\(\int \frac {h+i x}{(f+g x) (a+b \log (c (d+e x)^n))^2} \, dx\) [238]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {e^{-\frac {a}{b n}} i (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e g n^2}-\frac {i (d+e x)}{b e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {(g h-f i) \text {Int}\left (\frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{g} \]

[Out]

i*(e*x+d)*Ei((a+b*ln(c*(e*x+d)^n))/b/n)/b^2/e/exp(a/b/n)/g/n^2/((c*(e*x+d)^n)^(1/n))-i*(e*x+d)/b/e/g/n/(a+b*ln
(c*(e*x+d)^n))+(-f*i+g*h)*Unintegrable(1/(g*x+f)/(a+b*ln(c*(e*x+d)^n))^2,x)/g

Rubi [N/A]

Not integrable

Time = 0.14 (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 = {} \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

[In]

Int[(h + i*x)/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^2),x]

[Out]

(i*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(b^2*e*E^(a/(b*n))*g*n^2*(c*(d + e*x)^n)^n^(-1))
- (i*(d + e*x))/(b*e*g*n*(a + b*Log[c*(d + e*x)^n])) + ((g*h - f*i)*Defer[Int][1/((f + g*x)*(a + b*Log[c*(d +
e*x)^n])^2), 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 )^2}+\frac {g h-f i}{g (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\right ) \, dx \\ & = \frac {i \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, 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 )^2} \, dx}{g} \\ & = \frac {i \text {Subst}\left (\int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, 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 )^2} \, dx}{g} \\ & = -\frac {i (d+e x)}{b e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {(g h-f i) \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{g}+\frac {i \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e g n} \\ & = -\frac {i (d+e x)}{b e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {(g h-f i) \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, 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 )}{b e g n^2} \\ & = \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^2 e g n^2}-\frac {i (d+e x)}{b e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {(g h-f i) \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{g} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

[In]

Integrate[(h + i*x)/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^2),x]

[Out]

Integrate[(h + i*x)/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^2), x]

Maple [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00

\[\int \frac {i x +h}{\left (g x +f \right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}d x\]

[In]

int((i*x+h)/(g*x+f)/(a+b*ln(c*(e*x+d)^n))^2,x)

[Out]

int((i*x+h)/(g*x+f)/(a+b*ln(c*(e*x+d)^n))^2,x)

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.38 \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {i x + h}{{\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((i*x+h)/(g*x+f)/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="fricas")

[Out]

integral((i*x + h)/(a^2*g*x + a^2*f + (b^2*g*x + b^2*f)*log((e*x + d)^n*c)^2 + 2*(a*b*g*x + a*b*f)*log((e*x +
d)^n*c)), x)

Sympy [N/A]

Not integrable

Time = 7.96 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {h + i x}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2} \left (f + g x\right )}\, dx \]

[In]

integrate((i*x+h)/(g*x+f)/(a+b*ln(c*(e*x+d)**n))**2,x)

[Out]

Integral((h + i*x)/((a + b*log(c*(d + e*x)**n))**2*(f + g*x)), x)

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 223, normalized size of antiderivative = 7.69 \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {i x + h}{{\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((i*x+h)/(g*x+f)/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="maxima")

[Out]

-(e*i*x^2 + d*h + (e*h + d*i)*x)/(b^2*e*f*n*log(c) + a*b*e*f*n + (b^2*e*g*n*log(c) + a*b*e*g*n)*x + (b^2*e*g*n
*x + b^2*e*f*n)*log((e*x + d)^n)) + integrate((e*g*i*x^2 + 2*e*f*i*x + e*f*h - (g*h - f*i)*d)/(b^2*e*f^2*n*log
(c) + a*b*e*f^2*n + (b^2*e*g^2*n*log(c) + a*b*e*g^2*n)*x^2 + 2*(b^2*e*f*g*n*log(c) + a*b*e*f*g*n)*x + (b^2*e*g
^2*n*x^2 + 2*b^2*e*f*g*n*x + b^2*e*f^2*n)*log((e*x + d)^n)), x)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {i x + h}{{\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((i*x+h)/(g*x+f)/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="giac")

[Out]

integrate((i*x + h)/((g*x + f)*(b*log((e*x + d)^n*c) + a)^2), x)

Mupad [N/A]

Not integrable

Time = 1.43 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {h+i\,x}{\left (f+g\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \]

[In]

int((h + i*x)/((f + g*x)*(a + b*log(c*(d + e*x)^n))^2),x)

[Out]

int((h + i*x)/((f + g*x)*(a + b*log(c*(d + e*x)^n))^2), x)