\(\int \frac {(d+e x)^m}{(f+g x) \sqrt {a+b x+c x^2}} \, dx\) [952]

Optimal result

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

[Out]

Unintegrable((e*x+d)^m/(g*x+f)/(c*x^2+b*x+a)^(1/2),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 = {} \[ \int \frac {(d+e x)^m}{(f+g x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {(d+e x)^m}{(f+g x) \sqrt {a+b x+c x^2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 5.77 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^m}{(f+g x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {(d+e x)^m}{(f+g x) \sqrt {a+b x+c x^2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {(d+e x)^m}{(f+g x) \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{\sqrt {c x^{2} + b x + a} {\left (g x + f\right )}} \,d x } \]

[In]

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

[Out]

integral(sqrt(c*x^2 + b*x + a)*(e*x + d)^m/(c*g*x^3 + (c*f + b*g)*x^2 + a*f + (b*f + a*g)*x), x)

Sympy [N/A]

Not integrable

Time = 1.82 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x)^m}{(f+g x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{m}}{\left (f + g x\right ) \sqrt {a + b x + c x^{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^m}{(f+g x) \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{\sqrt {c x^{2} + b x + a} {\left (g x + f\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^m}{(f+g x) \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{\sqrt {c x^{2} + b x + a} {\left (g x + f\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 73.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^m}{(f+g x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{\left (f+g\,x\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \]

[In]

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

[Out]

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