3.15 \(\int \frac {d e+c f+2 d f x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\)

Optimal. Leaf size=1090 \[ -\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b d e+b c f-2 a d f)}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {4 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 d^2 f^2 h a^3-b d f (d f g+4 d e h+4 c f h) a^2+b^2 \left (e (f g+2 e h) d^2+c f (f g+3 e h) d+2 c^2 f^2 h\right ) a-b^3 \left (f (f g+e h) c^2-d e (f g-e h) c+d^2 e^2 g\right )\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {2 (d e-c f) \left (\left (f h c^2-d f g c+d e h c+2 d^2 e g\right ) b^2-a d (d f g+3 d e h+2 c f h) b+3 a^2 d^2 f h\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {4 b \left (3 d^2 f^2 h a^3-b d f (d f g+4 d e h+4 c f h) a^2+b^2 \left (e (f g+2 e h) d^2+c f (f g+3 e h) d+2 c^2 f^2 h\right ) a-b^3 \left (f (f g+e h) c^2-d e (f g-e h) c+d^2 e^2 g\right )\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}+\frac {4 d \left (3 d^2 f^2 h a^3-b d f (d f g+4 d e h+4 c f h) a^2+b^2 \left (e (f g+2 e h) d^2+c f (f g+3 e h) d+2 c^2 f^2 h\right ) a-b^3 \left (f (f g+e h) c^2-d e (f g-e h) c+d^2 e^2 g\right )\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}} \]

[Out]

4/3*d*(3*a^3*d^2*f^2*h-a^2*b*d*f*(4*c*f*h+4*d*e*h+d*f*g)-b^3*(d^2*e^2*g-c*d*e*(-e*h+f*g)+c^2*f*(e*h+f*g))+a*b^
2*(2*c^2*f^2*h+d^2*e*(2*e*h+f*g)+c*d*f*(3*e*h+f*g)))*(b*x+a)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)^2/(-
a*f+b*e)^2/(-a*h+b*g)^2/(d*x+c)^(1/2)-2/3*b*(-2*a*d*f+b*c*f+b*d*e)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(
-a*d+b*c)/(-a*f+b*e)/(-a*h+b*g)/(b*x+a)^(3/2)-4/3*b*(3*a^3*d^2*f^2*h-a^2*b*d*f*(4*c*f*h+4*d*e*h+d*f*g)-b^3*(d^
2*e^2*g-c*d*e*(-e*h+f*g)+c^2*f*(e*h+f*g))+a*b^2*(2*c^2*f^2*h+d^2*e*(2*e*h+f*g)+c*d*f*(3*e*h+f*g)))*(d*x+c)^(1/
2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)^2/(-a*f+b*e)^2/(-a*h+b*g)^2/(b*x+a)^(1/2)+2/3*(-c*f+d*e)*(3*a^2*d^2*
f*h-a*b*d*(2*c*f*h+3*d*e*h+d*f*g)+b^2*(c^2*f*h+c*d*e*h-c*d*f*g+2*d^2*e*g))*EllipticF((-a*h+b*g)^(1/2)*(f*x+e)^
(1/2)/(-e*h+f*g)^(1/2)/(b*x+a)^(1/2),(-(-a*d+b*c)*(-e*h+f*g)/(-c*f+d*e)/(-a*h+b*g))^(1/2))*((-a*f+b*e)*(d*x+c)
/(-c*f+d*e)/(b*x+a))^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)^2/(-a*f+b*e)/(-a*h+b*g)^(3/2)/(-e*h+f*g)^(1/2)/(d*x+c)^(1/
2)/(-(-a*f+b*e)*(h*x+g)/(-e*h+f*g)/(b*x+a))^(1/2)-4/3*(3*a^3*d^2*f^2*h-a^2*b*d*f*(4*c*f*h+4*d*e*h+d*f*g)-b^3*(
d^2*e^2*g-c*d*e*(-e*h+f*g)+c^2*f*(e*h+f*g))+a*b^2*(2*c^2*f^2*h+d^2*e*(2*e*h+f*g)+c*d*f*(3*e*h+f*g)))*EllipticE
((-c*h+d*g)^(1/2)*(f*x+e)^(1/2)/(-e*h+f*g)^(1/2)/(d*x+c)^(1/2),((-a*d+b*c)*(-e*h+f*g)/(-a*f+b*e)/(-c*h+d*g))^(
1/2))*(-c*h+d*g)^(1/2)*(-e*h+f*g)^(1/2)*(b*x+a)^(1/2)*(-(-c*f+d*e)*(h*x+g)/(-e*h+f*g)/(d*x+c))^(1/2)/(-a*d+b*c
)^2/(-a*f+b*e)^2/(-a*h+b*g)^2/((-c*f+d*e)*(b*x+a)/(-a*f+b*e)/(d*x+c))^(1/2)/(h*x+g)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 3.02, antiderivative size = 1090, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1599, 1602, 12, 170, 419, 176, 424} \[ -\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b d e+b c f-2 a d f)}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {4 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 d^2 f^2 h a^3-b d f (d f g+4 d e h+4 c f h) a^2+b^2 \left (e (f g+2 e h) d^2+c f (f g+3 e h) d+2 c^2 f^2 h\right ) a-b^3 \left (f (f g+e h) c^2-d e (f g-e h) c+d^2 e^2 g\right )\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {2 (d e-c f) \left (\left (f h c^2-d f g c+d e h c+2 d^2 e g\right ) b^2-a d (d f g+3 d e h+2 c f h) b+3 a^2 d^2 f h\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} F\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {4 b \left (3 d^2 f^2 h a^3-b d f (d f g+4 d e h+4 c f h) a^2+b^2 \left (e (f g+2 e h) d^2+c f (f g+3 e h) d+2 c^2 f^2 h\right ) a-b^3 \left (f (f g+e h) c^2-d e (f g-e h) c+d^2 e^2 g\right )\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}+\frac {4 d \left (3 d^2 f^2 h a^3-b d f (d f g+4 d e h+4 c f h) a^2+b^2 \left (e (f g+2 e h) d^2+c f (f g+3 e h) d+2 c^2 f^2 h\right ) a-b^3 \left (f (f g+e h) c^2-d e (f g-e h) c+d^2 e^2 g\right )\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}} \]

