3.43 \(\int \frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{a+b x} \, dx\)

Optimal. Leaf size=570 \[ \frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (3 a^2 d f h^2-3 a b h^2 (c f+d e)-\left (b^2 (d g (f g-e h)-c h (2 e h+f g))\right )\right ) \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {h (d e-c f)}{f (d g-c h)}\right )}{3 b^3 d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 (b e-a f) (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^3 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} (3 a d f h-b (c f h+d e h+d f g)) E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 b^2 d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.29, antiderivative size = 570, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {161, 1607, 169, 538, 537, 158, 114, 113, 121, 120} \[ \frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (3 a^2 d f h^2-3 a b h^2 (c f+d e)+b^2 (-(d g (f g-e h)-c h (2 e h+f g)))\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 b^3 d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} (3 a d f h-b (c f h+d e h+d f g)) E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 b^2 d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 (b e-a f) (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^3 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}+\frac {2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*b) - (2*Sqrt[-(d*e) + c*f]*(3*a*d*f*h - b*(d*f*g + d*e*h + 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^2*d*Sqrt[f]*h*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) + (2*S
qrt[-(d*e) + c*f]*(3*a^2*d*f*h^2 - 3*a*b*(d*e + c*f)*h^2 - b^2*(d*g*(f*g - e*h) - c*h*(f*g + 2*e*h)))*Sqrt[(d*
(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) +
c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(3*b^3*d*Sqrt[f]*h*Sqrt[e + f*x]*Sqrt[g + h*x]) - (2*(b*e - a*f)*Sqr
t[-(d*e) + c*f]*(b*g - a*h)*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^3*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x])

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 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 161

Int[((a_.) + (b_.)*(x_))^(m_)*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)], x_Sy
mbol] :> Simp[(2*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(b*(2*m + 5)), x] + Dist[1/(b*(2
*m + 5)), Int[((a + b*x)^m*Simp[3*b*c*e*g - a*(d*e*g + c*f*g + c*e*h) + 2*(b*(d*e*g + c*f*g + c*e*h) - a*(d*f*
g + d*e*h + c*f*h))*x - (3*a*d*f*h - b*(d*f*g + d*e*h + c*f*h))*x^2, x])/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g +
h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] &&  !LtQ[m, -1]

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 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]

Rule 1607

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.)*((g_.) + (h_.)*(x_)
)^(q_.), x_Symbol] :> Dist[PolynomialRemainder[Px, a + b*x, x], Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h
*x)^q, x], x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q,
x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && PolyQ[Px, x] && EqQ[m, -1]

