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

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

[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 = 0.22, 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 = {} \[ \int \frac {1}{\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2 \left (d^2-e^2 x^2\right )} \, dx \]

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 = 1.40, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2 \left (d^2-e^2 x^2\right )} \, dx \]

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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fricas [A]  time = 3.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{a^{2} e^{2} x^{2} - a^{2} d^{2} + \frac {2 \, {\left (a b e^{2} x^{2} - a b d^{2}\right )}}{F^{\frac {\sqrt {-e f x + d f} \sqrt {e x + d} c}{e f x - d f}}} + \frac {b^{2} e^{2} x^{2} - b^{2} d^{2}}{F^{\frac {2 \, \sqrt {-e f x + d f} \sqrt {e x + d} c}{e f x - d f}}}}, x\right ) \]

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*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)

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

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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maple [A]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,F^{\frac {\sqrt {e x +d}\, c}{\sqrt {-e f x +d f}}}+a \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\left (d^2-e^2\,x^2\right )\,{\left (a+b\,{\mathrm {e}}^{\frac {c\,\ln \relax (F)\,\sqrt {d+e\,x}}{\sqrt {d\,f-e\,f\,x}}}\right )}^2} \,d x \]

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

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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