3.1.19 \(\int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [19]

Optimal. Leaf size=309 \[ \frac {2 C \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 (b B-2 a C) \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}} \]

[Out]

2*C*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/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)-2*(B*b-2*C*a)*Ellipti
cPi(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)/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(
1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.53, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {24, 1621, 175, 552, 551, 12, 122, 121} \begin {gather*} \frac {2 C \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 )}{d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 (b B-2 a C) \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};\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 )}{\sqrt {f} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

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 175

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 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 552

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

Rule 1621

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

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 20.97, size = 249, normalized size = 0.81 \begin {gather*} \frac {2 i \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \left (-\left ((b c C-b B d+a C d) 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 )\right )+(-b B+2 a C) d \Pi \left (-\frac {b c f-a d f}{b d e-b c f};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 )\right )}{(-b c+a d) \sqrt {-c+\frac {d e}{f}} f \sqrt {\frac {d (e+f x)}{f (c+d 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)/((a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(665\) vs. \(2(275)=550\).
time = 0.12, size = 666, normalized size = 2.16

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 C \left (-\frac {e}{f}+\frac {g}{h}\right ) \sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 \left (B b -2 a C \right ) \left (-\frac {e}{f}+\frac {g}{h}\right ) \sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {a}{b}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{b \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}\, \left (-\frac {g}{h}+\frac {a}{b}\right )}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\)	\(475\)
default	\(-\frac {2 \sqrt {h x +g}\, \sqrt {f x +e}\, \sqrt {d x +c}\, \sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}\, \sqrt {\frac {\left (d x +c \right ) h}{c h -d g}}\, \sqrt {\frac {\left (f x +e \right ) h}{e h -f g}}\, \left (B \EllipticPi \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \frac {\left (e h -f g \right ) b}{f \left (a h -b g \right )}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) b e \,h^{2}-B \EllipticPi \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \frac {\left (e h -f g \right ) b}{f \left (a h -b g \right )}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) b f g h +C \EllipticF \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) a e \,h^{2}-C \EllipticF \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) a f g h -C \EllipticF \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) b e g h +C \EllipticF \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) b f \,g^{2}-2 C \EllipticPi \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \frac {\left (e h -f g \right ) b}{f \left (a h -b g \right )}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) a e \,h^{2}+2 C \EllipticPi \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \frac {\left (e h -f g \right ) b}{f \left (a h -b g \right )}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) a f g h \right )}{f h \left (a h -b g \right ) \left (d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g \right )}\)	\(666\)

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)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,method=_RETURNVERBOS
E)

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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}\,{\left (a+b\,x\right )}^2\,\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)*(a + b*x)^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)*(a + b*x)^2*(c + d*x)^(1/2)), x)

________________________________________________________________________________________