Optimal. Leaf size=152 \[ \frac{a^3 \log \left (\frac{\sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{3 a^2 b \text{ExpIntegralEi}\left (\frac{c \log (F) \sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{3 a b^2 \text{ExpIntegralEi}\left (\frac{2 c \log (F) \sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{b^3 \text{ExpIntegralEi}\left (\frac{3 c \log (F) \sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e} \]
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Rubi [A] time = 0.514673, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.064 \[ \frac{a^3 \log \left (\frac{\sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{3 a^2 b \text{ExpIntegralEi}\left (\frac{c \log (F) \sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{3 a b^2 \text{ExpIntegralEi}\left (\frac{2 c \log (F) \sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{b^3 \text{ExpIntegralEi}\left (\frac{3 c \log (F) \sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e} \]
Antiderivative was successfully verified.
[In] Int[(a + b*F^((c*Sqrt[d + e*x])/Sqrt[d*f - e*f*x]))^3/(d^2 - e^2*x^2),x]
[Out]
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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)))**3/(-e**2*x**2+d**2),x)
[Out]
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Mathematica [A] time = 0.876311, size = 0, normalized size = 0. \[ \int \frac{\left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{d f-e f x}}}\right )^3}{d^2-e^2 x^2} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(a + b*F^((c*Sqrt[d + e*x])/Sqrt[d*f - e*f*x]))^3/(d^2 - e^2*x^2),x]
[Out]
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Maple [F] time = 0.024, size = 0, normalized size = 0. \[ \int{\frac{1}{-{e}^{2}{x}^{2}+{d}^{2}} \left ( a+b{F}^{{c\sqrt{ex+d}{\frac{1}{\sqrt{-efx+df}}}}} \right ) ^{3}}\, dx \]
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)))^3/(-e^2*x^2+d^2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{1}{2} \, a^{3}{\left (\frac{\log \left (e x + d\right )}{d e} - \frac{\log \left (e x - d\right )}{d e}\right )} - b^{3} \int \frac{F^{\frac{3 \, \sqrt{e x + d} c}{\sqrt{-e x + d} \sqrt{f}}}}{e^{2} x^{2} - d^{2}}\,{d x} - 3 \, a b^{2} \int \frac{F^{\frac{2 \, \sqrt{e x + d} c}{\sqrt{-e x + d} \sqrt{f}}}}{e^{2} x^{2} - d^{2}}\,{d x} - 3 \, a^{2} b \int \frac{F^{\frac{\sqrt{e x + d} c}{\sqrt{-e x + d} \sqrt{f}}}}{e^{2} x^{2} - d^{2}}\,{d x} \]
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)^3/(e^2*x^2 - d^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{3 \, F^{\frac{\sqrt{e x + d} c}{\sqrt{-e f x + d f}}} a^{2} b + 3 \, F^{\frac{2 \, \sqrt{e x + d} c}{\sqrt{-e f x + d f}}} a b^{2} + F^{\frac{3 \, \sqrt{e x + d} c}{\sqrt{-e f x + d f}}} b^{3} + a^{3}}{e^{2} x^{2} - d^{2}}, x\right ) \]
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)^3/(e^2*x^2 - d^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{a^{3}}{- d^{2} + e^{2} x^{2}}\, dx - \int \frac{F^{\frac{3 c \sqrt{d + e x}}{\sqrt{d f - e f x}}} b^{3}}{- d^{2} + e^{2} x^{2}}\, dx - \int \frac{3 F^{\frac{c \sqrt{d + e x}}{\sqrt{d f - e f x}}} a^{2} b}{- d^{2} + e^{2} x^{2}}\, dx - \int \frac{3 F^{\frac{2 c \sqrt{d + e x}}{\sqrt{d f - e f x}}} a b^{2}}{- d^{2} + e^{2} x^{2}}\, dx \]
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)))**3/(-e**2*x**2+d**2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (F^{\frac{\sqrt{e x + d} c}{\sqrt{-e f x + d f}}} b + a\right )}^{3}}{e^{2} x^{2} - d^{2}}\,{d x} \]
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)^3/(e^2*x^2 - d^2),x, algorithm="giac")
[Out]