3.714 \(\int \frac{e^{3 x}}{\sqrt{25+16 e^{2 x}}} \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{32} e^x \sqrt{16 e^{2 x}+25}-\frac{25}{128} \sinh ^{-1}\left (\frac{4 e^x}{5}\right ) \]

[Out]

(E^x*Sqrt[25 + 16*E^(2*x)])/32 - (25*ArcSinh[(4*E^x)/5])/128

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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]

[Out]

(E^x*Sqrt[25 + 16*E^(2*x)])/32 - (25*ArcSinh[(4*E^x)/5])/128

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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]

sqrt(16*exp(2*x) + 25)*exp(x)/32 - 25*asinh(4*exp(x)/5)/128

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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]

[Out]

(E^x*Sqrt[25 + 16*E^(2*x)])/32 - (25*ArcSinh[(4*E^x)/5])/128

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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]

1/32*exp(x)*(25+16*exp(x)^2)^(1/2)-25/128*arcsinh(4/5*exp(x))

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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")

[Out]

25/32*sqrt(16*e^(2*x) + 25)*e^(-x)/((16*e^(2*x) + 25)*e^(-2*x) - 16) - 25/256*lo
g(sqrt(16*e^(2*x) + 25)*e^(-x) + 4) + 25/256*log(sqrt(16*e^(2*x) + 25)*e^(-x) -
4)

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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")

[Out]

1/128*(25*(8*sqrt(16*e^(2*x) + 25)*e^x - 32*e^(2*x) - 25)*log(sqrt(16*e^(2*x) +
25) - 4*e^x) - 4*(32*e^(3*x) + 25*e^x)*sqrt(16*e^(2*x) + 25) + 512*e^(4*x) + 800
*e^(2*x))/(8*sqrt(16*e^(2*x) + 25)*e^x - 32*e^(2*x) - 25)

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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]

Integral(exp(3*x)/sqrt(16*exp(2*x) + 25), x)

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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]

1/32*sqrt(16*e^(2*x) + 25)*e^x + 25/128*ln(sqrt(16*e^(2*x) + 25) - 4*e^x)