3.1.23 \(\int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [23]

Optimal. Leaf size=436 \[ \frac {2 (b B-2 a C) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} F\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {b g-a h} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {2 C \sqrt {-d g+c h} (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac {b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{\sqrt {b c-a d} h \sqrt {c+d x} \sqrt {e+f x}} \]

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

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Rubi [A]
time = 0.35, antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {24, 1612, 176, 430, 171, 551} \begin {gather*} \frac {2 \sqrt {g+h x} (b B-2 a C) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} 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 )}{\sqrt {c+d x} \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac {2 C (a+b x) \sqrt {c h-d g} \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt {\frac {(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \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 )}{h \sqrt {c+d x} \sqrt {e+f x} \sqrt {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)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 23.88, size = 583, normalized size = 1.34 \begin {gather*} \frac {2 (a+b x)^{3/2} \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \left (-\frac {b B \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} (g+h x) F\left (\sin ^{-1}\left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right )|\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )}{(b g-a h) (a+b x) \sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {2 a C \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} (g+h x) F\left (\sin ^{-1}\left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right )|\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )}{(-b g+a h) (a+b x) \sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {C (-f g+e h) \sqrt {-\frac {(b e-a f) (b g-a h) (e+f x) (g+h x)}{(f g-e h)^2 (a+b x)^2}} \Pi \left (\frac {b (-f g+e h)}{(b e-a f) h};\sin ^{-1}\left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right )|\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )}{(b e-a f) h}\right )}{\sqrt {c+d x} \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)/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(2*(a + b*x)^(3/2)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*(-((b*B*Sqrt[((b*g - a*h)*(e + f*x))/
((f*g - e*h)*(a + b*x))]*(g + h*x)*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]],
((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))])/((b*g - a*h)*(a + b*x)*Sqrt[((-(b*e) + a*f)*(g +
h*x))/((f*g - e*h)*(a + b*x))])) - (2*a*C*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]*(g + h*x)*Elli
pticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e
- a*f)*(d*g - c*h))])/((-(b*g) + a*h)*(a + b*x)*Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]) + (C
*(-(f*g) + e*h)*Sqrt[-(((b*e - a*f)*(b*g - a*h)*(e + f*x)*(g + h*x))/((f*g - e*h)^2*(a + b*x)^2))]*EllipticPi[
(b*(-(f*g) + e*h))/((b*e - a*f)*h), ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c)
+ a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))])/((b*e - a*f)*h)))/(Sqrt[c + d*x]*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. \(2954\) vs. \(2(398)=796\).
time = 0.12, size = 2955, normalized size = 6.78

method	result	size
elliptic	\(\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 \left (B b -a C \right ) \left (-\frac {a}{b}+\frac {g}{h}\right ) \sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \left (x +\frac {c}{d}\right )^{2} \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {e}{f}\right )}{\left (-\frac {e}{f}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {g}{h}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}, \sqrt {\frac {\left (-\frac {c}{d}+\frac {e}{f}\right ) \left (-\frac {a}{b}+\frac {g}{h}\right )}{\left (-\frac {a}{b}+\frac {e}{f}\right ) \left (-\frac {c}{d}+\frac {g}{h}\right )}}\right )}{\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (-\frac {c}{d}+\frac {a}{b}\right ) \sqrt {b d f h \left (x +\frac {a}{b}\right ) \left (x +\frac {c}{d}\right ) \left (x +\frac {e}{f}\right ) \left (x +\frac {g}{h}\right )}}+\frac {2 C b \left (-\frac {a}{b}+\frac {g}{h}\right ) \sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \left (x +\frac {c}{d}\right )^{2} \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {e}{f}\right )}{\left (-\frac {e}{f}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {g}{h}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \left (-\frac {c \EllipticF \left (\sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}, \sqrt {\frac {\left (-\frac {c}{d}+\frac {e}{f}\right ) \left (-\frac {a}{b}+\frac {g}{h}\right )}{\left (-\frac {a}{b}+\frac {e}{f}\right ) \left (-\frac {c}{d}+\frac {g}{h}\right )}}\right )}{d}+\left (-\frac {a}{b}+\frac {c}{d}\right ) \EllipticPi \left (\sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}, \frac {-\frac {g}{h}+\frac {a}{b}}{-\frac {g}{h}+\frac {c}{d}}, \sqrt {\frac {\left (-\frac {c}{d}+\frac {e}{f}\right ) \left (-\frac {a}{b}+\frac {g}{h}\right )}{\left (-\frac {a}{b}+\frac {e}{f}\right ) \left (-\frac {c}{d}+\frac {g}{h}\right )}}\right )\right )}{\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (-\frac {c}{d}+\frac {a}{b}\right ) \sqrt {b d f h \left (x +\frac {a}{b}\right ) \left (x +\frac {c}{d}\right ) \left (x +\frac {e}{f}\right ) \left (x +\frac {g}{h}\right )}}\right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\)	\(856\)
default	\(\text {Expression too large to display}\)	\(2955\)

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)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,method=_RETURNVE
RBOSE)

[Out]

