Integrand size = 21, antiderivative size = 160 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=-\frac {b \left (5 c^2 d-e\right ) e^2 x^2}{10 c^3}-\frac {b e^3 x^4}{20 c}-\frac {d^3 (a+b \arctan (c x))}{x}+3 d^2 e x (a+b \arctan (c x))+d e^2 x^3 (a+b \arctan (c x))+\frac {1}{5} e^3 x^5 (a+b \arctan (c x))+b c d^3 \log (x)-\frac {b \left (5 c^6 d^3+15 c^4 d^2 e-5 c^2 d e^2+e^3\right ) \log \left (1+c^2 x^2\right )}{10 c^5} \]
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Time = 0.16 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {276, 5096, 1813, 1634} \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=-\frac {d^3 (a+b \arctan (c x))}{x}+3 d^2 e x (a+b \arctan (c x))+d e^2 x^3 (a+b \arctan (c x))+\frac {1}{5} e^3 x^5 (a+b \arctan (c x))-\frac {b e^2 x^2 \left (5 c^2 d-e\right )}{10 c^3}-\frac {b \left (5 c^6 d^3+15 c^4 d^2 e-5 c^2 d e^2+e^3\right ) \log \left (c^2 x^2+1\right )}{10 c^5}+b c d^3 \log (x)-\frac {b e^3 x^4}{20 c} \]
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Rule 276
Rule 1634
Rule 1813
Rule 5096
Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 (a+b \arctan (c x))}{x}+3 d^2 e x (a+b \arctan (c x))+d e^2 x^3 (a+b \arctan (c x))+\frac {1}{5} e^3 x^5 (a+b \arctan (c x))-(b c) \int \frac {-d^3+3 d^2 e x^2+d e^2 x^4+\frac {e^3 x^6}{5}}{x \left (1+c^2 x^2\right )} \, dx \\ & = -\frac {d^3 (a+b \arctan (c x))}{x}+3 d^2 e x (a+b \arctan (c x))+d e^2 x^3 (a+b \arctan (c x))+\frac {1}{5} e^3 x^5 (a+b \arctan (c x))-\frac {1}{2} (b c) \text {Subst}\left (\int \frac {-d^3+3 d^2 e x+d e^2 x^2+\frac {e^3 x^3}{5}}{x \left (1+c^2 x\right )} \, dx,x,x^2\right ) \\ & = -\frac {d^3 (a+b \arctan (c x))}{x}+3 d^2 e x (a+b \arctan (c x))+d e^2 x^3 (a+b \arctan (c x))+\frac {1}{5} e^3 x^5 (a+b \arctan (c x))-\frac {1}{2} (b c) \text {Subst}\left (\int \left (\frac {\left (5 c^2 d-e\right ) e^2}{5 c^4}-\frac {d^3}{x}+\frac {e^3 x}{5 c^2}+\frac {5 c^6 d^3+15 c^4 d^2 e-5 c^2 d e^2+e^3}{5 c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = -\frac {b \left (5 c^2 d-e\right ) e^2 x^2}{10 c^3}-\frac {b e^3 x^4}{20 c}-\frac {d^3 (a+b \arctan (c x))}{x}+3 d^2 e x (a+b \arctan (c x))+d e^2 x^3 (a+b \arctan (c x))+\frac {1}{5} e^3 x^5 (a+b \arctan (c x))+b c d^3 \log (x)-\frac {b \left (5 c^6 d^3+15 c^4 d^2 e-5 c^2 d e^2+e^3\right ) \log \left (1+c^2 x^2\right )}{10 c^5} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.06 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=\frac {1}{20} \left (-\frac {20 a d^3}{x}+60 a d^2 e x+\frac {2 b e^2 \left (-5 c^2 d+e\right ) x^2}{c^3}+20 a d e^2 x^3-\frac {b e^3 x^4}{c}+4 a e^3 x^5+\frac {4 b \left (-5 d^3+15 d^2 e x^2+5 d e^2 x^4+e^3 x^6\right ) \arctan (c x)}{x}+20 b c d^3 \log (x)-\frac {2 b \left (5 c^6 d^3+15 c^4 d^2 e-5 c^2 d e^2+e^3\right ) \log \left (1+c^2 x^2\right )}{c^5}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.21
method | result | size |
parts | \(a \left (\frac {e^{3} x^{5}}{5}+d \,e^{2} x^{3}+3 d^{2} e x -\frac {d^{3}}{x}\right )+b c \left (\frac {\arctan \left (c x \right ) e^{3} x^{5}}{5 c}+\frac {\arctan \left (c x \right ) x^{3} d \,e^{2}}{c}+\frac {3 \arctan \left (c x \right ) x \,d^{2} e}{c}-\frac {\arctan \left (c x \right ) d^{3}}{c x}-\frac {\frac {e^{3} c^{4} x^{4}}{4}+\frac {5 d \,c^{4} e^{2} x^{2}}{2}-\frac {e^{3} c^{2} x^{2}}{2}-5 c^{6} d^{3} \ln \left (c x \right )+\frac {\left (5 c^{6} d^{3}+15 c^{4} d^{2} e -5 e^{2} d \,c^{2}+e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}}{5 c^{6}}\right )\) | \(193\) |
