3.25 \(\int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{7/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)

Optimal. Leaf size=1128 \[ -\frac{2 \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a-b^3 (2 B d e g-c (3 C e g-2 B f g-2 B e h))\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} b^2}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}-\frac{2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} b^2}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac{2 \sqrt{d g-c h} \sqrt{f g-e h} \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a-b^3 (2 B d e g-c (3 C e g-2 B f g-2 B e h))\right ) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) b}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{2 d \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a-b^3 (2 B d e g-c (3 C e g-2 B f g-2 B e h))\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} b}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}-\frac{2 \left (3 C d^2 f h a^3-3 b d (B d f h+C (d f g+d e h-c f h)) a^2+b^2 \left (3 B (f g+e h) d^2+C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a-b^3 \left (-B f h c^2-d (3 C e g-B f g-B e h) c+2 B d^2 e g\right )\right ) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right ),-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}} \]

[Out]

(2*b*d*(9*a^3*C*d*f*h - b^3*(2*B*d*e*g - c*(3*C*e*g - 2*B*f*g - 2*B*e*h)) + a*b^2*(C*(d*e*g + c*f*g + c*e*h) +
4*B*(d*f*g + d*e*h + c*f*h)) - a^2*b*(6*B*d*f*h + 5*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[a + b*x]*Sqrt[e + f*x]*S
qrt[g + h*x])/(3*(b*c - a*d)^2*(b*e - a*f)^2*(b*g - a*h)^2*Sqrt[c + d*x]) - (2*b^2*(b*B - 2*a*C)*Sqrt[c + d*x]
*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)^(3/2)) - (2*b^2*(9*a^3*C*d*f*h
- b^3*(2*B*d*e*g - c*(3*C*e*g - 2*B*f*g - 2*B*e*h)) + a*b^2*(C*(d*e*g + c*f*g + c*e*h) + 4*B*(d*f*g + d*e*h +
c*f*h)) - a^2*b*(6*B*d*f*h + 5*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c
- a*d)^2*(b*e - a*f)^2*(b*g - a*h)^2*Sqrt[a + b*x]) - (2*b*Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*(9*a^3*C*d*f*h - b^
3*(2*B*d*e*g - c*(3*C*e*g - 2*B*f*g - 2*B*e*h)) + a*b^2*(C*(d*e*g + c*f*g + c*e*h) + 4*B*(d*f*g + d*e*h + c*f*
h)) - a^2*b*(6*B*d*f*h + 5*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[a + b*x]*Sqrt[-(((d*e - c*f)*(g + h*x))/((f*g - e*
h)*(c + d*x)))]*EllipticE[ArcSin[(Sqrt[d*g - c*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h]*Sqrt[c + d*x])], ((b*c - a*d
)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))])/(3*(b*c - a*d)^2*(b*e - a*f)^2*(b*g - a*h)^2*Sqrt[((d*e - c*f)*(a +
b*x))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) - (2*(3*a^3*C*d^2*f*h - b^3*(2*B*d^2*e*g - B*c^2*f*h - c*d*(3*C
*e*g - B*f*g - B*e*h)) - 3*a^2*b*d*(B*d*f*h + C*(d*f*g + d*e*h - c*f*h)) + a*b^2*(3*B*d^2*(f*g + e*h) + C*(d^2
*e*g - c*d*f*g - c*d*e*h - 2*c^2*f*h)))*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]*El
lipticF[ArcSin[(Sqrt[b*g - a*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h]*Sqrt[a + b*x])], -(((b*c - a*d)*(f*g - e*h))/(
(d*e - c*f)*(b*g - a*h)))])/(3*(b*c - a*d)^2*(b*e - a*f)*(b*g - a*h)^(3/2)*Sqrt[f*g - e*h]*Sqrt[c + d*x]*Sqrt[
-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))])

