\(\int \frac {1}{(d+e x^2) \log (c (a+b x)^n)} \, dx\) [332]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=-\frac {\text {Int}\left (\frac {1}{\left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (c (a+b x)^n\right )},x\right )}{2 \sqrt {-d}}-\frac {\text {Int}\left (\frac {1}{\left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (c (a+b x)^n\right )},x\right )}{2 \sqrt {-d}} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int \frac {1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int \frac {1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )} \log \left ({\left (b x + a\right )}^{n} c\right )} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 21.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int \frac {1}{\left (d + e x^{2}\right ) \log {\left (c \left (a + b x\right )^{n} \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )} \log \left ({\left (b x + a\right )}^{n} c\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )} \log \left ({\left (b x + a\right )}^{n} c\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx=\int \frac {1}{\ln \left (c\,{\left (a+b\,x\right )}^n\right )\,\left (e\,x^2+d\right )} \,d x \]

[In]

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

[Out]

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