Integrand size = 32, antiderivative size = 124 \[ \int \frac {(h+i x)^2}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx=\frac {2 e^{-\frac {a}{b}} i (f h-e i) \operatorname {ExpIntegralEi}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c d f^3}+\frac {e^{-\frac {2 a}{b}} i^2 \operatorname {ExpIntegralEi}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )}{b c^2 d f^3}+\frac {(f h-e i)^2 \log (a+b \log (c (e+f x)))}{b d f^3} \]
2*i*(-e*i+f*h)*Ei((a+b*ln(c*(f*x+e)))/b)/b/c/d/exp(a/b)/f^3+i^2*Ei(2*(a+b* ln(c*(f*x+e)))/b)/b/c^2/d/exp(2*a/b)/f^3+(-e*i+f*h)^2*ln(a+b*ln(c*(f*x+e)) )/b/d/f^3
Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.90 \[ \int \frac {(h+i x)^2}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx=\frac {e^{-\frac {2 a}{b}} \left (2 c e^{a/b} i (f h-e i) \operatorname {ExpIntegralEi}\left (\frac {a}{b}+\log (c (e+f x))\right )+i^2 \operatorname {ExpIntegralEi}\left (2 \left (\frac {a}{b}+\log (c (e+f x))\right )\right )+c^2 e^{\frac {2 a}{b}} (f h-e i)^2 \log (a+b \log (c (e+f x)))\right )}{b c^2 d f^3} \]
(2*c*E^(a/b)*i*(f*h - e*i)*ExpIntegralEi[a/b + Log[c*(e + f*x)]] + i^2*Exp IntegralEi[2*(a/b + Log[c*(e + f*x)])] + c^2*E^((2*a)/b)*(f*h - e*i)^2*Log [a + b*Log[c*(e + f*x)]])/(b*c^2*d*E^((2*a)/b)*f^3)
Time = 0.51 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2858, 27, 2795, 2009}
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.
| \(\displaystyle \int \frac {(h+i x)^2}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {\int \frac {\left (f \left (h-\frac {e i}{f}\right )+i (e+f x)\right )^2}{d f^2 (e+f x) (a+b \log (c (e+f x)))}d(e+f x)}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(f h-e i+i (e+f x))^2}{(e+f x) (a+b \log (c (e+f x)))}d(e+f x)}{d f^3}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {\int \left (\frac {(e+f x) i^2}{a+b \log (c (e+f x))}+\frac {2 (f h-e i) i}{a+b \log (c (e+f x))}+\frac {(f h-e i)^2}{(e+f x) (a+b \log (c (e+f x)))}\right )d(e+f x)}{d f^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {i^2 e^{-\frac {2 a}{b}} \operatorname {ExpIntegralEi}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )}{b c^2}+\frac {2 i e^{-\frac {a}{b}} (f h-e i) \operatorname {ExpIntegralEi}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c}+\frac {(f h-e i)^2 \log (a+b \log (c (e+f x)))}{b}}{d f^3}\) |
((2*i*(f*h - e*i)*ExpIntegralEi[(a + b*Log[c*(e + f*x)])/b])/(b*c*E^(a/b)) + (i^2*ExpIntegralEi[(2*(a + b*Log[c*(e + f*x)]))/b])/(b*c^2*E^((2*a)/b)) + ((f*h - e*i)^2*Log[a + b*Log[c*(e + f*x)]])/b)/(d*f^3)
3.2.93.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b , c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 ] && IntegerQ[m] && IntegerQ[r]))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
Time = 2.12 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.61
| method | result | size |
| derivativedivides | \(\frac {-\frac {i^{2} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{b}+\frac {c^{2} e^{2} i^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {c^{2} f^{2} h^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {2 c e \,i^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {2 c f h i \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {2 c^{2} e f h i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}}{c^{2} f^{3} d}\) | \(200\) |
