\(\int \frac {(d+e x)^m (a+b x+c x^2)^p}{f+g x} \, dx\) [957]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )^p}{f+g x} \, dx=\text {Int}\left (\frac {(d+e x)^m \left (a+b x+c x^2\right )^p}{f+g x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 27, 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 = {} \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )^p}{f+g x} \, dx=\int \frac {(d+e x)^m \left (a+b x+c x^2\right )^p}{f+g x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )^p}{f+g x} \, dx=\int \frac {(d+e x)^m \left (a+b x+c x^2\right )^p}{f+g x} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )^p}{f+g x} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{p} {\left (e x + d\right )}^{m}}{g x + f} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )^p}{f+g x} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{p} {\left (e x + d\right )}^{m}}{g x + f} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )^p}{f+g x} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{p} {\left (e x + d\right )}^{m}}{g x + f} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 12.51 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )^p}{f+g x} \, dx=\int \frac {{\left (d+e\,x\right )}^m\,{\left (c\,x^2+b\,x+a\right )}^p}{f+g\,x} \,d x \]

[In]

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

[Out]

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