3.605 \(\int \frac{(d+e x)^{3/2} \sqrt{f+g x}}{a+c x^2} \, dx\)

Optimal. Leaf size=411 \[ \frac{\left (\frac{a \left (a e^2 g-c d (d g+2 e f)\right )}{\sqrt{c}}-\sqrt{-a} \left (c d^2 f-a e (2 d g+e f)\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{a c \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g}}+\frac{\left (\sqrt{-a} \left (c d^2 f-a e (2 d g+e f)\right )+\frac{a \left (a e^2 g-c d (d g+2 e f)\right )}{\sqrt{c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{a c \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f}}+\frac{e \sqrt{d+e x} \sqrt{f+g x}}{c}+\frac{\sqrt{e} (3 d g+e f) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{c \sqrt{g}} \]

[Out]

(e*Sqrt[d + e*x]*Sqrt[f + g*x])/c + (Sqrt[e]*(e*f + 3*d*g)*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g
*x])])/(c*Sqrt[g]) + (((a*(a*e^2*g - c*d*(2*e*f + d*g)))/Sqrt[c] - Sqrt[-a]*(c*d^2*f - a*e*(e*f + 2*d*g)))*Arc
Tanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[f + g*x])])/(a*c*Sqrt[Sqr
t[c]*d - Sqrt[-a]*e]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]) + (((a*(a*e^2*g - c*d*(2*e*f + d*g)))/Sqrt[c] + Sqrt[-a]*(c
*d^2*f - a*e*(e*f + 2*d*g)))*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e
]*Sqrt[f + g*x])])/(a*c*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[Sqrt[c]*f + Sqrt[-a]*g])

________________________________________________________________________________________

Rubi [A]  time = 2.51139, antiderivative size = 411, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {904, 80, 63, 217, 206, 6725, 93, 208} \[ \frac{\left (\frac{a \left (a e^2 g-c d (d g+2 e f)\right )}{\sqrt{c}}-\sqrt{-a} \left (c d^2 f-a e (2 d g+e f)\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{a c \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g}}+\frac{\left (\sqrt{-a} \left (c d^2 f-a e (2 d g+e f)\right )+\frac{a \left (a e^2 g-c d (d g+2 e f)\right )}{\sqrt{c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{a c \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f}}+\frac{e \sqrt{d+e x} \sqrt{f+g x}}{c}+\frac{\sqrt{e} (3 d g+e f) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{c \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*Sqrt[d + e*x]*Sqrt[f + g*x])/c + (Sqrt[e]*(e*f + 3*d*g)*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g
*x])])/(c*Sqrt[g]) + (((a*(a*e^2*g - c*d*(2*e*f + d*g)))/Sqrt[c] - Sqrt[-a]*(c*d^2*f - a*e*(e*f + 2*d*g)))*Arc
Tanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[f + g*x])])/(a*c*Sqrt[Sqr
t[c]*d - Sqrt[-a]*e]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]) + (((a*(a*e^2*g - c*d*(2*e*f + d*g)))/Sqrt[c] + Sqrt[-a]*(c
*d^2*f - a*e*(e*f + 2*d*g)))*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e
]*Sqrt[f + g*x])])/(a*c*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[Sqrt[c]*f + Sqrt[-a]*g])

