Integrand size = 19, antiderivative size = 237 \[ \int \frac {x^2 (a+b \arctan (c x))}{d+e x} \, dx=-\frac {a d x}{e^2}-\frac {b x}{2 c e}+\frac {b \arctan (c x)}{2 c^2 e}-\frac {b d x \arctan (c x)}{e^2}+\frac {x^2 (a+b \arctan (c x))}{2 e}-\frac {d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{e^3}+\frac {d^2 (a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}+\frac {b d \log \left (1+c^2 x^2\right )}{2 c e^2}+\frac {i b d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e^3}-\frac {i b d^2 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3} \]
-a*d*x/e^2-1/2*b*x/c/e+1/2*b*arctan(c*x)/c^2/e-b*d*x*arctan(c*x)/e^2+1/2*x ^2*(a+b*arctan(c*x))/e-d^2*(a+b*arctan(c*x))*ln(2/(1-I*c*x))/e^3+d^2*(a+b* arctan(c*x))*ln(2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/e^3+1/2*b*d*ln(c^2*x^2+1) /c/e^2+1/2*I*b*d^2*polylog(2,1-2/(1-I*c*x))/e^3-1/2*I*b*d^2*polylog(2,1-2* c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/e^3
Time = 1.29 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.70 \[ \int \frac {x^2 (a+b \arctan (c x))}{d+e x} \, dx=\frac {-2 a d e x-\frac {b e^2 x}{c}+a e^2 x^2+\frac {b e^2 \arctan (c x)}{c^2}+i b d^2 \pi \arctan (c x)-2 b d e x \arctan (c x)+b e^2 x^2 \arctan (c x)-2 i b d^2 \arctan \left (\frac {c d}{e}\right ) \arctan (c x)+i b d^2 \arctan (c x)^2+\frac {b d e \arctan (c x)^2}{c}-\frac {b d \sqrt {1+\frac {c^2 d^2}{e^2}} e e^{i \arctan \left (\frac {c d}{e}\right )} \arctan (c x)^2}{c}+b d^2 \pi \log \left (1+e^{-2 i \arctan (c x)}\right )-2 b d^2 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+2 b d^2 \arctan \left (\frac {c d}{e}\right ) \log \left (1-e^{2 i \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )}\right )+2 b d^2 \arctan (c x) \log \left (1-e^{2 i \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )}\right )+2 a d^2 \log (d+e x)+\frac {b d e \log \left (1+c^2 x^2\right )}{c}+\frac {1}{2} b d^2 \pi \log \left (1+c^2 x^2\right )-2 b d^2 \arctan \left (\frac {c d}{e}\right ) \log \left (\sin \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )\right )+i b d^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-i b d^2 \operatorname {PolyLog}\left (2,e^{2 i \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )}\right )}{2 e^3} \]
(-2*a*d*e*x - (b*e^2*x)/c + a*e^2*x^2 + (b*e^2*ArcTan[c*x])/c^2 + I*b*d^2* Pi*ArcTan[c*x] - 2*b*d*e*x*ArcTan[c*x] + b*e^2*x^2*ArcTan[c*x] - (2*I)*b*d ^2*ArcTan[(c*d)/e]*ArcTan[c*x] + I*b*d^2*ArcTan[c*x]^2 + (b*d*e*ArcTan[c*x ]^2)/c - (b*d*Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^ 2)/c + b*d^2*Pi*Log[1 + E^((-2*I)*ArcTan[c*x])] - 2*b*d^2*ArcTan[c*x]*Log[ 1 + E^((2*I)*ArcTan[c*x])] + 2*b*d^2*ArcTan[(c*d)/e]*Log[1 - E^((2*I)*(Arc Tan[(c*d)/e] + ArcTan[c*x]))] + 2*b*d^2*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcT an[(c*d)/e] + ArcTan[c*x]))] + 2*a*d^2*Log[d + e*x] + (b*d*e*Log[1 + c^2*x ^2])/c + (b*d^2*Pi*Log[1 + c^2*x^2])/2 - 2*b*d^2*ArcTan[(c*d)/e]*Log[Sin[A rcTan[(c*d)/e] + ArcTan[c*x]]] + I*b*d^2*PolyLog[2, -E^((2*I)*ArcTan[c*x]) ] - I*b*d^2*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))])/(2*e^3)
Time = 0.42 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5411, 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.
