3.1176 \(\int \sqrt{d+e x^2} (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right ),x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0238289, 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 = {} \[ \int \sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 4.702, size = 0, normalized size = 0. \[ \int \sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 1.524, size = 0, normalized size = 0. \begin{align*} \int \sqrt{e{x}^{2}+d} \left ( a+b\arctan \left ( cx \right ) \right ) \, 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)

[Out]

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

________________________________________________________________________________________

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, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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