-2/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)/d*((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2)*((a*d
-b*c)*(f*x+e)/(a*f-b*e)/(d*x+c))^(1/2)*((a*d-b*c)*(h*x+g)/(a*h-b*g)/(d*x+c))^(1/2)*(-B*EllipticF(((c*h-d*g)*(b
*x+a)/(a*h-b*g)/(d*x+c))^(1/2),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*b^2*d^3*g*x^2-C*EllipticF(((c*
h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*a^2*d^3*h*x^2-C*Ellip
ticPi(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),(a*h-b*g)*d/b/(c*h-d*g),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-
d*g))^(1/2))*a^2*d^3*h*x^2-B*EllipticF(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),((c*f-d*e)*(a*h-b*g)/(a*f-b
*e)/(c*h-d*g))^(1/2))*b^2*c^2*d*g-C*EllipticF(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),((c*f-d*e)*(a*h-b*g)
/(a*f-b*e)/(c*h-d*g))^(1/2))*a^2*c^2*d*h-C*EllipticF(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),((c*f-d*e)*(a
*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*a*b*c^3*h-C*EllipticPi(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),(a*h-b*
g)*d/b/(c*h-d*g),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*a^2*c^2*d*h+C*EllipticPi(((c*h-d*g)*(b*x+a)/
(a*h-b*g)/(d*x+c))^(1/2),(a*h-b*g)*d/b/(c*h-d*g),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*a*b*c^3*h-2*
B*EllipticF(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*b^2*c
*d^2*g*x-2*C*EllipticF(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(
1/2))*a^2*c*d^2*h*x+2*C*EllipticF(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(
c*h-d*g))^(1/2))*b^2*c^2*d*g*x-2*C*EllipticPi(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),(a*h-b*g)*d/b/(c*h-d
*g),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*a^2*c*d^2*h*x-2*C*EllipticPi(((c*h-d*g)*(b*x+a)/(a*h-b*g)
/(d*x+c))^(1/2),(a*h-b*g)*d/b/(c*h-d*g),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*b^2*c^2*d*g*x+B*Ellip
ticF(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*a*b*c^2*d*h+
C*EllipticF(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*a*b*c
^2*d*g+C*EllipticPi(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),(a*h-b*g)*d/b/(c*h-d*g),((c*f-d*e)*(a*h-b*g)/(
a*f-b*e)/(c*h-d*g))^(1/2))*a*b*c^2*d*g+B*EllipticF(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),((c*f-d*e)*(a*h
-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*a*b*d^3*h*x^2+C*EllipticF(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),((c*f-
d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*a*b*d^3*g*x^2+C*EllipticF(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/
2),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*b^2*c*d^2*g*x^2+C*EllipticPi(((c*h-d*g)*(b*x+a)/(a*h-b*g)/
(d*x+c))^(1/2),(a*h-b*g)*d/b/(c*h-d*g),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*a*b*d^3*g*x^2-C*Ellipt
icPi(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),(a*h-b*g)*d/b/(c*h-d*g),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d
*g))^(1/2))*b^2*c*d^2*g*x^2-C*EllipticF(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),((c*f-d*e)*(a*h-b*g)/(a*f-
b*e)/(c*h-d*g))^(1/2))*a*b*c*d^2*h*x^2+C*EllipticPi(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),(a*h-b*g)*d/b/
(c*h-d*g),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*a*b*c*d^2*h*x^2+2*B*EllipticF(((c*h-d*g)*(b*x+a)/(a
*h-b*g)/(d*x+c))^(1/2),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*a*b*c*d^2*h*x-2*C*EllipticF(((c*h-d*g)
*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*a*b*c^2*d*h*x+2*C*EllipticF
(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*a*b*c*d^2*g*x+2*
C*EllipticPi(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),(a*h-b*g)*d/b/(c*h-d*g),((c*f-d*e)*(a*h-b*g)/(a*f-b*e
)/(c*h-d*g))^(1/2))*a*b*c^2*d*h*x+2*C*EllipticPi(((c*h-d*g)*(b*x+a)/(a*h-b*g)/(d*x+c))^(1/2),(a*h-b*g)*d/b/(c*
h-d*g),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*a*b*c*d^2*g*x+C*EllipticF(((c*h-d*g)*(b*x+a)/(a*h-b*g)
/(d*x+c))^(1/2),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*b^2*c^3*g-C*EllipticPi(((c*h-d*g)*(b*x+a)/(a*
h-b*g)/(d*x+c))^(1/2),(a*h-b*g)*d/b/(c*h-d*g),((c*f-d*e)*(a*h-b*g)/(a*f-b*e)/(c*h-d*g))^(1/2))*b^2*c^3*g)/(c*h
-d*g)/(a*d-b*c)

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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)/(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*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((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*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(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*b**2*x**2+B*b**2*x+B*a*b-C*a**2)/(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: RuntimeError} \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)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm
="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:index.cc index_m
operator + Error: Bad Argument Valueindex.cc index_m operator + Error: Bad Argument Valueintegrate(1/abs(sageV
ARb)*sageVARb^

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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}\,{\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((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)^(3/2)*(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)^(3/2)*(c + d*x)^(1/2)), x
)

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