Rubi steps

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

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Mathematica [C]  time = 14.19, size = 1820, normalized size = 3.19 \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*b) + ((c + d*x)^(3/2)*(2*b^2*d*f^2*g*Sqrt[-c + (d*g)/h]*h + 2
*b^2*d*e*f*Sqrt[-c + (d*g)/h]*h^2 + 2*b^2*c*f^2*Sqrt[-c + (d*g)/h]*h^2 - 6*a*b*d*f^2*Sqrt[-c + (d*g)/h]*h^2 +
(2*b^2*d^3*e*f*g^2*Sqrt[-c + (d*g)/h])/(c + d*x)^2 - (2*b^2*c*d^2*f^2*g^2*Sqrt[-c + (d*g)/h])/(c + d*x)^2 + (2
*b^2*d^3*e^2*g*Sqrt[-c + (d*g)/h]*h)/(c + d*x)^2 - (2*b^2*c*d^2*e*f*g*Sqrt[-c + (d*g)/h]*h)/(c + d*x)^2 - (6*a
*b*d^3*e*f*g*Sqrt[-c + (d*g)/h]*h)/(c + d*x)^2 + (6*a*b*c*d^2*f^2*g*Sqrt[-c + (d*g)/h]*h)/(c + d*x)^2 - (2*b^2
*c*d^2*e^2*Sqrt[-c + (d*g)/h]*h^2)/(c + d*x)^2 + (6*a*b*c*d^2*e*f*Sqrt[-c + (d*g)/h]*h^2)/(c + d*x)^2 + (2*b^2
*c^3*f^2*Sqrt[-c + (d*g)/h]*h^2)/(c + d*x)^2 - (6*a*b*c^2*d*f^2*Sqrt[-c + (d*g)/h]*h^2)/(c + d*x)^2 + (2*b^2*d
^2*f^2*g^2*Sqrt[-c + (d*g)/h])/(c + d*x) + (4*b^2*d^2*e*f*g*Sqrt[-c + (d*g)/h]*h)/(c + d*x) - (2*b^2*c*d*f^2*g
*Sqrt[-c + (d*g)/h]*h)/(c + d*x) - (6*a*b*d^2*f^2*g*Sqrt[-c + (d*g)/h]*h)/(c + d*x) + (2*b^2*d^2*e^2*Sqrt[-c +
(d*g)/h]*h^2)/(c + d*x) - (2*b^2*c*d*e*f*Sqrt[-c + (d*g)/h]*h^2)/(c + d*x) - (6*a*b*d^2*e*f*Sqrt[-c + (d*g)/h
]*h^2)/(c + d*x) - (4*b^2*c^2*f^2*Sqrt[-c + (d*g)/h]*h^2)/(c + d*x) + (12*a*b*c*d*f^2*Sqrt[-c + (d*g)/h]*h^2)/
(c + d*x) - ((2*I)*b*f*(-(d*g) + c*h)*(-3*a*d*f*h + b*(d*f*g + d*e*h + c*f*h))*Sqrt[1 - c/(c + d*x) + (d*e)/(f
*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (d*g)/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]]
, (d*e*h - c*f*h)/(d*f*g - c*f*h)])/Sqrt[c + d*x] - ((2*I)*b*d*f*(-(d*g) + c*h)*(-(b*f*g) - 2*b*e*h + 3*a*f*h)
*Sqrt[1 - c/(c + d*x) + (d*e)/(f*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (d*g)/(h*(c + d*x))]*EllipticF[I*ArcSinh[S
qrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)])/Sqrt[c + d*x] + ((6*I)*b^2*d^2*e*f*g*h*Sqr
t[1 - c/(c + d*x) + (d*e)/(f*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (d*g)/(h*(c + d*x))]*EllipticPi[-((b*c*h - a*d
*h)/(b*d*g - b*c*h)), I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)])/Sqrt[c +
d*x] - ((6*I)*a*b*d^2*f^2*g*h*Sqrt[1 - c/(c + d*x) + (d*e)/(f*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (d*g)/(h*(c +
d*x))]*EllipticPi[-((b*c*h - a*d*h)/(b*d*g - b*c*h)), I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c
*f*h)/(d*f*g - c*f*h)])/Sqrt[c + d*x] - ((6*I)*a*b*d^2*e*f*h^2*Sqrt[1 - c/(c + d*x) + (d*e)/(f*(c + d*x))]*Sqr
t[1 - c/(c + d*x) + (d*g)/(h*(c + d*x))]*EllipticPi[-((b*c*h - a*d*h)/(b*d*g - b*c*h)), I*ArcSinh[Sqrt[-c + (d
*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)])/Sqrt[c + d*x] + ((6*I)*a^2*d^2*f^2*h^2*Sqrt[1 - c/(c
+ d*x) + (d*e)/(f*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (d*g)/(h*(c + d*x))]*EllipticPi[-((b*c*h - a*d*h)/(b*d*g
- b*c*h)), I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)])/Sqrt[c + d*x]))/(3*b
^3*d^2*f*Sqrt[-c + (d*g)/h]*h*Sqrt[e + ((c + d*x)*(f - (c*f)/(c + d*x)))/d]*Sqrt[g + ((c + d*x)*(h - (c*h)/(c
+ d*x)))/d])

