3.1.17 \(\int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [17]

Optimal. Leaf size=410 \[ \frac {2 b^2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}+\frac {2 b^2 \sqrt {-d e+c f} (3 B d f h-2 C (d f g+d e h+c f h)) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {2 \sqrt {-d e+c f} \left (3 a b B d f h^2-3 a^2 C d f h^2-b^2 (3 B d f g h-C (c h (f g-e h)+d g (2 f g+e h)))\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt {e+f x} \sqrt {g+h x}} \]

[Out]

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

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Rubi [A]
time = 0.36, antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.113, Rules used = {1629, 164, 115, 114, 122, 121} \begin {gather*} \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}} F\left (\text {ArcSin}\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 ) \left (-3 a^2 C d f h^2+3 a b B d f h^2-\left (b^2 (3 B d f g h-c C h (f g-e h)-C d g (e h+2 f g))\right )\right )}{3 d^2 f^{3/2} h^2 \sqrt {e+f x} \sqrt {g+h x}}+\frac {2 b^2 \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} (3 B d f h-2 C (c f h+d e h+d f g)) E\left (\text {ArcSin}\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 d^2 f^{3/2} h^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {2 b^2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*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)))], 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 115

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 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*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)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && Si
mplerQ[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 122

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 164

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 1629

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[
{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^
(p + 1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Dist[1/(d*f*b^q*(m + n + p + q + 1)), Int[(a + b*x)^m*(c +
d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a +
b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*
(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; F
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 24.17, size = 442, normalized size = 1.08 \begin {gather*} \frac {\sqrt {c+d x} \left (2 b^2 C d^2 f h (e+f x) (g+h x)+\frac {2 b^2 d^2 (3 B d f h-2 C (d f g+d e h+c f h)) (e+f x) (g+h x)}{c+d x}+2 i b^2 \sqrt {-c+\frac {d e}{f}} f h (3 B d f h-2 C (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )+\frac {2 i d h \left (3 a b B d f^2 h-3 a^2 C d f^2 h+b^2 (-3 B d e f h+c C f (-f g+e h)+C d e (f g+2 e h))\right ) \sqrt {c+d x} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )}{\sqrt {-c+\frac {d e}{f}}}\right )}{3 d^3 f^2 h^2 \sqrt {e+f x} \sqrt {g+h x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d*x]*(2*b^2*C*d^2*f*h*(e + f*x)*(g + h*x) + (2*b^2*d^2*(3*B*d*f*h - 2*C*(d*f*g + d*e*h + c*f*h))*(e
+ f*x)*(g + h*x))/(c + d*x) + (2*I)*b^2*Sqrt[-c + (d*e)/f]*f*h*(3*B*d*f*h - 2*C*(d*f*g + d*e*h + c*f*h))*Sqrt[
c + d*x]*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*
e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)] + ((2*I)*d*h*(3*a*b*B*d*f^2*h - 3*a^2*C*d*f^2*h + b^2*(
-3*B*d*e*f*h + c*C*f*(-(f*g) + e*h) + C*d*e*(f*g + 2*e*h)))*Sqrt[c + d*x]*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sq
rt[(d*(g + h*x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h
- c*f*h)])/Sqrt[-c + (d*e)/f]))/(3*d^3*f^2*h^2*Sqrt[e + f*x]*Sqrt[g + h*x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2824\) vs. \(2(364)=728\).
time = 0.12, size = 2825, normalized size = 6.89

