Integrand size = 21, antiderivative size = 130 \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b c x}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}-\frac {b \arctan (c x)}{4 d \left (c^2 d-e\right )^2}+\frac {x^4 (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \left (c^2 d-3 e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {d} \left (c^2 d-e\right )^2 e^{3/2}} \]
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Time = 0.12 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {270, 5096, 12, 481, 536, 211} \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {x^4 (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \left (c^2 d-3 e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {d} e^{3/2} \left (c^2 d-e\right )^2}-\frac {b \arctan (c x)}{4 d \left (c^2 d-e\right )^2}+\frac {b c x}{8 e \left (c^2 d-e\right ) \left (d+e x^2\right )} \]
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Rule 12
Rule 211
Rule 270
Rule 481
Rule 536
Rule 5096
Rubi steps \begin{align*} \text {integral}& = \frac {x^4 (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}-(b c) \int \frac {x^4}{4 \left (d+c^2 d x^2\right ) \left (d+e x^2\right )^2} \, dx \\ & = \frac {x^4 (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}-\frac {1}{4} (b c) \int \frac {x^4}{\left (d+c^2 d x^2\right ) \left (d+e x^2\right )^2} \, dx \\ & = \frac {b c x}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}+\frac {x^4 (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}-\frac {(b c) \int \frac {d^2+d \left (c^2 d-2 e\right ) x^2}{\left (d+c^2 d x^2\right ) \left (d+e x^2\right )} \, dx}{8 d \left (c^2 d-e\right ) e} \\ & = \frac {b c x}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}+\frac {x^4 (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}-\frac {(b c) \int \frac {1}{d+c^2 d x^2} \, dx}{4 \left (c^2 d-e\right )^2}-\frac {\left (b c \left (c^2 d-3 e\right )\right ) \int \frac {1}{d+e x^2} \, dx}{8 \left (c^2 d-e\right )^2 e} \\ & = \frac {b c x}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}-\frac {b \arctan (c x)}{4 d \left (c^2 d-e\right )^2}+\frac {x^4 (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \left (c^2 d-3 e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {d} \left (c^2 d-e\right )^2 e^{3/2}} \\ \end{align*}
Time = 2.46 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.22 \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {\frac {2 a d}{\left (d+e x^2\right )^2}+\frac {-4 a c^2 d+4 a e+b c e x}{\left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {2 b c^2 \left (c^2 d-2 e\right ) \arctan (c x)}{\left (-c^2 d+e\right )^2}-\frac {2 b \left (d+2 e x^2\right ) \arctan (c x)}{\left (d+e x^2\right )^2}-\frac {b c \left (c^2 d-3 e\right ) \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \left (-c^2 d+e\right )^2}}{8 e^2} \]
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Time = 1.12 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.62
method | result | size |
parts | \(a \left (\frac {d}{4 e^{2} \left (e \,x^{2}+d \right )^{2}}-\frac {1}{2 e^{2} \left (e \,x^{2}+d \right )}\right )+\frac {b \left (\frac {\arctan \left (c x \right ) c^{8} d}{4 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}-\frac {\arctan \left (c x \right ) c^{6}}{2 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {c^{6} \left (\frac {e^{2} \left (-\frac {\left (c^{2} d -e \right ) c x}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\left (c^{2} d -3 e \right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 e c \sqrt {e d}}\right )}{\left (c^{2} d -e \right )^{2}}+\frac {\left (-c^{2} d +2 e \right ) \arctan \left (c x \right )}{\left (c^{2} d -e \right )^{2}}\right )}{4 e^{2}}\right )}{c^{4}}\) | \(211\) |
derivativedivides | \(\frac {a \,c^{6} \left (-\frac {1}{2 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {d \,c^{2}}{4 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}\right )+b \,c^{6} \left (-\frac {\arctan \left (c x \right )}{2 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\arctan \left (c x \right ) d \,c^{2}}{4 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}-\frac {\frac {e^{2} \left (-\frac {\left (c^{2} d -e \right ) c x}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\left (c^{2} d -3 e \right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 e c \sqrt {e d}}\right )}{\left (c^{2} d -e \right )^{2}}+\frac {\left (-c^{2} d +2 e \right ) \arctan \left (c x \right )}{\left (c^{2} d -e \right )^{2}}}{4 e^{2}}\right )}{c^{4}}\) | \(229\) |
default | \(\frac {a \,c^{6} \left (-\frac {1}{2 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {d \,c^{2}}{4 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}\right )+b \,c^{6} \left (-\frac {\arctan \left (c x \right )}{2 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\arctan \left (c x \right ) d \,c^{2}}{4 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}-\frac {\frac {e^{2} \left (-\frac {\left (c^{2} d -e \right ) c x}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\left (c^{2} d -3 e \right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 e c \sqrt {e d}}\right )}{\left (c^{2} d -e \right )^{2}}+\frac {\left (-c^{2} d +2 e \right ) \arctan \left (c x \right )}{\left (c^{2} d -e \right )^{2}}}{4 e^{2}}\right )}{c^{4}}\) | \(229\) |
risch | \(\text {Expression too large to display}\) | \(1153\) |
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Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (114) = 228\).
