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

Optimal. Leaf size=49 \[ \text{Unintegrable}\left (\frac{1}{\left (d^2-e^2 x^2\right ) \left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{d f-e f x}}}\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.23251, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{\left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{d f-e f x}}}\right ) \left (d^2-e^2 x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

Rubi steps

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

Mathematica [A]  time = 0.4697, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{d f-e f x}}}\right ) \left (d^2-e^2 x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{e}^{2}{x}^{2}+{d}^{2}} \left ( a+b{F}^{{c\sqrt{ex+d}{\frac{1}{\sqrt{-efx+df}}}}} \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\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 )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{1}{a e^{2} x^{2} - a d^{2} + \frac{b e^{2} x^{2} - b d^{2}}{F^{\frac{\sqrt{-e f x + d f} \sqrt{e x + d} c}{e f x - d f}}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{- F^{\frac{c \sqrt{d + e x}}{\sqrt{d f - e f x}}} b d^{2} + F^{\frac{c \sqrt{d + e x}}{\sqrt{d f - e f x}}} b e^{2} x^{2} - a d^{2} + a e^{2} x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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