________________________________________________________________________________________

Rubi [A]  time = 4.29887, antiderivative size = 1119, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, integrand size = 62, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {24, 1599, 1602, 12, 170, 419, 176, 424} \[ -\frac{2 \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} b^2}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}-\frac{2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} b^2}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac{2 \sqrt{d g-c h} \sqrt{f g-e h} \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) b}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{2 d \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} b}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}-\frac{2 \left (3 C d^2 f h a^3-3 b d (B d f h+C (d f g+d e h-c f h)) a^2+b^2 \left (3 B (f g+e h) d^2+C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a-b^3 \left (-B f h c^2-d (3 C e g-B f g-B e h) c+2 B d^2 e g\right )\right ) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}} \]

Antiderivative was successfully verified.

[In]

Int[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^(7/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(2*b*d*(9*a^3*C*d*f*h + b^3*(3*c*C*e*g - 2*B*d*e*g - 2*B*c*(f*g + e*h)) + a*b^2*(C*(d*e*g + c*f*g + c*e*h) + 4
*B*(d*f*g + d*e*h + c*f*h)) - a^2*b*(6*B*d*f*h + 5*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqr
t[g + h*x])/(3*(b*c - a*d)^2*(b*e - a*f)^2*(b*g - a*h)^2*Sqrt[c + d*x]) - (2*b^2*(b*B - 2*a*C)*Sqrt[c + d*x]*S
qrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)^(3/2)) - (2*b^2*(9*a^3*C*d*f*h +
b^3*(3*c*C*e*g - 2*B*d*e*g - 2*B*c*(f*g + e*h)) + a*b^2*(C*(d*e*g + c*f*g + c*e*h) + 4*B*(d*f*g + d*e*h + c*f*
h)) - a^2*b*(6*B*d*f*h + 5*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*
d)^2*(b*e - a*f)^2*(b*g - a*h)^2*Sqrt[a + b*x]) - (2*b*Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*(9*a^3*C*d*f*h + b^3*(3
*c*C*e*g - 2*B*d*e*g - 2*B*c*(f*g + e*h)) + a*b^2*(C*(d*e*g + c*f*g + c*e*h) + 4*B*(d*f*g + d*e*h + c*f*h)) -
a^2*b*(6*B*d*f*h + 5*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[a + b*x]*Sqrt[-(((d*e - c*f)*(g + h*x))/((f*g - e*h)*(c
+ d*x)))]*EllipticE[ArcSin[(Sqrt[d*g - c*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h]*Sqrt[c + d*x])], ((b*c - a*d)*(f*g
- e*h))/((b*e - a*f)*(d*g - c*h))])/(3*(b*c - a*d)^2*(b*e - a*f)^2*(b*g - a*h)^2*Sqrt[((d*e - c*f)*(a + b*x))
/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) - (2*(3*a^3*C*d^2*f*h - b^3*(2*B*d^2*e*g - B*c^2*f*h - c*d*(3*C*e*g -
B*f*g - B*e*h)) - 3*a^2*b*d*(B*d*f*h + C*(d*f*g + d*e*h - c*f*h)) + a*b^2*(3*B*d^2*(f*g + e*h) + C*(d^2*e*g -
c*d*f*g - c*d*e*h - 2*c^2*f*h)))*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]*Elliptic
F[ArcSin[(Sqrt[b*g - a*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h]*Sqrt[a + b*x])], -(((b*c - a*d)*(f*g - e*h))/((d*e -
c*f)*(b*g - a*h)))])/(3*(b*c - a*d)^2*(b*e - a*f)*(b*g - a*h)^(3/2)*Sqrt[f*g - e*h]*Sqrt[c + d*x]*Sqrt[-(((b*
e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*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 1599

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 1602

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 170

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Dist[(2*Sqrt[g + h*x]*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/((f*g - e*h)*Sqrt[c +
d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))]), Subst[Int[1/(Sqrt[1 + ((b*c - a*d)*x^2)/(d*e
- c*f)]*Sqrt[1 - ((b*g - a*h)*x^2)/(f*g - e*h)]), x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d
, e, f, g, h}, x]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 176

Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Dist[(-2*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))])/((b*e - a*f)*Sqrt
[g + h*x]*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]), Subst[Int[Sqrt[1 + ((b*c - a*d)*x^2)/(d*e -
c*f)]/Sqrt[1 - ((b*g - a*h)*x^2)/(f*g - e*h)], x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e
, f, g, h}, x]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

\begin{align*} \int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{7/2} \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)^{5/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{b^2}\\ &=-\frac{2 b^2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac{\int \frac{b^2 \left (b^2 C (3 b c e g-a (d e g+c f g+c e h))-(b B-a C) \left (3 a^2 d f h+2 b^2 (d e g+c f g+c e h)-3 a b (d f g+d e h+c f h)\right )\right )+b^3 (b B-2 a C) (3 a d f h-b (d f g+d e h+c f h)) x}{(a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{3 b^2 (b c-a d) (b e-a f) (b g-a h)}\\ &=-\frac{2 b^2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac{2 b^2 \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}+\frac{\int \frac{b^2 \left (b^2 (b B-2 a C) (b c e g-a (d e g+c f g+c e h)) (3 a d f h-b (d f g+d e h+c f h))-a (a d f h-b (d f g+d e h+c f h)) \left (b^2 C (3 b c e g-a (d e g+c f g+c e h))-(b B-a C) \left (3 a^2 d f h+2 b^2 (d e g+c f g+c e h)-3 a b (d f g+d e h+c f h)\right )\right )\right )+b^3 (a d f h+b (d f g+d e h+c f h)) \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) x+2 b^4 d f h \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) x^2}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{3 b^2 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2}\\ &=\frac{2 b d \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}-\frac{2 b^2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac{2 b^2 \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}+\frac{\int -\frac{2 b^3 d f (b e-a f) h (b g-a h) \left (3 a^3 C d^2 f h-3 a^2 b d (B d f h+C (d f g+d e h-c f h))-b^3 \left (2 B d^2 e g-B c^2 f h-c d (3 C e g-B (f g+e h))\right )+a b^2 \left (3 B d^2 (f g+e h)+C \left (d^2 e g-2 c^2 f h-c d (f g+e h)\right )\right )\right )}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{6 b^3 d (b c-a d)^2 f (b e-a f)^2 h (b g-a h)^2}+\frac{\left (b (d e-c f) (d g-c h) \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right )\right ) \int \frac{\sqrt{a+b x}}{(c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2}\\ &=\frac{2 b d \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}-\frac{2 b^2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac{2 b^2 \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}-\frac{\left (3 a^3 C d^2 f h-3 a^2 b d (B d f h+C (d f g+d e h-c f h))-b^3 \left (2 B d^2 e g-B c^2 f h-c d (3 C e g-B (f g+e h))\right )+a b^2 \left (3 B d^2 (f g+e h)+C \left (d^2 e g-2 c^2 f h-c d (f g+e h)\right )\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{3 (b c-a d)^2 (b e-a f) (b g-a h)}-\frac{\left (2 b (d g-c h) \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{(-b c+a d) x^2}{b e-a f}}}{\sqrt{1-\frac{(d g-c h) x^2}{f g-e h}}} \, dx,x,\frac{\sqrt{e+f x}}{\sqrt{c+d x}}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}\\ &=\frac{2 b d \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}-\frac{2 b^2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac{2 b^2 \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}-\frac{2 b \sqrt{d g-c h} \sqrt{f g-e h} \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{\left (2 \left (3 a^3 C d^2 f h-3 a^2 b d (B d f h+C (d f g+d e h-c f h))-b^3 \left (2 B d^2 e g-B c^2 f h-c d (3 C e g-B (f g+e h))\right )+a b^2 \left (3 B d^2 (f g+e h)+C \left (d^2 e g-2 c^2 f h-c d (f g+e h)\right )\right )\right ) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{(b c-a d) x^2}{d e-c f}} \sqrt{1-\frac{(b g-a h) x^2}{f g-e h}}} \, dx,x,\frac{\sqrt{e+f x}}{\sqrt{a+b x}}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h) (f g-e h) \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}\\ &=\frac{2 b d \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}-\frac{2 b^2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac{2 b^2 \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}-\frac{2 b \sqrt{d g-c h} \sqrt{f g-e h} \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{2 \left (3 a^3 C d^2 f h-b^3 \left (2 B d^2 e g-B c^2 f h-c d (3 C e g-B f g-B e h)\right )-3 a^2 b d (B d f h+C (d f g+d e h-c f h))+a b^2 \left (3 B d^2 (f g+e h)+C \left (d^2 e g-c d f g-c d e h-2 c^2 f h\right )\right )\right ) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}\\ \end{align*}

