\(\int x^2 (d+e x^2)^{5/2} (a+b \arctan (c x)) \, dx\) [1194]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int x^2 \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\frac {5 a d^3 x \sqrt {d+e x^2}}{128 e}+\frac {5}{64} a d^2 x^3 \sqrt {d+e x^2}+\frac {5}{48} a d x^3 \left (d+e x^2\right )^{3/2}+\frac {1}{8} a x^3 \left (d+e x^2\right )^{5/2}-\frac {5 a d^4 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{128 e^{3/2}}+b \text {Int}\left (x^2 \left (d+e x^2\right )^{5/2} \arctan (c x),x\right ) \]

[Out]

5/48*a*d*x^3*(e*x^2+d)^(3/2)+1/8*a*x^3*(e*x^2+d)^(5/2)-5/128*a*d^4*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/e^(3/2)+
5/128*a*d^3*x*(e*x^2+d)^(1/2)/e+5/64*a*d^2*x^3*(e*x^2+d)^(1/2)+b*Unintegrable(x^2*(e*x^2+d)^(5/2)*arctan(c*x),
x)

Rubi [N/A]

Not integrable

Time = 0.14 (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 = {} \[ \int x^2 \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\int x^2 \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx \]

[In]

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

[Out]

(5*a*d^3*x*Sqrt[d + e*x^2])/(128*e) + (5*a*d^2*x^3*Sqrt[d + e*x^2])/64 + (5*a*d*x^3*(d + e*x^2)^(3/2))/48 + (a
*x^3*(d + e*x^2)^(5/2))/8 - (5*a*d^4*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(128*e^(3/2)) + b*Defer[Int][x^2*(d
+ e*x^2)^(5/2)*ArcTan[c*x], x]

Rubi steps \begin{align*} \text {integral}& = a \int x^2 \left (d+e x^2\right )^{5/2} \, dx+b \int x^2 \left (d+e x^2\right )^{5/2} \arctan (c x) \, dx \\ & = \frac {1}{8} a x^3 \left (d+e x^2\right )^{5/2}+b \int x^2 \left (d+e x^2\right )^{5/2} \arctan (c x) \, dx+\frac {1}{8} (5 a d) \int x^2 \left (d+e x^2\right )^{3/2} \, dx \\ & = \frac {5}{48} a d x^3 \left (d+e x^2\right )^{3/2}+\frac {1}{8} a x^3 \left (d+e x^2\right )^{5/2}+b \int x^2 \left (d+e x^2\right )^{5/2} \arctan (c x) \, dx+\frac {1}{16} \left (5 a d^2\right ) \int x^2 \sqrt {d+e x^2} \, dx \\ & = \frac {5}{64} a d^2 x^3 \sqrt {d+e x^2}+\frac {5}{48} a d x^3 \left (d+e x^2\right )^{3/2}+\frac {1}{8} a x^3 \left (d+e x^2\right )^{5/2}+b \int x^2 \left (d+e x^2\right )^{5/2} \arctan (c x) \, dx+\frac {1}{64} \left (5 a d^3\right ) \int \frac {x^2}{\sqrt {d+e x^2}} \, dx \\ & = \frac {5 a d^3 x \sqrt {d+e x^2}}{128 e}+\frac {5}{64} a d^2 x^3 \sqrt {d+e x^2}+\frac {5}{48} a d x^3 \left (d+e x^2\right )^{3/2}+\frac {1}{8} a x^3 \left (d+e x^2\right )^{5/2}+b \int x^2 \left (d+e x^2\right )^{5/2} \arctan (c x) \, dx-\frac {\left (5 a d^4\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{128 e} \\ & = \frac {5 a d^3 x \sqrt {d+e x^2}}{128 e}+\frac {5}{64} a d^2 x^3 \sqrt {d+e x^2}+\frac {5}{48} a d x^3 \left (d+e x^2\right )^{3/2}+\frac {1}{8} a x^3 \left (d+e x^2\right )^{5/2}+b \int x^2 \left (d+e x^2\right )^{5/2} \arctan (c x) \, dx-\frac {\left (5 a d^4\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{128 e} \\ & = \frac {5 a d^3 x \sqrt {d+e x^2}}{128 e}+\frac {5}{64} a d^2 x^3 \sqrt {d+e x^2}+\frac {5}{48} a d x^3 \left (d+e x^2\right )^{3/2}+\frac {1}{8} a x^3 \left (d+e x^2\right )^{5/2}-\frac {5 a d^4 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{128 e^{3/2}}+b \int x^2 \left (d+e x^2\right )^{5/2} \arctan (c x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 14.77 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int x^2 \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\int x^2 \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.47 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.91 \[ \int x^2 \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arctan \left (c x\right ) + a\right )} x^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 176.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int x^2 \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\int x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {5}{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(x^2*(e*x^2+d)^(5/2)*(a+b*arctan(c*x)),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 = 98.12 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int x^2 \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arctan \left (c x\right ) + a\right )} x^{2} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.79 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int x^2 \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{5/2} \,d x \]

[In]

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

[Out]

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