\(\int \frac {(a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}})^n}{d f+(e f+d g) x+e g x^2} \, dx\) [543]

Optimal result

Integrand size = 50, antiderivative size = 50 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}\right )^n}{d f+(e f+d g) x+e g x^2} \, dx=\text {Int}\left (\frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}\right )^n}{d f+(e f+d g) x+e g x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 50, 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 = {} \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}\right )^n}{d f+(e f+d g) x+e g x^2} \, dx=\int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}\right )^n}{d f+(e f+d g) x+e g x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}\right )^n}{d f+(e f+d g) x+e g x^2} \, dx=\int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}\right )^n}{d f+(e f+d g) x+e g x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   alglogextint: unimplemented

Sympy [N/A]

Not integrable

Time = 76.72 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}\right )^n}{d f+(e f+d g) x+e g x^2} \, dx=\int \frac {\left (F^{\frac {c \sqrt {d + e x}}{\sqrt {f + g x}}} b + a\right )^{n}}{\left (d + e x\right ) \left (f + g x\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.61 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}\right )^n}{d f+(e f+d g) x+e g x^2} \, dx=\int { \frac {{\left (F^{\frac {\sqrt {e x + d} c}{\sqrt {g x + f}}} b + a\right )}^{n}}{e g x^{2} + d f + {\left (e f + d g\right )} x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.79 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}\right )^n}{d f+(e f+d g) x+e g x^2} \, dx=\int { \frac {{\left (F^{\frac {\sqrt {e x + d} c}{\sqrt {g x + f}}} b + a\right )}^{n}}{e g x^{2} + d f + {\left (e f + d g\right )} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.58 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}\right )^n}{d f+(e f+d g) x+e g x^2} \, dx=\int \frac {{\left (a+F^{\frac {c\,\sqrt {d+e\,x}}{\sqrt {f+g\,x}}}\,b\right )}^n}{e\,g\,x^2+\left (d\,g+e\,f\right )\,x+d\,f} \,d x \]

[In]

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

[Out]

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