Rule 904

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[g/c, Int[Si
mp[2*e*f + d*g + e*g*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 2), x], x] + Dist[1/c, Int[(Simp[c*d*f^2 - 2*a*e*f
*g - a*d*g^2 + (c*e*f^2 + 2*c*d*f*g - a*e*g^2)*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 2))/(a + c*x^2), x], x]
/; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[m, 0] && GtQ
[n, 1]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [B]  time = 6.06434, size = 1587, normalized size = 3.86 \[ \frac{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (\frac{\left (\sqrt{c} d+\sqrt{-a} e\right ) \left (\frac{2 \left (\sqrt{c} f+\sqrt{-a} g\right ) \tan ^{-1}\left (\frac{\sqrt{-\sqrt{c} f-\sqrt{-a} g} \sqrt{d+e x}}{\sqrt{\sqrt{c} d+\sqrt{-a} e} \sqrt{f+g x}}\right )}{\sqrt{c} \sqrt{\sqrt{c} d+\sqrt{-a} e} \sqrt{-\sqrt{c} f-\sqrt{-a} g}}-\frac{2 \sqrt{g} \sqrt{e f-d g} \sqrt{\frac{e}{\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}}} \sqrt{\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}} \sqrt{\frac{e (f+g x)}{e f-d g}} \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g} \sqrt{\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}}}\right )}{\sqrt{c} e^{3/2} \sqrt{f+g x}}\right )}{\sqrt{c}}-\frac{2 \sqrt{d+e x} \sqrt{f+g x} \left (\frac{e g (d+e x)}{(e f-d g) \left (\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}\right )}+1\right )^{3/2} \left (\frac{\sqrt{e f-d g} \sqrt{\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}} \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g} \sqrt{\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}}}\right )}{2 \sqrt{e} \sqrt{g} \sqrt{d+e x} \left (\frac{e g (d+e x)}{(e f-d g) \left (\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}\right )}+1\right )^{3/2}}+\frac{1}{2 \left (\frac{e g (d+e x)}{(e f-d g) \left (\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}\right )}+1\right )}\right )}{\sqrt{c} \sqrt{\frac{e}{\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}}} \sqrt{\frac{e (f+g x)}{e f-d g}}}\right )}{2 a \sqrt{c}}-\frac{\sqrt{-a} \left (\sqrt{-a} e-\sqrt{c} d\right ) \left (\frac{2 \sqrt{d+e x} \sqrt{f+g x} \left (\frac{e g (d+e x)}{(e f-d g) \left (\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}\right )}+1\right )^{3/2} \left (\frac{\sqrt{e f-d g} \sqrt{\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}} \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g} \sqrt{\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}}}\right )}{2 \sqrt{e} \sqrt{g} \sqrt{d+e x} \left (\frac{e g (d+e x)}{(e f-d g) \left (\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}\right )}+1\right )^{3/2}}+\frac{1}{2 \left (\frac{e g (d+e x)}{(e f-d g) \left (\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}\right )}+1\right )}\right )}{\sqrt{c} \sqrt{\frac{e}{\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}}} \sqrt{\frac{e (f+g x)}{e f-d g}}}-\frac{\left (\sqrt{-a} e-\sqrt{c} d\right ) \left (\frac{2 \sqrt{g} \sqrt{e f-d