3.884 \(\int \frac{1}{(d+e x) (f+g x)^2 (a+b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=642 \[ \frac{g^2 \sqrt{a+b x+c x^2} \left (-4 c g (2 a g+b f)+3 b^2 g^2+4 c^2 f^2\right )}{\left (b^2-4 a c\right ) (f+g x) (e f-d g) \left (a g^2-b f g+c f^2\right )^2}-\frac{2 e^2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (e f-d g)^2 \left (a e^2-b d e+c d^2\right )}+\frac{2 e g \left (2 a c g+b^2 (-g)+c x (2 c f-b g)+b c f\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (e f-d g)^2 \left (a g^2-b f g+c f^2\right )}+\frac{2 g \left (2 a c g+b^2 (-g)+c x (2 c f-b g)+b c f\right )}{\left (b^2-4 a c\right ) (f+g x) \sqrt{a+b x+c x^2} (e f-d g) \left (a g^2-b f g+c f^2\right )}+\frac{e^4 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{(e f-d g)^2 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{e g^3 \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{(e f-d g)^2 \left (a g^2-b f g+c f^2\right )^{3/2}}-\frac{3 g^3 (2 c f-b g) \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{2 (e f-d g) \left (a g^2-b f g+c f^2\right )^{5/2}} \]

[Out]

(-2*e^2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2*Sq
rt[a + b*x + c*x^2]) + (2*e*g*(b*c*f - b^2*g + 2*a*c*g + c*(2*c*f - b*g)*x))/((b^2 - 4*a*c)*(e*f - d*g)^2*(c*f
^2 - b*f*g + a*g^2)*Sqrt[a + b*x + c*x^2]) + (2*g*(b*c*f - b^2*g + 2*a*c*g + c*(2*c*f - b*g)*x))/((b^2 - 4*a*c
)*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*(f + g*x)*Sqrt[a + b*x + c*x^2]) + (g^2*(4*c^2*f^2 + 3*b^2*g^2 - 4*c*g*(
b*f + 2*a*g))*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^2*(f + g*x)) + (e^4*Ar
cTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/((c*d^2 - b*d*e
+ a*e^2)^(3/2)*(e*f - d*g)^2) - (3*g^3*(2*c*f - b*g)*ArcTanh[(b*f - 2*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b
*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/(2*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^(5/2)) - (e*g^3*ArcTanh[(b*f - 2
*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/((e*f - d*g)^2*(c*f^2 - b*f*g
+ a*g^2)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.910685, antiderivative size = 642, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {960, 740, 12, 724, 206, 806} \[ \frac{g^2 \sqrt{a+b x+c x^2} \left (-4 c g (2 a g+b f)+3 b^2 g^2+4 c^2 f^2\right )}{\left (b^2-4 a c\right ) (f+g x) (e f-d g) \left (a g^2-b f g+c f^2\right )^2}-\frac{2 e^2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (e f-d g)^2 \left (a e^2-b d e+c d^2\right )}+\frac{2 e g \left (2 a c g+b^2 (-g)+c x (2 c f-b g)+b c f\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (e f-d g)^2 \left (a g^2-b f g+c f^2\right )}+\frac{2 g \left (2 a c g+b^2 (-g)+c x (2 c f-b g)+b c f\right )}{\left (b^2-4 a c\right ) (f+g x) \sqrt{a+b x+c x^2} (e f-d g) \left (a g^2-b f g+c f^2\right )}+\frac{e^4 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{(e f-d g)^2 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{e g^3 \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{(e f-d g)^2 \left (a g^2-b f g+c f^2\right )^{3/2}}-\frac{3 g^3 (2 c f-b g) \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{2 (e f-d g) \left (a g^2-b f g+c f^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(f + g*x)^2*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(-2*e^2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2*Sq
rt[a + b*x + c*x^2]) + (2*e*g*(b*c*f - b^2*g + 2*a*c*g + c*(2*c*f - b*g)*x))/((b^2 - 4*a*c)*(e*f - d*g)^2*(c*f
^2 - b*f*g + a*g^2)*Sqrt[a + b*x + c*x^2]) + (2*g*(b*c*f - b^2*g + 2*a*c*g + c*(2*c*f - b*g)*x))/((b^2 - 4*a*c
)*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*(f + g*x)*Sqrt[a + b*x + c*x^2]) + (g^2*(4*c^2*f^2 + 3*b^2*g^2 - 4*c*g*(
b*f + 2*a*g))*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^2*(f + g*x)) + (e^4*Ar
cTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/((c*d^2 - b*d*e
