Integrand size = 19, antiderivative size = 105 \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^7} \, dx=-\frac {b c d}{30 x^5}+\frac {b c \left (2 c^2 d-3 e\right )}{36 x^3}-\frac {b c^3 \left (2 c^2 d-3 e\right )}{12 x}-\frac {1}{12} b c^4 \left (2 c^2 d-3 e\right ) \arctan (c x)-\frac {d (a+b \arctan (c x))}{6 x^6}-\frac {e (a+b \arctan (c x))}{4 x^4} \]
-1/30*b*c*d/x^5+1/36*b*c*(2*c^2*d-3*e)/x^3-1/12*b*c^3*(2*c^2*d-3*e)/x-1/12 *b*c^4*(2*c^2*d-3*e)*arctan(c*x)-1/6*d*(a+b*arctan(c*x))/x^6-1/4*e*(a+b*ar ctan(c*x))/x^4
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.92 \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^7} \, dx=-\frac {a d}{6 x^6}-\frac {a e}{4 x^4}-\frac {b d \arctan (c x)}{6 x^6}-\frac {b e \arctan (c x)}{4 x^4}-\frac {b c d \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-c^2 x^2\right )}{30 x^5}-\frac {b c e \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-c^2 x^2\right )}{12 x^3} \]
-1/6*(a*d)/x^6 - (a*e)/(4*x^4) - (b*d*ArcTan[c*x])/(6*x^6) - (b*e*ArcTan[c *x])/(4*x^4) - (b*c*d*Hypergeometric2F1[-5/2, 1, -3/2, -(c^2*x^2)])/(30*x^ 5) - (b*c*e*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)])/(12*x^3)
Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.82, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5511, 27, 359, 264, 264, 216}
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.
\(\displaystyle \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^7} \, dx\) |
\(\Big \downarrow \) 5511 |
\(\displaystyle -b c \int -\frac {3 e x^2+2 d}{12 x^6 \left (c^2 x^2+1\right )}dx-\frac {d (a+b \arctan (c x))}{6 x^6}-\frac {e (a+b \arctan (c x))}{4 x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{12} b c \int \frac {3 e x^2+2 d}{x^6 \left (c^2 x^2+1\right )}dx-\frac {d (a+b \arctan (c x))}{6 x^6}-\frac {e (a+b \arctan (c x))}{4 x^4}\) |
\(\Big \downarrow \) 359 |
\(\displaystyle \frac {1}{12} b c \left (-\left (2 c^2 d-3 e\right ) \int \frac {1}{x^4 \left (c^2 x^2+1\right )}dx-\frac {2 d}{5 x^5}\right )-\frac {d (a+b \arctan (c x))}{6 x^6}-\frac {e (a+b \arctan (c x))}{4 x^4}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {1}{12} b c \left (-\left (2 c^2 d-3 e\right ) \left (c^2 \left (-\int \frac {1}{x^2 \left (c^2 x^2+1\right )}dx\right )-\frac {1}{3 x^3}\right )-\frac {2 d}{5 x^5}\right )-\frac {d (a+b \arctan (c x))}{6 x^6}-\frac {e (a+b \arctan (c x))}{4 x^4}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {1}{12} b c \left (-\left (2 c^2 d-3 e\right ) \left (-\left (c^2 \left (c^2 \left (-\int \frac {1}{c^2 x^2+1}dx\right )-\frac {1}{x}\right )\right )-\frac {1}{3 x^3}\right )-\frac {2 d}{5 x^5}\right )-\frac {d (a+b \arctan (c x))}{6 x^6}-\frac {e (a+b \arctan (c x))}{4 x^4}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {d (a+b \arctan (c x))}{6 x^6}-\frac {e (a+b \arctan (c x))}{4 x^4}+\frac {1}{12} b c \left (-\left (-\left (c^2 \left (-c \arctan (c x)-\frac {1}{x}\right )\right )-\frac {1}{3 x^3}\right ) \left (2 c^2 d-3 e\right )-\frac {2 d}{5 x^5}\right )\) |