g} \sqrt{\frac{e}{\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}}} \sqrt{\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}} \sqrt{\frac{e (f+g x)}{e f-d g}} \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g} \sqrt{\frac{e^2 f}{e f-d g}-\frac{d e g}{e f-d g}}}\right )}{\sqrt{c} e^{3/2} \sqrt{f+g x}}-\frac{2 \left (\sqrt{-a} g-\sqrt{c} f\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{c} f-\sqrt{-a} g} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e-\sqrt{c} d} \sqrt{f+g x}}\right )}{\sqrt{c} \sqrt{\sqrt{-a} e-\sqrt{c} d} \sqrt{\sqrt{c} f-\sqrt{-a} g}}\right )}{\sqrt{c}}\right )}{2 a \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-a]*(Sqrt[c]*d + Sqrt[-a]*e)*((-2*Sqrt[d + e*x]*Sqrt[f + g*x]*(1 + (e*g*(d + e*x))/((e*f - d*g)*((e^2*f)
/(e*f - d*g) - (d*e*g)/(e*f - d*g))))^(3/2)*(1/(2*(1 + (e*g*(d + e*x))/((e*f - d*g)*((e^2*f)/(e*f - d*g) - (d*
e*g)/(e*f - d*g))))) + (Sqrt[e*f - d*g]*Sqrt[(e^2*f)/(e*f - d*g) - (d*e*g)/(e*f - d*g)]*ArcSinh[(Sqrt[e]*Sqrt[
g]*Sqrt[d + e*x])/(Sqrt[e*f - d*g]*Sqrt[(e^2*f)/(e*f - d*g) - (d*e*g)/(e*f - d*g)])])/(2*Sqrt[e]*Sqrt[g]*Sqrt[
d + e*x]*(1 + (e*g*(d + e*x))/((e*f - d*g)*((e^2*f)/(e*f - d*g) - (d*e*g)/(e*f - d*g))))^(3/2))))/(Sqrt[c]*Sqr
t[e/((e^2*f)/(e*f - d*g) - (d*e*g)/(e*f - d*g))]*Sqrt[(e*(f + g*x))/(e*f - d*g)]) + ((Sqrt[c]*d + Sqrt[-a]*e)*
((-2*Sqrt[g]*Sqrt[e*f - d*g]*Sqrt[e/((e^2*f)/(e*f - d*g) - (d*e*g)/(e*f - d*g))]*Sqrt[(e^2*f)/(e*f - d*g) - (d
*e*g)/(e*f - d*g)]*Sqrt[(e*(f + g*x))/(e*f - d*g)]*ArcSinh[(Sqrt[e]*Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e*f - d*g]*Sq
rt[(e^2*f)/(e*f - d*g) - (d*e*g)/(e*f - d*g)])])/(Sqrt[c]*e^(3/2)*Sqrt[f + g*x]) + (2*(Sqrt[c]*f + Sqrt[-a]*g)
*ArcTan[(Sqrt[-(Sqrt[c]*f) - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[c
]*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[-(Sqrt[c]*f) - Sqrt[-a]*g])))/Sqrt[c]))/(2*a*Sqrt[c]) - (Sqrt[-a]*(-(Sqrt[
c]*d) + Sqrt[-a]*e)*((2*Sqrt[d + e*x]*Sqrt[f + g*x]*(1 + (e*g*(d + e*x))/((e*f - d*g)*((e^2*f)/(e*f - d*g) - (
d*e*g)/(e*f - d*g))))^(3/2)*(1/(2*(1 + (e*g*(d + e*x))/((e*f - d*g)*((e^2*f)/(e*f - d*g) - (d*e*g)/(e*f - d*g)
)))) + (Sqrt[e*f - d*g]*Sqrt[(e^2*f)/(e*f - d*g) - (d*e*g)/(e*f - d*g)]*ArcSinh[(Sqrt[e]*Sqrt[g]*Sqrt[d + e*x]
)/(Sqrt[e*f - d*g]*Sqrt[(e^2*f)/(e*f - d*g) - (d*e*g)/(e*f - d*g)])])/(2*Sqrt[e]*Sqrt[g]*Sqrt[d + e*x]*(1 + (e
*g*(d + e*x))/((e*f - d*g)*((e^2*f)/(e*f - d*g) - (d*e*g)/(e*f - d*g))))^(3/2))))/(Sqrt[c]*Sqrt[e/((e^2*f)/(e*
f - d*g) - (d*e*g)/(e*f - d*g))]*Sqrt[(e*(f + g*x))/(e*f - d*g)]) - ((-(Sqrt[c]*d) + Sqrt[-a]*e)*((2*Sqrt[g]*S
qrt[e*f - d*g]*Sqrt[e/((e^2*f)/(e*f - d*g) - (d*e*g)/(e*f - d*g))]*Sqrt[(e^2*f)/(e*f - d*g) - (d*e*g)/(e*f - d
*g)]*Sqrt[(e*(f + g*x))/(e*f - d*g)]*ArcSinh[(Sqrt[e]*Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e*f - d*g]*Sqrt[(e^2*f)/(e*
f - d*g) - (d*e*g)/(e*f - d*g)])])/(Sqrt[c]*e^(3/2)*Sqrt[f + g*x]) - (2*(-(Sqrt[c]*f) + Sqrt[-a]*g)*ArcTan[(Sq
rt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[c]*Sqrt[-(Sq
rt[c]*d) + Sqrt[-a]*e]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g])))/Sqrt[c]))/(2*a*Sqrt[c])

