\(\int \frac {1}{(d x+e x^2) \log (c (a+b x)^n)} \, dx\) [346]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{\left (d x+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\frac {\text {Int}\left (\frac {1}{x \log \left (c (a+b x)^n\right )},x\right )}{d}-\frac {e \text {Int}\left (\frac {1}{(d+e x) \log \left (c (a+b x)^n\right )},x\right )}{d} \]

[Out]

Unintegrable(1/x/ln(c*(b*x+a)^n),x)/d-e*Unintegrable(1/(e*x+d)/ln(c*(b*x+a)^n),x)/d

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {1}{\left (d x+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int \frac {1}{\left (d x+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]

[In]

Int[1/((d*x + e*x^2)*Log[c*(a + b*x)^n]),x]

[Out]

Defer[Int][1/(x*Log[c*(a + b*x)^n]), x]/d - (e*Defer[Int][1/((d + e*x)*Log[c*(a + b*x)^n]), x])/d

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

Mathematica [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (d x+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int \frac {1}{\left (d x+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]

[In]

Integrate[1/((d*x + e*x^2)*Log[c*(a + b*x)^n]),x]

[Out]

Integrate[1/((d*x + e*x^2)*Log[c*(a + b*x)^n]), x]

Maple [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (e \,x^{2}+d x \right ) \ln \left (c \left (b x +a \right )^{n}\right )}d x\]

[In]

int(1/(e*x^2+d*x)/ln(c*(b*x+a)^n),x)

[Out]

int(1/(e*x^2+d*x)/ln(c*(b*x+a)^n),x)

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (d x+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int { \frac {1}{{\left (e x^{2} + d x\right )} \log \left ({\left (b x + a\right )}^{n} c\right )} \,d x } \]

[In]

integrate(1/(e*x^2+d*x)/log(c*(b*x+a)^n),x, algorithm="fricas")

[Out]

integral(1/((e*x^2 + d*x)*log((b*x + a)^n*c)), x)

Sympy [N/A]

Not integrable

Time = 0.98 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (d x+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int \frac {1}{x \left (d + e x\right ) \log {\left (c \left (a + b x\right )^{n} \right )}}\, dx \]

[In]

integrate(1/(e*x**2+d*x)/ln(c*(b*x+a)**n),x)

[Out]

Integral(1/(x*(d + e*x)*log(c*(a + b*x)**n)), x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (d x+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int { \frac {1}{{\left (e x^{2} + d x\right )} \log \left ({\left (b x + a\right )}^{n} c\right )} \,d x } \]

[In]

integrate(1/(e*x^2+d*x)/log(c*(b*x+a)^n),x, algorithm="maxima")

[Out]

integrate(1/((e*x^2 + d*x)*log((b*x + a)^n*c)), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (d x+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int { \frac {1}{{\left (e x^{2} + d x\right )} \log \left ({\left (b x + a\right )}^{n} c\right )} \,d x } \]

[In]

integrate(1/(e*x^2+d*x)/log(c*(b*x+a)^n),x, algorithm="giac")

[Out]

integrate(1/((e*x^2 + d*x)*log((b*x + a)^n*c)), x)

Mupad [N/A]

Not integrable

Time = 1.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (d x+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int \frac {1}{\ln \left (c\,{\left (a+b\,x\right )}^n\right )\,\left (e\,x^2+d\,x\right )} \,d x \]

[In]

int(1/(log(c*(a + b*x)^n)*(d*x + e*x^2)),x)

[Out]

int(1/(log(c*(a + b*x)^n)*(d*x + e*x^2)), x)