3.1.34 \(\int \frac {A+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [34]

Optimal. Leaf size=867 \[ \frac {2 \left (A b^2+a^2 C\right ) d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {c+d x}}-\frac {2 \left (A b^2+a^2 C\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {2 \left (A b^2+a^2 C\right ) \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 \left (2 a b c C+A b^2 d-a^2 C d\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} F\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{b^2 (b c-a d) \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {2 C \sqrt {-d g+c h} (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac {b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{b^2 \sqrt {b c-a d} h \sqrt {c+d x} \sqrt {e+f x}} \]

[Out]

2*C*(b*x+a)*EllipticPi((-a*d+b*c)^(1/2)*(h*x+g)^(1/2)/(c*h-d*g)^(1/2)/(b*x+a)^(1/2),-b*(-c*h+d*g)/(-a*d+b*c)/h
,((-a*f+b*e)*(-c*h+d*g)/(-a*d+b*c)/(-e*h+f*g))^(1/2))*(c*h-d*g)^(1/2)*((-a*h+b*g)*(d*x+c)/(-c*h+d*g)/(b*x+a))^
(1/2)*((-a*h+b*g)*(f*x+e)/(-e*h+f*g)/(b*x+a))^(1/2)/b^2/h/(-a*d+b*c)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)+2*(A*b^
2+C*a^2)*d*(b*x+a)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/b/(-a*d+b*c)/(-a*f+b*e)/(-a*h+b*g)/(d*x+c)^(1/2)-2*(A*b^2
+C*a^2)*(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)^(1/2)-2*(A*b^2*d-C*
a^2*d+2*C*a*b*c)*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)/b^2/(-a*d+b*c)/(-
a*h+b*g)^(1/2)/(-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)-2*(A*b^2+C*a^2)*E
llipticE((-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)/b
/(-a*d+b*c)/(-a*f+b*e)/(-a*h+b*g)/((-c*f+d*e)*(b*x+a)/(-a*f+b*e)/(d*x+c))^(1/2)/(h*x+g)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.18, antiderivative size = 867, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.204, Rules used = {1619, 1616, 1612, 176, 430, 171, 551, 182, 435} \begin {gather*} -\frac {2 \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\text {ArcSin}\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 ) \left (C a^2+A b^2\right )}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C a^2+A b^2\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (C a^2+A b^2\right )}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {c+d x}}-\frac {2 \left (-C d a^2+2 b c C a+A b^2 d\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} F\left (\text {ArcSin}\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 )}{b^2 (b c-a d) \sqrt {b g-a h} \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 {2 C \sqrt {c h-d g} (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};\text {ArcSin}\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 )}{b^2 \sqrt {b c-a d} h \sqrt {c+d x} \sqrt {e+f x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(A*b^2 + a^2*C)*d*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(b*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)*Sqrt[c
+ d*x]) - (2*(A*b^2 + a^2*C)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)*(b*e - a*f)*(b*g - a*h)*S
qrt[a + b*x]) - (2*(A*b^2 + a^2*C)*Sqrt[d*g - c*h]*Sqrt[f*g - 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))])/(b*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)*Sqrt[((d*e - c*f
)*(a + b*x))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) - (2*(2*a*b*c*C + A*b^2*d - a^2*C*d)*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)))])/(b^2*(b*c - a*d)*Sqrt[b*g - a*h]
*Sqrt[f*g - e*h]*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))]) + (2*C*Sqrt[-(d*g) +
c*h]*(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))])/(b^2*Sqrt[b*c - a*d]*h*Sqrt[c
+ d*x]*Sqrt[e + f*x])

Rule 171

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 176

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 182

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 430

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

Rule 435

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

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], 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 1612

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 1616

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/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*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], 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 1619

Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (C_.)*(x_)^2))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt
[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[(A*b^2 + a^2*C)*(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)) + a*C*(a*(d*e*g + c*f*g + c*e*h) + 2
*b*c*e*g*(m + 1)) - 2*(A*b*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)) - C*(a^2*(d*f*g + d*e*h + c*f
*h) - b^2*c*e*g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g + c*e*h)))*x + d*f*h*(2*m + 5)*(A*b^2 + a^2*C)*x^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, C}, x] && IntegerQ[2*m] && LtQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 33.83, size = 1272, normalized size = 1.47 \begin {gather*} -\frac {2 \left (A b^2+a^2 C\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}+\frac {2 (a+b x)^{5/2} \left (-\left (\left (A b^2+a^2 C\right ) \left (d+\frac {b c}{a+b x}-\frac {a d}{a+b x}\right ) \left (f+\frac {b e}{a+b x}-\frac {a f}{a+b x}\right ) \left (h+\frac {b g}{a+b x}-\frac {a h}{a+b x}\right )\right )+\frac {(b e-a f) \sqrt {\frac {(b g-a h) \left (d+\frac {b c}{a+b x}-\frac {a d}{a+b x}\right )}{b (d g-c h)}} \left (-2 a C (b c-a d) h (b g-a h) \left (f+\frac {b e}{a+b x}-\frac {a f}{a+b x}\right ) \left (h+\frac {b g-a h}{a+b x}\right ) F\left (\sin ^{-1}\left (\sqrt {-\frac {(b e-a f) \left (h+\frac {b g}{a+b x}-\frac {a h}{a+b x}\right )}{b (f g-e h)}}\right )|\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )+A b^2 h \left (f+\frac {b e-a f}{a+b x}\right ) \left (h+\frac {b g}{a+b x}-\frac {a h}{a+b x}\right ) \left (b (d g-c h) E\left (\sin ^{-1}\left (\sqrt {\frac {(b e-a f) \left (h+\frac {b g}{a+b x}-\frac {a h}{a+b x}\right )}{b (-f g+e h)}}\right )|\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )+d (-b g+a h) F\left (\sin ^{-1}\left (\sqrt {\frac {(b e-a f) \left (h+\frac {b g}{a+b x}-\frac {a h}{a+b x}\right )}{b (-f g+e h)}}\right )|\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )\right )+a^2 C h \left (f+\frac {b e-a f}{a+b x}\right ) \left (h+\frac {b g}{a+b x}-\frac {a h}{a+b x}\right ) \left (b (d g-c h) E\left (\sin ^{-1}\left (\sqrt {\frac {(b e-a f) \left (h+\frac {b g}{a+b x}-\frac {a h}{a+b x}\right )}{b (-f g+e h)}}\right )|\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )+d (-b g+a h) F\left (\sin ^{-1}\left (\sqrt {\frac {(b e-a f) \left (h+\frac {b g}{a+b x}-\frac {a h}{a+b x}\right )}{b (-f g+e h)}}\right )|\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )\right )-C (b c-a d) (b g-a h)^2 \left (f+\frac {b e}{a+b x}-\frac {a f}{a+b x}\right ) \left (h+\frac {b g}{a+b x}-\frac {a h}{a+b x}\right ) \Pi \left (\frac {b (-f g+e h)}{(b e-a f) h};\sin ^{-1}\left (\sqrt {-\frac {(b e-a f) \left (h+\frac {b g}{a+b x}-\frac {a h}{a+b x}\right )}{b (f g-e h)}}\right )|\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )\right )}{b h (f g-e h) (a+b x) \sqrt {-\frac {(b e-a f) (b g-a h) \left (f+\frac {b e}{a+b x}-\frac {a f}{a+b x}\right ) \left (h+\frac {b g}{a+b x}-\frac {a h}{a+b x}\right )}{b^2 (f g-e h)^2}}}\right )}{b^3 (-b c+a d) (-b e+a f) (-b g+a h) \sqrt {c+\frac {(a+b x) \left (d-\frac {a d}{a+b x}\right )}{b}} \sqrt {e+\frac {(a+b x) \left (f-\frac {a f}{a+b x}\right )}{b}} \sqrt {g+\frac {(a+b x) \left (h-\frac {a h}{a+b x}\right )}{b}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(33893\) vs. \(2(794)=1588\).
time = 0.16, size = 33894, normalized size = 39.09

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2286\)
default	\(\text {Expression too large to display}\)	\(33894\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+A)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+A)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + A)/((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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+A)/(b*x+a)^(3/2)/(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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x**2+A)/(b*x+a)**(3/2)/(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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+A)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m operator + Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {C\,x^2+A}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + C*x^2)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)^(3/2)*(c + d*x)^(1/2)),x)

[Out]

int((A + C*x^2)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)^(3/2)*(c + d*x)^(1/2)), x)

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