3.11 \(\int \frac{(a+b x)^{3/2} (d e+c f+2 d f x)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)

Optimal. Leaf size=898 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} b}{h}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} (5 a d f h-b (3 d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{2 d f h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{(5 a d f h-b (3 d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{2 f h^2 \sqrt{c+d x}}-\frac{(b e-a f) \sqrt{b g-a h} (3 a d f h+b (c f h-d (3 f g+e h))) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right ),-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{2 f h^2 \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}} b}-\frac{\sqrt{c h-d g} \left (\left (-\left (3 f^2 g^2+e^2 h^2\right ) d^2+2 c e f h^2 d-c^2 f^2 h^2\right ) b^2+6 a d^2 f^2 g h b-3 a^2 d^2 f^2 h^2\right ) (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{2 d \sqrt{b c-a d} f h^3 \sqrt{c+d x} \sqrt{e+f x} b} \]

[Out]

((5*a*d*f*h - b*(3*d*f*g + d*e*h + c*f*h))*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(2*f*h^2*Sqrt[c + d*x])
+ (b*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/h - (Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*(5*a*d*f*h
- b*(3*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)))]*Elliptic
E[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))])/(2*d*f*h^2*Sqrt[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) - ((b*e - a
*f)*Sqrt[b*g - a*h]*(3*a*d*f*h + b*(c*f*h - d*(3*f*g + e*h)))*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b
*x))]*Sqrt[g + h*x]*EllipticF[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)))])/(2*b*f*h^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)))]) - (Sqrt[-(d*g) + c*h]*(6*a*b*d^2*f^2*g*h - 3*a^2*d^2*f^2*h^2 + b^2*(2*c
*d*e*f*h^2 - c^2*f^2*h^2 - d^2*(3*f^2*g^2 + e^2*h^2)))*(a + b*x)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a
+ b*x))]*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]*EllipticPi[-((b*(d*g - c*h))/((b*c - a*d)*h)),
ArcSin[(Sqrt[b*c - a*d]*Sqrt[g + h*x])/(Sqrt[-(d*g) + c*h]*Sqrt[a + b*x])], ((b*e - a*f)*(d*g - c*h))/((b*c -
a*d)*(f*g - e*h))])/(2*b*d*Sqrt[b*c - a*d]*f*h^3*Sqrt[c + d*x]*Sqrt[e + f*x])

________________________________________________________________________________________

Rubi [A]  time = 2.51844, antiderivative size = 897, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.204, Rules used = {1597, 1600, 1602, 1598, 170, 419, 165, 537, 176, 424} \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} b}{h}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} (5 a d f h-b (3 d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{2 d f h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{(5 a d f h-b (3 d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{2 f h^2 \sqrt{c+d x}}-\frac{(b e-a f) \sqrt{b g-a h} (b c f h+3 a d f h-b d (3 f g+e h)) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{2 f h^2 \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}} b}-\frac{\sqrt{c h-d g} \left (\left (-\left (3 f^2 g^2+e^2 h^2\right ) d^2+2 c e f h^2 d-c^2 f^2 h^2\right ) b^2+6 a d^2 f^2 g h b-3 a^2 d^2 f^2 h^2\right ) (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{2 d \sqrt{b c-a d} f h^3 \sqrt{c+d x} \sqrt{e+f x} b} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(d*e + c*f + 2*d*f*x))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

((5*a*d*f*h - b*(3*d*f*g + d*e*h + c*f*h))*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(2*f*h^2*Sqrt[c + d*x])
+ (b*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/h - (Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*(5*a*d*f*h
- b*(3*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)))]*Elliptic
E[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))])/(2*d*f*h^2*Sqrt[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) - ((b*e - a
*f)*Sqrt[b*g - a*h]*(b*c*f*h + 3*a*d*f*h - b*d*(3*f*g + e*h))*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b
*x))]*Sqrt[g + h*x]*EllipticF[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)))])/(2*b*f*h^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)))]) - (Sqrt[-(d*g) + c*h]*(6*a*b*d^2*f^2*g*h - 3*a^2*d^2*f^2*h^2 + b^2*(2*c
*d*e*f*h^2 - c^2*f^2*h^2 - d^2*(3*f^2*g^2 + e^2*h^2)))*(a + b*x)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a
+ b*x))]*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]*EllipticPi[-((b*(d*g - c*h))/((b*c - a*d)*h)),
ArcSin[(Sqrt[b*c - a*d]*Sqrt[g + h*x])/(Sqrt[-(d*g) + c*h]*Sqrt[a + b*x])], ((b*e - a*f)*(d*g - c*h))/((b*c -
a*d)*(f*g - e*h))])/(2*b*d*Sqrt[b*c - a*d]*f*h^3*Sqrt[c + d*x]*Sqrt[e + f*x])

Rule 1597

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

Rule 1600

Int[(((a_.) + (b_.)*(x_))^(m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f
_.)*(x_)]*Sqrt[(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]*Sqr
t[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 + 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 + (b*B*d*f*h*(2*m + 3) + 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, B, C}, x]
&& IntegerQ[2*m] && GtQ[m, 0]