+ a*e^2)^(3/2)*(e*f - d*g)^2) - (3*g^3*(2*c*f - b*g)*ArcTanh[(b*f - 2*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b
*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/(2*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^(5/2)) - (e*g^3*ArcTanh[(b*f - 2
*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/((e*f - d*g)^2*(c*f^2 - b*f*g
+ a*g^2)^(3/2))

Rule 960

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (ILtQ[m, 0] &&
ILtQ[n, 0])) &&  !(IGtQ[m, 0] || IGtQ[n, 0])

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, 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 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 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 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x) (f+g x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx &=\int \left (\frac{e^2}{(e f-d g)^2 (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac{g}{(e f-d g) (f+g x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{e g}{(e f-d g)^2 (f+g x) \left (a+b x+c x^2\right )^{3/2}}\right ) \, dx\\ &=\frac{e^2 \int \frac{1}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx}{(e f-d g)^2}-\frac{(e g) \int \frac{1}{(f+g x) \left (a+b x+c x^2\right )^{3/2}} \, dx}{(e f-d g)^2}-\frac{g \int \frac{1}{(f+g x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx}{e f-d g}\\ &=-\frac{2 e^2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt{a+b x+c x^2}}+\frac{2 e g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt{a+b x+c x^2}}+\frac{2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x) \sqrt{a+b x+c x^2}}-\frac{\left (2 e^2\right ) \int -\frac{\left (b^2-4 a c\right ) e^2}{2 (d+e x) \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (e f-d g)^2}+\frac{(2 e g) \int -\frac{\left (b^2-4 a c\right ) g^2}{2 (f+g x) \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right )}+\frac{(2 g) \int \frac{\frac{1}{2} g \left (2 b c f-3 b^2 g+8 a c g\right )+c g (2 c f-b g) x}{(f+g x)^2 \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )}\\ &=-\frac{2 e^2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt{a+b x+c x^2}}+\frac{2 e g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt{a+b x+c x^2}}+\frac{2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x) \sqrt{a+b x+c x^2}}+\frac{g^2 \left (4 c^2 f^2+3 b^2 g^2-4 c g (b f+2 a g)\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )^2 (f+g x)}+\frac{e^4 \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{\left (c d^2-b d e+a e^2\right ) (e f-d g)^2}-\frac{\left (3 g^3 (2 c f-b g)\right ) \int \frac{1}{(f+g x) \sqrt{a+b x+c x^2}} \, dx}{2 (e f-d g) \left (c f^2-b f g+a g^2\right )^2}-\frac{\left (e g^3\right ) \int \frac{1}{(f+g x) \sqrt{a+b x+c x^2}} \, dx}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right )}\\ &=-\frac{2 e^2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt{a+b x+c x^2}}+\frac{2 e g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt{a+b x+c x^2}}+\frac{2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x) \sqrt{a+b x+c x^2}}+\frac{g^2 \left (4 c^2 f^2+3 b^2 g^2-4 c g (b f+2 a g)\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )^2 (f+g x)}-\frac{\left (2 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x}{\sqrt{a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right ) (e f-d g)^2}+\frac{\left (3 g^3 (2 c f-b g)\right ) \operatorname{Subst}\left (\int \frac{1}{4 c f^2-4 b f g+4 a g^2-x^2} \, dx,x,\frac{-b f+2 a g-(2 c f-b g) x}{\sqrt{a+b x+c x^2}}\right )}{(e f-d g) \left (c f^2-b f g+a g^2\right )^2}+\frac{\left (2 e g^3\right ) \operatorname{Subst}\left (\int \frac{1}{4 c f^2-4 b f g+4 a g^2-x^2} \, dx,x,\frac{-b f+2 a g-(2 c f-b g) x}{\sqrt{a+b x+c x^2}}\right )}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right )}\\ &=-\frac{2 e^2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt{a+b