3.295 \(\int \left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right ) \, dx\)

Optimal. Leaf size=68 \[ \frac{1}{2} f x \sqrt{a+\frac{e^2 x^2}{f^2}}+\frac{a f^2 \tanh ^{-1}\left (\frac{e x}{f \sqrt{a+\frac{e^2 x^2}{f^2}}}\right )}{2 e}+d x+\frac{e x^2}{2} \]

[Out]

d*x + (e*x^2)/2 + (f*x*Sqrt[a + (e^2*x^2)/f^2])/2 + (a*f^2*ArcTanh[(e*x)/(f*Sqrt
[a + (e^2*x^2)/f^2])])/(2*e)

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Rubi [A]  time = 0.0699131, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{1}{2} f x \sqrt{a+\frac{e^2 x^2}{f^2}}+\frac{a f^2 \tanh ^{-1}\left (\frac{e x}{f \sqrt{a+\frac{e^2 x^2}{f^2}}}\right )}{2 e}+d x+\frac{e x^2}{2} \]

Antiderivative was successfully verified.

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

[Out]

d*x + (e*x^2)/2 + (f*x*Sqrt[a + (e^2*x^2)/f^2])/2 + (a*f^2*ArcTanh[(e*x)/(f*Sqrt
[a + (e^2*x^2)/f^2])])/(2*e)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a f^{2} \operatorname{atanh}{\left (\frac{e x}{f \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}} \right )}}{2 e} + e \int x\, dx + \frac{f x \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}}{2} + \int d\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(d+e*x+f*(a+e**2*x**2/f**2)**(1/2),x)

[Out]

a*f**2*atanh(e*x/(f*sqrt(a + e**2*x**2/f**2)))/(2*e) + e*Integral(x, x) + f*x*sq
rt(a + e**2*x**2/f**2)/2 + Integral(d, x)

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Mathematica [A]  time = 0.0794617, size = 81, normalized size = 1.19 \[ \frac{1}{2} f x \sqrt{\frac{a f^2+e^2 x^2}{f^2}}+\frac{a f^2 \log \left (e f \sqrt{\frac{a f^2+e^2 x^2}{f^2}}+e^2 x\right )}{2 e}+d x+\frac{e x^2}{2} \]

Antiderivative was successfully verified.

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

[Out]

d*x + (e*x^2)/2 + (f*x*Sqrt[(a*f^2 + e^2*x^2)/f^2])/2 + (a*f^2*Log[e^2*x + e*f*S
qrt[(a*f^2 + e^2*x^2)/f^2]])/(2*e)

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Maple [A]  time = 0.006, size = 75, normalized size = 1.1 \[ dx+{\frac{e{x}^{2}}{2}}+{\frac{fx}{2}\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}}}+{\frac{fa}{2}\ln \left ({\frac{{e}^{2}x}{{f}^{2}}{\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}}+\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ){\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(d+e*x+f*(a+e^2*x^2/f^2)^(1/2),x)

[Out]

d*x+1/2*e*x^2+1/2*f*x*(a+e^2*x^2/f^2)^(1/2)+1/2*f*a*ln(e^2*x/f^2/(1/f^2*e^2)^(1/
2)+(a+e^2*x^2/f^2)^(1/2))/(1/f^2*e^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282706, size = 100, normalized size = 1.47 \[ \frac{e^{2} x^{2} - a f^{2} \log \left (-e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + e f x \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + 2 \, d e x}{2 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(e^2*x^2 - a*f^2*log(-e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2)) + e*f*x*sqrt((e^2
*x^2 + a*f^2)/f^2) + 2*d*e*x)/e

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Sympy [A]  time = 7.18205, size = 54, normalized size = 0.79 \[ d x + \frac{e x^{2}}{2} + f \left (\frac{\sqrt{a} x \sqrt{1 + \frac{e^{2} x^{2}}{a f^{2}}}}{2} + \frac{a f \operatorname{asinh}{\left (\frac{e x}{\sqrt{a} f} \right )}}{2 e}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d+e*x+f*(a+e**2*x**2/f**2)**(1/2),x)

[Out]

d*x + e*x**2/2 + f*(sqrt(a)*x*sqrt(1 + e**2*x**2/(a*f**2))/2 + a*f*asinh(e*x/(sq
rt(a)*f))/(2*e))

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GIAC/XCAS [A]  time = 0.282809, size = 88, normalized size = 1.29 \[ \frac{1}{2} \, x^{2} e + d x - \frac{{\left (a f^{2} e^{\left (-1\right )}{\rm ln}\left ({\left | -x e + \sqrt{a f^{2} + x^{2} e^{2}} \right |}\right ) - \sqrt{a f^{2} + x^{2} e^{2}} x\right )}{\left | f \right |}}{2 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2*e + d*x - 1/2*(a*f^2*e^(-1)*ln(abs(-x*e + sqrt(a*f^2 + x^2*e^2))) - sqrt
(a*f^2 + x^2*e^2)*x)*abs(f)/f