Integrand size = 29, antiderivative size = 484 \[ \int \frac {x^2}{a+b f^{c+d x}+c f^{2 c+2 d x}} \, dx=-\frac {2 c x^3}{3 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right )}-\frac {2 c x^3}{3 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right )}-\frac {2 c x^2 \log \left (1+\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) d \log (f)}+\frac {2 c x^2 \log \left (1+\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) d \log (f)}-\frac {4 c x \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}+\frac {4 c x \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}+\frac {4 c \operatorname {PolyLog}\left (3,-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) d^3 \log ^3(f)}-\frac {4 c \operatorname {PolyLog}\left (3,-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) d^3 \log ^3(f)} \]
-2*c*x^2*ln(1+2*c*f^(d*x+c)/(b-(-4*a*c+b^2)^(1/2)))/d/ln(f)/(b-(-4*a*c+b^2 )^(1/2))/(-4*a*c+b^2)^(1/2)-4*c*x*polylog(2,-2*c*f^(d*x+c)/(b-(-4*a*c+b^2) ^(1/2)))/d^2/ln(f)^2/(b-(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2)+4*c*polylog (3,-2*c*f^(d*x+c)/(b-(-4*a*c+b^2)^(1/2)))/d^3/ln(f)^3/(b-(-4*a*c+b^2)^(1/2 ))/(-4*a*c+b^2)^(1/2)+2*c*x^2*ln(1+2*c*f^(d*x+c)/(b+(-4*a*c+b^2)^(1/2)))/d /ln(f)/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))+4*c*x*polylog(2,-2*c*f^(d *x+c)/(b+(-4*a*c+b^2)^(1/2)))/d^2/ln(f)^2/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^ 2)^(1/2))-4*c*polylog(3,-2*c*f^(d*x+c)/(b+(-4*a*c+b^2)^(1/2)))/d^3/ln(f)^3 /(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))-2/3*c*x^3/(b^2-4*a*c-b*(-4*a*c+ b^2)^(1/2))-2/3*c*x^3/(b^2-4*a*c+b*(-4*a*c+b^2)^(1/2))
Time = 0.27 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.69 \[ \int \frac {x^2}{a+b f^{c+d x}+c f^{2 c+2 d x}} \, dx=\frac {2 c \left (\frac {x^2 \log \left (1+\frac {\left (b-\sqrt {b^2-4 a c}\right ) f^{-c-d x}}{2 c}\right )}{-b+\sqrt {b^2-4 a c}}+\frac {x^2 \log \left (1+\frac {\left (b+\sqrt {b^2-4 a c}\right ) f^{-c-d x}}{2 c}\right )}{b+\sqrt {b^2-4 a c}}-\frac {2 \left (d x \log (f) \operatorname {PolyLog}\left (2,\frac {\left (-b+\sqrt {b^2-4 a c}\right ) f^{-c-d x}}{2 c}\right )+\operatorname {PolyLog}\left (3,\frac {\left (-b+\sqrt {b^2-4 a c}\right ) f^{-c-d x}}{2 c}\right )\right )}{\left (-b+\sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}-\frac {2 \left (d x \log (f) \operatorname {PolyLog}\left (2,-\frac {\left (b+\sqrt {b^2-4 a c}\right ) f^{-c-d x}}{2 c}\right )+\operatorname {PolyLog}\left (3,-\frac {\left (b+\sqrt {b^2-4 a c}\right ) f^{-c-d x}}{2 c}\right )\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}\right )}{\sqrt {b^2-4 a c} d \log (f)} \]
(2*c*((x^2*Log[1 + ((b - Sqrt[b^2 - 4*a*c])*f^(-c - d*x))/(2*c)])/(-b + Sq rt[b^2 - 4*a*c]) + (x^2*Log[1 + ((b + Sqrt[b^2 - 4*a*c])*f^(-c - d*x))/(2* c)])/(b + Sqrt[b^2 - 4*a*c]) - (2*(d*x*Log[f]*PolyLog[2, ((-b + Sqrt[b^2 - 4*a*c])*f^(-c - d*x))/(2*c)] + PolyLog[3, ((-b + Sqrt[b^2 - 4*a*c])*f^(-c - d*x))/(2*c)]))/((-b + Sqrt[b^2 - 4*a*c])*d^2*Log[f]^2) - (2*(d*x*Log[f] *PolyLog[2, -1/2*((b + Sqrt[b^2 - 4*a*c])*f^(-c - d*x))/c] + PolyLog[3, -1 /2*((b + Sqrt[b^2 - 4*a*c])*f^(-c - d*x))/c]))/((b + Sqrt[b^2 - 4*a*c])*d^ 2*Log[f]^2)))/(Sqrt[b^2 - 4*a*c]*d*Log[f])
