\(\int \frac {1}{(d+e x+f x^2) \log (c (a+b x)^n)} \, dx\) [350]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {1}{\left (d+e x+f x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\frac {2 f \text {Int}\left (\frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \log \left (c (a+b x)^n\right )},x\right )}{\sqrt {e^2-4 d f}}-\frac {2 f \text {Int}\left (\frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \log \left (c (a+b x)^n\right )},x\right )}{\sqrt {e^2-4 d f}} \]

[Out]

2*f*Unintegrable(1/ln(c*(b*x+a)^n)/(e+2*f*x-(-4*d*f+e^2)^(1/2)),x)/(-4*d*f+e^2)^(1/2)-2*f*Unintegrable(1/ln(c*
(b*x+a)^n)/(e+2*f*x+(-4*d*f+e^2)^(1/2)),x)/(-4*d*f+e^2)^(1/2)

Rubi [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {1}{\left (d+e x+f x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int \frac {1}{\left (d+e x+f x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]

[In]

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

[Out]

(2*f*Defer[Int][1/((e - Sqrt[e^2 - 4*d*f] + 2*f*x)*Log[c*(a + b*x)^n]), x])/Sqrt[e^2 - 4*d*f] - (2*f*Defer[Int
][1/((e + Sqrt[e^2 - 4*d*f] + 2*f*x)*Log[c*(a + b*x)^n]), x])/Sqrt[e^2 - 4*d*f]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 f}{\sqrt {e^2-4 d f} \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \log \left (c (a+b x)^n\right )}-\frac {2 f}{\sqrt {e^2-4 d f} \left (e+\sqrt {e^2-4 d f}+2 f x\right ) \log \left (c (a+b x)^n\right )}\right ) \, dx \\ & = \frac {(2 f) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \log \left (c (a+b x)^n\right )} \, dx}{\sqrt {e^2-4 d f}}-\frac {(2 f) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \log \left (c (a+b x)^n\right )} \, dx}{\sqrt {e^2-4 d f}} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.62 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (d+e x+f x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int \frac {1}{\left (d+e x+f x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.62 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (d+e x+f x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int { \frac {1}{{\left (f x^{2} + e x + d\right )} \log \left ({\left (b x + a\right )}^{n} c\right )} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x+f x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (d+e x+f x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int { \frac {1}{{\left (f x^{2} + e x + d\right )} \log \left ({\left (b x + a\right )}^{n} c\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (d+e x+f x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int { \frac {1}{{\left (f x^{2} + e x + d\right )} \log \left ({\left (b x + a\right )}^{n} c\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (d+e x+f x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int \frac {1}{\ln \left (c\,{\left (a+b\,x\right )}^n\right )\,\left (f\,x^2+e\,x+d\right )} \,d x \]

[In]

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

[Out]

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