Integrand size = 25, antiderivative size = 74 \[ \int x^2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {d \sqrt {d+e x^2}}{3 (-e)^{3/2}}-\frac {\left (d+e x^2\right )^{3/2}}{9 (-e)^{3/2}}+\frac {1}{3} x^3 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \] Output:
1/3*d*(e*x^2+d)^(1/2)/(-e)^(3/2)-1/9*(e*x^2+d)^(3/2)/(-e)^(3/2)+1/3*x^3*ar ctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))
Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.81 \[ \int x^2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {1}{9} \left (\frac {\left (2 d-e x^2\right ) \sqrt {d+e x^2}}{(-e)^{3/2}}+3 x^3 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )\right ) \] Input:
Integrate[x^2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]
Output:
(((2*d - e*x^2)*Sqrt[d + e*x^2])/(-e)^(3/2) + 3*x^3*ArcTan[(Sqrt[-e]*x)/Sq rt[d + e*x^2]])/9
Time = 0.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5674, 243, 53, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx\) |
\(\Big \downarrow \) 5674 |
\(\displaystyle \frac {1}{3} x^3 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{3} \sqrt {-e} \int \frac {x^3}{\sqrt {e x^2+d}}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{3} x^3 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {-e} \int \frac {x^2}{\sqrt {e x^2+d}}dx^2\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {1}{3} x^3 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {-e} \int \left (\frac {\sqrt {e x^2+d}}{e}-\frac {d}{e \sqrt {e x^2+d}}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} x^3 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {-e} \left (\frac {2 \left (d+e x^2\right )^{3/2}}{3 e^2}-\frac {2 d \sqrt {d+e x^2}}{e^2}\right )\) |
Input:
Int[x^2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]
Output:
-1/6*(Sqrt[-e]*((-2*d*Sqrt[d + e*x^2])/e^2 + (2*(d + e*x^2)^(3/2))/(3*e^2) )) + (x^3*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/3
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[ArcTan[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]*((d_.)*(x_))^(m_.), x_S ymbol] :> Simp[(d*x)^(m + 1)*(ArcTan[(c*x)/Sqrt[a + b*x^2]]/(d*(m + 1))), x ] - Simp[c/(d*(m + 1)) Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x], x] /; FreeQ [{a, b, c, d, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(134\) vs. \(2(56)=112\).
Time = 0.02 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.82
method | result | size |
default | \(\frac {x^{3} \arctan \left (\frac {\sqrt {-e}\, x}{\sqrt {e \,x^{2}+d}}\right )}{3}+\frac {\sqrt {-e}\, e \left (\frac {x^{4} \sqrt {e \,x^{2}+d}}{5 e}-\frac {4 d \left (\frac {x^{2} \sqrt {e \,x^{2}+d}}{3 e}-\frac {2 d \sqrt {e \,x^{2}+d}}{3 e^{2}}\right )}{5 e}\right )}{3 d}-\frac {\sqrt {-e}\, \left (\frac {x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 e}-\frac {2 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 e^{2}}\right )}{3 d}\) | \(135\) |
parts | \(\frac {x^{3} \arctan \left (\frac {\sqrt {-e}\, x}{\sqrt {e \,x^{2}+d}}\right )}{3}+\frac {\sqrt {-e}\, e \left (\frac {x^{4} \sqrt {e \,x^{2}+d}}{5 e}-\frac {4 d \left (\frac {x^{2} \sqrt {e \,x^{2}+d}}{3 e}-\frac {2 d \sqrt {e \,x^{2}+d}}{3 e^{2}}\right )}{5 e}\right )}{3 d}-\frac {\sqrt {-e}\, \left (\frac {x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 e}-\frac {2 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 e^{2}}\right )}{3 d}\) | \(135\) |
Input:
int(x^2*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x,method=_RETURNVERBOSE)
Output:
1/3*x^3*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))+1/3*(-e)^(1/2)*e/d*(1/5*x^4/e *(e*x^2+d)^(1/2)-4/5*d/e*(1/3*x^2/e*(e*x^2+d)^(1/2)-2/3*d/e^2*(e*x^2+d)^(1 /2)))-1/3*(-e)^(1/2)/d*(1/5*x^2*(e*x^2+d)^(3/2)/e-2/15*d/e^2*(e*x^2+d)^(3/ 2))
Time = 0.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.76 \[ \int x^2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {3 \, e^{2} x^{3} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - \sqrt {e x^{2} + d} {\left (e x^{2} - 2 \, d\right )} \sqrt {-e}}{9 \, e^{2}} \] Input:
integrate(x^2*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x, algorithm="fricas")
Output:
1/9*(3*e^2*x^3*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - sqrt(e*x^2 + d)*(e*x^2 - 2*d)*sqrt(-e))/e^2
Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.01 \[ \int x^2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\begin {cases} \frac {2 d \sqrt {- e} \sqrt {d + e x^{2}}}{9 e^{2}} + \frac {x^{3} \operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{3} - \frac {x^{2} \sqrt {- e} \sqrt {d + e x^{2}}}{9 e} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x**2*atan((-e)**(1/2)*x/(e*x**2+d)**(1/2)),x)
Output:
Piecewise((2*d*sqrt(-e)*sqrt(d + e*x**2)/(9*e**2) + x**3*atan(x*sqrt(-e)/s qrt(d + e*x**2))/3 - x**2*sqrt(-e)*sqrt(d + e*x**2)/(9*e), Ne(e, 0)), (0, True))
Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.50 \[ \int x^2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {1}{3} \, x^{3} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - \frac {{\left (3 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} - 5 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d\right )} \sqrt {-e}}{45 \, d e^{2}} + \frac {{\left (3 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x^{2} + d} d^{2}\right )} \sqrt {-e}}{45 \, d e^{2}} \] Input:
integrate(x^2*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x, algorithm="maxima")
Output:
1/3*x^3*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - 1/45*(3*(e*x^2 + d)^(5/2) - 5 *(e*x^2 + d)^(3/2)*d)*sqrt(-e)/(d*e^2) + 1/45*(3*(e*x^2 + d)^(5/2) - 10*(e *x^2 + d)^(3/2)*d + 15*sqrt(e*x^2 + d)*d^2)*sqrt(-e)/(d*e^2)
Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.86 \[ \int x^2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {1}{3} \, x^{3} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + \frac {\sqrt {-e^{2} x^{2} - d e} d}{3 \, e^{2}} + \frac {{\left (-e^{2} x^{2} - d e\right )}^{\frac {3}{2}}}{9 \, e^{3}} \] Input:
integrate(x^2*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x, algorithm="giac")
Output:
1/3*x^3*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + 1/3*sqrt(-e^2*x^2 - d*e)*d/e^ 2 + 1/9*(-e^2*x^2 - d*e)^(3/2)/e^3
Timed out. \[ \int x^2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int x^2\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \] Input:
int(x^2*atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2)),x)
Output:
int(x^2*atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2)), x)
Time = 0.19 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.19 \[ \int x^2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {3 \mathit {atan} \left (\frac {\sqrt {e}\, \sqrt {e \,x^{2}+d}\, i x +e i \,x^{2}}{\sqrt {e}\, \sqrt {e \,x^{2}+d}\, x +d +e \,x^{2}}\right ) e^{2} x^{3}+2 \sqrt {e}\, \sqrt {e \,x^{2}+d}\, d i -\sqrt {e}\, \sqrt {e \,x^{2}+d}\, e i \,x^{2}}{9 e^{2}} \] Input:
int(x^2*atan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x)
Output:
(3*atan((sqrt(e)*sqrt(d + e*x**2)*i*x + e*i*x**2)/(sqrt(e)*sqrt(d + e*x**2 )*x + d + e*x**2))*e**2*x**3 + 2*sqrt(e)*sqrt(d + e*x**2)*d*i - sqrt(e)*sq rt(d + e*x**2)*e*i*x**2)/(9*e**2)