Integrand size = 21, antiderivative size = 239 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=-\frac {b \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) x^2}{630 c^7}-\frac {b e \left (189 c^4 d^2-135 c^2 d e+35 e^2\right ) x^4}{1260 c^5}-\frac {b \left (27 c^2 d-7 e\right ) e^2 x^6}{378 c^3}-\frac {b e^3 x^8}{72 c}+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))+\frac {3}{5} d^2 e x^5 (a+b \arctan (c x))+\frac {3}{7} d e^2 x^7 (a+b \arctan (c x))+\frac {1}{9} e^3 x^9 (a+b \arctan (c x))+\frac {b \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) \log \left (1+c^2 x^2\right )}{630 c^9} \]
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Time = 0.24 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {276, 5096, 12, 1813, 1634} \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {1}{3} d^3 x^3 (a+b \arctan (c x))+\frac {3}{5} d^2 e x^5 (a+b \arctan (c x))+\frac {3}{7} d e^2 x^7 (a+b \arctan (c x))+\frac {1}{9} e^3 x^9 (a+b \arctan (c x))-\frac {b e^2 x^6 \left (27 c^2 d-7 e\right )}{378 c^3}-\frac {b e x^4 \left (189 c^4 d^2-135 c^2 d e+35 e^2\right )}{1260 c^5}+\frac {b \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) \log \left (c^2 x^2+1\right )}{630 c^9}-\frac {b x^2 \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right )}{630 c^7}-\frac {b e^3 x^8}{72 c} \]
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Rule 12
Rule 276
Rule 1634
Rule 1813
Rule 5096
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} d^3 x^3 (a+b \arctan (c x))+\frac {3}{5} d^2 e x^5 (a+b \arctan (c x))+\frac {3}{7} d e^2 x^7 (a+b \arctan (c x))+\frac {1}{9} e^3 x^9 (a+b \arctan (c x))-(b c) \int \frac {x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right )}{315 \left (1+c^2 x^2\right )} \, dx \\ & = \frac {1}{3} d^3 x^3 (a+b \arctan (c x))+\frac {3}{5} d^2 e x^5 (a+b \arctan (c x))+\frac {3}{7} d e^2 x^7 (a+b \arctan (c x))+\frac {1}{9} e^3 x^9 (a+b \arctan (c x))-\frac {1}{315} (b c) \int \frac {x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right )}{1+c^2 x^2} \, dx \\ & = \frac {1}{3} d^3 x^3 (a+b \arctan (c x))+\frac {3}{5} d^2 e x^5 (a+b \arctan (c x))+\frac {3}{7} d e^2 x^7 (a+b \arctan (c x))+\frac {1}{9} e^3 x^9 (a+b \arctan (c x))-\frac {1}{630} (b c) \text {Subst}\left (\int \frac {x \left (105 d^3+189 d^2 e x+135 d e^2 x^2+35 e^3 x^3\right )}{1+c^2 x} \, dx,x,x^2\right ) \\ & = \frac {1}{3} d^3 x^3 (a+b \arctan (c x))+\frac {3}{5} d^2 e x^5 (a+b \arctan (c x))+\frac {3}{7} d e^2 x^7 (a+b \arctan (c x))+\frac {1}{9} e^3 x^9 (a+b \arctan (c x))-\frac {1}{630} (b c) \text {Subst}\left (\int \left (\frac {105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3}{c^8}+\frac {e \left (189 c^4 d^2-135 c^2 d e+35 e^2\right ) x}{c^6}+\frac {5 \left (27 c^2 d-7 e\right ) e^2 x^2}{c^4}+\frac {35 e^3 x^3}{c^2}+\frac {-105 c^6 d^3+189 c^4 d^2 e-135 c^2 d e^2+35 e^3}{c^8 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = -\frac {b \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) x^2}{630 c^7}-\frac {b e \left (189 c^4 d^2-135 c^2 d e+35 e^2\right ) x^4}{1260 c^5}-\frac {b \left (27 c^2 d-7 e\right ) e^2 x^6}{378 c^3}-\frac {b e^3 x^8}{72 c}+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))+\frac {3}{5} d^2 e x^5 (a+b \arctan (c x))+\frac {3}{7} d e^2 x^7 (a+b \arctan (c x))+\frac {1}{9} e^3 x^9 (a+b \arctan (c x))+\frac {b \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) \log \left (1+c^2 x^2\right )}{630 c^9} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.05 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {1}{3} d^3 x^3 (a+b \arctan (c x))+\frac {3}{5} d^2 e x^5 (a+b \arctan (c x))+\frac {3}{7} d e^2 x^7 (a+b \arctan (c x))+\frac {1}{9} e^3 x^9 (a+b \arctan (c x))+\frac {1}{216} b e^3 \left (\frac {12 x^2}{c^7}-\frac {6 x^4}{c^5}+\frac {4 x^6}{c^3}-\frac {3 x^8}{c}-\frac {12 \log \left (1+c^2 x^2\right )}{c^9}\right )-\frac {1}{28} b d e^2 \left (\frac {6 x^2}{c^5}-\frac {3 x^4}{c^3}+\frac {2 x^6}{c}-\frac {6 \log \left (1+c^2 x^2\right )}{c^7}\right )+\frac {3}{20} b d^2 e \left (\frac {2 x^2}{c^3}-\frac {x^4}{c}-\frac {2 \log \left (1+c^2 x^2\right )}{c^5}\right )-\frac {1}{6} b d^3 \left (\frac {x^2}{c}-\frac {\log \left (1+c^2 x^2\right )}{c^3}\right ) \]
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Time = 0.32 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.14
method | result | size |
parts | \(a \left (\frac {1}{9} e^{3} x^{9}+\frac {3}{7} e^{2} d \,x^{7}+\frac {3}{5} e \,d^{2} x^{5}+\frac {1}{3} d^{3} x^{3}\right )+\frac {b \left (\frac {\arctan \left (c x \right ) c^{3} e^{3} x^{9}}{9}+\frac {3 \arctan \left (c x \right ) c^{3} e^{2} d \,x^{7}}{7}+\frac {3 \arctan \left (c x \right ) c^{3} d^{2} e \,x^{5}}{5}+\frac {\arctan \left (c x \right ) d^{3} c^{3} x^{3}}{3}-\frac {\frac {105 d^{3} c^{8} x^{2}}{2}+\frac {189 d^{2} c^{8} e \,x^{4}}{4}+\frac {45 d \,c^{8} e^{2} x^{6}}{2}-\frac {189 d^{2} c^{6} e \,x^{2}}{2}+\frac {35 e^{3} c^{8} x^{8}}{8}-\frac {135 d \,c^{6} e^{2} x^{4}}{4}-\frac {35 e^{3} c^{6} x^{6}}{6}+\frac {135 d \,c^{4} e^{2} x^{2}}{2}+\frac {35 e^{3} c^{4} x^{4}}{4}-\frac {35 e^{3} c^{2} x^{2}}{2}+\frac {\left (-105 c^{6} d^{3}+189 c^{4} d^{2} e -135 e^{2} d \,c^{2}+35 e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}}{315 c^{6}}\right )}{c^{3}}\) | \(272\) |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{3} d^{3} c^{9} x^{3}+\frac {3}{5} d^{2} c^{9} e \,x^{5}+\frac {3}{7} d \,c^{9} e^{2} x^{7}+\frac {1}{9} e^{3} c^{9} x^{9}\right )}{c^{6}}+\frac {b \left (\frac {\arctan \left (c x \right ) d^{3} c^{9} x^{3}}{3}+\frac {3 \arctan \left (c x \right ) d^{2} c^{9} e \,x^{5}}{5}+\frac {3 \arctan \left (c x \right ) d \,c^{9} e^{2} x^{7}}{7}+\frac {\arctan \left (c x \right ) e^{3} c^{9} x^{9}}{9}-\frac {d^{3} c^{8} x^{2}}{6}-\frac {3 d^{2} c^{8} e \,x^{4}}{20}+\frac {3 d^{2} c^{6} e \,x^{2}}{10}-\frac {d \,c^{8} e^{2} x^{6}}{14}+\frac {3 d \,c^{6} e^{2} x^{4}}{28}-\frac {e^{3} c^{8} x^{8}}{72}-\frac {3 d \,c^{4} e^{2} x^{2}}{14}+\frac {e^{3} c^{6} x^{6}}{54}-\frac {e^{3} c^{4} x^{4}}{36}+\frac {e^{3} c^{2} x^{2}}{18}-\frac {\left (-105 c^{6} d^{3}+189 c^{4} d^{2} e -135 e^{2} d \,c^{2}+35 e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{630}\right )}{c^{6}}}{c^{3}}\) | \(285\) |
default | \(\frac {\frac {a \left (\frac {1}{3} d^{3} c^{9} x^{3}+\frac {3}{5} d^{2} c^{9} e \,x^{5}+\frac {3}{7} d \,c^{9} e^{2} x^{7}+\frac {1}{9} e^{3} c^{9} x^{9}\right )}{c^{6}}+\frac {b \left (\frac {\arctan \left (c x \right ) d^{3} c^{9} x^{3}}{3}+\frac {3 \arctan \left (c x \right ) d^{2} c^{9} e \,x^{5}}{5}+\frac {3 \arctan \left (c x \right ) d \,c^{9} e^{2} x^{7}}{7}+\frac {\arctan \left (c x \right ) e^{3} c^{9} x^{9}}{9}-\frac {d^{3} c^{8} x^{2}}{6}-\frac {3 d^{2} c^{8} e \,x^{4}}{20}+\frac {3 d^{2} c^{6} e \,x^{2}}{10}-\frac {d \,c^{8} e^{2} x^{6}}{14}+\frac {3 d \,c^{6} e^{2} x^{4}}{28}-\frac {e^{3} c^{8} x^{8}}{72}-\frac {3 d \,c^{4} e^{2} x^{2}}{14}+\frac {e^{3} c^{6} x^{6}}{54}-\frac {e^{3} c^{4} x^{4}}{36}+\frac {e^{3} c^{2} x^{2}}{18}-\frac {\left (-105 c^{6} d^{3}+189 c^{4} d^{2} e -135 e^{2} d \,c^{2}+35 e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{630}\right )}{c^{6}}}{c^{3}}\) | \(285\) |
parallelrisch | \(\frac {840 x^{9} \arctan \left (c x \right ) b \,c^{9} e^{3}+840 a \,c^{9} e^{3} x^{9}+3240 x^{7} \arctan \left (c x \right ) b \,c^{9} d \,e^{2}-105 b \,c^{8} e^{3} x^{8}+3240 a \,c^{9} d \,e^{2} x^{7}+4536 x^{5} \arctan \left (c x \right ) b \,c^{9} d^{2} e -540 b \,c^{8} d \,e^{2} x^{6}+4536 a \,c^{9} d^{2} e \,x^{5}+2520 x^{3} \arctan \left (c x \right ) b \,c^{9} d^{3}+140 b \,c^{6} e^{3} x^{6}-1134 b \,c^{8} d^{2} e \,x^{4}+2520 a \,c^{9} d^{3} x^{3}+810 b \,c^{6} d \,e^{2} x^{4}-1260 b \,c^{8} d^{3} x^{2}-210 b \,c^{4} e^{3} x^{4}+2268 b \,c^{6} d^{2} e \,x^{2}+1260 \ln \left (c^{2} x^{2}+1\right ) b \,c^{6} d^{3}-1620 b \,c^{4} d \,e^{2} x^{2}-2268 \ln \left (c^{2} x^{2}+1\right ) b \,c^{4} d^{2} e +420 b \,c^{2} e^{3} x^{2}+1620 \ln \left (c^{2} x^{2}+1\right ) b \,c^{2} d \,e^{2}-420 \ln \left (c^{2} x^{2}+1\right ) b \,e^{3}}{7560 c^{9}}\) | \(323\) |
