\(\int \frac {(d+e x^2)^{3/2} (a+b \arctan (c x))}{x^5} \, dx\) [1191]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^5} \, dx=-\frac {3 a e \sqrt {d+e x^2}}{8 x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{4 x^4}-\frac {3 a e^2 \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{8 \sqrt {d}}+b \text {Int}\left (\frac {\left (d+e x^2\right )^{3/2} \arctan (c x)}{x^5},x\right ) \]

[Out]

-1/4*a*(e*x^2+d)^(3/2)/x^4-3/8*a*e^2*arctanh((e*x^2+d)^(1/2)/d^(1/2))/d^(1/2)-3/8*a*e*(e*x^2+d)^(1/2)/x^2+b*Un
integrable((e*x^2+d)^(3/2)*arctan(c*x)/x^5,x)

Rubi [N/A]

Not integrable

Time = 0.13 (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 = {} \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^5} \, dx=\int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^5} \, dx \]

[In]

Int[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^5,x]

[Out]

(-3*a*e*Sqrt[d + e*x^2])/(8*x^2) - (a*(d + e*x^2)^(3/2))/(4*x^4) - (3*a*e^2*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/
(8*Sqrt[d]) + b*Defer[Int][((d + e*x^2)^(3/2)*ArcTan[c*x])/x^5, x]

Rubi steps \begin{align*} \text {integral}& = a \int \frac {\left (d+e x^2\right )^{3/2}}{x^5} \, dx+b \int \frac {\left (d+e x^2\right )^{3/2} \arctan (c x)}{x^5} \, dx \\ & = \frac {1}{2} a \text {Subst}\left (\int \frac {(d+e x)^{3/2}}{x^3} \, dx,x,x^2\right )+b \int \frac {\left (d+e x^2\right )^{3/2} \arctan (c x)}{x^5} \, dx \\ & = -\frac {a \left (d+e x^2\right )^{3/2}}{4 x^4}+b \int \frac {\left (d+e x^2\right )^{3/2} \arctan (c x)}{x^5} \, dx+\frac {1}{8} (3 a e) \text {Subst}\left (\int \frac {\sqrt {d+e x}}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {3 a e \sqrt {d+e x^2}}{8 x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{4 x^4}+b \int \frac {\left (d+e x^2\right )^{3/2} \arctan (c x)}{x^5} \, dx+\frac {1}{16} \left (3 a e^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right ) \\ & = -\frac {3 a e \sqrt {d+e x^2}}{8 x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{4 x^4}+b \int \frac {\left (d+e x^2\right )^{3/2} \arctan (c x)}{x^5} \, dx+\frac {1}{8} (3 a e) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right ) \\ & = -\frac {3 a e \sqrt {d+e x^2}}{8 x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{4 x^4}-\frac {3 a e^2 \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{8 \sqrt {d}}+b \int \frac {\left (d+e x^2\right )^{3/2} \arctan (c x)}{x^5} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 13.74 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^5} \, dx=\int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^5} \, dx \]

[In]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^5,x]

[Out]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^5, x]

Maple [N/A] (verified)

Not integrable

Time = 0.40 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )}{x^{5}}d x\]

[In]

int((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^5,x)

[Out]

int((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^5,x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74 \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^5} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{5}} \,d x } \]

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^5,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^5, x)

Sympy [N/A]

Not integrable

Time = 15.59 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^5} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{5}}\, dx \]

[In]

integrate((e*x**2+d)**(3/2)*(a+b*atan(c*x))/x**5,x)

[Out]

Integral((a + b*atan(c*x))*(d + e*x**2)**(3/2)/x**5, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^5} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^5,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. \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^5} \, dx=\text {Timed out} \]

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^5,x, algorithm="giac")

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 1.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^5} \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2}}{x^5} \,d x \]

[In]

int(((a + b*atan(c*x))*(d + e*x^2)^(3/2))/x^5,x)

[Out]

int(((a + b*atan(c*x))*(d + e*x^2)^(3/2))/x^5, x)