Integrand size = 23, antiderivative size = 109 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {b c}{3 \left (c^2 d-e\right ) e \sqrt {d+e x^2}}+\frac {x^3 (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b \text {arctanh}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d \left (c^2 d-e\right )^{3/2}} \]
1/3*x^3*(a+b*arctan(c*x))/d/(e*x^2+d)^(3/2)-1/3*b*arctanh(c*(e*x^2+d)^(1/2 )/(c^2*d-e)^(1/2))/d/(c^2*d-e)^(3/2)+1/3*b*c/(c^2*d-e)/e/(e*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 0.77 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.31 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {\frac {2 a d x}{e \left (d+e x^2\right )^{3/2}}-\frac {2 \left (b c d+a \left (c^2 d-e\right ) x\right )}{\left (c^2 d-e\right ) e \sqrt {d+e x^2}}-\frac {2 b x^3 \arctan (c x)}{\left (d+e x^2\right )^{3/2}}+\frac {b \log \left (\frac {12 c d \sqrt {c^2 d-e} \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b (i+c x)}\right )}{\left (c^2 d-e\right )^{3/2}}+\frac {b \log \left (\frac {12 c d \sqrt {c^2 d-e} \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b (-i+c x)}\right )}{\left (c^2 d-e\right )^{3/2}}}{6 d} \]
-1/6*((2*a*d*x)/(e*(d + e*x^2)^(3/2)) - (2*(b*c*d + a*(c^2*d - e)*x))/((c^ 2*d - e)*e*Sqrt[d + e*x^2]) - (2*b*x^3*ArcTan[c*x])/(d + e*x^2)^(3/2) + (b *Log[(12*c*d*Sqrt[c^2*d - e]*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2 ]))/(b*(I + c*x))])/(c^2*d - e)^(3/2) + (b*Log[(12*c*d*Sqrt[c^2*d - e]*(c* d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(-I + c*x))])/(c^2*d - e) ^(3/2))/d
Time = 0.34 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5511, 27, 354, 87, 73, 221}
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.
| \(\displaystyle \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5511 |
| \(\displaystyle \frac {x^3 (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}-b c \int \frac {x^3}{3 d \left (c^2 x^2+1\right ) \left (e x^2+d\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^3 (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c \int \frac {x^3}{\left (c^2 x^2+1\right ) \left (e x^2+d\right )^{3/2}}dx}{3 d}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {x^3 (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c \int \frac {x^2}{\left (c^2 x^2+1\right ) \left (e x^2+d\right )^{3/2}}dx^2}{6 d}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {x^3 (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c \left (-\frac {\int \frac {1}{\left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx^2}{c^2 d-e}-\frac {2 d}{e \left (c^2 d-e\right ) \sqrt {d+e x^2}}\right )}{6 d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {x^3 (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c \left (-\frac {2 \int \frac {1}{\frac {c^2 x^4}{e}-\frac {c^2 d}{e}+1}d\sqrt {e x^2+d}}{e \left (c^2 d-e\right )}-\frac {2 d}{e \left (c^2 d-e\right ) \sqrt {d+e x^2}}\right )}{6 d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x^3 (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c \left (\frac {2 \text {arctanh}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{c \left (c^2 d-e\right )^{3/2}}-\frac {2 d}{e \left (c^2 d-e\right ) \sqrt {d+e x^2}}\right )}{6 d}\) |
(x^3*(a + b*ArcTan[c*x]))/(3*d*(d + e*x^2)^(3/2)) - (b*c*((-2*d)/((c^2*d - e)*e*Sqrt[d + e*x^2]) + (2*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/ (c*(c^2*d - e)^(3/2))))/(6*d)
3.13.19.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x _)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Sim p[(a + b*ArcTan[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(1 + c^2 *x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ((IGtQ[q, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[q, 0] && GtQ[m + 2*q + 3, 0])) || (ILtQ[(m + 2*q + 1)/2, 0] && !ILt Q[(m - 1)/2, 0]))
\[\int \frac {x^{2} \left (a +b \arctan \left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (93) = 186\).