risch | \(\frac {i b \,d^{3} x^{3} \ln \left (-i c x +1\right )}{6}+\frac {i b \,e^{3} x^{9} \ln \left (-i c x +1\right )}{18}+\frac {3 i b d \,e^{2} x^{7} \ln \left (-i c x +1\right )}{14}+\frac {x^{9} e^{3} a}{9}-\frac {i b \left (35 e^{3} x^{9}+135 e^{2} d \,x^{7}+189 e \,d^{2} x^{5}+105 d^{3} x^{3}\right ) \ln \left (i c x +1\right )}{630}+\frac {3 x^{7} e^{2} d a}{7}-\frac {b \,e^{3} x^{8}}{72 c}+\frac {3 i b \,d^{2} e \,x^{5} \ln \left (-i c x +1\right )}{10}+\frac {3 x^{5} e \,d^{2} a}{5}-\frac {b d \,e^{2} x^{6}}{14 c}+\frac {x^{3} d^{3} a}{3}-\frac {3 b \,d^{2} e \,x^{4}}{20 c}+\frac {b \,e^{3} x^{6}}{54 c^{3}}-\frac {b \,d^{3} x^{2}}{6 c}+\frac {3 b d \,e^{2} x^{4}}{28 c^{3}}+\frac {3 b \,d^{2} e \,x^{2}}{10 c^{3}}-\frac {b \,e^{3} x^{4}}{36 c^{5}}+\frac {\ln \left (-c^{2} x^{2}-1\right ) b \,d^{3}}{6 c^{3}}-\frac {3 b d \,e^{2} x^{2}}{14 c^{5}}-\frac {3 \ln \left (-c^{2} x^{2}-1\right ) b \,d^{2} e}{10 c^{5}}+\frac {b \,e^{3} x^{2}}{18 c^{7}}+\frac {3 \ln \left (-c^{2} x^{2}-1\right ) b d \,e^{2}}{14 c^{7}}-\frac {\ln \left (-c^{2} x^{2}-1\right ) b \,e^{3}}{18 c^{9}}\) | \(368\) |
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Time = 0.27 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.16 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {840 \, a c^{9} e^{3} x^{9} + 3240 \, a c^{9} d e^{2} x^{7} - 105 \, b c^{8} e^{3} x^{8} + 4536 \, a c^{9} d^{2} e x^{5} + 2520 \, a c^{9} d^{3} x^{3} - 20 \, {\left (27 \, b c^{8} d e^{2} - 7 \, b c^{6} e^{3}\right )} x^{6} - 6 \, {\left (189 \, b c^{8} d^{2} e - 135 \, b c^{6} d e^{2} + 35 \, b c^{4} e^{3}\right )} x^{4} - 12 \, {\left (105 \, b c^{8} d^{3} - 189 \, b c^{6} d^{2} e + 135 \, b c^{4} d e^{2} - 35 \, b c^{2} e^{3}\right )} x^{2} + 24 \, {\left (35 \, b c^{9} e^{3} x^{9} + 135 \, b c^{9} d e^{2} x^{7} + 189 \, b c^{9} d^{2} e x^{5} + 105 \, b c^{9} d^{3} x^{3}\right )} \arctan \left (c x\right ) + 12 \, {\left (105 \, b c^{6} d^{3} - 189 \, b c^{4} d^{2} e + 135 \, b c^{2} d e^{2} - 35 \, b e^{3}\right )} \log \left (c^{2} x^{2} + 1\right )}{7560 \, c^{9}} \]
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Time = 0.72 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.63 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\begin {cases} \frac {a d^{3} x^{3}}{3} + \frac {3 a d^{2} e x^{5}}{5} + \frac {3 a d e^{2} x^{7}}{7} + \frac {a e^{3} x^{9}}{9} + \frac {b d^{3} x^{3} \operatorname {atan}{\left (c x \right )}}{3} + \frac {3 b d^{2} e x^{5} \operatorname {atan}{\left (c x \right )}}{5} + \frac {3 b d e^{2} x^{7} \operatorname {atan}{\left (c x \right )}}{7} + \frac {b e^{3} x^{9} \operatorname {atan}{\left (c x \right )}}{9} - \frac {b d^{3} x^{2}}{6 c} - \frac {3 b d^{2} e x^{4}}{20 c} - \frac {b d e^{2} x^{6}}{14 c} - \frac {b e^{3} x^{8}}{72 c} + \frac {b d^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{6 