derivativedivides | \(c \left (\frac {a \left (3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-\frac {c^{5} d^{3}}{x}\right )}{c^{6}}+\frac {b \left (3 \arctan \left (c x \right ) c^{5} d^{2} e x +\arctan \left (c x \right ) c^{5} d \,e^{2} x^{3}+\frac {\arctan \left (c x \right ) e^{3} c^{5} x^{5}}{5}-\frac {\arctan \left (c x \right ) c^{5} d^{3}}{x}-\frac {d \,c^{4} e^{2} x^{2}}{2}-\frac {e^{3} c^{4} x^{4}}{20}+\frac {e^{3} c^{2} x^{2}}{10}-\frac {\left (5 c^{6} d^{3}+15 c^{4} d^{2} e -5 e^{2} d \,c^{2}+e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{10}+c^{6} d^{3} \ln \left (c x \right )\right )}{c^{6}}\right )\) | \(205\) |
default | \(c \left (\frac {a \left (3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-\frac {c^{5} d^{3}}{x}\right )}{c^{6}}+\frac {b \left (3 \arctan \left (c x \right ) c^{5} d^{2} e x +\arctan \left (c x \right ) c^{5} d \,e^{2} x^{3}+\frac {\arctan \left (c x \right ) e^{3} c^{5} x^{5}}{5}-\frac {\arctan \left (c x \right ) c^{5} d^{3}}{x}-\frac {d \,c^{4} e^{2} x^{2}}{2}-\frac {e^{3} c^{4} x^{4}}{20}+\frac {e^{3} c^{2} x^{2}}{10}-\frac {\left (5 c^{6} d^{3}+15 c^{4} d^{2} e -5 e^{2} d \,c^{2}+e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{10}+c^{6} d^{3} \ln \left (c x \right )\right )}{c^{6}}\right )\) | \(205\) |
parallelrisch | \(\frac {4 x^{6} \arctan \left (c x \right ) b \,c^{5} e^{3}+4 a \,c^{5} e^{3} x^{6}+20 x^{4} \arctan \left (c x \right ) b \,c^{5} d \,e^{2}-b \,c^{4} e^{3} x^{5}+20 a \,c^{5} d \,e^{2} x^{4}+20 b \,c^{6} d^{3} \ln \left (x \right ) x -10 \ln \left (c^{2} x^{2}+1\right ) b \,c^{6} d^{3} x +60 x^{2} \arctan \left (c x \right ) b \,c^{5} d^{2} e -10 b \,c^{4} d \,e^{2} x^{3}+60 a \,c^{5} d^{2} e \,x^{2}-30 \ln \left (c^{2} x^{2}+1\right ) b \,c^{4} d^{2} e x -20 \arctan \left (c x \right ) b \,c^{5} d^{3}+2 b \,c^{2} e^{3} x^{3}-20 a \,c^{5} d^{3}+10 \ln \left (c^{2} x^{2}+1\right ) b \,c^{2} d \,e^{2} x -2 \ln \left (c^{2} x^{2}+1\right ) b \,e^{3} x}{20 x \,c^{5}}\) | \(248\) |
risch | \(\frac {i b \left (-e^{3} x^{6}-5 x^{4} e^{2} d -15 e \,d^{2} x^{2}+5 d^{3}\right ) \ln \left (i c x +1\right )}{10 x}+\frac {2 i b \,c^{5} e^{3} x^{6} \ln \left (-i c x +1\right )+30 i b \,c^{5} d^{2} e \,x^{2} \ln \left (-i c x +1\right )+4 a \,c^{5} e^{3} x^{6}-10 i b \,c^{5} d^{3} \ln \left (-i c x +1\right )+20 a \,c^{5} d \,e^{2} x^{4}-b \,c^{4} e^{3} x^{5}+20 b \,c^{6} d^{3} \ln \left (x \right ) x -10 \ln \left (c^{2} x^{2}+1\right ) b \,c^{6} d^{3} x +10 i b \,c^{5} d \,e^{2} x^{4} \ln \left (-i c x +1\right )+60 a \,c^{5} d^{2} e \,x^{2}-10 b \,c^{4} d \,e^{2} x^{3}-30 \ln \left (c^{2} x^{2}+1\right ) b \,c^{4} d^{2} e x -20 a \,c^{5} d^{3}+2 b \,c^{2} e^{3} x^{3}+10 \ln \left (c^{2} x^{2}+1\right ) b \,c^{2} d \,e^{2} x -2 \ln \left (c^{2} x^{2}+1\right ) b \,e^{3} x}{20 c^{5} x}\) | \(316\) |
