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

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

[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)

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Rubi [A]
time = 0.15, 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 )^2 \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]))^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*} \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\\ \end {align*}

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Mathematica [A]
time = 0.94, 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 )^2 \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]))^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]

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

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)))^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)

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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)))^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)

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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)))^2/(-e^2*x^2+d^2),x, algorithm="fricas")

[Out]

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

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Sympy [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)))**2/(-e**2*x**2+d**2),x)

[Out]

Timed out

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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)))^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 )}^2} \,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)^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)

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