Integrand size = 18, antiderivative size = 315 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=-\frac {b d e \arctan \left (\sqrt [3]{c} x\right )}{c^{2/3}}-\frac {b d^3 \arctan \left (c x^3\right )}{3 e}+\frac {(d+e x)^3 \left (a+b \arctan \left (c x^3\right )\right )}{3 e}+\frac {b d e \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d e \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {\sqrt {3} b d^2 \arctan \left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}+\frac {b d^2 \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} b d e \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d e \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}-\frac {b d^2 \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c} \]
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Time = 0.47 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {4980, 1845, 281, 298, 31, 648, 631, 210, 642, 301, 632, 209, 1483, 649, 266} \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {(d+e x)^3 \left (a+b \arctan \left (c x^3\right )\right )}{3 e}+\frac {\sqrt {3} b d^2 \arctan \left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {b d e \arctan \left (\sqrt [3]{c} x\right )}{c^{2/3}}+\frac {b d e \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d e \arctan \left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{2 c^{2/3}}-\frac {b d^3 \arctan \left (c x^3\right )}{3 e}+\frac {b d^2 \log \left (c^{2/3} x^2+1\right )}{2 \sqrt [3]{c}}-\frac {b d^2 \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}-\frac {\sqrt {3} b d e \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d e \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{4 c^{2/3}}-\frac {b e^2 \log \left (c^2 x^6+1\right )}{6 c} \]
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Rule 31
Rule 209
Rule 210
Rule 266
Rule 281
Rule 298
Rule 301
Rule 631
Rule 632
Rule 642
Rule 648
Rule 649
Rule 1483
Rule 1845
Rule 4980
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^3 \left (a+b \arctan \left (c x^3\right )\right )}{3 e}-\frac {(b c) \int \frac {x^2 (d+e x)^3}{1+c^2 x^6} \, dx}{e} \\ & = \frac {(d+e x)^3 \left (a+b \arctan \left (c x^3\right )\right )}{3 e}-\frac {(b c) \int \left (\frac {3 d^2 e x^3}{1+c^2 x^6}+\frac {3 d e^2 x^4}{1+c^2 x^6}+\frac {x^2 \left (d^3+e^3 x^3\right )}{1+c^2 x^6}\right ) \, dx}{e} \\ & = \frac {(d+e x)^3 \left (a+b \arctan \left (c x^3\right )\right )}{3 e}-\left (3 b c d^2\right ) \int \frac {x^3}{1+c^2 x^6} \, dx-\frac {(b c) \int \frac {x^2 \left (d^3+e^3 x^3\right )}{1+c^2 x^6} \, dx}{e}-(3 b c d e) \int \frac {x^4}{1+c^2 x^6} \, dx \\ & = \frac {(d+e x)^3 \left (a+b \arctan \left (c x^3\right )\right )}{3 e}-\frac {1}{2} \left (3 b c d^2\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x^3} \, dx,x,x^2\right )-\frac {(b c) \text {Subst}\left (\int \frac {d^3+e^3 x}{1+c^2 x^2} \, dx,x,x^3\right )}{3 e}-\frac {(b d e) \int \frac {1}{1+c^{2/3} x^2} \, dx}{\sqrt [3]{c}}-\frac {(b d e) \int \frac {-\frac {1}{2}+\frac {1}{2} \sqrt {3} \sqrt [3]{c} x}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{\sqrt [3]{c}}-\frac {(b d e) \int \frac {-\frac {1}{2}-\frac {1}{2} \sqrt {3} \sqrt [3]{c} x}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{\sqrt [3]{c}} \\ & = -\frac {b d e \arctan \left (\sqrt [3]{c} x\right )}{c^{2/3}}+\frac {(d+e x)^3 \left (a+b \arctan \left (c x^3\right )\right )}{3 e}+\frac {1}{2} \left (b \sqrt [3]{c} d^2\right ) \text {Subst}\left (\int \frac {1}{1+c^{2/3} x} \, dx,x,x^2\right )-\frac {1}{2} \left (b \sqrt [3]{c} d^2\right ) \text {Subst}\left (\int \frac {1+c^{2/3} x}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )-\frac {\left (b c d^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,x^3\right )}{3 e}-\frac {\left (\sqrt {3} b d e\right ) \int \frac {-\sqrt {3} \sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 c^{2/3}}+\frac {\left (\sqrt {3} b d e\right ) \int \frac {\sqrt {3} \sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 c^{2/3}}-\frac {(b d e) \int \frac {1}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \sqrt [3]{c}}-\frac {(b d e) \int \frac {1}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \sqrt [3]{c}}-\frac {1}{3} \left (b c e^2\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x^2} \, dx,x,x^3\right ) \\ & = -\frac {b d e \arctan \left (\sqrt [3]{c} x\right )}{c^{2/3}}-\frac {b d^3 \arctan \left (c x^3\right )}{3 e}+\frac {(d+e x)^3 \left (a+b \arctan \left (c x^3\right )\right )}{3 e}+\frac {b d^2 \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} b d e \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d e \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c}-\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {-c^{2/3}+2 c^{4/3} x}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{4 \sqrt [3]{c}}-\frac {1}{4} \left (3 b \sqrt [3]{c} d^2\right ) \text {Subst}\left (\int \frac {1}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )-\frac {(b d e) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{2 \sqrt {3} c^{2/3}}+\frac {(b d e) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{2 \sqrt {3} c^{2/3}} \\ & = -\frac {b d e \arctan \left (\sqrt [3]{c} x\right )}{c^{2/3}}-\frac {b d^3 \arctan \left (c x^3\right )}{3 e}+\frac {(d+e x)^3 \left (a+b \arctan \left (c x^3\right )\right )}{3 e}+\frac {b d e \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d e \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {b d^2 \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} b d e \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d e \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}-\frac {b d^2 \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c}-\frac {\left (3 b d^2\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 c^{2/3} x^2\right )}{2 \sqrt [3]{c}} \\ & = -\frac {b d e \arctan \left (\sqrt [3]{c} x\right )}{c^{2/3}}-\frac {b d^3 \arctan \left (c x^3\right )}{3 e}+\frac {(d+e x)^3 \left (a+b \arctan \left (c x^3\right )\right )}{3 e}+\frac {b d e \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d e \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {\sqrt {3} b d^2 \arctan \left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}+\frac {b d^2 \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} b d e \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d e \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}-\frac {b d^2 \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c} \\ \end{align*}
Time = 106.47 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.94 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {12 a c d^2 x+12 a c d e x^2+4 a c e^2 x^3-12 b \sqrt [3]{c} d e \arctan \left (\sqrt [3]{c} x\right )+4 b c x \left (3 d^2+3 d e x+e^2 x^2\right ) \arctan \left (c x^3\right )+6 b \sqrt [3]{c} d \left (\sqrt {3} \sqrt [3]{c} d+e\right ) \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )+6 b \sqrt [3]{c} d \left (\sqrt {3} \sqrt [3]{c} d-e\right ) \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )+6 b c^{2/3} d^2 \log \left (1+c^{2/3} x^2\right )-3 b \sqrt [3]{c} d \left (\sqrt [3]{c} d+\sqrt {3} e\right ) \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )-3 b \sqrt [3]{c} d \left (\sqrt [3]{c} d-\sqrt {3} e\right ) \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )-2 b e^2 \log \left (1+c^2 x^6\right )}{12 c} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(502\) vs. \(2(244)=488\).
Time = 33.36 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.60
method | result | size |
default | \(\frac {a \left (e x +d \right )^{3}}{3 e}+b \left (\frac {e^{2} \arctan \left (c \,x^{3}\right ) x^{3}}{3}+e \arctan \left (c \,x^{3}\right ) d \,x^{2}+\arctan \left (c \,x^{3}\right ) d^{2} x +\frac {\arctan \left (c \,x^{3}\right ) d^{3}}{3 e}-\frac {c \left (-\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} d \,e^{2}}{4}+\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d^{2} e}{4}+\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) e^{3}}{6 c^{2}}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d \,e^{2}}{2 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, d^{2} e}{2}+\frac {\sqrt {\frac {1}{c^{2}}}\, \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d^{3}}{3}+\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} d \,e^{2}}{4}+\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d^{2} e}{4}+\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) e^{3}}{6 c^{2}}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d \,e^{2}}{2 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\frac {\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, d^{2} e}{2}+\frac {\sqrt {\frac {1}{c^{2}}}\, \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d^{3}}{3}-\frac {\ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d^{2} e}{2}+\frac {\ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) e^{3}}{6 c^{2}}-\frac {\sqrt {\frac {1}{c^{2}}}\, \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right ) d^{3}}{3}+\frac {\arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right ) d \,e^{2}}{c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{e}\right )\) | \(503\) |
