\(\int \frac {x^4 (a+b \arctan (c x))}{(d+e x^2)^{5/2}} \, dx\) [1217]

Optimal result

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

[Out]

-1/3*a*x^3/e/(e*x^2+d)^(3/2)+a*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/e^(5/2)-a*x/e^2/(e*x^2+d)^(1/2)+b*Unintegrab
le(x^4*arctan(c*x)/(e*x^2+d)^(5/2),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 = {} \[ \int \frac {x^4 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x^4 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx \]

[In]

Int[(x^4*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2),x]

[Out]

-1/3*(a*x^3)/(e*(d + e*x^2)^(3/2)) - (a*x)/(e^2*Sqrt[d + e*x^2]) + (a*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/e^
(5/2) + b*Defer[Int][(x^4*ArcTan[c*x])/(d + e*x^2)^(5/2), x]

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

Mathematica [N/A]

Not integrable

Time = 17.78 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {x^4 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x^4 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx \]

[In]

Integrate[(x^4*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2),x]

[Out]

Integrate[(x^4*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.59 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

\[\int \frac {x^{4} \left (a +b \arctan \left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]

[In]

int(x^4*(a+b*arctan(c*x))/(e*x^2+d)^(5/2),x)

[Out]

int(x^4*(a+b*arctan(c*x))/(e*x^2+d)^(5/2),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.52 \[ \int \frac {x^4 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^4*(a+b*arctan(c*x))/(e*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral((b*x^4*arctan(c*x) + a*x^4)*sqrt(e*x^2 + d)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(x**4*(a+b*atan(c*x))/(e*x**2+d)**(5/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^4*(a+b*arctan(c*x))/(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 = 168.21 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {x^4 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^4*(a+b*arctan(c*x))/(e*x^2+d)^(5/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.99 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^4 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]

[In]

int((x^4*(a + b*atan(c*x)))/(d + e*x^2)^(5/2),x)

[Out]

int((x^4*(a + b*atan(c*x)))/(d + e*x^2)^(5/2), x)