Antiderivative was successfully verified.

[In]

Int[(d*e + c*f + 2*d*f*x)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(4*d*(3*a^3*d^2*f^2*h - a^2*b*d*f*(d*f*g + 4*d*e*h + 4*c*f*h) - b^3*(d^2*e^2*g - c*d*e*(f*g - e*h) + c^2*f*(f*
g + e*h)) + a*b^2*(2*c^2*f^2*h + d^2*e*(f*g + 2*e*h) + c*d*f*(f*g + 3*e*h)))*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]) - (2*b*(b*d*e + b*c*f - 2*a*d*f)*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)) - (4*b*(3*a^3*d^2*
f^2*h - a^2*b*d*f*(d*f*g + 4*d*e*h + 4*c*f*h) - b^3*(d^2*e^2*g - c*d*e*(f*g - e*h) + c^2*f*(f*g + e*h)) + a*b^
2*(2*c^2*f^2*h + d^2*e*(f*g + 2*e*h) + c*d*f*(f*g + 3*e*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]) - (4*Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*(3*a^3*d^2*f^2*h -
a^2*b*d*f*(d*f*g + 4*d*e*h + 4*c*f*h) - b^3*(d^2*e^2*g - c*d*e*(f*g - e*h) + c^2*f*(f*g + e*h)) + a*b^2*(2*c^2
*f^2*h + d^2*e*(f*g + 2*e*h) + c*d*f*(f*g + 3*e*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*(d*e - c*f)*(3*a^2*d^2*f*h - a*b*d*(d*f*g + 3*d*e*h + 2*c*f*
h) + b^2*(2*d^2*e*g - c*d*f*g + c*d*e*h + c^2*f*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)))])/(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 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 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 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 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]

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]

