\(\int \frac {(d+e x^2)^{3/2} (a+b \arctan (c x))}{x^3} \, dx\) [1189]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.12 (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 = {} \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^3} \, dx=\int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^3} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 12.44 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^3} \, dx=\int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.42 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74 \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 10.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^3} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((e*x^2+d)^(3/2)*(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 [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 1.14 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2}}{x^3} \,d x \]

[In]

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

[Out]

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