}{d g-c h}+\frac{d h x}{d g-c h}}}{\sqrt{c+d x} \sqrt{\frac{d e}{d e-c f}+\frac{d f x}{d e-c f}}} \, dx}{3 d f h^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}\\ &=\frac{2 b^2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h}+\frac{2 b^2 \sqrt{-d e+c f} (3 B d f h-2 C (d f g+d e h+c f h)) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{-d e+c f}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{\left (\left (3 a b B-3 a^2 C-\frac{b^2 (3 B d f g h-c C h (f g-e h)-C d g (2 f g+e h))}{d f h^2}\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}{3 \sqrt{e+f x} \sqrt{g+h x}}\\ &=\frac{2 b^2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h}+\frac{2 b^2 \sqrt{-d e+c f} (3 B d f h-2 C (d f g+d e h+c f h)) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{-d e+c f}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 \sqrt{-d e+c f} \left (3 a b B d f h^2-3 a^2 C d f h^2-b^2 (3 B d f g h-c C h (f g-e h)-C d g (2 f g+e h))\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 )}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{g+h x}}\\ \end{align*}

Mathematica [C]  time = 6.55086, size = 442, normalized size = 1.08 \[ \frac{\sqrt{c+d x} \left (\frac{2 i d h \sqrt{c+d x} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right ),\frac{d f g-c f h}{d e h-c f h}\right ) \left (-3 a^2 C d f^2 h+3 a b B d f^2 h+b^2 (-3 B d e f h+c C f (e h-f g)+C d e (2 e h+f g))\right )}{\sqrt{\frac{d e}{f}-c}}+\frac{2 b^2 d^2 (e+f x) (g+h x) (3 B d f h-2 C (c f h+d e h+d f g))}{c+d x}+2 i b^2 f h \sqrt{c+d x} \sqrt{\frac{d e}{f}-c} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} (3 B d f h-2 C (c f h+d e h+d f g)) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right )|\frac{d f g-c f h}{d e h-c f h}\right )+2 b^2 C d^2 f h (e+f x) (g+h x)\right )}{3 d^3 f^2 h^2 \sqrt{e+f x} \sqrt{g+h x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.032, size = 2825, normalized size = 6.9 \begin{align*} \text{output too large to display} \end{align*}

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

[Out]

-2/3*(-C*x^2*b^2*c*d^2*f^2*h^2-C*x^2*b^2*d^3*e*f*h^2-C*b^2*c*d^2*e*f*g*h-C*x*b^2*c*d^2*f^2*g*h-C*x*b^2*d^3*e*f
*g*h-2*C*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticE(((d*x
+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c^3*f^2*h^2-C*x^2*b^2*d^3*f^2*g*h+2*C*((d*x+c)*f/(
c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2
),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*e*f*g*h-C*x*b^2*c*d^2*e*f*h^2-3*B*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(
h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c
*h-d*g))^(1/2))*b^2*c*d^2*e*f*h^2-3*B*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*
f-d*e))^(1/2)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*f^2*g*h-3*B*((d
*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticF(((d*x+c)*f/(c*f-d
*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a*b*c*d^2*f^2*h^2+3*B*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h
-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2
))*a*b*d^3*e*f*h^2+3*B*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*E
llipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*f^2*g*h-3*B*((d*x+c)*f/(c*f-d*
e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*
f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*d^3*e*f*g*h+3*B*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f
*x+e)*d/(c*f-d*e))^(1/2)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*d^3*e*f*g*
h+C*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticF(((d*x+c)*f
/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c^2*d*e*f*h^2-C*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d
/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))
^(1/2))*b^2*c^2*d*f^2*g*h+3*B*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^
(1/2)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c^2*d*f^2*h^2+3*C*((d*x+c)*f/
(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticF(((d*x+c)*f/(c*f-d*e))^(1/
2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a^2*c*d^2*f^2*h^2-3*C*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(
1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a^2*d
^3*e*f*h^2-C*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticF((
(d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*e^2*h^2-2*C*((d*x+c)*f/(c*f-d*e))^(1/2)*
(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f
/(c*h-d*g))^(1/2))*b^2*c*d^2*f^2*g^2+C*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c
*f-d*e))^(1/2)*EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*d^3*e^2*g*h+2*C*((d*
x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticF(((d*x+c)*f/(c*f-d*
e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*d^3*e*f*g^2+2*C*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*
g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*
b^2*c*d^2*e^2*h^2-C*x^3*b^2*d^3*f^2*h^2+2*C*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)
*d/(c*f-d*e))^(1/2)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*f^2*g^2-2
*C*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticE(((d*x+c)*f/
(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*d^3*e^2*g*h-2*C*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/
(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^
(1/2))*b^2*d^3*e*f*g^2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/h^2/f^2/d^3/(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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \end{align*}

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b\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 ) \end{align*}

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

[Out]

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

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)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \end{align*}

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

[Out]

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