Optimal. Leaf size=33 \[ \frac{1}{32} e^x \sqrt{16 e^{2 x}+25}-\frac{25}{128} \sinh ^{-1}\left (\frac{4 e^x}{5}\right ) \]
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Rubi [A] time = 0.0574667, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{1}{32} e^x \sqrt{16 e^{2 x}+25}-\frac{25}{128} \sinh ^{-1}\left (\frac{4 e^x}{5}\right ) \]
Antiderivative was successfully verified.
[In] Int[E^(3*x)/Sqrt[25 + 16*E^(2*x)],x]
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Rubi in Sympy [A] time = 8.219, size = 27, normalized size = 0.82 \[ \frac{\sqrt{16 e^{2 x} + 25} e^{x}}{32} - \frac{25 \operatorname{asinh}{\left (\frac{4 e^{x}}{5} \right )}}{128} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(exp(3*x)/(25+16*exp(2*x))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0252025, size = 33, normalized size = 1. \[ \frac{1}{32} e^x \sqrt{16 e^{2 x}+25}-\frac{25}{128} \sinh ^{-1}\left (\frac{4 e^x}{5}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[E^(3*x)/Sqrt[25 + 16*E^(2*x)],x]
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Maple [A] time = 0.016, size = 23, normalized size = 0.7 \[{\frac{{{\rm e}^{x}}}{32}\sqrt{25+16\, \left ({{\rm e}^{x}} \right ) ^{2}}}-{\frac{25}{128}{\it Arcsinh} \left ({\frac{4\,{{\rm e}^{x}}}{5}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(exp(3*x)/(25+16*exp(2*x))^(1/2),x)
[Out]
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Maxima [A] time = 0.792142, size = 100, normalized size = 3.03 \[ \frac{25 \, \sqrt{16 \, e^{\left (2 \, x\right )} + 25} e^{\left (-x\right )}}{32 \,{\left ({\left (16 \, e^{\left (2 \, x\right )} + 25\right )} e^{\left (-2 \, x\right )} - 16\right )}} - \frac{25}{256} \, \log \left (\sqrt{16 \, e^{\left (2 \, x\right )} + 25} e^{\left (-x\right )} + 4\right ) + \frac{25}{256} \, \log \left (\sqrt{16 \, e^{\left (2 \, x\right )} + 25} e^{\left (-x\right )} - 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(3*x)/sqrt(16*e^(2*x) + 25),x, algorithm="maxima")
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Fricas [A] time = 0.259502, size = 138, normalized size = 4.18 \[ \frac{25 \,{\left (8 \, \sqrt{16 \, e^{\left (2 \, x\right )} + 25} e^{x} - 32 \, e^{\left (2 \, x\right )} - 25\right )} \log \left (\sqrt{16 \, e^{\left (2 \, x\right )} + 25} - 4 \, e^{x}\right ) - 4 \,{\left (32 \, e^{\left (3 \, x\right )} + 25 \, e^{x}\right )} \sqrt{16 \, e^{\left (2 \, x\right )} + 25} + 512 \, e^{\left (4 \, x\right )} + 800 \, e^{\left (2 \, x\right )}}{128 \,{\left (8 \, \sqrt{16 \, e^{\left (2 \, x\right )} + 25} e^{x} - 32 \, e^{\left (2 \, x\right )} - 25\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(3*x)/sqrt(16*e^(2*x) + 25),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{3 x}}{\sqrt{16 e^{2 x} + 25}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(exp(3*x)/(25+16*exp(2*x))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.223246, size = 45, normalized size = 1.36 \[ \frac{1}{32} \, \sqrt{16 \, e^{\left (2 \, x\right )} + 25} e^{x} + \frac{25}{128} \,{\rm ln}\left (\sqrt{16 \, e^{\left (2 \, x\right )} + 25} - 4 \, e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(3*x)/sqrt(16*e^(2*x) + 25),x, algorithm="giac")
[Out]