Rubi steps

\begin {align*} \int \frac {d e+c f+2 d f x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=-\frac {2 b (b d e+b c f-2 a d f) \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 {2 b d f (3 b c e g-a (d e g+c f g+c e h))-(d e+c f) \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 )+(b d e+b c f-2 a d f) (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 c-a d) (b e-a f) (b g-a h)}\\ &=-\frac {2 b (b d e+b c f-2 a d f) \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 {4 b \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\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 (b d e+b c f-2 a d f) (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 (2 b d f (3 b c e g-a (d e g+c f g+c e h))-(d e+c f) \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 )+2 (a d f h+b (d f g+d e h+c f h)) \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) x+4 b d f h \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) x^2}{\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)^2 (b g-a h)^2}\\ &=\frac {4 d \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\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 (b d e+b c f-2 a d f) \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 {4 b \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\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 d f (b e-a f) (d e-c f) h (b g-a h) \left (3 a^2 d^2 f h-a b d (d f g+3 d e h+2 c f h)+b^2 \left (2 d^2 e g-c d f g+c d e h+c^2 f h\right )\right )}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{6 b d (b c-a d)^2 f (b e-a f)^2 h (b g-a h)^2}+\frac {\left (2 (d e-c f) (d g-c h) \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\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 {4 d \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\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 (b d e+b c f-2 a d f) \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 {4 b \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\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 ((d e-c f) \left (3 a^2 d^2 f h-a b d (d f g+3 d e h+2 c f h)+b^2 \left (2 d^2 e g-c d f g+c d e h+c^2 f 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 (4 (d g-c h) \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\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 {4 d \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\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 (b d e+b c f-2 a d f) \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 {4 b \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\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 {4 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\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 (d e-c f) \left (3 a^2 d^2 f h-a b d (d f g+3 d e h+2 c f h)+b^2 \left (2 d^2 e g-c d f g+c d e h+c^2 f h\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 {4 d \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\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 (b d e+b c f-2 a d f) \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 {4 b \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\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 {4 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\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 (d e-c f) \left (3 a^2 d^2 f h-a b d (d f g+3 d e h+2 c f h)+b^2 \left (2 d^2 e g-c d f g+c d e h+c^2 f h\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*}

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Mathematica [B]  time = 20.65, size = 10629, normalized size = 9.75 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(d*e + c*f + 2*d*f*x)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

Result too large to show

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fricas [F]  time = 8.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (2 \, d f x + d e + c f\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}{b^{3} d f h x^{6} + a^{3} c e g + {\left (b^{3} d f g + {\left (b^{3} d e + {\left (b^{3} c + 3 \, a b^{2} d\right )} f\right )} h\right )} x^{5} + {\left ({\left (b^{3} d e + {\left (b^{3} c + 3 \, a b^{2} d\right )} f\right )} g + {\left ({\left (b^{3} c + 3 \, a b^{2} d\right )} e + 3 \, {\left (a b^{2} c + a^{2} b d\right )} f\right )} h\right )} x^{4} + {\left ({\left ({\left (b^{3} c + 3 \, a b^{2} d\right )} e + 3 \, {\left (a b^{2} c + a^{2} b d\right )} f\right )} g + {\left (3 \, {\left (a b^{2} c + a^{2} b d\right )} e + {\left (3 \, a^{2} b c + a^{3} d\right )} f\right )} h\right )} x^{3} + {\left ({\left (3 \, {\left (a b^{2} c + a^{2} b d\right )} e + {\left (3 \, a^{2} b c + a^{3} d\right )} f\right )} g + {\left (a^{3} c f + {\left (3 \, a^{2} b c + a^{3} d\right )} e\right )} h\right )} x^{2} + {\left (a^{3} c e h + {\left (a^{3} c f + {\left (3 \, a^{2} b c + a^{3} d\right )} e\right )} g\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*d*f*x+c*f+d*e)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")

[Out]

integral((2*d*f*x + d*e + c*f)*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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2 \, d f x + d e + c f}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*d*f*x+c*f+d*e)/(b*x+a)^(5/2)/(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)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)

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maple [B]  time = 0.87, size = 65243, normalized size = 59.86 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*d*f*x+c*f+d*e)/(b*x+a)^(5/2)/(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2 \, d f x + d e + c f}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*d*f*x+c*f+d*e)/(b*x+a)^(5/2)/(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)^(5/2)*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 \[ \int \frac {c\,f+d\,e+2\,d\,f\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*f + d*e + 2*d*f*x)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)^(5/2)*(c + d*x)^(1/2)),x)

[Out]

int((c*f + d*e + 2*d*f*x)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)^(5/2)*(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((2*d*f*x+c*f+d*e)/(b*x+a)**(5/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

Timed out

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