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}{b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.07, size = 3678, normalized size = 6.45 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x*b^2*c*d^2*e*f*h^2-3*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/
2)*EllipticPi(((d*x+c)/(c*f-d*e)*f)^(1/2),-(c*f-d*e)*b/f/(a*d-b*c),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*
e*f*g*h+((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticE(((d*x+
c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*e*f*g*h+3*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+
g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticPi(((d*x+c)/(c*f-d*e)*f)^(1/2),-(c*f-d*e)*b/f/(a*d-
b*c),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a*b*c*d^2*e*f*h^2-3*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(
1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a*b*c
*d^2*e*f*h^2-3*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticE
(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a*b*c*d^2*f^2*g*h+3*((d*x+c)/(c*f-d*e)*f)^(1/2)*
(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f
/(c*h-d*g))^(1/2))*a*b*d^3*e*f*g*h+3*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d
*e)*d)^(1/2)*EllipticPi(((d*x+c)/(c*f-d*e)*f)^(1/2),-(c*f-d*e)*b/f/(a*d-b*c),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*
a*b*c*d^2*f^2*g*h-3*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*Elli
pticPi(((d*x+c)/(c*f-d*e)*f)^(1/2),-(c*f-d*e)*b/f/(a*d-b*c),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a*b*d^3*e*f*g*h+x
^2*b^2*c*d^2*f^2*h^2+x^2*b^2*d^3*e*f*h^2+x^2*b^2*d^3*f^2*g*h-((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g)
)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^
2*c^3*f^2*h^2+x*b^2*c*d^2*f^2*g*h+x*b^2*d^3*e*f*g*h+x^3*b^2*d^3*f^2*h^2-3*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g
)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d*
g))^(1/2))*a*b*c^2*d*f^2*h^2+2*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)
^(1/2)*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c^2*d*e*f*h^2+((d*x+c)/(c*f-
d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),
((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c^2*d*f^2*g*h+3*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*
(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a*b*c^2*d*
f^2*h^2+3*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticPi(((d
*x+c)/(c*f-d*e)*f)^(1/2),-(c*f-d*e)*b/f/(a*d-b*c),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a^2*d^3*e*f*h^2-3*((d*x+c)/
(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticPi(((d*x+c)/(c*f-d*e)*f)^
(1/2),-(c*f-d*e)*b/f/(a*d-b*c),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a*b*d^3*e^2*h^2+3*((d*x+c)/(c*f-d*e)*f)^(1/2)*
(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticPi(((d*x+c)/(c*f-d*e)*f)^(1/2),-(c*f-d*e)*b/
f/(a*d-b*c),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*d^3*e^2*g*h-((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g)
)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^
2*c*d^2*f^2*g^2-((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*Elliptic
F(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*d^3*e^2*g*h+((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(
h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c
*h-d*g))^(1/2))*b^2*d^3*e*f*g^2+((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d
)^(1/2)*EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*e^2*h^2+((d*x+c)/(c*f
-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2)
,((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*f^2*g^2-((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(
-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*d^3*e^2
*g*h-((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticE(((d*x+c)/
(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*d^3*e*f*g^2-3*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/
(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticPi(((d*x+c)/(c*f-d*e)*f)^(1/2),-(c*f-d*e)*b/f/(a*d-b*c),
((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a^2*c*d^2*f^2*h^2+3*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*
(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a^2*c*d^2*
f^2*h^2-3*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticF(((d*
x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a^2*d^3*e*f*h^2+3*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+
g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d
*g))^(1/2))*a*b*d^3*e^2*h^2-2*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^
(1/2)*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*e^2*h^2+b^2*c*d^2*e*f*g
*h)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/b^3/h/f/d^2/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f
*g*x+d*e*g*x+c*e*g)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}{b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}}{a+b\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}{a + b x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*x)/(a + b*x), x)

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