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Time = 0.29 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.29 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=\frac {4 \, a c^{5} e^{3} x^{6} + 20 \, a c^{5} d e^{2} x^{4} - b c^{4} e^{3} x^{5} + 20 \, b c^{6} d^{3} x \log \left (x\right ) + 60 \, a c^{5} d^{2} e x^{2} - 20 \, a c^{5} d^{3} - 2 \, {\left (5 \, b c^{4} d e^{2} - b c^{2} e^{3}\right )} x^{3} - 2 \, {\left (5 \, b c^{6} d^{3} + 15 \, b c^{4} d^{2} e - 5 \, b c^{2} d e^{2} + b e^{3}\right )} x \log \left (c^{2} x^{2} + 1\right ) + 4 \, {\left (b c^{5} e^{3} x^{6} + 5 \, b c^{5} d e^{2} x^{4} + 15 \, b c^{5} d^{2} e x^{2} - 5 \, b c^{5} d^{3}\right )} \arctan \left (c x\right )}{20 \, c^{5} x} \]
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Time = 0.62 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.61 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=\begin {cases} - \frac {a d^{3}}{x} + 3 a d^{2} e x + a d e^{2} x^{3} + \frac {a e^{3} x^{5}}{5} + b c d^{3} \log {\left (x \right )} - \frac {b c d^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2} - \frac {b d^{3} \operatorname {atan}{\left (c x \right )}}{x} + 3 b d^{2} e x \operatorname {atan}{\left (c x \right )} + b d e^{2} x^{3} \operatorname {atan}{\left (c x \right )} + \frac {b e^{3} x^{5} \operatorname {atan}{\left (c x \right )}}{5} - \frac {3 b d^{2} e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c} - \frac {b d e^{2} x^{2}}{2 c} - \frac {b e^{3} x^{4}}{20 c} + \frac {b d e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c^{3}} + \frac {b e^{3} x^{2}}{10 c^{3}} - \frac {b e^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{10 c^{5}} & \text {for}\: c \neq 0 \\a \left (- \frac {d^{3}}{x} + 3 d^{2} e x + d e^{2} x^{3} + \frac {e^{3} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.23 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=\frac {1}{5} \, a e^{3} x^{5} + a d e^{2} x^{3} - \frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b d^{3} + \frac {1}{2} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b d e^{2} + \frac {1}{20} \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b e^{3} + 3 \, a d^{2} e x + \frac {3 \, {\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d^{2} e}{2 \, c} - \frac {a d^{3}}{x} \]
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\[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
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Time = 0.71 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.48 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=x\,\left (\frac {\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}}{c^2}+\frac {3\,a\,d\,e\,\left (d\,c^2+e\right )}{c^2}\right )-x^3\,\left (\frac {a\,e^3}{3\,c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{3\,c^2}\right )+x^2\,\left (\frac {b\,e^3}{10\,c^3}-\frac {b\,d\,e^2}{2\,c}\right )-\frac {a\,d^3}{x}+\frac {a\,e^3\,x^5}{5}-\frac {\ln \left (c^2\,x^2+1\right )\,\left (5\,b\,c^6\,d^3+15\,b\,c^4\,d^2\,e-5\,b\,c^2\,d\,e^2+b\,e^3\right )}{10\,c^5}+\frac {\mathrm {atan}\left (c\,x\right )\,\left (-b\,d^3+3\,b\,d^2\,e\,x^2+b\,d\,e^2\,x^4+\frac {b\,e^3\,x^6}{5}\right )}{x}-\frac {b\,e^3\,x^4}{20\,c}+b\,c\,d^3\,\ln \left (x\right ) \]
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