Mathematica [B]  time = 26.6129, size = 10645, normalized size = 9.44 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^(7/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

Result too large to show

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Maple [B]  time = 1.269, size = 75992, normalized size = 67.4 \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)/(b*x+a)^(7/2)/(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., size = 0, normalized size = 0. \begin{align*} \int \frac{C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{\frac{7}{2}} \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)/(b*x+a)^(7/2)/(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)/((b*x + a)^(7/2)*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 x - C a + B b\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}{b^{3} d f h x^{6} + a^{3} c e g +{\left (b^{3} d f g +{\left (b^{3} d e +{\left (b^{3} c + 3 \, a b^{2} d\right )} f\right )} h\right )} x^{5} +{\left ({\left (b^{3} d e +{\left (b^{3} c + 3 \, a b^{2} d\right )} f\right )} g +{\left ({\left (b^{3} c + 3 \, a b^{2} d\right )} e + 3 \,{\left (a b^{2} c + a^{2} b d\right )} f\right )} h\right )} x^{4} +{\left ({\left ({\left (b^{3} c + 3 \, a b^{2} d\right )} e + 3 \,{\left (a b^{2} c + a^{2} b d\right )} f\right )} g +{\left (3 \,{\left (a b^{2} c + a^{2} b d\right )} e +{\left (3 \, a^{2} b c + a^{3} d\right )} f\right )} h\right )} x^{3} +{\left ({\left (3 \,{\left (a b^{2} c + a^{2} b d\right )} e +{\left (3 \, a^{2} b c + a^{3} d\right )} f\right )} g +{\left (a^{3} c f +{\left (3 \, a^{2} b c + a^{3} d\right )} e\right )} h\right )} x^{2} +{\left (a^{3} c e h +{\left (a^{3} c f +{\left (3 \, a^{2} b c + a^{3} d\right )} e\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)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm
="fricas")

[Out]

integral((C*b*x - C*a + B*b)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)/(b^3*d*f*h*x^6 + a^3*c*e*
g + (b^3*d*f*g + (b^3*d*e + (b^3*c + 3*a*b^2*d)*f)*h)*x^5 + ((b^3*d*e + (b^3*c + 3*a*b^2*d)*f)*g + ((b^3*c + 3
*a*b^2*d)*e + 3*(a*b^2*c + a^2*b*d)*f)*h)*x^4 + (((b^3*c + 3*a*b^2*d)*e + 3*(a*b^2*c + a^2*b*d)*f)*g + (3*(a*b
^2*c + a^2*b*d)*e + (3*a^2*b*c + a^3*d)*f)*h)*x^3 + ((3*(a*b^2*c + a^2*b*d)*e + (3*a^2*b*c + a^3*d)*f)*g + (a^
3*c*f + (3*a^2*b*c + a^3*d)*e)*h)*x^2 + (a^3*c*e*h + (a^3*c*f + (3*a^2*b*c + a^3*d)*e)*g)*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)/(b*x+a)**(7/2)/(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., size = 0, normalized size = 0. \begin{align*} \int \frac{C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{\frac{7}{2}} \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)/(b*x+a)^(7/2)/(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)/((b*x + a)^(7/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x
)