Integrand size = 21, antiderivative size = 224 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=-\frac {b c d^3}{42 x^6}+\frac {b c d^2 \left (5 c^2 d-21 e\right )}{140 x^4}-\frac {b c d \left (5 c^4 d^2-21 c^2 d e+35 e^2\right )}{70 x^2}-\frac {d^3 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}-\frac {1}{35} b c \left (5 c^6 d^3-21 c^4 d^2 e+35 c^2 d e^2-35 e^3\right ) \log (x)+\frac {1}{70} b c \left (5 c^6 d^3-21 c^4 d^2 e+35 c^2 d e^2-35 e^3\right ) \log \left (1+c^2 x^2\right ) \]
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Time = 0.21 (sec) , antiderivative size = 224, 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} \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=-\frac {d^3 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}+\frac {b c d^2 \left (5 c^2 d-21 e\right )}{140 x^4}-\frac {b c d \left (5 c^4 d^2-21 c^2 d e+35 e^2\right )}{70 x^2}+\frac {1}{70} b c \left (5 c^6 d^3-21 c^4 d^2 e+35 c^2 d e^2-35 e^3\right ) \log \left (c^2 x^2+1\right )-\frac {1}{35} b c \log (x) \left (5 c^6 d^3-21 c^4 d^2 e+35 c^2 d e^2-35 e^3\right )-\frac {b c d^3}{42 x^6} \]
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Rule 12
Rule 276
Rule 1634
Rule 1813
Rule 5096
Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}-(b c) \int \frac {-5 d^3-21 d^2 e x^2-35 d e^2 x^4-35 e^3 x^6}{35 x^7 \left (1+c^2 x^2\right )} \, dx \\ & = -\frac {d^3 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}-\frac {1}{35} (b c) \int \frac {-5 d^3-21 d^2 e x^2-35 d e^2 x^4-35 e^3 x^6}{x^7 \left (1+c^2 x^2\right )} \, dx \\ & = -\frac {d^3 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}-\frac {1}{70} (b c) \text {Subst}\left (\int \frac {-5 d^3-21 d^2 e x-35 d e^2 x^2-35 e^3 x^3}{x^4 \left (1+c^2 x\right )} \, dx,x,x^2\right ) \\ & = -\frac {d^3 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}-\frac {1}{70} (b c) \text {Subst}\left (\int \left (-\frac {5 d^3}{x^4}+\frac {d^2 \left (5 c^2 d-21 e\right )}{x^3}-\frac {d \left (5 c^4 d^2-21 c^2 d e+35 e^2\right )}{x^2}+\frac {5 c^6 d^3-21 c^4 d^2 e+35 c^2 d e^2-35 e^3}{x}+\frac {-5 c^8 d^3+21 c^6 d^2 e-35 c^4 d e^2+35 c^2 e^3}{1+c^2 x}\right ) \, dx,x,x^2\right ) \\ & = -\frac {b c d^3}{42 x^6}+\frac {b c d^2 \left (5 c^2 d-21 e\right )}{140 x^4}-\frac {b c d \left (5 c^4 d^2-21 c^2 d e+35 e^2\right )}{70 x^2}-\frac {d^3 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}-\frac {1}{35} b c \left (5 c^6 d^3-21 c^4 d^2 e+35 c^2 d e^2-35 e^3\right ) \log (x)+\frac {1}{70} b c \left (5 c^6 d^3-21 c^4 d^2 e+35 c^2 d e^2-35 e^3\right ) \log \left (1+c^2 x^2\right ) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.03 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=-\frac {d^3 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}+\frac {1}{2} b c e^3 \left (2 \log (x)-\log \left (1+c^2 x^2\right )\right )-\frac {1}{2} b c d e^2 \left (\frac {1}{x^2}+2 c^2 \log (x)-c^2 \log \left (1+c^2 x^2\right )\right )-\frac {3}{20} b c d^2 e \left (\frac {1}{x^4}-\frac {2 c^2}{x^2}-4 c^4 \log (x)+2 c^4 \log \left (1+c^2 x^2\right )\right )-\frac {1}{84} b c d^3 \left (\frac {2}{x^6}-\frac {3 c^2}{x^4}+\frac {6 c^4}{x^2}+12 c^6 \log (x)-6 c^6 \log \left (1+c^2 x^2\right )\right ) \]
