\(\int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^3} \, dx\) [1179]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^3} \, dx=-\frac {a \sqrt {d+e x^2}}{2 x^2}-\frac {a e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 \sqrt {d}}+b \text {Int}\left (\frac {\sqrt {d+e x^2} \arctan (c x)}{x^3},x\right ) \]

[Out]

-1/2*a*e*arctanh((e*x^2+d)^(1/2)/d^(1/2))/d^(1/2)-1/2*a*(e*x^2+d)^(1/2)/x^2+b*Unintegrable(arctan(c*x)*(e*x^2+
d)^(1/2)/x^3,x)

Rubi [N/A]

Not integrable

Time = 0.11 (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 = {} \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^3} \, dx=\int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^3} \, dx \]

[In]

Int[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x^3,x]

[Out]

-1/2*(a*Sqrt[d + e*x^2])/x^2 - (a*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*Sqrt[d]) + b*Defer[Int][(Sqrt[d + e*x
^2]*ArcTan[c*x])/x^3, x]

Rubi steps \begin{align*} \text {integral}& = a \int \frac {\sqrt {d+e x^2}}{x^3} \, dx+b \int \frac {\sqrt {d+e x^2} \arctan (c x)}{x^3} \, dx \\ & = \frac {1}{2} a \text {Subst}\left (\int \frac {\sqrt {d+e x}}{x^2} \, dx,x,x^2\right )+b \int \frac {\sqrt {d+e x^2} \arctan (c x)}{x^3} \, dx \\ & = -\frac {a \sqrt {d+e x^2}}{2 x^2}+b \int \frac {\sqrt {d+e x^2} \arctan (c x)}{x^3} \, dx+\frac {1}{4} (a e) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right ) \\ & = -\frac {a \sqrt {d+e x^2}}{2 x^2}+\frac {1}{2} a \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )+b \int \frac {\sqrt {d+e x^2} \arctan (c x)}{x^3} \, dx \\ & = -\frac {a \sqrt {d+e x^2}}{2 x^2}-\frac {a e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 \sqrt {d}}+b \int \frac {\sqrt {d+e x^2} \arctan (c x)}{x^3} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 11.96 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^3} \, dx=\int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^3} \, dx \]

[In]

Integrate[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x^3,x]

[Out]

Integrate[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x^3, x]

Maple [N/A] (verified)

Not integrable

Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

\[\int \frac {\sqrt {e \,x^{2}+d}\, \left (a +b \arctan \left (c x \right )\right )}{x^{3}}d x\]

[In]

int((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x^3,x)

[Out]

int((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x^3,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^3} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]

[In]

integrate((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x^3,x, algorithm="fricas")

[Out]

integral(sqrt(e*x^2 + d)*(b*arctan(c*x) + a)/x^3, x)

Sympy [N/A]

Not integrable

Time = 3.78 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^3} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{3}}\, dx \]

[In]

integrate((e*x**2+d)**(1/2)*(a+b*atan(c*x))/x**3,x)

[Out]

Integral((a + b*atan(c*x))*sqrt(d + e*x**2)/x**3, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x^3,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 = 288.61 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^3} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]

[In]

integrate((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 1.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\sqrt {e\,x^2+d}}{x^3} \,d x \]

[In]

int(((a + b*atan(c*x))*(d + e*x^2)^(1/2))/x^3,x)

[Out]

int(((a + b*atan(c*x))*(d + e*x^2)^(1/2))/x^3, x)