Integrand size = 20, antiderivative size = 144 \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {b c}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {b \left (3 c^2 d-2 e\right ) \text {arctanh}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d^2 \left (c^2 d-e\right )^{3/2}} \]
1/3*x*(a+b*arctan(c*x))/d/(e*x^2+d)^(3/2)+1/3*b*(3*c^2*d-2*e)*arctanh(c*(e *x^2+d)^(1/2)/(c^2*d-e)^(1/2))/d^2/(c^2*d-e)^(3/2)-1/3*b*c/d/(c^2*d-e)/(e* x^2+d)^(1/2)+2/3*x*(a+b*arctan(c*x))/d^2/(e*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.20 \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {c^2 d-e} \left (-b c d \left (d+e x^2\right )+a \left (c^2 d-e\right ) x \left (3 d+2 e x^2\right )\right )+2 b \left (c^2 d-e\right )^{3/2} x \left (3 d+2 e x^2\right ) \arctan (c x)+b \left (3 c^2 d-2 e\right ) \left (d+e x^2\right )^{3/2} \log \left (-\frac {12 c d^2 \sqrt {c^2 d-e} \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (3 c^2 d-2 e\right ) (i+c x)}\right )+b \left (3 c^2 d-2 e\right ) \left (d+e x^2\right )^{3/2} \log \left (-\frac {12 c d^2 \sqrt {c^2 d-e} \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (3 c^2 d-2 e\right ) (-i+c x)}\right )}{6 d^2 \left (c^2 d-e\right )^{3/2} \left (d+e x^2\right )^{3/2}} \]
(2*Sqrt[c^2*d - e]*(-(b*c*d*(d + e*x^2)) + a*(c^2*d - e)*x*(3*d + 2*e*x^2) ) + 2*b*(c^2*d - e)^(3/2)*x*(3*d + 2*e*x^2)*ArcTan[c*x] + b*(3*c^2*d - 2*e )*(d + e*x^2)^(3/2)*Log[(-12*c*d^2*Sqrt[c^2*d - e]*(c*d - I*e*x + Sqrt[c^2 *d - e]*Sqrt[d + e*x^2]))/(b*(3*c^2*d - 2*e)*(I + c*x))] + b*(3*c^2*d - 2* e)*(d + e*x^2)^(3/2)*Log[(-12*c*d^2*Sqrt[c^2*d - e]*(c*d + I*e*x + Sqrt[c^ 2*d - e]*Sqrt[d + e*x^2]))/(b*(3*c^2*d - 2*e)*(-I + c*x))])/(6*d^2*(c^2*d - e)^(3/2)*(d + e*x^2)^(3/2))
Time = 0.33 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5447, 27, 435, 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.
| \(\displaystyle \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5447 |
| \(\displaystyle -b c \int \frac {x \left (2 e x^2+3 d\right )}{3 d^2 \left (c^2 x^2+1\right ) \left (e x^2+d\right )^{3/2}}dx+\frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c \int \frac {x \left (2 e x^2+3 d\right )}{\left (c^2 x^2+1\right ) \left (e x^2+d\right )^{3/2}}dx}{3 d^2}+\frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 435 |
| \(\displaystyle -\frac {b c \int \frac {2 e x^2+3 d}{\left (c^2 x^2+1\right ) \left (e x^2+d\right )^{3/2}}dx^2}{6 d^2}+\frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {b c \left (\frac {\left (3 c^2 d-2 e\right ) \int \frac {1}{\left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx^2}{c^2 d-e}+\frac {2 d}{\left (c^2 d-e\right ) \sqrt {d+e x^2}}\right )}{6 d^2}+\frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {b c \left (\frac {2 \left (3 c^2 d-2 e\right ) \int \frac {1}{\frac {c^2 x^4}{e}-\frac {c^2 d}{e}+1}d\sqrt {e x^2+d}}{e \left (c^2 d-e\right )}+\frac {2 d}{\left (c^2 d-e\right ) \sqrt {d+e x^2}}\right )}{6 d^2}+\frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c \left (\frac {2 d}{\left (c^2 d-e\right ) \sqrt {d+e x^2}}-\frac {2 \left (3 c^2 d-2 e\right ) \text {arctanh}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{c \left (c^2 d-e\right )^{3/2}}\right )}{6 d^2}\) |
(x*(a + b*ArcTan[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcTan[c*x]) )/(3*d^2*Sqrt[d + e*x^2]) - (b*c*((2*d)/((c^2*d - e)*Sqrt[d + e*x^2]) - (2 *(3*c^2*d - 2*e)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(c*(c^2*d - e)^(3/2))))/(6*d^2)
3.13.21.3.1 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_.)*(( e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2) *(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^q, x]}, Simp[(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}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
\[\int \frac {a +b \arctan \left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (124) = 248\).