-1/6*(d*(a + b*ArcTan[c*x]))/x^6 - (e*(a + b*ArcTan[c*x]))/(4*x^4) + (b*c* ((-2*d)/(5*x^5) - (2*c^2*d - 3*e)*(-1/3*1/x^3 - c^2*(-x^(-1) - c*ArcTan[c* x]))))/12
3.12.24.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
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]))
Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(-\frac {30 x^{6} \arctan \left (c x \right ) b \,c^{6} d -45 x^{6} \arctan \left (c x \right ) b \,c^{4} e +30 b \,c^{5} d \,x^{5}-45 b \,c^{3} e \,x^{5}-10 b \,c^{3} d \,x^{3}+15 b c e \,x^{3}+45 \arctan \left (c x \right ) b e \,x^{2}+45 a e \,x^{2}+6 b c d x +30 \arctan \left (c x \right ) b d +30 a d}{180 x^{6}}\) | \(109\) |
parts | \(a \left (-\frac {d}{6 x^{6}}-\frac {e}{4 x^{4}}\right )+b \,c^{6} \left (-\frac {\arctan \left (c x \right ) d}{6 c^{6} x^{6}}-\frac {\arctan \left (c x \right ) e}{4 c^{6} x^{4}}-\frac {-\frac {-2 c^{2} d +3 e}{c x}-\frac {2 c^{2} d -3 e}{3 c^{3} x^{3}}+\frac {2 d}{5 c^{3} x^{5}}+\left (2 c^{2} d -3 e \right ) \arctan \left (c x \right )}{12 c^{2}}\right )\) | \(115\) |
derivativedivides | \(c^{6} \left (\frac {a \left (-\frac {e}{4 c^{4} x^{4}}-\frac {d}{6 c^{4} x^{6}}\right )}{c^{2}}+\frac {b \left (-\frac {\arctan \left (c x \right ) e}{4 c^{4} x^{4}}-\frac {\arctan \left (c x \right ) d}{6 c^{4} x^{6}}-\frac {\left (2 c^{2} d -3 e \right ) \arctan \left (c x \right )}{12}+\frac {-2 c^{2} d +3 e}{12 c x}+\frac {2 c^{2} d -3 e}{36 c^{3} x^{3}}-\frac {d}{30 c^{3} x^{5}}\right )}{c^{2}}\right )\) | \(123\) |
default | \(c^{6} \left (\frac {a \left (-\frac {e}{4 c^{4} x^{4}}-\frac {d}{6 c^{4} x^{6}}\right )}{c^{2}}+\frac {b \left (-\frac {\arctan \left (c x \right ) e}{4 c^{4} x^{4}}-\frac {\arctan \left (c x \right ) d}{6 c^{4} x^{6}}-\frac {\left (2 c^{2} d -3 e \right ) \arctan \left (c x \right )}{12}+\frac {-2 c^{2} d +3 e}{12 c x}+\frac {2 c^{2} d -3 e}{36 c^{3} x^{3}}-\frac {d}{30 c^{3} x^{5}}\right )}{c^{2}}\right )\) | \(123\) |
risch | \(\frac {i b \left (3 e \,x^{2}+2 d \right ) \ln \left (i c x +1\right )}{24 x^{6}}-\frac {30 i \ln \left (-c x -i\right ) b \,c^{6} d \,x^{6}-30 i \ln \left (-c x +i\right ) b \,c^{6} d \,x^{6}-45 i \ln \left (-c x -i\right ) b \,c^{4} e \,x^{6}+45 i \ln \left (-c x +i\right ) b \,c^{4} e \,x^{6}+60 b \,c^{5} d \,x^{5}-90 b \,c^{3} e \,x^{5}-20 b \,c^{3} d \,x^{3}+45 i b e \ln \left (-i c x +1\right ) x^{2}+30 b c e \,x^{3}+30 i b d \ln \left (-i c x +1\right )+90 a e \,x^{2}+12 b c d x +60 a d}{360 x^{6}}\) | \(193\) |
-1/180*(30*x^6*arctan(c*x)*b*c^6*d-45*x^6*arctan(c*x)*b*c^4*e+30*b*c^5*d*x ^5-45*b*c^3*e*x^5-10*b*c^3*d*x^3+15*b*c*e*x^3+45*arctan(c*x)*b*e*x^2+45*a* e*x^2+6*b*c*d*x+30*arctan(c*x)*b*d+30*a*d)/x^6
Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.93 \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^7} \, dx=-\frac {15 \, {\left (2 \, b c^{5} d - 3 \, b c^{3} e\right )} x^{5} + 6 \, b c d x + 45 \, a e x^{2} - 5 \, {\left (2 \, b c^{3} d - 3 \, b c e\right )} x^{3} + 30 \, a d + 15 \, {\left ({\left (2 \, b c^{6} d - 3 \, b c^{4} e\right )} x^{6} + 3 \, b e x^{2} + 2 \, b d\right )} \arctan \left (c x\right )}{180 \, x^{6}} \]
-1/180*(15*(2*b*c^5*d - 3*b*c^3*e)*x^5 + 6*b*c*d*x + 45*a*e*x^2 - 5*(2*b*c ^3*d - 3*b*c*e)*x^3 + 30*a*d + 15*((2*b*c^6*d - 3*b*c^4*e)*x^6 + 3*b*e*x^2 + 2*b*d)*arctan(c*x))/x^6
Time = 0.37 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.16 \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^7} \, dx=- \frac {a d}{6 x^{6}} - \frac {a e}{4 x^{4}} - \frac {b c^{6} d \operatorname {atan}{\left (c x \right )}}{6} - \frac {b c^{5} d}{6 x} + \frac {b c^{4} e \operatorname {atan}{\left (c x \right )}}{4} + \frac {b c^{3} d}{18 x^{3}} + \frac {b c^{3} e}{4 x} - \frac {b c d}{30 x^{5}} - \frac {b c e}{12 x^{3}} - \frac {b d \operatorname {atan}{\left (c x \right )}}{6 x^{6}} - \frac {b e \operatorname {atan}{\left (c x \right )}}{4 x^{4}} \]
-a*d/(6*x**6) - a*e/(4*x**4) - b*c**6*d*atan(c*x)/6 - b*c**5*d/(6*x) + b*c **4*e*atan(c*x)/4 + b*c**3*d/(18*x**3) + b*c**3*e/(4*x) - b*c*d/(30*x**5) - b*c*e/(12*x**3) - b*d*atan(c*x)/(6*x**6) - b*e*atan(c*x)/(4*x**4)
Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.98 \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^7} \, dx=-\frac {1}{90} \, {\left ({\left (15 \, c^{5} \arctan \left (c x\right ) + \frac {15 \, c^{4} x^{4} - 5 \, c^{2} x^{2} + 3}{x^{5}}\right )} c + \frac {15 \, \arctan \left (c x\right )}{x^{6}}\right )} b d + \frac {1}{12} \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b e - \frac {a e}{4 \, x^{4}} - \frac {a d}{6 \, x^{6}} \]
-1/90*((15*c^5*arctan(c*x) + (15*c^4*x^4 - 5*c^2*x^2 + 3)/x^5)*c + 15*arct an(c*x)/x^6)*b*d + 1/12*((3*c^3*arctan(c*x) + (3*c^2*x^2 - 1)/x^3)*c - 3*a rctan(c*x)/x^4)*b*e - 1/4*a*e/x^4 - 1/6*a*d/x^6
\[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^7} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{7}} \,d x } \]
Time = 0.71 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^7} \, dx=\frac {b\,c^4\,\mathrm {atan}\left (\frac {b\,c^2\,x\,\left (3\,e-2\,c^2\,d\right )}{3\,b\,c\,e-2\,b\,c^3\,d}\right )\,\left (3\,e-2\,c^2\,d\right )}{12}-\frac {\mathrm {atan}\left (c\,x\right )\,\left (\frac {b\,e\,x^2}{4}+\frac {b\,d}{6}\right )}{x^6}-\frac {x^3\,\left (b\,c\,e-\frac {2\,b\,c^3\,d}{3}\right )+2\,a\,d-c^2\,x^5\,\left (3\,b\,c\,e-2\,b\,c^3\,d\right )+3\,a\,e\,x^2+\frac {2\,b\,c\,d\,x}{5}}{12\,x^6} \]