\(\int \frac {(a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}})^n}{d^2-e^2 x^2} \, dx\) [550]

Optimal result

Integrand size = 47, antiderivative size = 47 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^n}{d^2-e^2 x^2} \, dx=\text {Int}\left (\frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^n}{d^2-e^2 x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 47, 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 = {} \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^n}{d^2-e^2 x^2} \, dx=\int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^n}{d^2-e^2 x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^n}{d^2-e^2 x^2} \, dx=\int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^n}{d^2-e^2 x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 73.52 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^n}{d^2-e^2 x^2} \, dx=- \int \frac {\left (F^{\frac {c \sqrt {d + e x}}{\sqrt {d f - e f x}}} b + a\right )^{n}}{- d^{2} + e^{2} x^{2}}\, dx \]

[In]

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

[Out]

-Integral((F**(c*sqrt(d + e*x)/sqrt(d*f - e*f*x))*b + a)**n/(-d**2 + e**2*x**2), x)

Maxima [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^n}{d^2-e^2 x^2} \, dx=\int { -\frac {{\left (F^{\frac {\sqrt {e x + d} c}{\sqrt {-e f x + d f}}} b + a\right )}^{n}}{e^{2} x^{2} - d^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.44 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^n}{d^2-e^2 x^2} \, dx=\int { -\frac {{\left (F^{\frac {\sqrt {e x + d} c}{\sqrt {-e f x + d f}}} b + a\right )}^{n}}{e^{2} x^{2} - d^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^n}{d^2-e^2 x^2} \, dx=\int \frac {{\left (a+b\,{\mathrm {e}}^{\frac {c\,\ln \left (F\right )\,\sqrt {d+e\,x}}{\sqrt {d\,f-e\,f\,x}}}\right )}^n}{d^2-e^2\,x^2} \,d x \]

[In]

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

[Out]

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