Integrand size = 32, antiderivative size = 177 \[ \int \frac {(h+i x)^3}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx=\frac {3 e^{-\frac {a}{b}} i (f h-e i)^2 \operatorname {ExpIntegralEi}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c d f^4}+\frac {3 e^{-\frac {2 a}{b}} i^2 (f h-e i) \operatorname {ExpIntegralEi}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )}{b c^2 d f^4}+\frac {e^{-\frac {3 a}{b}} i^3 \operatorname {ExpIntegralEi}\left (\frac {3 (a+b \log (c (e+f x)))}{b}\right )}{b c^3 d f^4}+\frac {(f h-e i)^3 \log (a+b \log (c (e+f x)))}{b d f^4} \]
3*i*(-e*i+f*h)^2*Ei((a+b*ln(c*(f*x+e)))/b)/b/c/d/exp(a/b)/f^4+3*i^2*(-e*i+ f*h)*Ei(2*(a+b*ln(c*(f*x+e)))/b)/b/c^2/d/exp(2*a/b)/f^4+i^3*Ei(3*(a+b*ln(c *(f*x+e)))/b)/b/c^3/d/exp(3*a/b)/f^4+(-e*i+f*h)^3*ln(a+b*ln(c*(f*x+e)))/b/ d/f^4
\[ \int \frac {(h+i x)^3}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx=\int \frac {(h+i x)^3}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx \]
Time = 0.61 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.90, 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.
| \(\displaystyle \int \frac {(h+i x)^3}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {\int \frac {\left (f \left (h-\frac {e i}{f}\right )+i (e+f x)\right )^3}{d f^3 (e+f x) (a+b \log (c (e+f x)))}d(e+f x)}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(f h-e i+i (e+f x))^3}{(e+f x) (a+b \log (c (e+f x)))}d(e+f x)}{d f^4}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {\int \left (\frac {(e+f x)^2 i^3}{a+b \log (c (e+f x))}+\frac {3 (f h-e i) (e+f x) i^2}{a+b \log (c (e+f x))}+\frac {3 (f h-e i)^2 i}{a+b \log (c (e+f x))}+\frac {(f h-e i)^3}{(e+f x) (a+b \log (c (e+f x)))}\right )d(e+f x)}{d f^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {i^3 e^{-\frac {3 a}{b}} \operatorname {ExpIntegralEi}\left (\frac {3 (a+b \log (c (e+f x)))}{b}\right )}{b c^3}+\frac {3 i^2 e^{-\frac {2 a}{b}} (f h-e i) \operatorname {ExpIntegralEi}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )}{b c^2}+\frac {3 i e^{-\frac {a}{b}} (f h-e i)^2 \operatorname {ExpIntegralEi}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c}+\frac {(f h-e i)^3 \log (a+b \log (c (e+f x)))}{b}}{d f^4}\) |
((3*i*(f*h - e*i)^2*ExpIntegralEi[(a + b*Log[c*(e + f*x)])/b])/(b*c*E^(a/b )) + (3*i^2*(f*h - e*i)*ExpIntegralEi[(2*(a + b*Log[c*(e + f*x)]))/b])/(b* c^2*E^((2*a)/b)) + (i^3*ExpIntegralEi[(3*(a + b*Log[c*(e + f*x)]))/b])/(b* c^3*E^((3*a)/b)) + ((f*h - e*i)^3*Log[a + b*Log[c*(e + f*x)]])/b)/(d*f^4)
3.2.92.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(360\) vs. \(2(179)=358\).
Time = 2.46 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.04
| method | result | size |
| derivativedivides | \(-\frac {\frac {i^{3} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \ln \left (c f x +c e \right )-\frac {3 a}{b}\right )}{b}-\frac {c^{3} f^{3} h^{3} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {c^{3} e^{3} i^{3} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}-\frac {3 c e \,i^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{b}+\frac {3 c^{2} e^{2} i^{3} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}+\frac {3 c f h \,i^{2} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{b}+\frac {3 c^{2} f^{2} h^{2} i \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}+\frac {3 c^{3} e \,f^{2} h^{2} i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}-\frac {3 c^{3} e^{2} f h \,i^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}-\frac {6 c^{2} e f h \,i^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}}{c^{3} f^{4} d}\) | \(361\) |
| default | \(-\frac {\frac {i^{3} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \ln \left (c f x +c e \right )-\frac {3 a}{b}\right )}{b}-\frac {c^{3} f^{3} h^{3} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {c^{3} e^{3} i^{3} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}-\frac {3 c e \,i^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{b}+\frac {3 c^{2} e^{2} i^{3} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}+\frac {3 c f h \,i^{2} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{b}+\frac {3 c^{2} f^{2} h^{2} i \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}+\frac {3 c^{3} e \,f^{2} h^{2} i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}-\frac {3 c^{3} e^{2} f h \,i^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}-\frac {6 c^{2} e f h \,i^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}}{c^{3} f^{4} d}\) | \(361\) |
| risch | \(-\frac {e^{3} i^{3} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{f^{4} d b}+\frac {3 e^{2} h \,i^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{f^{3} d b}-\frac {3 e \,h^{2} i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{f^{2} d b}+\frac {h^{3} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{f d b}-\frac {3 e^{2} i^{3} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{c \,f^{4} d b}+\frac {6 e h \,i^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{c \,f^{3} d b}-\frac {3 h^{2} i \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{c \,f^{2} d b}+\frac {3 e \,i^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{c^{2} f^{4} d b}-\frac {3 h \,i^{2} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{c^{2} f^{3} d b}-\frac {i^{3} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \ln \left (c f x +c e \right )-\frac {3 a}{b}\right )}{c^{3} f^{4} d b}\) | \(394\) |
-1/c^3/f^4/d*(i^3/b*exp(-3*a/b)*Ei(1,-3*ln(c*f*x+c*e)-3*a/b)-c^3*f^3*h^3*l n(a+b*ln(c*f*x+c*e))/b+c^3*e^3*i^3*ln(a+b*ln(c*f*x+c*e))/b-3*c*e*i^3/b*exp (-2*a/b)*Ei(1,-2*ln(c*f*x+c*e)-2*a/b)+3*c^2*e^2*i^3/b*exp(-a/b)*Ei(1,-ln(c *f*x+c*e)-a/b)+3*c*f*h*i^2/b*exp(-2*a/b)*Ei(1,-2*ln(c*f*x+c*e)-2*a/b)+3*c^ 2*f^2*h^2*i/b*exp(-a/b)*Ei(1,-ln(c*f*x+c*e)-a/b)+3*c^3*e*f^2*h^2*i*ln(a+b* ln(c*f*x+c*e))/b-3*c^3*e^2*f*h*i^2*ln(a+b*ln(c*f*x+c*e))/b-6*c^2*e*f*h*i^2 /b*exp(-a/b)*Ei(1,-ln(c*f*x+c*e)-a/b))
Time = 0.31 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.47 \[ \int \frac {(h+i x)^3}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx=\frac {{\left (i^{3} \operatorname {log\_integral}\left ({\left (c^{3} f^{3} x^{3} + 3 \, c^{3} e f^{2} x^{2} + 3 \, c^{3} e^{2} f x + c^{3} e^{3}\right )} e^{\left (\frac {3 \, a}{b}\right )}\right ) + {\left (c^{3} f^{3} h^{3} - 3 \, c^{3} e f^{2} h^{2} i + 3 \, c^{3} e^{2} f h i^{2} - c^{3} e^{3} i^{3}\right )} e^{\left (\frac {3 \, a}{b}\right )} \log \left (b \log \left (c f x + c e\right ) + a\right ) + 3 \, {\left (c f h i^{2} - c e i^{3}\right )} e^{\frac {a}{b}} \operatorname {log\_integral}\left ({\left (c^{2} f^{2} x^{2} + 2 \, c^{2} e f x + c^{2} e^{2}\right )} e^{\left (\frac {2 \, a}{b}\right )}\right ) + 3 \, {\left (c^{2} f^{2} h^{2} i - 2 \, c^{2} e f h i^{2} + c^{2} e^{2} i^{3}\right )} e^{\left (\frac {2 \, a}{b}\right )} \operatorname {log\_integral}\left ({\left (c f x + c e\right )} e^{\frac {a}{b}}\right )\right )} e^{\left (-\frac {3 \, a}{b}\right )}}{b c^{3} d f^{4}} \]
(i^3*log_integral((c^3*f^3*x^3 + 3*c^3*e*f^2*x^2 + 3*c^3*e^2*f*x + c^3*e^3 )*e^(3*a/b)) + (c^3*f^3*h^3 - 3*c^3*e*f^2*h^2*i + 3*c^3*e^2*f*h*i^2 - c^3* e^3*i^3)*e^(3*a/b)*log(b*log(c*f*x + c*e) + a) + 3*(c*f*h*i^2 - c*e*i^3)*e ^(a/b)*log_integral((c^2*f^2*x^2 + 2*c^2*e*f*x + c^2*e^2)*e^(2*a/b)) + 3*( c^2*f^2*h^2*i - 2*c^2*e*f*h*i^2 + c^2*e^2*i^3)*e^(2*a/b)*log_integral((c*f *x + c*e)*e^(a/b)))*e^(-3*a/b)/(b*c^3*d*f^4)
\[ \int \frac {(h+i x)^3}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx=\frac {\int \frac {h^{3}}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx + \int \frac {i^{3} x^{3}}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx + \int \frac {3 h i^{2} x^{2}}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx + \int \frac {3 h^{2} i x}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx}{d} \]
(Integral(h**3/(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**3*x**3/(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(3*h*i**2*x**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(3*h**2*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
\[ \int \frac {(h+i x)^3}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx=\int { \frac {{\left (i x + h\right )}^{3}}{{\left (d f x + d e\right )} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}} \,d x } \]
h^3*log((b*log(f*x + e) + b*log(c) + a)/b)/(b*d*f) + integrate((i^3*x^3 + 3*h*i^2*x^2 + 3*h^2*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)
\[ \int \frac {(h+i x)^3}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx=\int { \frac {{\left (i x + h\right )}^{3}}{{\left (d f x + d e\right )} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}} \,d x } \]
Timed out. \[ \int \frac {(h+i x)^3}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx=\int \frac {{\left (h+i\,x\right )}^3}{\left (d\,e+d\,f\,x\right )\,\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )} \,d x \]