x+c x^2}}+\frac{2 e g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt{a+b x+c x^2}}+\frac{2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x) \sqrt{a+b x+c x^2}}+\frac{g^2 \left (4 c^2 f^2+3 b^2 g^2-4 c g (b f+2 a g)\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )^2 (f+g x)}+\frac{e^4 \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{3/2} (e f-d g)^2}-\frac{3 g^3 (2 c f-b g) \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b f g+a g^2} \sqrt{a+b x+c x^2}}\right )}{2 (e f-d g) \left (c f^2-b f g+a g^2\right )^{5/2}}-\frac{e g^3 \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b f g+a g^2} \sqrt{a+b x+c x^2}}\right )}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 6.21851, size = 868, normalized size = 1.35 \[ \frac{\left (c x^2+b x+a\right )^{3/2} \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-e (b d-a e)} \sqrt{c x^2+b x+a}}\right ) e^4}{\left (c d^2-e (b d-a e)\right )^{3/2} (e f-d g)^2 (a+x (b+c x))^{3/2}}-\frac{2 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right ) \left (c x^2+b x+a\right ) e^2}{\left (b^2-4 a c\right ) \left (c d^2-e (b d-a e)\right ) (e f-d g)^2 (a+x (b+c x))^{3/2}}-\frac{g^3 \left (c x^2+b x+a\right )^{3/2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-g (b f-a g)} \sqrt{c x^2+b x+a}}\right ) e}{(e f-d g)^2 \left (c f^2-g (b f-a g)\right )^{3/2} (a+x (b+c x))^{3/2}}+\frac{2 g \left (-g b^2+c f b+2 a c g+c (2 c f-b g) x\right ) \left (c x^2+b x+a\right ) e}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-g (b f-a g)\right ) (a+x (b+c x))^{3/2}}-\frac{2 g \left (c x^2+b x+a\right )^{3/2} \left (\frac{2 \left (b \left (\frac{1}{2} \left (-3 g b^2+2 c f b+8 a c g\right ) g^2+c f (2 c f-b g) g\right )-2 \left (a c (2 c f-b g) g^2+\frac{1}{2} c f \left (-3 g b^2+2 c f b+8 a c g\right ) g\right )\right ) \tanh ^{-1}\left (\frac{-b f+2 a g-(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right )}{\sqrt{c f^2-b g f+a g^2} \left (4 c f^2-4 b g f+4 a g^2\right )}-\frac{\left (\frac{1}{2} g^2 \left (-3 g b^2+2 c f b+8 a c g\right )-c f g (2 c f-b g)\right ) \sqrt{c x^2+b x+a}}{\left (c f^2-b g f+a g^2\right ) (f+g x)}\right )}{\left (b^2-4 a c\right ) (d g-e f) \left (c f^2-b g f+a g^2\right ) (a+x (b+c x))^{3/2}}+\frac{2 g \left (-g b^2+c f b+2 a c g+c (2 c f-b g) x\right ) \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-g (b f-a g)\right ) (f+g x) (a+x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(f + g*x)^2*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(-2*e^2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2))/((b^2 - 4*a*c)*(c*d^2 - e*(b*d - a*e)
)*(e*f - d*g)^2*(a + x*(b + c*x))^(3/2)) + (2*e*g*(b*c*f - b^2*g + 2*a*c*g + c*(2*c*f - b*g)*x)*(a + b*x + c*x
^2))/((b^2 - 4*a*c)*(e*f - d*g)^2*(c*f^2 - g*(b*f - a*g))*(a + x*(b + c*x))^(3/2)) + (2*g*(b*c*f - b^2*g + 2*a
*c*g + c*(2*c*f - b*g)*x)*(a + b*x + c*x^2))/((b^2 - 4*a*c)*(e*f - d*g)*(c*f^2 - g*(b*f - a*g))*(f + g*x)*(a +
x*(b + c*x))^(3/2)) + (e^4*(a + b*x + c*x^2)^(3/2)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - e*
(b*d - a*e)]*Sqrt[a + b*x + c*x^2])])/((c*d^2 - e*(b*d - a*e))^(3/2)*(e*f - d*g)^2*(a + x*(b + c*x))^(3/2)) -
(2*g*(a + b*x + c*x^2)^(3/2)*(-(((-(c*f*g*(2*c*f - b*g)) + (g^2*(2*b*c*f - 3*b^2*g + 8*a*c*g))/2)*Sqrt[a + b*x
+ c*x^2])/((c*f^2 - b*f*g + a*g^2)*(f + g*x))) + (2*(-2*(a*c*g^2*(2*c*f - b*g) + (c*f*g*(2*b*c*f - 3*b^2*g +
8*a*c*g))/2) + b*(c*f*g*(2*c*f - b*g) + (g^2*(2*b*c*f - 3*b^2*g + 8*a*c*g))/2))*ArcTanh[(-(b*f) + 2*a*g - (2*c
*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c*f^2 - b*f*g + a*g^2]*(4*c*f^2 - 4
*b*f*g + 4*a*g^2))))/((b^2 - 4*a*c)*(-(e*f) + d*g)*(c*f^2 - b*f*g + a*g^2)*(a + x*(b + c*x))^(3/2)) - (e*g^3*(
a + b*x + c*x^2)^(3/2)*ArcTanh[(b*f - 2*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - g*(b*f - a*g)]*Sqrt[a + b*x + c
