\[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx \]
Optimal antiderivative \[ -\frac {b f}{2 d \left (-e i +f h \right )^{2} \left (i x +h \right )}-\frac {b \,f^{2} \ln \left (f x +e \right )}{2 d \left (-e i +f h \right )^{3}}+\frac {a +b \ln \left (c \left (f x +e \right )\right )}{2 d \left (-e i +f h \right ) \left (i x +h \right )^{2}}-\frac {f i \left (f x +e \right ) \left (a +b \ln \left (c \left (f x +e \right )\right )\right )}{d \left (-e i +f h \right )^{3} \left (i x +h \right )}+\frac {3 b \,f^{2} \ln \left (i x +h \right )}{2 d \left (-e i +f h \right )^{3}}-\frac {f^{2} \left (a +b \ln \left (c \left (f x +e \right )\right )\right ) \ln \left (1+\frac {-e i +f h}{i \left (f x +e \right )}\right )}{d \left (-e i +f h \right )^{3}}+\frac {b \,f^{2} \polylog \left (2, \frac {e i -f h}{i \left (f x +e \right )}\right )}{d \left (-e i +f h \right )^{3}} \]
command
integrate((a+b*log(c*(f*x+e)))/(d*f*x+d*e)/(i*x+h)^3,x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {i \, {\left (\log \left (f x + e\right ) \log \left (-\frac {f x + e}{i \, f h + e} + 1\right ) + {\rm Li}_2\left (\frac {f x + e}{i \, f h + e}\right )\right )} b f^{2}}{-i \, d f^{3} h^{3} - 3 \, d f^{2} h^{2} e + 3 i \, d f h e^{2} + d e^{3}} + \frac {{\left (2 i \, a f^{2} + {\left (2 i \, f^{2} \log \left (c\right ) - 3 i \, f^{2}\right )} b\right )} \log \left (-2 i \, h + 2 \, x\right )}{-2 i \, d f^{3} h^{3} - 6 \, d f^{2} h^{2} e + 6 i \, d f h e^{2} + 2 \, d e^{3}} + \frac {16 \, {\left (3 i \, a f^{2} h^{2} + {\left (i \, b f^{2} h^{2} - 2 \, b f^{2} h x - i \, b f^{2} x^{2}\right )} \log \left (f x + e\right )^{2} + {\left (3 i \, f^{2} h^{2} \log \left (c\right ) - i \, f^{2} h^{2}\right )} b - {\left (2 \, a f^{2} h + {\left (2 \, f^{2} h \log \left (c\right ) - f^{2} h\right )} b - {\left ({\left (2 i \, f \log \left (c\right ) - i \, f\right )} b + 2 i \, a f\right )} e\right )} x + {\left (-i \, b \log \left (c\right ) - i \, a\right )} e^{2} + {\left (4 \, a f h + {\left (4 \, f h \log \left (c\right ) - f h\right )} b\right )} e + {\left (2 i \, b f^{2} h^{2} \log \left (c\right ) + 2 i \, a f^{2} h^{2} + 4 \, b f h e + {\left (-2 i \, a f^{2} + {\left (-2 i \, f^{2} \log \left (c\right ) + 3 i \, f^{2}\right )} b\right )} x^{2} - 2 \, {\left (2 \, a f^{2} h - i \, b f e + 2 \, {\left (f^{2} h \log \left (c\right ) - f^{2} h\right )} b\right )} x - i \, b e^{2}\right )} \log \left (f x + e\right )\right )}}{32 i \, d f^{3} h^{5} + 96 \, d f^{2} h^{4} e - 96 i \, d f h^{3} e^{2} - 32 \, d h^{2} e^{3} - 32 \, {\left (i \, d f^{3} h^{3} + 3 \, d f^{2} h^{2} e - 3 i \, d f h e^{2} - d e^{3}\right )} x^{2} - 64 \, {\left (d f^{3} h^{4} - 3 i \, d f^{2} h^{3} e - 3 \, d f h^{2} e^{2} + i \, d h e^{3}\right )} x} \]
Maxima 5.44 via sagemath 9.3 output
\[ \frac {1}{2} \, {\left (\frac {2 \, f^{2} \log \left (f x + e\right )}{d f^{3} h^{3} - 3 \, d e f^{2} h^{2} i + 3 \, d e^{2} f h i^{2} - d e^{3} i^{3}} - \frac {2 \, f^{2} \log \left (i x + h\right )}{d f^{3} h^{3} - 3 \, d e f^{2} h^{2} i + 3 \, d e^{2} f h i^{2} - d e^{3} i^{3}} + \frac {2 \, f i x + 3 \, f h - e i}{d f^{2} h^{4} - 2 \, d e f h^{3} i + d e^{2} h^{2} i^{2} + {\left (d f^{2} h^{2} i^{2} - 2 \, d e f h i^{3} + d e^{2} i^{4}\right )} x^{2} + 2 \, {\left (d f^{2} h^{3} i - 2 \, d e f h^{2} i^{2} + d e^{2} h i^{3}\right )} x}\right )} a + b \int \frac {\log \left (f x + e\right ) + \log \left (c\right )}{d f i^{3} x^{4} + d e h^{3} + {\left (3 \, f h i^{2} + e i^{3}\right )} d x^{3} + 3 \, {\left (f h^{2} i + e h i^{2}\right )} d x^{2} + {\left (f h^{3} + 3 \, e h^{2} i\right )} d x}\,{d x} \]________________________________________________________________________________________