### 3.155 $$\int \frac{(f x)^m (a+b x^n+c x^{2 n})^p}{d+e x^n} \, dx$$

Optimal. Leaf size=33 $\text{Unintegrable}\left (\frac{(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{d+e x^n},x\right )$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0255954, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0., Rules used = {} $\int \frac{(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{d+e x^n} \, dx$

Veriﬁcation is Not applicable to the result.

[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*} \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\\ \end{align*}

Mathematica [A]  time = 0.208149, size = 0, normalized size = 0. $\int \frac{(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{d+e x^n} \, dx$

Veriﬁcation is Not applicable to the result.

[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 [A]  time = 0.073, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{m} \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{p}}{d+e{x}^{n}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (f x\right )^{m}}{e x^{n} + d}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (f x\right )^{m}}{e x^{n} + d}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (f x\right )^{m}}{e x^{n} + d}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)