\(\int \frac {(d+e x)^m \sqrt {a+b x+c x^2}}{f+g x} \, dx\) [948]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {(d+e x)^m \sqrt {a+b x+c x^2}}{f+g x} \, dx=\text {Int}\left (\frac {(d+e x)^m \sqrt {a+b x+c x^2}}{f+g x},x\right ) \]

[Out]

Unintegrable((e*x+d)^m*(c*x^2+b*x+a)^(1/2)/(g*x+f),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 29, 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 = {} \[ \int \frac {(d+e x)^m \sqrt {a+b x+c x^2}}{f+g x} \, dx=\int \frac {(d+e x)^m \sqrt {a+b x+c x^2}}{f+g x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^m \sqrt {a+b x+c x^2}}{f+g x} \, dx=\int \frac {(d+e x)^m \sqrt {a+b x+c x^2}}{f+g x} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93

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

[In]

int((e*x+d)^m*(c*x^2+b*x+a)^(1/2)/(g*x+f),x)

[Out]

int((e*x+d)^m*(c*x^2+b*x+a)^(1/2)/(g*x+f),x)

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^m \sqrt {a+b x+c x^2}}{f+g x} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{m}}{g x + f} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.41 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x)^m \sqrt {a+b x+c x^2}}{f+g x} \, dx=\int \frac {\left (d + e x\right )^{m} \sqrt {a + b x + c x^{2}}}{f + g x}\, dx \]

[In]

integrate((e*x+d)**m*(c*x**2+b*x+a)**(1/2)/(g*x+f),x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^m \sqrt {a+b x+c x^2}}{f+g x} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{m}}{g x + f} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^m \sqrt {a+b x+c x^2}}{f+g x} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{m}}{g x + f} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 12.56 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^m \sqrt {a+b x+c x^2}}{f+g x} \, dx=\int \frac {{\left (d+e\,x\right )}^m\,\sqrt {c\,x^2+b\,x+a}}{f+g\,x} \,d x \]

[In]

int(((d + e*x)^m*(a + b*x + c*x^2)^(1/2))/(f + g*x),x)

[Out]

int(((d + e*x)^m*(a + b*x + c*x^2)^(1/2))/(f + g*x), x)