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 C \,b^{2} \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}{3 d f h}+\frac {2 \left (a b B -C \,a^{2}-\frac {2 C \,b^{2} \left (\frac {1}{2} c e h +\frac {1}{2} c f g +\frac {1}{2} d e g \right )}{3 d f h}\right ) \left (-\frac {e}{f}+\frac {g}{h}\right ) \sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 \left (B \,b^{2}-\frac {2 C \,b^{2} \left (h f c +d e h +g f d \right )}{3 d f h}\right ) \left (-\frac {e}{f}+\frac {g}{h}\right ) \sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \left (\left (-\frac {g}{h}+\frac {c}{d}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )-\frac {c \EllipticF \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{d}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\)	\(637\)
default	\(\text {Expression too large to display}\)	\(2825\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(-2*C*(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticE((-(h
*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*b^2*d^2*f^2*g^3-C*b^2*d^2*f^2*h^3*x^3-3*B*(-(h*x+g)*
f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticE((-(h*x+g)*f/(e*h-f*g))^(1
/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*b^2*d^2*e*f*g*h^2-C*(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1
/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticF((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*b^2*c*
d*f^2*g^2*h+C*(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticF((
-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*b^2*d^2*e*f*g^2*h-3*C*(-(h*x+g)*f/(e*h-f*g))^(1/2
)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticF((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/
f/(c*h-d*g))^(1/2))*a^2*d^2*e*f*h^3+3*C*(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(e
*h-f*g))^(1/2)*EllipticF((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*a^2*d^2*f^2*g*h^2+2*C*(
-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticF((-(h*x+g)*f/(e*h
-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*b^2*c^2*e*f*h^3-2*C*(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h
-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticF((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2
))*b^2*c^2*f^2*g*h^2+C*(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*El
lipticF((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*b^2*c*d*e^2*h^3-C*(-(h*x+g)*f/(e*h-f*g))
^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticF((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*
g)*d/f/(c*h-d*g))^(1/2))*b^2*d^2*e^2*g*h^2-2*C*(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+
e)*h/(e*h-f*g))^(1/2)*EllipticE((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*b^2*c^2*e*f*h^3+
2*C*(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticE((-(h*x+g)*f
/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*b^2*c^2*f^2*g*h^2+3*B*(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)
*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticE((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g
))^(1/2))*b^2*d^2*f^2*g^2*h-2*C*(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))
^(1/2)*EllipticE((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*b^2*c*d*e^2*h^3+2*C*(-(h*x+g)*f
/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticE((-(h*x+g)*f/(e*h-f*g))^(1/
2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*b^2*d^2*e^2*g*h^2+3*B*(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(
1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticF((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*a*b*d
^2*e*f*h^3-3*B*(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticF(
(-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*a*b*d^2*f^2*g*h^2-3*B*(-(h*x+g)*f/(e*h-f*g))^(1/
2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticF((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d
/f/(c*h-d*g))^(1/2))*b^2*c*d*e*f*h^3+3*B*(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(
e*h-f*g))^(1/2)*EllipticF((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*b^2*c*d*f^2*g*h^2+3*B*
(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticE((-(h*x+g)*f/(e*
h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*b^2*c*d*e*f*h^3-3*B*(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*
h-d*g))^(1/2)*((f*x+e)*h/(e*h-f*g))^(1/2)*EllipticE((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/
2))*b^2*c*d*f^2*g*h^2-C*b^2*c*d*e*f*h^3*x+2*C*(-(h*x+g)*f/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*((f*x+e
)*h/(e*h-f*g))^(1/2)*EllipticE((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*b^2*c*d*e*f*g*h^2
