\(\int \frac {h+i x}{(f+g x) (a+b \log (c (d+e x)^n))} \, dx\) [234]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\frac {e^{-\frac {a}{b n}} i (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e g n}+\frac {(g h-f i) \text {Int}\left (\frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right )}{g} \]

[Out]

i*(e*x+d)*Ei((a+b*ln(c*(e*x+d)^n))/b/n)/b/e/exp(a/b/n)/g/n/((c*(e*x+d)^n)^(1/n))+(-f*i+g*h)*Unintegrable(1/(g*
x+f)/(a+b*ln(c*(e*x+d)^n)),x)/g

Rubi [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

[In]

Int[(h + i*x)/((f + g*x)*(a + b*Log[c*(d + e*x)^n])),x]

[Out]

(i*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(b*e*E^(a/(b*n))*g*n*(c*(d + e*x)^n)^n^(-1)) + (
(g*h - f*i)*Defer[Int][1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])), x])/g

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {i}{g \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g h-f i}{g (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx \\ & = \frac {i \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{g}+\frac {(g h-f i) \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx}{g} \\ & = \frac {i \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e g}+\frac {(g h-f i) \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx}{g} \\ & = \frac {(g h-f i) \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx}{g}+\frac {\left (i (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e g n} \\ & = \frac {e^{-\frac {a}{b n}} i (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e g n}+\frac {(g h-f i) \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx}{g} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

[In]

Integrate[(h + i*x)/((f + g*x)*(a + b*Log[c*(d + e*x)^n])),x]

[Out]

Integrate[(h + i*x)/((f + g*x)*(a + b*Log[c*(d + e*x)^n])), x]

Maple [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00

\[\int \frac {i x +h}{\left (g x +f \right ) \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}d x\]

[In]

int((i*x+h)/(g*x+f)/(a+b*ln(c*(e*x+d)^n)),x)

[Out]

int((i*x+h)/(g*x+f)/(a+b*ln(c*(e*x+d)^n)),x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int { \frac {i x + h}{{\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}} \,d x } \]

[In]

integrate((i*x+h)/(g*x+f)/(a+b*log(c*(e*x+d)^n)),x, algorithm="fricas")

[Out]

integral((i*x + h)/(a*g*x + a*f + (b*g*x + b*f)*log((e*x + d)^n*c)), x)

Sympy [N/A]

Not integrable

Time = 2.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int \frac {h + i x}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right ) \left (f + g x\right )}\, dx \]

[In]

integrate((i*x+h)/(g*x+f)/(a+b*ln(c*(e*x+d)**n)),x)

[Out]

Integral((h + i*x)/((a + b*log(c*(d + e*x)**n))*(f + g*x)), x)

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int { \frac {i x + h}{{\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}} \,d x } \]

[In]

integrate((i*x+h)/(g*x+f)/(a+b*log(c*(e*x+d)^n)),x, algorithm="maxima")

[Out]

integrate((i*x + h)/((g*x + f)*(b*log((e*x + d)^n*c) + a)), x)

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int { \frac {i x + h}{{\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}} \,d x } \]

[In]

integrate((i*x+h)/(g*x+f)/(a+b*log(c*(e*x+d)^n)),x, algorithm="giac")

[Out]

integrate((i*x + h)/((g*x + f)*(b*log((e*x + d)^n*c) + a)), x)

Mupad [N/A]

Not integrable

Time = 1.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int \frac {h+i\,x}{\left (f+g\,x\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )} \,d x \]

[In]

int((h + i*x)/((f + g*x)*(a + b*log(c*(d + e*x)^n))),x)

[Out]

int((h + i*x)/((f + g*x)*(a + b*log(c*(d + e*x)^n))), x)