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

Optimal. Leaf size=110 \[ \frac{a^2 \log \left (\frac{\sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{2 a b \text{ExpIntegralEi}\left (\frac{c \log (F) \sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{b^2 \text{ExpIntegralEi}\left (\frac{2 c \log (F) \sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e} \]

[Out]

(2*a*b*ExpIntegralEi[(c*Sqrt[d + e*x]*Log[F])/Sqrt[d*f - e*f*x]])/(d*e) + (b^2*E
xpIntegralEi[(2*c*Sqrt[d + e*x]*Log[F])/Sqrt[d*f - e*f*x]])/(d*e) + (a^2*Log[Sqr
t[d + e*x]/Sqrt[d*f - e*f*x]])/(d*e)

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Rubi [A]  time = 0.483039, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.064 \[ \frac{a^2 \log \left (\frac{\sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{2 a b \text{ExpIntegralEi}\left (\frac{c \log (F) \sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{b^2 \text{ExpIntegralEi}\left (\frac{2 c \log (F) \sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*b*ExpIntegralEi[(c*Sqrt[d + e*x]*Log[F])/Sqrt[d*f - e*f*x]])/(d*e) + (b^2*E
xpIntegralEi[(2*c*Sqrt[d + e*x]*Log[F])/Sqrt[d*f - e*f*x]])/(d*e) + (a^2*Log[Sqr
t[d + e*x]/Sqrt[d*f - e*f*x]])/(d*e)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((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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Mathematica [A]  time = 0.598847, size = 0, normalized size = 0. \[ \int \frac{\left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{d f-e f x}}}\right )^2}{d^2-e^2 x^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(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 [F]  time = 0.026, size = 0, normalized size = 0. \[ \int{\frac{1}{-{e}^{2}{x}^{2}+{d}^{2}} \left ( a+b{F}^{{c\sqrt{ex+d}{\frac{1}{\sqrt{-efx+df}}}}} \right ) ^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (F^{\frac{\sqrt{e x + d} c}{\sqrt{-e f x + d f}}} b + a\right )}^{2}}{e^{2} x^{2} - d^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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