-C*b^2*c*d*f^2*h^3*x^2-C*b^2*d^2*e*f*h^3*x^2-C*b^2*d^2*f^2*g*h^2*x^2-C*b^2*c*d*f^2*g*h^2*x-C*b^2*d^2*e*f*g*h^2
*x-C*b^2*c*d*e*f*g*h^2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/h^3/f^2/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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.36, size = 921, normalized size = 2.25 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g} C b^{2} d^{2} f^{2} h^{2} + {\left (2 \, C b^{2} d^{2} f^{2} g^{2} + 2 \, C b^{2} d^{2} h^{2} e^{2} + {\left (C b^{2} c d - 3 \, B b^{2} d^{2}\right )} f^{2} g h + {\left (2 \, C b^{2} c^{2} - 3 \, B b^{2} c d - 9 \, {\left (C a^{2} - B a b\right )} d^{2}\right )} f^{2} h^{2} + {\left (C b^{2} d^{2} f g h + {\left (C b^{2} c d - 3 \, B b^{2} d^{2}\right )} f h^{2}\right )} e\right )} \sqrt {d f h} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - c d f^{2} g h + c^{2} f^{2} h^{2} + d^{2} h^{2} e^{2} - {\left (d^{2} f g h + c d f h^{2}\right )} e\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, c d^{2} f^{3} g^{2} h - 3 \, c^{2} d f^{3} g h^{2} + 2 \, c^{3} f^{3} h^{3} + 2 \, d^{3} h^{3} e^{3} - 3 \, {\left (d^{3} f g h^{2} + c d^{2} f h^{3}\right )} e^{2} - 3 \, {\left (d^{3} f^{2} g^{2} h - 4 \, c d^{2} f^{2} g h^{2} + c^{2} d f^{2} h^{3}\right )} e\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + c f h + d h e}{3 \, d f h}\right ) + 3 \, {\left (2 \, C b^{2} d^{2} f^{2} g h + 2 \, C b^{2} d^{2} f h^{2} e + {\left (2 \, C b^{2} c d - 3 \, B b^{2} d^{2}\right )} f^{2} h^{2}\right )} \sqrt {d f h} {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - c d f^{2} g h + c^{2} f^{2} h^{2} + d^{2} h^{2} e^{2} - {\left (d^{2} f g h + c d f h^{2}\right )} e\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, c d^{2} f^{3} g^{2} h - 3 \, c^{2} d f^{3} g h^{2} + 2 \, c^{3} f^{3} h^{3} + 2 \, d^{3} h^{3} e^{3} - 3 \, {\left (d^{3} f g h^{2} + c d^{2} f h^{3}\right )} e^{2} - 3 \, {\left (d^{3} f^{2} g^{2} h - 4 \, c d^{2} f^{2} g h^{2} + c^{2} d f^{2} h^{3}\right )} e\right )}}{27 \, d^{3} f^{3} h^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - c d f^{2} g h + c^{2} f^{2} h^{2} + d^{2} h^{2} e^{2} - {\left (d^{2} f g h + c d f h^{2}\right )} e\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, c d^{2} f^{3} g^{2} h - 3 \, c^{2} d f^{3} g h^{2} + 2 \, c^{3} f^{3} h^{3} + 2 \, d^{3} h^{3} e^{3} - 3 \, {\left (d^{3} f g h^{2} + c d^{2} f h^{3}\right )} e^{2} - 3 \, {\left (d^{3} f^{2} g^{2} h - 4 \, c d^{2} f^{2} g h^{2} + c^{2} d f^{2} h^{3}\right )} e\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + c f h + d h e}{3 \, d f h}\right )\right )\right )}}{9 \, d^{3} f^{3} h^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(3*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)*C*b^2*d^2*f^2*h^2 + (2*C*b^2*d^2*f^2*g^2 + 2*C*b^2*d^2*h^2*e^
2 + (C*b^2*c*d - 3*B*b^2*d^2)*f^2*g*h + (2*C*b^2*c^2 - 3*B*b^2*c*d - 9*(C*a^2 - B*a*b)*d^2)*f^2*h^2 + (C*b^2*d
^2*f*g*h + (C*b^2*c*d - 3*B*b^2*d^2)*f*h^2)*e)*sqrt(d*f*h)*weierstrassPInverse(4/3*(d^2*f^2*g^2 - c*d*f^2*g*h
+ c^2*f^2*h^2 + d^2*h^2*e^2 - (d^2*f*g*h + c*d*f*h^2)*e)/(d^2*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*c*d^2*f^3*g^2
*h - 3*c^2*d*f^3*g*h^2 + 2*c^3*f^3*h^3 + 2*d^3*h^3*e^3 - 3*(d^3*f*g*h^2 + c*d^2*f*h^3)*e^2 - 3*(d^3*f^2*g^2*h
- 4*c*d^2*f^2*g*h^2 + c^2*d*f^2*h^3)*e)/(d^3*f^3*h^3), 1/3*(3*d*f*h*x + d*f*g + c*f*h + d*h*e)/(d*f*h)) + 3*(2
*C*b^2*d^2*f^2*g*h + 2*C*b^2*d^2*f*h^2*e + (2*C*b^2*c*d - 3*B*b^2*d^2)*f^2*h^2)*sqrt(d*f*h)*weierstrassZeta(4/
3*(d^2*f^2*g^2 - c*d*f^2*g*h + c^2*f^2*h^2 + d^2*h^2*e^2 - (d^2*f*g*h + c*d*f*h^2)*e)/(d^2*f^2*h^2), -4/27*(2*
d^3*f^3*g^3 - 3*c*d^2*f^3*g^2*h - 3*c^2*d*f^3*g*h^2 + 2*c^3*f^3*h^3 + 2*d^3*h^3*e^3 - 3*(d^3*f*g*h^2 + c*d^2*f
*h^3)*e^2 - 3*(d^3*f^2*g^2*h - 4*c*d^2*f^2*g*h^2 + c^2*d*f^2*h^3)*e)/(d^3*f^3*h^3), weierstrassPInverse(4/3*(d
^2*f^2*g^2 - c*d*f^2*g*h + c^2*f^2*h^2 + d^2*h^2*e^2 - (d^2*f*g*h + c*d*f*h^2)*e)/(d^2*f^2*h^2), -4/27*(2*d^3*
f^3*g^3 - 3*c*d^2*f^3*g^2*h - 3*c^2*d*f^3*g*h^2 + 2*c^3*f^3*h^3 + 2*d^3*h^3*e^3 - 3*(d^3*f*g*h^2 + c*d^2*f*h^3
)*e^2 - 3*(d^3*f^2*g^2*h - 4*c*d^2*f^2*g*h^2 + c^2*d*f^2*h^3)*e)/(d^3*f^3*h^3), 1/3*(3*d*f*h*x + d*f*g + c*f*h
+ d*h*e)/(d*f*h))))/(d^3*f^3*h^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right ) \left (B b - C a + C b x\right )}{\sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/(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 \begin {gather*} \int \frac {-C\,a^2+B\,a\,b+C\,b^2\,x^2+B\,b^2\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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