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Time = 0.27 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.10
method | result | size |
parts | \(a \left (-\frac {e^{3}}{x}-\frac {d^{3}}{7 x^{7}}-\frac {d \,e^{2}}{x^{3}}-\frac {3 d^{2} e}{5 x^{5}}\right )+b \,c^{7} \left (-\frac {\arctan \left (c x \right ) e^{3}}{c^{7} x}-\frac {\arctan \left (c x \right ) d^{3}}{7 c^{7} x^{7}}-\frac {\arctan \left (c x \right ) e^{2} d}{c^{7} x^{3}}-\frac {3 \arctan \left (c x \right ) d^{2} e}{5 c^{7} x^{5}}-\frac {\left (5 c^{6} d^{3}-21 c^{4} d^{2} e +35 e^{2} d \,c^{2}-35 e^{3}\right ) \ln \left (c x \right )-\frac {d^{2} \left (5 c^{2} d -21 e \right )}{4 x^{4}}+\frac {5 d^{3}}{6 x^{6}}+\frac {d \left (5 c^{4} d^{2}-21 c^{2} d e +35 e^{2}\right )}{2 x^{2}}+\frac {\left (-5 c^{6} d^{3}+21 c^{4} d^{2} e -35 e^{2} d \,c^{2}+35 e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}}{35 c^{6}}\right )\) | \(247\) |
derivativedivides | \(c^{7} \left (\frac {a \left (-\frac {3 d^{2} e}{5 c \,x^{5}}-\frac {e^{3}}{c x}-\frac {d^{3}}{7 c \,x^{7}}-\frac {d \,e^{2}}{c \,x^{3}}\right )}{c^{6}}+\frac {b \left (-\frac {3 \arctan \left (c x \right ) d^{2} e}{5 c \,x^{5}}-\frac {\arctan \left (c x \right ) e^{3}}{c x}-\frac {\arctan \left (c x \right ) d^{3}}{7 c \,x^{7}}-\frac {\arctan \left (c x \right ) d \,e^{2}}{c \,x^{3}}-\frac {\left (-5 c^{6} d^{3}+21 c^{4} d^{2} e -35 e^{2} d \,c^{2}+35 e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{70}-\frac {\left (5 c^{6} d^{3}-21 c^{4} d^{2} e +35 e^{2} d \,c^{2}-35 e^{3}\right ) \ln \left (c x \right )}{35}+\frac {d^{2} \left (5 c^{2} d -21 e \right )}{140 x^{4}}-\frac {d^{3}}{42 x^{6}}-\frac {d \left (5 c^{4} d^{2}-21 c^{2} d e +35 e^{2}\right )}{70 x^{2}}\right )}{c^{6}}\right )\) | \(261\) |
default | \(c^{7} \left (\frac {a \left (-\frac {3 d^{2} e}{5 c \,x^{5}}-\frac {e^{3}}{c x}-\frac {d^{3}}{7 c \,x^{7}}-\frac {d \,e^{2}}{c \,x^{3}}\right )}{c^{6}}+\frac {b \left (-\frac {3 \arctan \left (c x \right ) d^{2} e}{5 c \,x^{5}}-\frac {\arctan \left (c x \right ) e^{3}}{c x}-\frac {\arctan \left (c x \right ) d^{3}}{7 c \,x^{7}}-\frac {\arctan \left (c x \right ) d \,e^{2}}{c \,x^{3}}-\frac {\left (-5 c^{6} d^{3}+21 c^{4} d^{2} e -35 e^{2} d \,c^{2}+35 e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{70}-\frac {\left (5 c^{6} d^{3}-21 c^{4} d^{2} e +35 e^{2} d \,c^{2}-35 e^{3}\right ) \ln \left (c x \right )}{35}+\frac {d^{2} \left (5 c^{2} d -21 e \right )}{140 x^{4}}-\frac {d^{3}}{42 x^{6}}-\frac {d \left (5 c^{4} d^{2}-21 c^{2} d e +35 e^{2}\right )}{70 x^{2}}\right )}{c^{6}}\right )\) | \(261\) |
