\(\int \frac {1}{(d+e x^3)^2 (a+b \log (c x^n))^2} \, dx\) [328]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{\left (d+e x^3\right )^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{\left (d+e x^3\right )^2 \left (a+b \log \left (c x^n\right )\right )^2},x\right ) \]

[Out]

Unintegrable(1/(e*x^3+d)^2/(a+b*ln(c*x^n))^2,x)

Rubi [N/A]

Not integrable

Time = 0.02 (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 = {} \[ \int \frac {1}{\left (d+e x^3\right )^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{\left (d+e x^3\right )^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx \]

[In]

Int[1/((d + e*x^3)^2*(a + b*Log[c*x^n])^2),x]

[Out]

Defer[Int][1/((d + e*x^3)^2*(a + b*Log[c*x^n])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (d+e x^3\right )^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 17.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (d+e x^3\right )^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{\left (d+e x^3\right )^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx \]

[In]

Integrate[1/((d + e*x^3)^2*(a + b*Log[c*x^n])^2),x]

[Out]

Integrate[1/((d + e*x^3)^2*(a + b*Log[c*x^n])^2), x]

Maple [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (e \,x^{3}+d \right )^{2} {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}d x\]

[In]

int(1/(e*x^3+d)^2/(a+b*ln(c*x^n))^2,x)

[Out]

int(1/(e*x^3+d)^2/(a+b*ln(c*x^n))^2,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.64 \[ \int \frac {1}{\left (d+e x^3\right )^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (e x^{3} + d\right )}^{2} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(e*x^3+d)^2/(a+b*log(c*x^n))^2,x, algorithm="fricas")

[Out]

integral(1/(a^2*e^2*x^6 + 2*a^2*d*e*x^3 + a^2*d^2 + (b^2*e^2*x^6 + 2*b^2*d*e*x^3 + b^2*d^2)*log(c*x^n)^2 + 2*(
a*b*e^2*x^6 + 2*a*b*d*e*x^3 + a*b*d^2)*log(c*x^n)), x)

Sympy [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.05 \[ \int \frac {1}{\left (d+e x^3\right )^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \log {\left (c x^{n} \right )}\right )^{2} \left (d + e x^{3}\right )^{2}} \, dx \]

[In]

integrate(1/(e*x**3+d)**2/(a+b*ln(c*x**n))**2,x)

[Out]

integrate(1/(e*x**3+d)**2/(a+b*ln(c*x**n))**2,x)

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 261, normalized size of antiderivative = 11.86 \[ \int \frac {1}{\left (d+e x^3\right )^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (e x^{3} + d\right )}^{2} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(e*x^3+d)^2/(a+b*log(c*x^n))^2,x, algorithm="maxima")

[Out]

-x/((b^2*e^2*n*log(c) + a*b*e^2*n)*x^6 + b^2*d^2*n*log(c) + a*b*d^2*n + 2*(b^2*d*e*n*log(c) + a*b*d*e*n)*x^3 +
(b^2*e^2*n*x^6 + 2*b^2*d*e*n*x^3 + b^2*d^2*n)*log(x^n)) - integrate((5*e*x^3 - d)/((b^2*e^3*n*log(c) + a*b*e^
3*n)*x^9 + 3*(b^2*d*e^2*n*log(c) + a*b*d*e^2*n)*x^6 + b^2*d^3*n*log(c) + a*b*d^3*n + 3*(b^2*d^2*e*n*log(c) + a
*b*d^2*e*n)*x^3 + (b^2*e^3*n*x^9 + 3*b^2*d*e^2*n*x^6 + 3*b^2*d^2*e*n*x^3 + b^2*d^3*n)*log(x^n)), x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (d+e x^3\right )^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (e x^{3} + d\right )}^{2} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(e*x^3+d)^2/(a+b*log(c*x^n))^2,x, algorithm="giac")

[Out]

integrate(1/((e*x^3 + d)^2*(b*log(c*x^n) + a)^2), x)

Mupad [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (d+e x^3\right )^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{{\left (e\,x^3+d\right )}^2\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]

[In]

int(1/((d + e*x^3)^2*(a + b*log(c*x^n))^2),x)

[Out]

int(1/((d + e*x^3)^2*(a + b*log(c*x^n))^2), x)