| \(\displaystyle \int \frac {x^2 (a+b \arctan (c x))}{d+e x} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (\frac {d^2 (a+b \arctan (c x))}{e^2 (d+e x)}-\frac {d (a+b \arctan (c x))}{e^2}+\frac {x (a+b \arctan (c x))}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^2 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{e^3}+\frac {d^2 (a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^3}+\frac {x^2 (a+b \arctan (c x))}{2 e}-\frac {a d x}{e^2}+\frac {b \arctan (c x)}{2 c^2 e}-\frac {b d x \arctan (c x)}{e^2}+\frac {b d \log \left (c^2 x^2+1\right )}{2 c e^2}+\frac {i b d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e^3}-\frac {i b d^2 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3}-\frac {b x}{2 c e}\) |
-((a*d*x)/e^2) - (b*x)/(2*c*e) + (b*ArcTan[c*x])/(2*c^2*e) - (b*d*x*ArcTan [c*x])/e^2 + (x^2*(a + b*ArcTan[c*x]))/(2*e) - (d^2*(a + b*ArcTan[c*x])*Lo g[2/(1 - I*c*x)])/e^3 + (d^2*(a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^3 + (b*d*Log[1 + c^2*x^2])/(2*c*e^2) + ((I/2)*b*d ^2*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^3 - ((I/2)*b*d^2*PolyLog[2, 1 - (2*c*( d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^3
3.2.35.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f* x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] & & IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
Time = 0.34 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.20
| method | result | size |
| parts | \(\frac {a \,x^{2}}{2 e}-\frac {a d x}{e^{2}}+\frac {a \,d^{2} \ln \left (e x +d \right )}{e^{3}}+\frac {b \left (\frac {c^{3} \arctan \left (c x \right ) x^{2}}{2 e}-\frac {c^{3} \arctan \left (c x \right ) d x}{e^{2}}+\frac {c^{3} \arctan \left (c x \right ) d^{2} \ln \left (e c x +c d \right )}{e^{3}}-\frac {c \left (\frac {c^{2} d^{2} \left (-\frac {i \ln \left (e c x +c d \right ) \left (\ln \left (\frac {-e c x +i e}{c d +i e}\right )-\ln \left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-e c x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{e}-\frac {c d \ln \left (c^{2} d^{2}-2 c d \left (e c x +c d \right )+e^{2}+\left (e c x +c d \right )^{2}\right )}{2 e}-\frac {\arctan \left (c x \right )}{2}+\frac {e c x +c d}{2 e}\right )}{e}\right )}{c^{3}}\) | \(284\) |
| derivativedivides | \(\frac {-\frac {a \,c^{3} d x}{e^{2}}+\frac {a \,c^{3} x^{2}}{2 e}+\frac {a \,c^{3} d^{2} \ln \left (e c x +c d \right )}{e^{3}}+b c \left (-\frac {\arctan \left (c x \right ) c^{2} d x}{e^{2}}+\frac {\arctan \left (c x \right ) c^{2} x^{2}}{2 e}+\frac {\arctan \left (c x \right ) c^{2} d^{2} \ln \left (e c x +c d \right )}{e^{3}}-\frac {\frac {c^{2} d^{2} \left (\frac {i \ln \left (e c x +c d \right ) \left (-\ln \left (\frac {-e c x +i e}{c d +i e}\right )+\ln \left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-e c x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{e}-\frac {c d \ln \left (c^{2} d^{2}-2 c d \left (e c x +c d \right )+e^{2}+\left (e c x +c d \right )^{2}\right )}{2 e}-\frac {\arctan \left (c x \right )}{2}+\frac {e c x +c d}{2 e}}{e}\right )}{c^{3}}\) | \(297\) |