parallelrisch | \(-\frac {420 a d \,e^{2} x^{4}+252 a \,d^{2} e \,x^{2}+420 x^{4} \arctan \left (c x \right ) b d \,e^{2}+252 x^{2} \arctan \left (c x \right ) b \,d^{2} e -15 x^{3} d^{3} c^{3} b +30 x^{5} c^{5} d^{3} b +10 b c \,d^{3} x +60 d^{3} a +420 a \,e^{3} x^{6}+60 b \,d^{3} \arctan \left (c x \right )-126 x^{5} c^{3} d^{2} e b -30 b \,c^{7} d^{3} x^{7}+60 \ln \left (x \right ) b \,c^{7} d^{3} x^{7}-30 \ln \left (c^{2} x^{2}+1\right ) b \,c^{7} d^{3} x^{7}-420 \ln \left (x \right ) b c \,e^{3} x^{7}+210 \ln \left (c^{2} x^{2}+1\right ) b c \,e^{3} x^{7}+126 b \,c^{5} d^{2} e \,x^{7}-210 b \,c^{3} d \,e^{2} x^{7}-252 \ln \left (x \right ) b \,c^{5} d^{2} e \,x^{7}+126 \ln \left (c^{2} x^{2}+1\right ) b \,c^{5} d^{2} e \,x^{7}+420 \ln \left (x \right ) b \,c^{3} d \,e^{2} x^{7}-210 \ln \left (c^{2} x^{2}+1\right ) b \,c^{3} d \,e^{2} x^{7}+420 x^{6} \arctan \left (c x \right ) b \,e^{3}+63 b c e \,d^{2} x^{3}+210 b c \,e^{2} d \,x^{5}}{420 x^{7}}\) | \(342\) |
risch | \(\frac {i b \left (35 e^{3} x^{6}+35 x^{4} e^{2} d +21 e \,d^{2} x^{2}+5 d^{3}\right ) \ln \left (i c x +1\right )}{70 x^{7}}-\frac {60 \ln \left (x \right ) b \,c^{7} d^{3} x^{7}-30 \ln \left (c^{2} x^{2}+1\right ) b \,c^{7} d^{3} x^{7}-252 \ln \left (x \right ) b \,c^{5} d^{2} e \,x^{7}+126 \ln \left (c^{2} x^{2}+1\right ) b \,c^{5} d^{2} e \,x^{7}+420 \ln \left (x \right ) b \,c^{3} d \,e^{2} x^{7}-210 \ln \left (c^{2} x^{2}+1\right ) b \,c^{3} d \,e^{2} x^{7}+30 x^{5} c^{5} d^{3} b -420 \ln \left (x \right ) b c \,e^{3} x^{7}+210 \ln \left (c^{2} x^{2}+1\right ) b c \,e^{3} x^{7}+30 i b \,d^{3} \ln \left (-i c x +1\right )-126 x^{5} c^{3} d^{2} e b +210 i b \,e^{2} d \ln \left (-i c x +1\right ) x^{4}+420 a \,e^{3} x^{6}-15 x^{3} d^{3} c^{3} b +210 b c \,e^{2} d \,x^{5}+210 i b \,e^{3} \ln \left (-i c x +1\right ) x^{6}+420 a d \,e^{2} x^{4}+63 b c e \,d^{2} x^{3}+126 i b \,d^{2} e \ln \left (-i c x +1\right ) x^{2}+252 a \,d^{2} e \,x^{2}+10 b c \,d^{3} x +60 d^{3} a}{420 x^{7}}\) | \(372\) |
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Time = 0.28 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.08 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=-\frac {420 \, a e^{3} x^{6} - 6 \, {\left (5 \, b c^{7} d^{3} - 21 \, b c^{5} d^{2} e + 35 \, b c^{3} d e^{2} - 35 \, b c e^{3}\right )} x^{7} \log \left (c^{2} x^{2} + 1\right ) + 12 \, {\left (5 \, b c^{7} d^{3} - 21 \, b c^{5} d^{2} e + 35 \, b c^{3} d e^{2} - 35 \, b c e^{3}\right )} x^{7} \log \left (x\right ) + 420 \, a d e^{2} x^{4} + 10 \, b c d^{3} x + 252 \, a d^{2} e x^{2} + 6 \, {\left (5 \, b c^{5} d^{3} - 21 \, b c^{3} d^{2} e + 35 \, b c d e^{2}\right )} x^{5} + 60 \, a d^{3} - 3 \, {\left (5 \, b c^{3} d^{3} - 21 \, b c d^{2} e\right )} x^{3} + 12 \, {\left (35 \, b e^{3} x^{6} + 35 \, b d e^{2} x^{4} + 21 \, b d^{2} e x^{2} + 5 \, b d^{3}\right )} \arctan \left (c x\right )}{420 \, x^{7}} \]
