3.6.55 \(\int \frac {1}{(a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}) (d^2-e^2 x^2)} \, dx\) [555]

Optimal. Leaf size=50 \[ \text {Int}\left (\frac {1}{\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right ) \left (d^2-e^2 x^2\right )},x\right ) \]

[Out]

Unintegrable(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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Rubi [A]
time = 0.16, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 {gather*}

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*}

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Mathematica [A]
time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \,F^{\frac {c \sqrt {e x +d}}{\sqrt {-e f x +d f}}}\right ) \left (-e^{2} x^{2}+d^{2}\right )}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 {gather*}

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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \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 )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)),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)))), x)

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