c^{3}} + \frac {3 b d^{2} e x^{2}}{10 c^{3}} + \frac {3 b d e^{2} x^{4}}{28 c^{3}} + \frac {b e^{3} x^{6}}{54 c^{3}} - \frac {3 b d^{2} e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{10 c^{5}} - \frac {3 b d e^{2} x^{2}}{14 c^{5}} - \frac {b e^{3} x^{4}}{36 c^{5}} + \frac {3 b d e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{14 c^{7}} + \frac {b e^{3} x^{2}}{18 c^{7}} - \frac {b e^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{18 c^{9}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{3} x^{3}}{3} + \frac {3 d^{2} e x^{5}}{5} + \frac {3 d e^{2} x^{7}}{7} + \frac {e^{3} x^{9}}{9}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.11 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {1}{9} \, a e^{3} x^{9} + \frac {3}{7} \, a d e^{2} x^{7} + \frac {3}{5} \, a d^{2} e x^{5} + \frac {1}{3} \, a d^{3} x^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b d^{3} + \frac {3}{20} \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b d^{2} e + \frac {1}{28} \, {\left (12 \, x^{7} \arctan \left (c x\right ) - c {\left (\frac {2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} - \frac {6 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )}\right )} b d e^{2} + \frac {1}{216} \, {\left (24 \, x^{9} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{6} x^{8} - 4 \, c^{4} x^{6} + 6 \, c^{2} x^{4} - 12 \, x^{2}}{c^{8}} + \frac {12 \, \log \left (c^{2} x^{2} + 1\right )}{c^{10}}\right )}\right )} b e^{3} \]
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\[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )} x^{2} \,d x } \]
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Time = 1.10 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.24 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {a\,d^3\,x^3}{3}+\frac {a\,e^3\,x^9}{9}+\frac {b\,d^3\,\ln \left (c^2\,x^2+1\right )}{6\,c^3}-\frac {b\,e^3\,\ln \left (c^2\,x^2+1\right )}{18\,c^9}-\frac {b\,d^3\,x^2}{6\,c}-\frac {b\,e^3\,x^8}{72\,c}+\frac {b\,e^3\,x^6}{54\,c^3}-\frac {b\,e^3\,x^4}{36\,c^5}+\frac {b\,e^3\,x^2}{18\,c^7}+\frac {3\,a\,d^2\,e\,x^5}{5}+\frac {3\,a\,d\,e^2\,x^7}{7}+\frac {b\,d^3\,x^3\,\mathrm {atan}\left (c\,x\right )}{3}+\frac {b\,e^3\,x^9\,\mathrm {atan}\left (c\,x\right )}{9}+\frac {3\,b\,d^2\,e\,x^5\,\mathrm {atan}\left (c\,x\right )}{5}+\frac {3\,b\,d\,e^2\,x^7\,\mathrm {atan}\left (c\,x\right )}{7}-\frac {3\,b\,d^2\,e\,\ln \left (c^2\,x^2+1\right )}{10\,c^5}+\frac {3\,b\,d\,e^2\,\ln \left (c^2\,x^2+1\right )}{14\,c^7}-\frac {3\,b\,d^2\,e\,x^4}{20\,c}+\frac {3\,b\,d^2\,e\,x^2}{10\,c^3}-\frac {b\,d\,e^2\,x^6}{14\,c}+\frac {3\,b\,d\,e^2\,x^4}{28\,c^3}-\frac {3\,b\,d\,e^2\,x^2}{14\,c^5} \]
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