3.553 \(\int \frac{d+e x}{\sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=43 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c}}+\frac{e \sqrt{a+c x^2}}{c} \]

[Out]

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

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Rubi [A]  time = 0.0479933, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c}}+\frac{e \sqrt{a+c x^2}}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.80699, size = 37, normalized size = 0.86 \[ \frac{e \sqrt{a + c x^{2}}}{c} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{\sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*sqrt(a + c*x**2)/c + d*atanh(sqrt(c)*x/sqrt(a + c*x**2))/sqrt(c)

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Mathematica [A]  time = 0.0312236, size = 46, normalized size = 1.07 \[ \frac{d \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{\sqrt{c}}+\frac{e \sqrt{a+c x^2}}{c} \]

Antiderivative was successfully verified.

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

[Out]

(e*Sqrt[a + c*x^2])/c + (d*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/Sqrt[c]

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Maple [A]  time = 0.006, size = 37, normalized size = 0.9 \[{d\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{e}{c}\sqrt{c{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^2+a)^(1/2),x)

[Out]

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

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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 + d)/sqrt(c*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242846, size = 1, normalized size = 0.02 \[ \left [\frac{c d \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \, \sqrt{c x^{2} + a} \sqrt{c} e}{2 \, c^{\frac{3}{2}}}, \frac{c d \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) + \sqrt{c x^{2} + a} \sqrt{-c} e}{\sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(c*d*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)) + 2*sqrt(c*x^2 + a
)*sqrt(c)*e)/c^(3/2), (c*d*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + sqrt(c*x^2 + a)*
sqrt(-c)*e)/(sqrt(-c)*c)]

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Sympy [A]  time = 2.75853, size = 102, normalized size = 2.37 \[ d \left (\begin{cases} \frac{\sqrt{- \frac{a}{c}} \operatorname{asin}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c < 0 \\\frac{\sqrt{\frac{a}{c}} \operatorname{asinh}{\left (x \sqrt{\frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c > 0 \\\frac{\sqrt{- \frac{a}{c}} \operatorname{acosh}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{- a}} & \text{for}\: c > 0 \wedge a < 0 \end{cases}\right ) + e \left (\begin{cases} \frac{x^{2}}{2 \sqrt{a}} & \text{for}\: c = 0 \\\frac{\sqrt{a + c x^{2}}}{c} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*Piecewise((sqrt(-a/c)*asin(x*sqrt(-c/a))/sqrt(a), (a > 0) & (c < 0)), (sqrt(a/
c)*asinh(x*sqrt(c/a))/sqrt(a), (a > 0) & (c > 0)), (sqrt(-a/c)*acosh(x*sqrt(-c/a
))/sqrt(-a), (c > 0) & (a < 0))) + e*Piecewise((x**2/(2*sqrt(a)), Eq(c, 0)), (sq
rt(a + c*x**2)/c, True))

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GIAC/XCAS [A]  time = 0.218666, size = 54, normalized size = 1.26 \[ -\frac{d{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{\sqrt{c}} + \frac{\sqrt{c x^{2} + a} e}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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