| default | \(\frac {-\frac {a \,c^{3} d x}{e^{2}}+\frac {a \,c^{3} x^{2}}{2 e}+\frac {a \,c^{3} d^{2} \ln \left (e c x +c d \right )}{e^{3}}+b c \left (-\frac {\arctan \left (c x \right ) c^{2} d x}{e^{2}}+\frac {\arctan \left (c x \right ) c^{2} x^{2}}{2 e}+\frac {\arctan \left (c x \right ) c^{2} d^{2} \ln \left (e c x +c d \right )}{e^{3}}-\frac {\frac {c^{2} d^{2} \left (\frac {i \ln \left (e c x +c d \right ) \left (-\ln \left (\frac {-e c x +i e}{c d +i e}\right )+\ln \left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-e c x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{e}-\frac {c d \ln \left (c^{2} d^{2}-2 c d \left (e c x +c d \right )+e^{2}+\left (e c x +c d \right )^{2}\right )}{2 e}-\frac {\arctan \left (c x \right )}{2}+\frac {e c x +c d}{2 e}}{e}\right )}{c^{3}}\) | \(297\) |
| risch | \(-\frac {b x}{2 c e}+\frac {b \arctan \left (c x \right )}{4 c^{2} e}+\frac {b d \ln \left (c^{2} x^{2}+1\right )}{4 c \,e^{2}}-\frac {i b \ln \left (i c x +1\right ) x^{2}}{4 e}-\frac {i b \ln \left (i c x +1\right )}{4 c^{2} e}+\frac {i b \ln \left (c^{2} x^{2}+1\right )}{8 c^{2} e}+\frac {i b \ln \left (i c x +1\right ) d x}{2 e^{2}}-\frac {i b d \arctan \left (c x \right )}{2 c \,e^{2}}+\frac {a \,d^{2} \ln \left (i c d -\left (-i c x +1\right ) e +e \right )}{e^{3}}-\frac {a d x}{e^{2}}+\frac {b \ln \left (i c x +1\right ) d}{2 c \,e^{2}}+\frac {i b \,d^{2} \operatorname {dilog}\left (\frac {-i c d +\left (-i c x +1\right ) e -e}{-i c d -e}\right )}{2 e^{3}}+\frac {i b \ln \left (-i c x +1\right ) x^{2}}{4 e}-\frac {i b \,d^{2} \ln \left (i c x +1\right ) \ln \left (\frac {i c d +\left (i c x +1\right ) e -e}{i c d -e}\right )}{2 e^{3}}-\frac {i b \ln \left (-i c x +1\right ) x d}{2 e^{2}}-\frac {i b \,d^{2} \operatorname {dilog}\left (\frac {i c d +\left (i c x +1\right ) e -e}{i c d -e}\right )}{2 e^{3}}-\frac {b d}{c \,e^{2}}+\frac {i b \,d^{2} \ln \left (-i c x +1\right ) \ln \left (\frac {-i c d +\left (-i c x +1\right ) e -e}{-i c d -e}\right )}{2 e^{3}}-\frac {i a d}{c \,e^{2}}+\frac {a}{2 c^{2} e}+\frac {a \,x^{2}}{2 e}\) | \(439\) |
1/2*a/e*x^2-a*d*x/e^2+a/e^3*d^2*ln(e*x+d)+b/c^3*(1/2*c^3*arctan(c*x)/e*x^2 -c^3*arctan(c*x)*d/e^2*x+c^3*arctan(c*x)*d^2/e^3*ln(c*e*x+c*d)-c/e*(1/e*c^ 2*d^2*(-1/2*I*ln(c*e*x+c*d)*(ln((I*e-e*c*x)/(c*d+I*e))-ln((I*e+e*c*x)/(I*e -c*d)))/e-1/2*I*(dilog((I*e-e*c*x)/(c*d+I*e))-dilog((I*e+e*c*x)/(I*e-c*d)) )/e)-1/2/e*c*d*ln(c^2*d^2-2*c*d*(c*e*x+c*d)+e^2+(c*e*x+c*d)^2)-1/2*arctan( c*x)+1/2/e*(c*e*x+c*d)))
\[ \int \frac {x^2 (a+b \arctan (c x))}{d+e x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{e x + d} \,d x } \]
\[ \int \frac {x^2 (a+b \arctan (c x))}{d+e x} \, dx=\int \frac {x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{d + e x}\, dx \]
\[ \int \frac {x^2 (a+b \arctan (c x))}{d+e x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{e x + d} \,d x } \]
1/2*a*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) + 2*b*integrate(1/2*x ^2*arctan(c*x)/(e*x + d), x)
\[ \int \frac {x^2 (a+b \arctan (c x))}{d+e x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))}{d+e x} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{d+e\,x} \,d x \]