3.1177 \(\int \frac{\sqrt{d+e x^2} (a+b \tan ^{-1}(c x))}{x} \, dx\)

Optimal. Leaf size=63 \[ b \text{Unintegrable}\left (\frac{\tan ^{-1}(c x) \sqrt{d+e x^2}}{x},x\right )+a \sqrt{d+e x^2}+a \left (-\sqrt{d}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \]

[Out]

a*Sqrt[d + e*x^2] - a*Sqrt[d]*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + b*Unintegrable[(Sqrt[d + e*x^2]*ArcTan[c*x])/
x, x]

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Rubi [A]  time = 0.16305, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

a*Sqrt[d + e*x^2] - a*Sqrt[d]*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + b*Defer[Int][(Sqrt[d + e*x^2]*ArcTan[c*x])/x,
x]

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x} \, dx &=a \int \frac{\sqrt{d+e x^2}}{x} \, dx+b \int \frac{\sqrt{d+e x^2} \tan ^{-1}(c x)}{x} \, dx\\ &=\frac{1}{2} a \operatorname{Subst}\left (\int \frac{\sqrt{d+e x}}{x} \, dx,x,x^2\right )+b \int \frac{\sqrt{d+e x^2} \tan ^{-1}(c x)}{x} \, dx\\ &=a \sqrt{d+e x^2}+b \int \frac{\sqrt{d+e x^2} \tan ^{-1}(c x)}{x} \, dx+\frac{1}{2} (a d) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )\\ &=a \sqrt{d+e x^2}+b \int \frac{\sqrt{d+e x^2} \tan ^{-1}(c x)}{x} \, dx+\frac{(a d) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{e}\\ &=a \sqrt{d+e x^2}-a \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )+b \int \frac{\sqrt{d+e x^2} \tan ^{-1}(c x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 71.1609, size = 0, normalized size = 0. \[ \int \frac{\sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.771, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arctan \left ( cx \right ) }{x}\sqrt{e{x}^{2}+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x,x)

[Out]

int((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x^{2} + d}{\left (b \arctan \left (c x\right ) + a\right )}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x,x, algorithm="fricas")

[Out]

integral(sqrt(e*x^2 + d)*(b*arctan(c*x) + a)/x, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{atan}{\left (c x \right )}\right ) \sqrt{d + e x^{2}}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**(1/2)*(a+b*atan(c*x))/x,x)

[Out]

Integral((a + b*atan(c*x))*sqrt(d + e*x**2)/x, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x^{2} + d}{\left (b \arctan \left (c x\right ) + a\right )}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x,x, algorithm="giac")

[Out]

integrate(sqrt(e*x^2 + d)*(b*arctan(c*x) + a)/x, x)