| default | \(\frac {-\frac {i^{2} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{b}+\frac {c^{2} e^{2} i^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {c^{2} f^{2} h^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {2 c e \,i^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {2 c f h i \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {2 c^{2} e f h i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}}{c^{2} f^{3} d}\) | \(200\) |
| risch | \(\frac {e^{2} i^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{f^{3} d b}-\frac {2 e h i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{f^{2} d b}+\frac {h^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{f d b}+\frac {2 e \,i^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{c \,f^{3} d b}-\frac {2 h i \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{c \,f^{2} d b}-\frac {i^{2} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{c^{2} f^{3} d b}\) | \(219\) |
1/c^2/f^3/d*(-i^2/b*exp(-2*a/b)*Ei(1,-2*ln(c*f*x+c*e)-2*a/b)+c^2*e^2*i^2*l n(a+b*ln(c*f*x+c*e))/b+c^2*f^2*h^2*ln(a+b*ln(c*f*x+c*e))/b+2*c*e*i^2/b*exp (-a/b)*Ei(1,-ln(c*f*x+c*e)-a/b)-2*c*f*h*i/b*exp(-a/b)*Ei(1,-ln(c*f*x+c*e)- a/b)-2*c^2*e*f*h*i*ln(a+b*ln(c*f*x+c*e))/b)
Time = 0.31 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.20 \[ \int \frac {(h+i x)^2}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx=\frac {{\left ({\left (c^{2} f^{2} h^{2} - 2 \, c^{2} e f h i + c^{2} e^{2} i^{2}\right )} e^{\left (\frac {2 \, a}{b}\right )} \log \left (b \log \left (c f x + c e\right ) + a\right ) + i^{2} \operatorname {log\_integral}\left ({\left (c^{2} f^{2} x^{2} + 2 \, c^{2} e f x + c^{2} e^{2}\right )} e^{\left (\frac {2 \, a}{b}\right )}\right ) + 2 \, {\left (c f h i - c e i^{2}\right )} e^{\frac {a}{b}} \operatorname {log\_integral}\left ({\left (c f x + c e\right )} e^{\frac {a}{b}}\right )\right )} e^{\left (-\frac {2 \, a}{b}\right )}}{b c^{2} d f^{3}} \]
((c^2*f^2*h^2 - 2*c^2*e*f*h*i + c^2*e^2*i^2)*e^(2*a/b)*log(b*log(c*f*x + c *e) + a) + i^2*log_integral((c^2*f^2*x^2 + 2*c^2*e*f*x + c^2*e^2)*e^(2*a/b )) + 2*(c*f*h*i - c*e*i^2)*e^(a/b)*log_integral((c*f*x + c*e)*e^(a/b)))*e^ (-2*a/b)/(b*c^2*d*f^3)
\[ \int \frac {(h+i x)^2}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx=\frac {\int \frac {h^{2}}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx + \int \frac {i^{2} x^{2}}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx + \int \frac {2 h i x}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx}{d} \]
(Integral(h**2/(a*e + a*f*x + b*e*log(c*e + c*f*x) + b*f*x*log(c*e + c*f*x )), x) + Integral(i**2*x**2/(a*e + a*f*x + b*e*log(c*e + c*f*x) + b*f*x*lo g(c*e + c*f*x)), x) + Integral(2*h*i*x/(a*e + a*f*x + b*e*log(c*e + c*f*x) + b*f*x*log(c*e + c*f*x)), x))/d
\[ \int \frac {(h+i x)^2}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx=\int { \frac {{\left (i x + h\right )}^{2}}{{\left (d f x + d e\right )} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}} \,d x } \]
h^2*log((b*log(f*x + e) + b*log(c) + a)/b)/(b*d*f) + integrate((i^2*x^2 + 2*h*i*x)/(b*d*e*log(c) + a*d*e + (b*d*f*log(c) + a*d*f)*x + (b*d*f*x + b*d *e)*log(f*x + e)), x)
\[ \int \frac {(h+i x)^2}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx=\int { \frac {{\left (i x + h\right )}^{2}}{{\left (d f x + d e\right )} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}} \,d x } \]
Timed out. \[ \int \frac {(h+i x)^2}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx=\int \frac {{\left (h+i\,x\right )}^2}{\left (d\,e+d\,f\,x\right )\,\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )} \,d x \]