Time = 1.41 (sec) , antiderivative size = 395, normalized size of antiderivative = 0.82, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2693, 2615, 2620, 3011, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2}{a+b f^{c+d x}+c f^{2 c+2 d x}} \, dx\) |
| \(\Big \downarrow \) 2693 |
| \(\displaystyle \frac {2 c \int \frac {x^2}{2 c f^{c+d x}+b-\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}-\frac {2 c \int \frac {x^2}{2 c f^{c+d x}+b+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 2615 |
| \(\displaystyle \frac {2 c \left (\frac {x^3}{3 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \int \frac {f^{c+d x} x^2}{2 c f^{c+d x}+b-\sqrt {b^2-4 a c}}dx}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {x^3}{3 \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \int \frac {f^{c+d x} x^2}{2 c f^{c+d x}+b+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}+b}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 c \left (\frac {x^3}{3 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \left (\frac {x^2 \log \left (\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c d \log (f)}-\frac {\int x \log \left (\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}+1\right )dx}{c d \log (f)}\right )}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {x^3}{3 \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \left (\frac {x^2 \log \left (\frac {2 c f^{c+d x}}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c d \log (f)}-\frac {\int x \log \left (\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}+1\right )dx}{c d \log (f)}\right )}{\sqrt {b^2-4 a c}+b}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {2 c \left (\frac {x^3}{3 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \left (\frac {x^2 \log \left (\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c d \log (f)}-\frac {\frac {\int \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )dx}{d \log (f)}-\frac {x \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{d \log (f)}}{c d \log (f)}\right )}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {x^3}{3 \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \left (\frac {x^2 \log \left (\frac {2 c f^{c+d x}}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c d \log (f)}-\frac {\frac {\int \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )dx}{d \log (f)}-\frac {x \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{d \log (f)}}{c d \log (f)}\right )}{\sqrt {b^2-4 a c}+b}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {2 c \left (\frac {x^3}{3 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \left (\frac {x^2 \log \left (\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c d \log (f)}-\frac {\frac {\int f^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )df^{c+d x}}{d^2 \log ^2(f)}-\frac {x \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{d \log (f)}}{c d \log (f)}\right )}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {x^3}{3 \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \left (\frac {x^2 \log \left (\frac {2 c f^{c+d x}}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c d \log (f)}-\frac {\frac {\int f^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )df^{c+d x}}{d^2 \log ^2(f)}-\frac {x \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{d \log (f)}}{c d \log (f)}\right )}{\sqrt {b^2-4 a c}+b}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {2 c \left (\frac {x^3}{3 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \left (\frac {x^2 \log \left (\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c d \log (f)}-\frac {\frac {\operatorname {PolyLog}\left (3,-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{d^2 \log ^2(f)}-\frac {x \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{d \log (f)}}{c d \log (f)}\right )}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {x^3}{3 \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \left (\frac {x^2 \log \left (\frac {2 c f^{c+d x}}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c d \log (f)}-\frac {\frac {\operatorname {PolyLog}\left (3,-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{d^2 \log ^2(f)}-\frac {x \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{d \log (f)}}{c d \log (f)}\right )}{\sqrt {b^2-4 a c}+b}\right )}{\sqrt {b^2-4 a c}}\) |
(2*c*(x^3/(3*(b - Sqrt[b^2 - 4*a*c])) - (2*c*((x^2*Log[1 + (2*c*f^(c + d*x ))/(b - Sqrt[b^2 - 4*a*c])])/(2*c*d*Log[f]) - (-((x*PolyLog[2, (-2*c*f^(c + d*x))/(b - Sqrt[b^2 - 4*a*c])])/(d*Log[f])) + PolyLog[3, (-2*c*f^(c + d* x))/(b - Sqrt[b^2 - 4*a*c])]/(d^2*Log[f]^2))/(c*d*Log[f])))/(b - Sqrt[b^2 - 4*a*c])))/Sqrt[b^2 - 4*a*c] - (2*c*(x^3/(3*(b + Sqrt[b^2 - 4*a*c])) - (2 *c*((x^2*Log[1 + (2*c*f^(c + d*x))/(b + Sqrt[b^2 - 4*a*c])])/(2*c*d*Log[f] ) - (-((x*PolyLog[2, (-2*c*f^(c + d*x))/(b + Sqrt[b^2 - 4*a*c])])/(d*Log[f ])) + PolyLog[3, (-2*c*f^(c + d*x))/(b + Sqrt[b^2 - 4*a*c])]/(d^2*Log[f]^2 ))/(c*d*Log[f])))/(b + Sqrt[b^2 - 4*a*c])))/Sqrt[b^2 - 4*a*c]
3.6.26.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x _))))^(n_.)), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(a*d*(m + 1)), x] - Simp[ b/a Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n)), x] , x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((f_.) + (g_.)*(x_))^(m_.)/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int[(f + g*x)^m /(b - q + 2*c*F^u), x], x] - Simp[2*(c/q) Int[(f + g*x)^m/(b + q + 2*c*F^ u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
\[\int \frac {x^{2}}{a +b \,f^{d x +c}+c \,f^{2 d x +2 c}}d x\]
Time = 0.33 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.43 \[ \int \frac {x^2}{a+b f^{c+d x}+c f^{2 c+2 d x}} \, dx=\frac {2 \, {\left (b^{2} - 4 \, a c\right )} d^{3} x^{3} \log \left (f\right )^{3} - 6 \, {\left (a b d x \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \left (f\right ) + {\left (b^{2} - 4 \, a c\right )} d x \log \left (f\right )\right )} {\rm Li}_2\left (-\frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right )} f^{d x + c} + 2 \, a}{2 \, a} + 1\right ) + 6 \, {\left (a b d x \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \left (f\right ) - {\left (b^{2} - 4 \, a c\right )} d x \log \left (f\right )\right )} {\rm Li}_2\left (\frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b\right )} f^{d x + c} - 2 \, a}{2 \, a} + 1\right ) + 3 \, {\left (a b c^{2} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \left (f\right )^{2} - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \log \left (f\right )^{2}\right )} \log \left (2 \, c f^{d x + c} + a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right ) - 3 \, {\left (a b c^{2} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \left (f\right )^{2} + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \log \left (f\right )^{2}\right )} \log \left (2 \, c f^{d x + c} - a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right ) - 3 \, {\left ({\left (a b d^{2} x^{2} - a b c^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \left (f\right )^{2} + {\left ({\left (b^{2} - 4 \, a c\right )} d^{2} x^{2} - b^{2} c^{2} + 4 \, a c^{3}\right )} \log \left (f\right )^{2}\right )} \log \left (\frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right )} f^{d x + c} + 2 \, a}{2 \, a}\right ) + 3 \, {\left ({\left (a b d^{2} x^{2} - a b c^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \left (f\right )^{2} - {\left ({\left (b^{2} - 4 \, a c\right )} d^{2} x^{2} - b^{2} c^{2} + 4 \, a c^{3}\right )} \log \left (f\right )^{2}\right )} \log \left (-\frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b\right )} f^{d x + c} - 2 \, a}{2 \, a}\right ) + 6 \, {\left (a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 4 \, a c\right )} {\rm polylog}\left (3, -\frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right )} f^{d x + c}}{2 \, a}\right ) - 6 \, {\left (a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b^{2} + 4 \, a c\right )} {\rm polylog}\left (3, \frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b\right )} f^{d x + c}}{2 \, a}\right )}{6 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d^{3} \log \left (f\right )^{3}} \]
1/6*(2*(b^2 - 4*a*c)*d^3*x^3*log(f)^3 - 6*(a*b*d*x*sqrt((b^2 - 4*a*c)/a^2) *log(f) + (b^2 - 4*a*c)*d*x*log(f))*dilog(-1/2*((a*sqrt((b^2 - 4*a*c)/a^2) + b)*f^(d*x + c) + 2*a)/a + 1) + 6*(a*b*d*x*sqrt((b^2 - 4*a*c)/a^2)*log(f ) - (b^2 - 4*a*c)*d*x*log(f))*dilog(1/2*((a*sqrt((b^2 - 4*a*c)/a^2) - b)*f ^(d*x + c) - 2*a)/a + 1) + 3*(a*b*c^2*sqrt((b^2 - 4*a*c)/a^2)*log(f)^2 - ( b^2*c^2 - 4*a*c^3)*log(f)^2)*log(2*c*f^(d*x + c) + a*sqrt((b^2 - 4*a*c)/a^ 2) + b) - 3*(a*b*c^2*sqrt((b^2 - 4*a*c)/a^2)*log(f)^2 + (b^2*c^2 - 4*a*c^3 )*log(f)^2)*log(2*c*f^(d*x + c) - a*sqrt((b^2 - 4*a*c)/a^2) + b) - 3*((a*b *d^2*x^2 - a*b*c^2)*sqrt((b^2 - 4*a*c)/a^2)*log(f)^2 + ((b^2 - 4*a*c)*d^2* x^2 - b^2*c^2 + 4*a*c^3)*log(f)^2)*log(1/2*((a*sqrt((b^2 - 4*a*c)/a^2) + b )*f^(d*x + c) + 2*a)/a) + 3*((a*b*d^2*x^2 - a*b*c^2)*sqrt((b^2 - 4*a*c)/a^ 2)*log(f)^2 - ((b^2 - 4*a*c)*d^2*x^2 - b^2*c^2 + 4*a*c^3)*log(f)^2)*log(-1 /2*((a*sqrt((b^2 - 4*a*c)/a^2) - b)*f^(d*x + c) - 2*a)/a) + 6*(a*b*sqrt((b ^2 - 4*a*c)/a^2) + b^2 - 4*a*c)*polylog(3, -1/2*(a*sqrt((b^2 - 4*a*c)/a^2) + b)*f^(d*x + c)/a) - 6*(a*b*sqrt((b^2 - 4*a*c)/a^2) - b^2 + 4*a*c)*polyl og(3, 1/2*(a*sqrt((b^2 - 4*a*c)/a^2) - b)*f^(d*x + c)/a))/((a*b^2 - 4*a^2* c)*d^3*log(f)^3)
\[ \int \frac {x^2}{a+b f^{c+d x}+c f^{2 c+2 d x}} \, dx=\int \frac {x^{2}}{a + b f^{c + d x} + c f^{2 c + 2 d x}}\, dx \]
Exception generated. \[ \int \frac {x^2}{a+b f^{c+d x}+c f^{2 c+2 d x}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
\[ \int \frac {x^2}{a+b f^{c+d x}+c f^{2 c+2 d x}} \, dx=\int { \frac {x^{2}}{c f^{2 \, d x + 2 \, c} + b f^{d x + c} + a} \,d x } \]
Timed out. \[ \int \frac {x^2}{a+b f^{c+d x}+c f^{2 c+2 d x}} \, dx=\int \frac {x^2}{a+b\,f^{c+d\,x}+c\,f^{2\,c+2\,d\,x}} \,d x \]