\(\int \frac {a+b \arctan (c x)}{x^3 (d+e x^2)^{3/2}} \, dx\) [1215]

Optimal result

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

[Out]

3/2*a*e*arctanh((e*x^2+d)^(1/2)/d^(1/2))/d^(5/2)-3/2*a*e/d^2/(e*x^2+d)^(1/2)-1/2*a/d/x^2/(e*x^2+d)^(1/2)+b*Uni
ntegrable(arctan(c*x)/x^3/(e*x^2+d)^(3/2),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 = {} \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 14.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.38 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]

[In]

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

[Out]

integral(sqrt(e*x^2 + d)*(b*arctan(c*x) + a)/(e^2*x^7 + 2*d*e*x^5 + d^2*x^3), x)

Sympy [N/A]

Not integrable

Time = 53.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {atan}{\left (c x \right )}}{x^{3} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((a+b*arctan(c*x))/x^3/(e*x^2+d)^(3/2),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 = 43.67 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 1.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^3\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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