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Time = 0.75 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.62 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=\begin {cases} - \frac {a d^{3}}{7 x^{7}} - \frac {3 a d^{2} e}{5 x^{5}} - \frac {a d e^{2}}{x^{3}} - \frac {a e^{3}}{x} - \frac {b c^{7} d^{3} \log {\left (x \right )}}{7} + \frac {b c^{7} d^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{14} - \frac {b c^{5} d^{3}}{14 x^{2}} + \frac {3 b c^{5} d^{2} e \log {\left (x \right )}}{5} - \frac {3 b c^{5} d^{2} e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{10} + \frac {b c^{3} d^{3}}{28 x^{4}} + \frac {3 b c^{3} d^{2} e}{10 x^{2}} - b c^{3} d e^{2} \log {\left (x \right )} + \frac {b c^{3} d e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2} - \frac {b c d^{3}}{42 x^{6}} - \frac {3 b c d^{2} e}{20 x^{4}} - \frac {b c d e^{2}}{2 x^{2}} + b c e^{3} \log {\left (x \right )} - \frac {b c e^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2} - \frac {b d^{3} \operatorname {atan}{\left (c x \right )}}{7 x^{7}} - \frac {3 b d^{2} e \operatorname {atan}{\left (c x \right )}}{5 x^{5}} - \frac {b d e^{2} \operatorname {atan}{\left (c x \right )}}{x^{3}} - \frac {b e^{3} \operatorname {atan}{\left (c x \right )}}{x} & \text {for}\: c \neq 0 \\a \left (- \frac {d^{3}}{7 x^{7}} - \frac {3 d^{2} e}{5 x^{5}} - \frac {d e^{2}}{x^{3}} - \frac {e^{3}}{x}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.10 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=\frac {1}{84} \, {\left ({\left (6 \, c^{6} \log \left (c^{2} x^{2} + 1\right ) - 6 \, c^{6} \log \left (x^{2}\right ) - \frac {6 \, c^{4} x^{4} - 3 \, c^{2} x^{2} + 2}{x^{6}}\right )} c - \frac {12 \, \arctan \left (c x\right )}{x^{7}}\right )} b d^{3} - \frac {3}{20} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac {2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac {4 \, \arctan \left (c x\right )}{x^{5}}\right )} b d^{2} e + \frac {1}{2} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b d e^{2} - \frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b e^{3} - \frac {a e^{3}}{x} - \frac {a d e^{2}}{x^{3}} - \frac {3 \, a d^{2} e}{5 \, x^{5}} - \frac {a d^{3}}{7 \, x^{7}} \]
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\[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{8}} \,d x } \]
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Time = 0.83 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.05 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=\ln \left (c^2\,x^2+1\right )\,\left (\frac {b\,c^7\,d^3}{14}-\frac {3\,b\,c^5\,d^2\,e}{10}+\frac {b\,c^3\,d\,e^2}{2}-\frac {b\,c\,e^3}{2}\right )-\ln \left (x\right )\,\left (\frac {b\,c^7\,d^3}{7}-\frac {3\,b\,c^5\,d^2\,e}{5}+b\,c^3\,d\,e^2-b\,c\,e^3\right )-\frac {5\,a\,d^3-x^3\,\left (\frac {5\,b\,c^3\,d^3}{4}-\frac {21\,b\,c\,d^2\,e}{4}\right )+x^5\,\left (\frac {5\,b\,c^5\,d^3}{2}-\frac {21\,b\,c^3\,d^2\,e}{2}+\frac {35\,b\,c\,d\,e^2}{2}\right )+35\,a\,e^3\,x^6+\frac {5\,b\,c\,d^3\,x}{6}+21\,a\,d^2\,e\,x^2+35\,a\,d\,e^2\,x^4}{35\,x^7}-\frac {\mathrm {atan}\left (c\,x\right )\,\left (\frac {b\,d^3}{7}+\frac {3\,b\,d^2\,e\,x^2}{5}+b\,d\,e^2\,x^4+b\,e^3\,x^6\right )}{x^7} \]
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