\(\int \frac {(f x)^m (a+b x^n+c x^{2 n})^p}{d+e x^n} \, dx\) [155]

Optimal result

Integrand size = 31, antiderivative size = 31 \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{d+e x^n} \, dx=\text {Int}\left (\frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{d+e x^n},x\right ) \]

[Out]

Unintegrable((f*x)^m*(a+b*x^n+c*x^(2*n))^p/(d+e*x^n),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 31, 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 = {} \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{d+e x^n} \, dx \]

[In]

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

[Out]

Defer[Int][((f*x)^m*(a + b*x^n + c*x^(2*n))^p)/(d + e*x^n), x]

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

Mathematica [N/A]

Not integrable

Time = 1.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{d+e x^n} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int { \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (f x\right )^{m}}{e x^{n} + d} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{d+e x^n} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int { \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (f x\right )^{m}}{e x^{n} + d} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int { \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (f x\right )^{m}}{e x^{n} + d} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 8.70 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int \frac {{\left (f\,x\right )}^m\,{\left (a+b\,x^n+c\,x^{2\,n}\right )}^p}{d+e\,x^n} \,d x \]

[In]

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

[Out]

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