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

Optimal. Leaf size=23 \[ \text {Int}\left (\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right ),x\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 5.10, size = 0, normalized size = 0.00 \[ \int \sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

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]

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )}, x\right ) \]

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)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

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maple [A]  time = 2.38, size = 0, normalized size = 0.00 \[ \int \sqrt {e \,x^{2}+d}\, \left (a +b \arctan \left (c x \right )\right )\, dx \]

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)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e-c^2*d>0)', see `assume?` for
more details)Is e-c^2*d positive or negative?

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\sqrt {e\,x^2+d} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {atan}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]

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)

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