Time = 0.42 (sec) , antiderivative size = 676, normalized size of antiderivative = 6.20 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx=\left [-\frac {{\left (b e^{3} x^{4} + 2 \, b d e^{2} x^{2} + b d^{2} e\right )} \sqrt {c^{2} d - e} \log \left (\frac {c^{4} e^{2} x^{4} + 8 \, c^{4} d^{2} - 8 \, c^{2} d e + 2 \, {\left (4 \, c^{4} d e - 3 \, c^{2} e^{2}\right )} x^{2} + 4 \, {\left (c^{3} e x^{2} + 2 \, c^{3} d - c e\right )} \sqrt {c^{2} d - e} \sqrt {e x^{2} + d} + e^{2}}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) - 4 \, {\left (b c^{3} d^{3} - b c d^{2} e + {\left (b c^{4} d^{2} e - 2 \, b c^{2} d e^{2} + b e^{3}\right )} x^{3} \arctan \left (c x\right ) + {\left (a c^{4} d^{2} e - 2 \, a c^{2} d e^{2} + a e^{3}\right )} x^{3} + {\left (b c^{3} d^{2} e - b c d e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d}}{12 \, {\left (c^{4} d^{5} e - 2 \, c^{2} d^{4} e^{2} + d^{3} e^{3} + {\left (c^{4} d^{3} e^{3} - 2 \, c^{2} d^{2} e^{4} + d e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{2}\right )}}, -\frac {{\left (b e^{3} x^{4} + 2 \, b d e^{2} x^{2} + b d^{2} e\right )} \sqrt {-c^{2} d + e} \arctan \left (-\frac {{\left (c^{2} e x^{2} + 2 \, c^{2} d - e\right )} \sqrt {-c^{2} d + e} \sqrt {e x^{2} + d}}{2 \, {\left (c^{3} d^{2} - c d e + {\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (b c^{3} d^{3} - b c d^{2} e + {\left (b c^{4} d^{2} e - 2 \, b c^{2} d e^{2} + b e^{3}\right )} x^{3} \arctan \left (c x\right ) + {\left (a c^{4} d^{2} e - 2 \, a c^{2} d e^{2} + a e^{3}\right )} x^{3} + {\left (b c^{3} d^{2} e - b c d e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d}}{6 \, {\left (c^{4} d^{5} e - 2 \, c^{2} d^{4} e^{2} + d^{3} e^{3} + {\left (c^{4} d^{3} e^{3} - 2 \, c^{2} d^{2} e^{4} + d e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{2}\right )}}\right ] \]
[-1/12*((b*e^3*x^4 + 2*b*d*e^2*x^2 + b*d^2*e)*sqrt(c^2*d - e)*log((c^4*e^2 *x^4 + 8*c^4*d^2 - 8*c^2*d*e + 2*(4*c^4*d*e - 3*c^2*e^2)*x^2 + 4*(c^3*e*x^ 2 + 2*c^3*d - c*e)*sqrt(c^2*d - e)*sqrt(e*x^2 + d) + e^2)/(c^4*x^4 + 2*c^2 *x^2 + 1)) - 4*(b*c^3*d^3 - b*c*d^2*e + (b*c^4*d^2*e - 2*b*c^2*d*e^2 + b*e ^3)*x^3*arctan(c*x) + (a*c^4*d^2*e - 2*a*c^2*d*e^2 + a*e^3)*x^3 + (b*c^3*d ^2*e - b*c*d*e^2)*x^2)*sqrt(e*x^2 + d))/(c^4*d^5*e - 2*c^2*d^4*e^2 + d^3*e ^3 + (c^4*d^3*e^3 - 2*c^2*d^2*e^4 + d*e^5)*x^4 + 2*(c^4*d^4*e^2 - 2*c^2*d^ 3*e^3 + d^2*e^4)*x^2), -1/6*((b*e^3*x^4 + 2*b*d*e^2*x^2 + b*d^2*e)*sqrt(-c ^2*d + e)*arctan(-1/2*(c^2*e*x^2 + 2*c^2*d - e)*sqrt(-c^2*d + e)*sqrt(e*x^ 2 + d)/(c^3*d^2 - c*d*e + (c^3*d*e - c*e^2)*x^2)) - 2*(b*c^3*d^3 - b*c*d^2 *e + (b*c^4*d^2*e - 2*b*c^2*d*e^2 + b*e^3)*x^3*arctan(c*x) + (a*c^4*d^2*e - 2*a*c^2*d*e^2 + a*e^3)*x^3 + (b*c^3*d^2*e - b*c*d*e^2)*x^2)*sqrt(e*x^2 + d))/(c^4*d^5*e - 2*c^2*d^4*e^2 + d^3*e^3 + (c^4*d^3*e^3 - 2*c^2*d^2*e^4 + d*e^5)*x^4 + 2*(c^4*d^4*e^2 - 2*c^2*d^3*e^3 + d^2*e^4)*x^2)]
Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
-1/3*a*(x/((e*x^2 + d)^(3/2)*e) - x/(sqrt(e*x^2 + d)*d*e)) + 2*b*integrate (1/2*x^2*arctan(c*x)/((e^2*x^4 + 2*d*e*x^2 + d^2)*sqrt(e*x^2 + d)), x)
\[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]