*x^2])])/((e*f - d*g)^2*(c*f^2 - g*(b*f - a*g))^(3/2)*(a + x*(b + c*x))^(3/2))

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Maple [B]  time = 0.416, size = 2807, normalized size = 4.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^3/(d*g-e*f)^2/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)
/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)
)/(d/e+x))-g/(d*g-e*f)/(a*g^2-b*f*g+c*f^2)/(x+f/g)/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)
^(1/2)+2*e/(d*g-e*f)^2*g^2/(a*g^2-b*f*g+c*f^2)/(4*a*c-b^2)/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f
^2)/g^2)^(1/2)*x*b*c-4*e/(d*g-e*f)^2*g/(a*g^2-b*f*g+c*f^2)/(4*a*c-b^2)/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g
^2-b*f*g+c*f^2)/g^2)^(1/2)*x*c^2*f-12*g^2/(d*g-e*f)/(a*g^2-b*f*g+c*f^2)^2/(4*a*c-b^2)/((x+f/g)^2*c+(b*g-2*c*f)
/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*x*b*c^2*f+6*g/(d*g-e*f)/(a*g^2-b*f*g+c*f^2)^2/(4*a*c-b^2)/((x+f/g)^2
*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*b*c^2*f^2+2*e^2/(d*g-e*f)^2/(a*e^2-b*d*e+c*d^2)/(4*a*c
-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c*d+3*g^3/(d*g-e*f)/(a*g^2-b*f*g+c*f
^2)^2/(4*a*c-b^2)/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*x*b^2*c+12*g/(d*g-e*f)/(a*
g^2-b*f*g+c*f^2)^2/(4*a*c-b^2)/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*x*c^3*f^2-6*g
^2/(d*g-e*f)/(a*g^2-b*f*g+c*f^2)^2/(4*a*c-b^2)/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/
2)*b^2*c*f-2*e^3/(d*g-e*f)^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c
*d^2)/e^2)^(1/2)*x*b*c+4*e^2/(d*g-e*f)^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a
*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^2*d-3/2*g^3/(d*g-e*f)/(a*g^2-b*f*g+c*f^2)^2/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g
)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*b-e/(d*g-e*f)^2/(a*g^2-b*f*g+c*f^2)*g^2/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a
*g^2-b*f*g+c*f^2)/g^2)^(1/2)-2*e/(d*g-e*f)^2*g/(a*g^2-b*f*g+c*f^2)/(4*a*c-b^2)/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f
/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*b*c*f+e^3/(d*g-e*f)^2/(a*e^2-b*d*e+c*d^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x
)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+e/(d*g-e*f)^2*g^2/(a*g^2-b*f*g+c*f^2)/(4*a*c-b^2)/((x+f/g)^2*c+(b*g-2*c*f)/g*
(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*b^2-3*g^2/(d*g-e*f)/(a*g^2-b*f*g+c*f^2)^2/((a*g^2-b*f*g+c*f^2)/g^2)^(1/
2)*ln((2*(a*g^2-b*f*g+c*f^2)/g^2+(b*g-2*c*f)/g*(x+f/g)+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2*c
*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))/(x+f/g))*c*f-8*g/(d*g-e*f)*c^2/(a*g^2-b*f*g+c*f^2)/(4*a*c-b^2)/(
(x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*x-4*g/(d*g-e*f)*c/(a*g^2-b*f*g+c*f^2)/(4*a*c-
b^2)/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*b-e^3/(d*g-e*f)^2/(a*e^2-b*d*e+c*d^2)/(
4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2+3*g^2/(d*g-e*f)/(a*g^2-b*f*g+
c*f^2)^2/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*c*f+3/2*g^3/(d*g-e*f)/(a*g^2-b*f*g+
c*f^2)^2/(4*a*c-b^2)/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*b^3+3/2*g^3/(d*g-e*f)/(
a*g^2-b*f*g+c*f^2)^2/((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*ln((2*(a*g^2-b*f*g+c*f^2)/g^2+(b*g-2*c*f)/g*(x+f/g)+2*((a
*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))/(x+f/g))*b+e/(
d*g-e*f)^2/(a*g^2-b*f*g+c*f^2)*g^2/((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*ln((2*(a*g^2-b*f*g+c*f^2)/g^2+(b*g-2*c*f)/g
*(x+f/g)+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))/
(x+f/g))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}{\left (g x + f\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}{\left (g x + f\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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