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Maple [B]  time = 0.493, size = 2497, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}} \sqrt{g x + f}}{c x^{2} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x + d)^(3/2)*sqrt(g*x + f)/(c*x^2 + a), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)^(1/2)/(c*x^2+a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{\frac{3}{2}} \sqrt{f + g x}}{a + c x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)*(g*x+f)**(1/2)/(c*x**2+a),x)

[Out]

Integral((d + e*x)**(3/2)*sqrt(f + g*x)/(a + c*x**2), x)

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Giac [B]  time = 48.7111, size = 4543, normalized size = 11.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(c^2*d^7*g^(13/2)*e^(9/2) - 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^2*d^
5*g^(9/2)*e^(5/2) + 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c^2*d^4*g^(7/2)*
e^(3/2) - (sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*c^2*d^3*g^(5/2)*e^(1/2) + c^
2*d^6*f*g^(11/2)*e^(11/2) - 8*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^2*d^4*
f*g^(7/2)*e^(7/2) + 8*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c^2*d^3*f*g^(5/2
)*e^(5/2) - (sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*c^2*d^2*f*g^(3/2)*e^(3/2)
- 12*c^2*d^5*f^2*g^(9/2)*e^(13/2) - 5*a*c*d^5*g^(13/2)*e^(13/2) - 28*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*
e + d)*g*e - d*g*e + f*e^2))^4*c^2*d^3*f^2*g^(5/2)*e^(9/2) - 36*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d
)*g*e - d*g*e + f*e^2))^4*a*c*d^3*g^(9/2)*e^(9/2) + 12*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d
*g*e + f*e^2))^6*c^2*d^2*f^2*g^(3/2)*e^(7/2) + 12*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e
+ f*e^2))^6*a*c*d^2*g^(7/2)*e^(7/2) - 2*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^
8*c^2*d*f^2*sqrt(g)*e^(5/2) - 3*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*a*c*d*
g^(5/2)*e^(5/2) + 18*c^2*d^4*f^3*g^(7/2)*e^(15/2) + 17*a*c*d^4*f*g^(11/2)*e^(15/2) - 16*(sqrt(x*e + d)*sqrt(g)
*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^2*d^2*f^3*g^(3/2)*e^(11/2) - 32*(sqrt(x*e + d)*sqrt(g)*e^(
1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*a*c*d^2*f*g^(7/2)*e^(11/2) + 8*(sqrt(x*e + d)*sqrt(g)*e^(1/2) -
sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c^2*d*f^3*sqrt(g)*e^(9/2) + 16*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x
*e + d)*g*e - d*g*e + f*e^2))^6*a*c*d*f*g^(5/2)*e^(9/2) - (sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e
- d*g*e + f*e^2))^8*a*c*f*g^(3/2)*e^(7/2) - 7*c^2*d^3*f^4*g^(5/2)*e^(17/2) - 18*a*c*d^3*f^2*g^(9/2)*e^(17/2) -
8*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^2*d*f^4*sqrt(g)*e^(13/2) - 52*(sq
rt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*a*c*d*f^2*g^(5/2)*e^(13/2) - 48*(sqrt(x*e
+ d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*a^2*d*g^(9/2)*e^(13/2) + 4*(sqrt(x*e + d)*sqrt(
g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*a*c*f^2*g^(3/2)*e^(11/2) - 3*c^2*d^2*f^5*g^(3/2)*e^(19/2)
+ 2*a*c*d^2*f^3*g^(7/2)*e^(19/2) - 8*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*a
*c*f^3*g^(3/2)*e^(15/2) - 16*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*a^2*f*g^(
7/2)*e^(15/2) + 2*c^2*d*f^6*sqrt(g)*e^(21/2) + 7*a*c*d*f^4*g^(5/2)*e^(21/2) - 3*a*c*f^5*g^(3/2)*e^(23/2))*log(
abs(c*d^4*g^4*e^4 - 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^2*c*d^3*g^3*e^3 +
6*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c*d^2*g^2*e^2 - 4*(sqrt(x*e + d)*sqr
t(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c*d*g*e + (sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d
)*g*e - d*g*e + f*e^2))^8*c - 4*c*d^3*f*g^3*e^5 + 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*
e + f*e^2))^2*c*d^2*f*g^2*e^4 + 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c*d*
f*g*e^3 - 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c*f*e^2 + 6*c*d^2*f^2*g^2*
e^6 + 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^2*c*d*f^2*g*e^5 + 6*(sqrt(x*e +
d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c*f^2*e^4 + 16*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sq
rt((x*e + d)*g*e - d*g*e + f*e^2))^4*a*g^2*e^4 - 4*c*d*f^3*g*e^7 - 4*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*
e + d)*g*e - d*g*e + f*e^2))^2*c*f^3*e^6 + c*f^4*e^8))/(c^3*d^6*g^6*e^4 - 6*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - s
qrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^3*d^4*g^4*e^2 + 8*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e
- d*g*e + f*e^2))^6*c^3*d^3*g^3*e - 3*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*
c^3*d^2*g^2 + 2*c^3*d^5*f*g^5*e^5 - 24*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4
*c^3*d^3*f*g^3*e^3 + 24*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c^3*d^2*f*g^2*
e^2 - 2*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*c^3*d*f*g*e - 17*c^3*d^4*f^2*g
^4*e^6 - 8*a*c^2*d^4*g^6*e^6 - 68*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^3*
d^2*f^2*g^2*e^4 - 96*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*a*c^2*d^2*g^4*e^4
+ 24*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c^3*d*f^2*g*e^3 + 32*(sqrt(x*e +
d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*a*c^2*d*g^3*e^3 - 3*(sqrt(x*e + d)*sqrt(g)*e^(1/2
) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*c^3*f^2*e^2 - 8*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e
- d*g*e + f*e^2))^8*a*c^2*g^2*e^2 + 28*c^3*d^3*f^3*g^3*e^7 + 32*a*c^2*d^3*f*g^5*e^7 - 24*(sqrt(x*e + d)*sqrt(
g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c^3*d*f^3*g*e^5 - 64*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt
((x*e + d)*g*e - d*g*e + f*e^2))^4*a*c^2*d*f*g^3*e^5 + 8*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e -
d*g*e + f*e^2))^6*c^3*f^3*e^4 + 32*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*a*
c^2*f*g^2*e^4 - 17*c^3*d^2*f^4*g^2*e^8 - 48*a*c^2*d^2*f^2*g^4*e^8 - 6*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x
*e + d)*g*e - d*g*e + f*e^2))^4*c^3*f^4*e^6 - 96*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e +
f*e^2))^4*a*c^2*f^2*g^2*e^6 - 128*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*a^2
*c*g^4*e^6 + 2*c^3*d*f^5*g*e^9 + 32*a*c^2*d*f^3*g^3*e^9 + c^3*f^6*e^10 - 8*a*c^2*f^4*g^2*e^10) + sqrt((x*e + d
)*g*e - d*g*e + f*e^2)*sqrt(x*e + d)/c - 1/2*(3*d*g^(3/2)*e^(1/2) + f*sqrt(g)*e^(3/2))*log((sqrt(x*e + d)*sqrt
(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^2)/(c*g)