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 1598

Int[((A_.) + (B_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.
) + (h_.)*(x_)]), x_Symbol] :> Dist[(A*b - a*B)/b, Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h
*x]), x], x] + Dist[B/b, Int[Sqrt[a + b*x]/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] /; FreeQ[{a, b,
c, d, e, f, g, h, A, 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 165

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

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 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 x)^{3/2} (d e+c f+2 d f x)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=\frac{\int \frac{\sqrt{a+b x} \left (6 a d f (d e+c f) h+6 d f (b d e+b c f+2 a d f) h x+12 b d^2 f^2 h x^2\right )}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{6 d f h}\\ &=\frac{b \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{h}+\frac{\int \frac{12 d^2 f^2 h \left (2 a^2 (d e+c f) h-b (b c e g+a (d e g+c f g+c e h))\right )+24 d^2 f^2 h \left (2 a^2 d f h-b^2 (d e g+c f g+c e h)-a b (d f g-d e h-c f h)\right ) x+12 b d^2 f^2 h (5 a d f h-b (3 d f g+d e h+c f h)) x^2}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{24 d^2 f^2 h^2}\\ &=\frac{(5 a d f h-b (3 d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{2 f h^2 \sqrt{c+d x}}+\frac{b \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{h}+\frac{\int \frac{12 b d^2 f^2 h \left (a^2 d f (4 d e-c f) h^2+b^2 d e g (3 d f g+d e h-c f h)-a b f h \left (7 d^2 e g-c^2 f h-c d (f g-e h)\right )\right )-12 b d^2 f^2 h \left (6 a b d^2 f^2 g h-3 a^2 d^2 f^2 h^2+b^2 \left (2 c d e f h^2-c^2 f^2 h^2-d^2 \left (3 f^2 g^2+e^2 h^2\right )\right )\right ) x}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{48 b d^3 f^3 h^3}+\frac{((d e-c f) (d g-c h) (5 a d f h-b (3 d f g+d e h+c f h))) \int \frac{\sqrt{a+b x}}{(c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{4 d f h^2}\\ &=\frac{(5 a d f h-b (3 d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{2 f h^2 \sqrt{c+d x}}+\frac{b \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{h}-\frac{1}{4} \left (-\frac{3 a^2 d f}{b}+b \left (2 c e-\frac{d e^2}{f}-\frac{c^2 f}{d}-\frac{3 d f g^2}{h^2}\right )+\frac{6 a d f g}{h}\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx-\frac{((b e-a f) (b g-a h) (b c f h+3 a d f h-b d (3 f g+e h))) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{4 b f h^2}-\frac{\left ((d g-c h) (5 a d f h-b (3 d f g+d e h+c f h)) \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 )}{2 d f h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}\\ &=\frac{(5 a d f h-b (3 d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{2 f h^2 \sqrt{c+d x}}+\frac{b \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{h}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} (5 a d f h-b (3 d f g+d e h+c f h)) \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 )}{2 d f h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{\left (\left (-\frac{3 a^2 d f}{b}+b \left (2 c e-\frac{d e^2}{f}-\frac{c^2 f}{d}-\frac{3 d f g^2}{h^2}\right )+\frac{6 a d f g}{h}\right ) (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (h-b x^2\right ) \sqrt{1+\frac{(b c-a d) x^2}{d g-c h}} \sqrt{1+\frac{(b e-a f) x^2}{f g-e h}}} \, dx,x,\frac{\sqrt{g+h x}}{\sqrt{a+b x}}\right )}{2 \sqrt{c+d x} \sqrt{e+f x}}-\frac{\left ((b e-a f) (b g-a h) (b c f h+3 a d f h-b d (3 f g+e h)) \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 )}{2 b f h^2 (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{(5 a d f h-b (3 d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{2 f h^2 \sqrt{c+d x}}+\frac{b \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{h}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} (5 a d f h-b (3 d f g+d e h+c f h)) \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 )}{2 d f h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{(b e-a f) \sqrt{b g-a h} (b c f h+3 a d f h-b d (3 f g+e h)) \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 )}{2 b f h^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)}}}+\frac{\left (\frac{3 a^2 d f}{b}-b \left (2 c e-\frac{d e^2}{f}-\frac{c^2 f}{d}-\frac{3 d f g^2}{h^2}\right )-\frac{6 a d f g}{h}\right ) \sqrt{-d g+c h} (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{-d g+c h} \sqrt{a+b x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{2 \sqrt{b c-a d} h \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}

Mathematica [B]  time = 16.3708, size = 14853, normalized size = 16.54 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((a + b*x)^(3/2)*(d*e + c*f + 2*d*f*x))/(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 = 0.107, size = 35480, normalized size = 39.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(2*d*f*x+c*f+d*e)/(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{{\left (2 \, d f x + d e + c f\right )}{\left (b x + a\right )}^{\frac{3}{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((b*x+a)^(3/2)*(2*d*f*x+c*f+d*e)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [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((b*x+a)^(3/2)*(2*d*f*x+c*f+d*e)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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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((b*x+a)**(3/2)*(2*d*f*x+c*f+d*e)/(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{{\left (2 \, d f x + d e + c f\right )}{\left (b x + a\right )}^{\frac{3}{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((b*x+a)^(3/2)*(2*d*f*x+c*f+d*e)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="giac")

[Out]

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