\(\int \frac {1}{(a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}})^2 (d^2-e^2 x^2)} \, dx\) [556]

Optimal result

Integrand size = 47, antiderivative size = 47 \[ \int \frac {1}{\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2 \left (d^2-e^2 x^2\right )} \, dx=\text {Int}\left (\frac {1}{\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2 \left (d^2-e^2 x^2\right )},x\right ) \]

[Out]

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

[In]

Int[1/((a + b*F^((c*Sqrt[d + e*x])/Sqrt[d*f - e*f*x]))^2*(d^2 - e^2*x^2)),x]

[Out]

Defer[Int][1/((a + b*F^((c*Sqrt[d + e*x])/Sqrt[d*f - e*f*x]))^2*(d^2 - e^2*x^2)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2 \left (d^2-e^2 x^2\right )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.00 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2 \left (d^2-e^2 x^2\right )} \, dx=\int \frac {1}{\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2 \left (d^2-e^2 x^2\right )} \, dx \]

[In]

Integrate[1/((a + b*F^((c*Sqrt[d + e*x])/Sqrt[d*f - e*f*x]))^2*(d^2 - e^2*x^2)),x]

[Out]

Integrate[1/((a + b*F^((c*Sqrt[d + e*x])/Sqrt[d*f - e*f*x]))^2*(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

\[\int \frac {1}{\left (a +b \,F^{\frac {c \sqrt {e x +d}}{\sqrt {-e f x +d f}}}\right )^{2} \left (-e^{2} x^{2}+d^{2}\right )}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 2.34 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.85 \[ \int \frac {1}{\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2 \left (d^2-e^2 x^2\right )} \, dx=\int { -\frac {1}{{\left (e^{2} x^{2} - d^{2}\right )} {\left (F^{\frac {\sqrt {e x + d} c}{\sqrt {-e f x + d f}}} b + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.02 \[ \int \frac {1}{\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2 \left (d^2-e^2 x^2\right )} \, dx=\int \frac {1}{\left (d^{2} - e^{2} x^{2}\right ) \left (F^{\frac {c \sqrt {d + e x}}{\sqrt {d f - e f x}}} b + a\right )^{2}} \, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 187, normalized size of antiderivative = 3.98 \[ \int \frac {1}{\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2 \left (d^2-e^2 x^2\right )} \, dx=\int { -\frac {1}{{\left (e^{2} x^{2} - d^{2}\right )} {\left (F^{\frac {\sqrt {e x + d} c}{\sqrt {-e f x + d f}}} b + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 6.72 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2 \left (d^2-e^2 x^2\right )} \, dx=\int { -\frac {1}{{\left (e^{2} x^{2} - d^{2}\right )} {\left (F^{\frac {\sqrt {e x + d} c}{\sqrt {-e f x + d f}}} b + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.34 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2 \left (d^2-e^2 x^2\right )} \, dx=\int \frac {1}{\left (d^2-e^2\,x^2\right )\,{\left (a+b\,{\mathrm {e}}^{\frac {c\,\ln \left (F\right )\,\sqrt {d+e\,x}}{\sqrt {d\,f-e\,f\,x}}}\right )}^2} \,d x \]

[In]

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

[Out]

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