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

Optimal. Leaf size=644 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (2 c e g \left (a^2 e^2 g^2+a b e g (d g+e f)-b^2 (d g+e f)^2\right )+b^2 e^2 g^2 (-2 a e g+b d g+b e f)-c^2 \left (4 a d e^2 f g^2-b \left (5 d^2 e f g^2+d^3 g^3+5 d e^2 f^2 g+e^3 f^3\right )\right )-2 c^3 d f \left (d^2 g^2+d e f g+e^2 f^2\right )\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )^2 \left (c f^2-g (b f-a g)\right )^2}-\frac{c x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )+b c (c d f-3 a e g)+2 a c^2 (d g+e f)-b^2 c (d g+e f)+b^3 e g}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right ) \left (c f^2-g (b f-a g)\right )}+\frac{2 c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right ) \left (c f^2-g (b f-a g)\right )}-\frac{\log \left (a+b x+c x^2\right ) (-b e g+c d g+c e f) \left (e g (2 a e g-b (d g+e f))+c \left (d^2 g^2+e^2 f^2\right )\right )}{2 \left (a e^2-b d e+c d^2\right )^2 \left (c f^2-g (b f-a g)\right )^2}+\frac{e^4 \log (d+e x)}{(e f-d g) \left (a e^2-b d e+c d^2\right )^2}-\frac{g^4 \log (f+g x)}{(e f-d g) \left (a g^2-b f g+c f^2\right )^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.05232, antiderivative size = 644, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {893, 638, 618, 206, 634, 628} \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (2 c e g \left (a^2 e^2 g^2+a b e g (d g+e f)-b^2 (d g+e f)^2\right )+b^2 e^2 g^2 (-2 a e g+b d g+b e f)-c^2 \left (4 a d e^2 f g^2-b \left (5 d^2 e f g^2+d^3 g^3+5 d e^2 f^2 g+e^3 f^3\right )\right )-2 c^3 d f \left (d^2 g^2+d e f g+e^2 f^2\right )\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )^2 \left (c f^2-g (b f-a g)\right )^2}-\frac{c x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )+b c (c d f-3 a e g)+2 a c^2 (d g+e f)-b^2 c (d g+e f)+b^3 e g}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right ) \left (c f^2-g (b f-a g)\right )}+\frac{2 c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right ) \left (c f^2-g (b f-a g)\right )}-\frac{\log \left (a+b x+c x^2\right ) (-b e g+c d g+c e f) \left (e g (2 a e g-b (d g+e f))+c \left (d^2 g^2+e^2 f^2\right )\right )}{2 \left (a e^2-b d e+c d^2\right )^2 \left (c f^2-g (b f-a g)\right )^2}+\frac{e^4 \log (d+e x)}{(e f-d g) \left (a e^2-b d e+c d^2\right )^2}-\frac{g^4 \log (f+g x)}{(e f-d g) \left (a g^2-b f g+c f^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 893

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] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 638

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 634

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [A]  time = 3.23609, size = 710, normalized size = 1.1 \[ \frac{\tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (-2 c^3 \left (2 a^2 e g \left (d^2 g^2-5 d e f g+e^2 f^2\right )+a b \left (11 d^2 e f g^2+3 d^3 g^3+11 d e^2 f^2 g+3 e^3 f^3\right )-4 b^2 d^2 e f^2 g\right )+c^2 \left (-6 a^2 b e^2 g^2 (d g+e f)-12 a^3 e^3 g^3+12 a b^2 e g \left (d^2 g^2+d e f g+e^2 f^2\right )+b^3 \left (d^2 e f g^2+d^3 g^3+d e^2 f^2 g+e^3 f^3\right )\right )-2 b^2 c e g \left (-6 a^2 e^2 g^2+2 a b e g (d g+e f)+b^2 \left (d^2 g^2+d e f g+e^2 f^2\right )\right )+b^4 e^2 g^2 (-2 a e g+b d g+b e f)+2 c^4 d f \left (2 a \left (3 d^2 g^2+d e f g+3 e^2 f^2\right )-3 b d f (d g+e f)\right )+4 c^5 d^3 f^3\right )}{\left (4 a c-b^2\right )^{3/2} \left (e (a e-b d)+c d^2\right )^2 \left (g (a g-b f)+c f^2\right )^2}+\frac{b c (3 a e g+c (-d f+d g x+e f x))-2 c^2 (a d g+a e (f-g x)+c d f x)+b^2 c (d g+e (f-g x))+b^3 (-e) g}{\left (b^2-4 a c\right ) (a+x (b+c x)) \left (e (b d-a e)-c d^2\right ) \left (g (b f-a g)-c f^2\right )}-\frac{\log (a+x (b+c x)) (-b e g+c d g+c e f) \left (e g (2 a e g-b (d g+e f))+c \left (d^2 g^2+e^2 f^2\right )\right )}{2 \left (e (a e-b d)+c d^2\right )^2 \left (g (a g-b f)+c f^2\right )^2}+\frac{e^4 \log (d+e x)}{(e f-d g) \left (e (a e-b d)+c d^2\right )^2}-\frac{g^4 \log (f+g x)}{(e f-d g) \left (g (a g-b f)+c f^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(b^3*e*g) + b^2*c*(d*g + e*(f - g*x)) - 2*c^2*(a*d*g + c*d*f*x + a*e*(f - g*x)) + b*c*(3*a*e*g + c*(-(d*f) +
e*f*x + d*g*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(-(c*f^2) + g*(b*f - a*g))*(a + x*(b + c*x))) + ((
4*c^5*d^3*f^3 + b^4*e^2*g^2*(b*e*f + b*d*g - 2*a*e*g) - 2*b^2*c*e*g*(-6*a^2*e^2*g^2 + 2*a*b*e*g*(e*f + d*g) +
b^2*(e^2*f^2 + d*e*f*g + d^2*g^2)) + 2*c^4*d*f*(-3*b*d*f*(e*f + d*g) + 2*a*(3*e^2*f^2 + d*e*f*g + 3*d^2*g^2))
+ c^2*(-12*a^3*e^3*g^3 - 6*a^2*b*e^2*g^2*(e*f + d*g) + 12*a*b^2*e*g*(e^2*f^2 + d*e*f*g + d^2*g^2) + b^3*(e^3*f
^3 + d*e^2*f^2*g + d^2*e*f*g^2 + d^3*g^3)) - 2*c^3*(-4*b^2*d^2*e*f^2*g + 2*a^2*e*g*(e^2*f^2 - 5*d*e*f*g + d^2*
g^2) + a*b*(3*e^3*f^3 + 11*d*e^2*f^2*g + 11*d^2*e*f*g^2 + 3*d^3*g^3)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])
/((-b^2 + 4*a*c)^(3/2)*(c*d^2 + e*(-(b*d) + a*e))^2*(c*f^2 + g*(-(b*f) + a*g))^2) + (e^4*Log[d + e*x])/((c*d^2
+ e*(-(b*d) + a*e))^2*(e*f - d*g)) - (g^4*Log[f + g*x])/((e*f - d*g)*(c*f^2 + g*(-(b*f) + a*g))^2) - ((c*e*f
+ c*d*g - b*e*g)*(c*(e^2*f^2 + d^2*g^2) + e*g*(2*a*e*g - b*(e*f + d*g)))*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(
-(b*d) + a*e))^2*(c*f^2 + g*(-(b*f) + a*g))^2)

________________________________________________________________________________________

Maple [B]  time = 0.256, size = 9103, normalized size = 14.1 \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)/(c*x^2+b*x+a)^2,x)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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)/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError