\(\int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^2} \, dx\) [1178]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^2} \, dx=-\frac {a \sqrt {d+e x^2}}{x}+a \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+b \text {Int}\left (\frac {\sqrt {d+e x^2} \arctan (c x)}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^2} \, dx=\int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = a \int \frac {\sqrt {d+e x^2}}{x^2} \, dx+b \int \frac {\sqrt {d+e x^2} \arctan (c x)}{x^2} \, dx \\ & = -\frac {a \sqrt {d+e x^2}}{x}+b \int \frac {\sqrt {d+e x^2} \arctan (c x)}{x^2} \, dx+(a e) \int \frac {1}{\sqrt {d+e x^2}} \, dx \\ & = -\frac {a \sqrt {d+e x^2}}{x}+b \int \frac {\sqrt {d+e x^2} \arctan (c x)}{x^2} \, dx+(a e) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right ) \\ & = -\frac {a \sqrt {d+e x^2}}{x}+a \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+b \int \frac {\sqrt {d+e x^2} \arctan (c x)}{x^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 10.45 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^2} \, dx=\int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.18 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

\[\int \frac {\sqrt {e \,x^{2}+d}\, \left (a +b \arctan \left (c x \right )\right )}{x^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^2} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^2} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x^2,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>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 281.84 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^2} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 1.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\sqrt {e\,x^2+d}}{x^2} \,d x \]

[In]

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

[Out]

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