3.72 \(\int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx\)

Optimal. Leaf size=875 \[ -\frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right ) b^2}{(b c-a d)^3 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 d \sqrt {h} \sqrt {e h-f g} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right ) b}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}+\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x} b}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {c+d x}}+\frac {4 d \sqrt {f} (d f g+d e h-2 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (c f-d e)^{3/2} (d g-c h)^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {f} (2 d f g+d e h-3 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (c f-d e)^{3/2} (d g-c h) \sqrt {e+f x} \sqrt {g+h x}}-\frac {4 d^2 (d f g+d e h-2 c f h) \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt {c+d x}}+\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.34, antiderivative size = 875, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 12, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.343, Rules used = {179, 104, 152, 158, 114, 113, 121, 120, 21, 169, 538, 537} \[ -\frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right ) b^2}{(b c-a d)^3 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 d \sqrt {h} \sqrt {e h-f g} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right ) b}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}+\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x} b}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {c+d x}}+\frac {4 d \sqrt {f} (d f g+d e h-2 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (c f-d e)^{3/2} (d g-c h)^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {f} (2 d f g+d e h-3 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (c f-d e)^{3/2} (d g-c h) \sqrt {e+f x} \sqrt {g+h x}}-\frac {4 d^2 (d f g+d e h-2 c f h) \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt {c+d x}}+\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)*(c + d*x)^(5/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 104

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegersQ[2*m, 2*n, 2*p]

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 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 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)
])

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 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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 169

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

Rule 179

Int[(((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_))/(Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)])
, x_Symbol] :> Int[ExpandIntegrand[1/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), (a + b*x)^m*(c + d*x)^(n + 1
/2), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IntegerQ[m] && IntegerQ[n + 1/2]

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 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rubi steps

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

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Mathematica [C]  time = 15.77, size = 721, normalized size = 0.82 \[ \frac {2 \left (d^2 (e+f x) (g+h x) (b c-a d) \sqrt {\frac {d g}{h}-c} \left ((c+d x) \left (2 a d (-2 c f h+d e h+d f g)+b \left (7 c^2 f h-5 c d (e h+f g)+3 d^2 e g\right )\right )+(b c-a d) (c f-d e) (c h-d g)\right )-(c+d x) \left (d^2 (e+f x) (g+h x) (b c-a d) \sqrt {\frac {d g}{h}-c} \left (2 a d (-2 c f h+d e h+d f g)+b \left (7 c^2 f h-5 c d (e h+f g)+3 d^2 e g\right )\right )+i (c+d x)^{3/2} (d g-c h) \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \left (\left (a^2 d^2 f (-3 c f h+d e h+2 d f g)+a b d f \left (9 c^2 f h-c d (5 e h+7 f g)+3 d^2 e g\right )+b^2 \left (-9 c^3 f^2 h+2 c^2 d f (5 e h+4 f g)-3 c d^2 e (e h+3 f g)+3 d^3 e^2 g\right )\right ) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {d g}{h}-c}}{\sqrt {c+d x}}\right ),\frac {d e h-c f h}{d f g-c f h}\right )-3 b^2 (d e-c f)^2 (d g-c h) \Pi \left (-\frac {b c h-a d h}{b d g-b c h};i \sinh ^{-1}\left (\frac {\sqrt {\frac {d g}{h}-c}}{\sqrt {c+d x}}\right )|\frac {d e h-c f h}{d f g-c f h}\right )+f (b c-a d) \left (2 a d (-2 c f h+d e h+d f g)+b \left (7 c^2 f h-5 c d (e h+f g)+3 d^2 e g\right )\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {d g}{h}-c}}{\sqrt {c+d x}}\right )|\frac {d e h-c f h}{d f g-c f h}\right )\right )\right )\right )}{3 (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)^3 (d e-c f)^2 \sqrt {\frac {d g}{h}-c} (d g-c h)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)*(c + d*x)^(5/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(2*(d^2*(b*c - a*d)*Sqrt[-c + (d*g)/h]*(e + f*x)*(g + h*x)*((b*c - a*d)*(-(d*e) + c*f)*(-(d*g) + c*h) + (2*a*d
*(d*f*g + d*e*h - 2*c*f*h) + b*(3*d^2*e*g + 7*c^2*f*h - 5*c*d*(f*g + e*h)))*(c + d*x)) - (c + d*x)*(d^2*(b*c -
a*d)*Sqrt[-c + (d*g)/h]*(2*a*d*(d*f*g + d*e*h - 2*c*f*h) + b*(3*d^2*e*g + 7*c^2*f*h - 5*c*d*(f*g + e*h)))*(e
+ f*x)*(g + h*x) + I*(d*g - c*h)*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c +
d*x))]*((b*c - a*d)*f*(2*a*d*(d*f*g + d*e*h - 2*c*f*h) + b*(3*d^2*e*g + 7*c^2*f*h - 5*c*d*(f*g + e*h)))*Ellipt
icE[I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)] + (a^2*d^2*f*(2*d*f*g + d*e*
h - 3*c*f*h) + b^2*(3*d^3*e^2*g - 9*c^3*f^2*h - 3*c*d^2*e*(3*f*g + e*h) + 2*c^2*d*f*(4*f*g + 5*e*h)) + a*b*d*f
*(3*d^2*e*g + 9*c^2*f*h - c*d*(7*f*g + 5*e*h)))*EllipticF[I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h
- c*f*h)/(d*f*g - c*f*h)] - 3*b^2*(d*e - c*f)^2*(d*g - c*h)*EllipticPi[-((b*c*h - a*d*h)/(b*d*g - b*c*h)), I*A
rcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)]))))/(3*(b*c - a*d)^3*(d*e - c*f)^2*
Sqrt[-c + (d*g)/h]*(d*g - c*h)^2*(c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x])

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

[Out]

Timed out

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

[Out]

Timed out

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maple [B]  time = 0.28, size = 17330, normalized size = 19.81 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)/(d*x+c)^(5/2)/(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 \[ \int \frac {1}{{\left (b x + a\right )} {\left (d x + c\right )}^{\frac {5}{2}} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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