Time = 0.37 (sec) , antiderivative size = 697, normalized size of antiderivative = 5.36 \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\left [-\frac {4 \, a c^{4} d^{4} - 8 \, a c^{2} d^{3} e + 4 \, a d^{2} e^{2} - 2 \, {\left (b c^{3} d^{2} e^{2} - b c d e^{3}\right )} x^{3} + 8 \, {\left (a c^{4} d^{3} e - 2 \, a c^{2} d^{2} e^{2} + a d e^{3}\right )} x^{2} - {\left (b c^{3} d^{3} - 3 \, b c d^{2} e + {\left (b c^{3} d e^{2} - 3 \, b c e^{3}\right )} x^{4} + 2 \, {\left (b c^{3} d^{2} e - 3 \, b c d e^{2}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - 2 \, {\left (b c^{3} d^{3} e - b c d^{2} e^{2}\right )} x + 4 \, {\left (2 \, b d e^{3} x^{2} + b d^{2} e^{2} - {\left (b c^{4} d^{2} e^{2} - 2 \, b c^{2} d e^{3}\right )} x^{4}\right )} \arctan \left (c x\right )}{16 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4} + {\left (c^{4} d^{3} e^{4} - 2 \, c^{2} d^{2} e^{5} + d e^{6}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{2}\right )}}, -\frac {2 \, a c^{4} d^{4} - 4 \, a c^{2} d^{3} e + 2 \, a d^{2} e^{2} - {\left (b c^{3} d^{2} e^{2} - b c d e^{3}\right )} x^{3} + 4 \, {\left (a c^{4} d^{3} e - 2 \, a c^{2} d^{2} e^{2} + a d e^{3}\right )} x^{2} + {\left (b c^{3} d^{3} - 3 \, b c d^{2} e + {\left (b c^{3} d e^{2} - 3 \, b c e^{3}\right )} x^{4} + 2 \, {\left (b c^{3} d^{2} e - 3 \, b c d e^{2}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - {\left (b c^{3} d^{3} e - b c d^{2} e^{2}\right )} x + 2 \, {\left (2 \, b d e^{3} x^{2} + b d^{2} e^{2} - {\left (b c^{4} d^{2} e^{2} - 2 \, b c^{2} d e^{3}\right )} x^{4}\right )} \arctan \left (c x\right )}{8 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4} + {\left (c^{4} d^{3} e^{4} - 2 \, c^{2} d^{2} e^{5} + d e^{6}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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Time = 3.71 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.10 \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b\,c^4\,d\,\mathrm {atan}\left (c\,x\right )}{4\,e^2\,{\left (e-c^2\,d\right )}^2}-\frac {a\,d}{4\,e^2\,{\left (e\,x^2+d\right )}^2}-\frac {b\,d\,\mathrm {atan}\left (c\,x\right )}{4\,e^2\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c\,x^3}{8\,\left (e-c^2\,d\right )\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c^2\,\mathrm {atan}\left (c\,x\right )}{2\,e\,{\left (e-c^2\,d\right )}^2}-\frac {b\,x^2\,\mathrm {atan}\left (c\,x\right )}{2\,e\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c^3\,\mathrm {atan}\left (\frac {x\,\sqrt {-d\,e^3}\,1{}\mathrm {i}}{d\,e}\right )\,\sqrt {-d\,e^3}\,1{}\mathrm {i}}{8\,e^3\,{\left (e-c^2\,d\right )}^2}-\frac {a\,x^2}{2\,e\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c\,d\,x}{8\,e\,\left (e-c^2\,d\right )\,{\left (e\,x^2+d\right )}^2}+\frac {b\,c\,\mathrm {atan}\left (\frac {x\,\sqrt {-d\,e^3}\,1{}\mathrm {i}}{d\,e}\right )\,\sqrt {-d\,e^3}\,3{}\mathrm {i}}{8\,d\,e^2\,{\left (e-c^2\,d\right )}^2} \]
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