\(\int \frac {a+b \arctan (c x)}{x^3 (d+e x^2)^{5/2}} \, dx\) [1224]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx=-\frac {5 a e}{6 d^2 \left (d+e x^2\right )^{3/2}}-\frac {a}{2 d x^2 \left (d+e x^2\right )^{3/2}}-\frac {5 a e}{2 d^3 \sqrt {d+e x^2}}+\frac {5 a e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 d^{7/2}}+b \text {Int}\left (\frac {\arctan (c x)}{x^3 \left (d+e x^2\right )^{5/2}},x\right ) \]

[Out]

-5/6*a*e/d^2/(e*x^2+d)^(3/2)-1/2*a/d/x^2/(e*x^2+d)^(3/2)+5/2*a*e*arctanh((e*x^2+d)^(1/2)/d^(1/2))/d^(7/2)-5/2*
a*e/d^3/(e*x^2+d)^(1/2)+b*Unintegrable(arctan(c*x)/x^3/(e*x^2+d)^(5/2),x)

Rubi [N/A]

Not integrable

Time = 0.15 (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 = {} \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx \]

[In]

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

[Out]

(-5*a*e)/(6*d^2*(d + e*x^2)^(3/2)) - a/(2*d*x^2*(d + e*x^2)^(3/2)) - (5*a*e)/(2*d^3*Sqrt[d + e*x^2]) + (5*a*e*
ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*d^(7/2)) + b*Defer[Int][ArcTan[c*x]/(x^3*(d + e*x^2)^(5/2)), x]

Rubi steps \begin{align*} \text {integral}& = a \int \frac {1}{x^3 \left (d+e x^2\right )^{5/2}} \, dx+b \int \frac {\arctan (c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx \\ & = \frac {1}{2} a \text {Subst}\left (\int \frac {1}{x^2 (d+e x)^{5/2}} \, dx,x,x^2\right )+b \int \frac {\arctan (c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx \\ & = -\frac {a}{2 d x^2 \left (d+e x^2\right )^{3/2}}+b \int \frac {\arctan (c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx-\frac {(5 a e) \text {Subst}\left (\int \frac {1}{x (d+e x)^{5/2}} \, dx,x,x^2\right )}{4 d} \\ & = -\frac {5 a e}{6 d^2 \left (d+e x^2\right )^{3/2}}-\frac {a}{2 d x^2 \left (d+e x^2\right )^{3/2}}+b \int \frac {\arctan (c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx-\frac {(5 a e) \text {Subst}\left (\int \frac {1}{x (d+e x)^{3/2}} \, dx,x,x^2\right )}{4 d^2} \\ & = -\frac {5 a e}{6 d^2 \left (d+e x^2\right )^{3/2}}-\frac {a}{2 d x^2 \left (d+e x^2\right )^{3/2}}-\frac {5 a e}{2 d^3 \sqrt {d+e x^2}}+b \int \frac {\arctan (c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx-\frac {(5 a e) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{4 d^3} \\ & = -\frac {5 a e}{6 d^2 \left (d+e x^2\right )^{3/2}}-\frac {a}{2 d x^2 \left (d+e x^2\right )^{3/2}}-\frac {5 a e}{2 d^3 \sqrt {d+e x^2}}+b \int \frac {\arctan (c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx-\frac {(5 a) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{2 d^3} \\ & = -\frac {5 a e}{6 d^2 \left (d+e x^2\right )^{3/2}}-\frac {a}{2 d x^2 \left (d+e x^2\right )^{3/2}}-\frac {5 a e}{2 d^3 \sqrt {d+e x^2}}+\frac {5 a e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 d^{7/2}}+b \int \frac {\arctan (c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 18.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.44 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]

[In]

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

[Out]

integral(sqrt(e*x^2 + d)*(b*arctan(c*x) + a)/(e^3*x^9 + 3*d*e^2*x^7 + 3*d^2*e*x^5 + d^3*x^3), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate((a+b*arctan(c*x))/x^3/(e*x^2+d)^(5/2),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 = 55.52 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

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

[In]

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

[Out]

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