Time = 0.63 (sec) , antiderivative size = 864, normalized size of antiderivative = 6.00 \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\left [\frac {{\left (3 \, b c^{2} d^{3} + {\left (3 \, b c^{2} d e^{2} - 2 \, b e^{3}\right )} x^{4} - 2 \, b d^{2} e + 2 \, {\left (3 \, b c^{2} d^{2} e - 2 \, b d e^{2}\right )} x^{2}\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 - 2 \, {\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} - 3 \, {\left (a c^{4} d^{3} - 2 \, a c^{2} d^{2} e + a d e^{2}\right )} x - {\left (2 \, {\left (b c^{4} d^{2} e - 2 \, b c^{2} d e^{2} + b e^{3}\right )} x^{3} + 3 \, {\left (b c^{4} d^{3} - 2 \, b c^{2} d^{2} e + b d e^{2}\right )} x\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{12 \, {\left (c^{4} d^{6} - 2 \, c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e - 2 \, c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}, \frac {{\left (3 \, b c^{2} d^{3} + {\left (3 \, b c^{2} d e^{2} - 2 \, b e^{3}\right )} x^{4} - 2 \, b d^{2} e + 2 \, {\left (3 \, b c^{2} d^{2} e - 2 \, b d e^{2}\right )} x^{2}\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 - 2 \, {\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} - 3 \, {\left (a c^{4} d^{3} - 2 \, a c^{2} d^{2} e + a d e^{2}\right )} x - {\left (2 \, {\left (b c^{4} d^{2} e - 2 \, b c^{2} d e^{2} + b e^{3}\right )} x^{3} + 3 \, {\left (b c^{4} d^{3} - 2 \, b c^{2} d^{2} e + b d e^{2}\right )} x\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{6 \, {\left (c^{4} d^{6} - 2 \, c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e - 2 \, c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}\right ] \]
[1/12*((3*b*c^2*d^3 + (3*b*c^2*d*e^2 - 2*b*e^3)*x^4 - 2*b*d^2*e + 2*(3*b*c ^2*d^2*e - 2*b*d*e^2)*x^2)*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 - 2*(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 - 3*(a*c^4*d^3 - 2*a*c^2*d^2*e + a*d*e^2)*x - (2*(b *c^4*d^2*e - 2*b*c^2*d*e^2 + b*e^3)*x^3 + 3*(b*c^4*d^3 - 2*b*c^2*d^2*e + b *d*e^2)*x)*arctan(c*x))*sqrt(e*x^2 + d))/(c^4*d^6 - 2*c^2*d^5*e + d^4*e^2 + (c^4*d^4*e^2 - 2*c^2*d^3*e^3 + d^2*e^4)*x^4 + 2*(c^4*d^5*e - 2*c^2*d^4*e ^2 + d^3*e^3)*x^2), 1/6*((3*b*c^2*d^3 + (3*b*c^2*d*e^2 - 2*b*e^3)*x^4 - 2* b*d^2*e + 2*(3*b*c^2*d^2*e - 2*b*d*e^2)*x^2)*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 - 2*(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 - 3*(a*c^4*d^3 - 2*a*c^2*d^2*e + a*d*e^2)*x - (2*(b*c^4*d^2*e - 2*b*c^2*d*e^2 + b*e^3)*x^ 3 + 3*(b*c^4*d^3 - 2*b*c^2*d^2*e + b*d*e^2)*x)*arctan(c*x))*sqrt(e*x^2 + d ))/(c^4*d^6 - 2*c^2*d^5*e + d^4*e^2 + (c^4*d^4*e^2 - 2*c^2*d^3*e^3 + d^2*e ^4)*x^4 + 2*(c^4*d^5*e - 2*c^2*d^4*e^2 + d^3*e^3)*x^2)]
Timed out. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
1/3*a*(2*x/(sqrt(e*x^2 + d)*d^2) + x/((e*x^2 + d)^(3/2)*d)) + 2*b*integrat e(1/2*arctan(c*x)/((e^2*x^4 + 2*d*e*x^2 + d^2)*sqrt(e*x^2 + d)), x)
\[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]