parts | \(\frac {a \left (e x +d \right )^{3}}{3 e}+b \left (\frac {e^{2} \arctan \left (c \,x^{3}\right ) x^{3}}{3}+e \arctan \left (c \,x^{3}\right ) d \,x^{2}+\arctan \left (c \,x^{3}\right ) d^{2} x +\frac {\arctan \left (c \,x^{3}\right ) d^{3}}{3 e}-\frac {c \left (-\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} d \,e^{2}}{4}+\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d^{2} e}{4}+\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) e^{3}}{6 c^{2}}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d \,e^{2}}{2 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, d^{2} e}{2}+\frac {\sqrt {\frac {1}{c^{2}}}\, \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d^{3}}{3}+\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} d \,e^{2}}{4}+\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d^{2} e}{4}+\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) e^{3}}{6 c^{2}}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d \,e^{2}}{2 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\frac {\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, d^{2} e}{2}+\frac {\sqrt {\frac {1}{c^{2}}}\, \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d^{3}}{3}-\frac {\ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d^{2} e}{2}+\frac {\ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) e^{3}}{6 c^{2}}-\frac {\sqrt {\frac {1}{c^{2}}}\, \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right ) d^{3}}{3}+\frac {\arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right ) d \,e^{2}}{c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{e}\right )\) | \(503\) |
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Exception generated. \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\text {Exception raised: RuntimeError} \]
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Time = 22.95 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.48 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} - 3 b c d^{2} \operatorname {RootSum} {\left (216 t^{3} c^{4} + 1, \left ( t \mapsto t \log {\left (36 t^{2} c^{2} + x^{2} \right )} \right )\right )} - 3 b c d e \operatorname {RootSum} {\left (46656 t^{6} c^{10} + 1, \left ( t \mapsto t \log {\left (7776 t^{5} c^{8} + x \right )} \right )\right )} + b d^{2} x \operatorname {atan}{\left (c x^{3} \right )} + b d e x^{2} \operatorname {atan}{\left (c x^{3} \right )} + b e^{2} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {x^{3} \operatorname {atan}{\left (c x^{3} \right )}}{3} - \frac {\log {\left (c^{2} x^{6} + 1 \right )}}{6 c} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.89 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} - \frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} - c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}} + \frac {\log \left (c^{\frac {4}{3}} x^{4} - c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {4}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} + 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}}\right )} - 4 \, x \arctan \left (c x^{3}\right )\right )} b d^{2} + \frac {1}{4} \, {\left (4 \, x^{2} \arctan \left (c x^{3}\right ) + c {\left (\frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {4 \, \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}}\right )}\right )} b d e + a d^{2} x + \frac {{\left (2 \, c x^{3} \arctan \left (c x^{3}\right ) - \log \left (c^{2} x^{6} + 1\right )\right )} b e^{2}}{6 \, c} \]
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Time = 9.73 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.99 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{3} \, b e^{2} x^{3} \arctan \left (c x^{3}\right ) + \frac {1}{3} \, a e^{2} x^{3} + b d e x^{2} \arctan \left (c x^{3}\right ) + a d e x^{2} + b d^{2} x \arctan \left (c x^{3}\right ) + a d^{2} x - \frac {b c d e \arctan \left (x {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {5}{3}}} + \frac {{\left (\sqrt {3} b c d^{2} {\left | c \right |}^{\frac {1}{3}} - b c d e\right )} \arctan \left ({\left (2 \, x + \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{2 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {{\left (\sqrt {3} b c d^{2} {\left | c \right |}^{\frac {1}{3}} + b c d e\right )} \arctan \left ({\left (2 \, x - \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{2 \, {\left | c \right |}^{\frac {5}{3}}} + \frac {{\left (3 \, \sqrt {3} b c d e {\left | c \right |}^{\frac {1}{3}} - 3 \, b c d^{2} {\left | c \right |}^{\frac {2}{3}} - 2 \, b c e^{2}\right )} \log \left (x^{2} + \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{12 \, c^{2}} - \frac {{\left (3 \, \sqrt {3} b c d e {\left | c \right |}^{\frac {1}{3}} + 3 \, b c d^{2} {\left | c \right |}^{\frac {2}{3}} + 2 \, b c e^{2}\right )} \log \left (x^{2} - \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{12 \, c^{2}} + \frac {{\left (3 \, b c d^{2} {\left | c \right |}^{\frac {2}{3}} - b c e^{2}\right )} \log \left (x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{6 \, c^{2}} \]
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Time = 0.59 (sec) , antiderivative size = 988, normalized size of antiderivative = 3.14 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\mathrm {atan}\left (c\,x^3\right )\,\left (b\,d^2\,x+b\,d\,e\,x^2+\frac {b\,e^2\,x^3}{3}\right )+\left (\sum _{k=1}^6\ln \left (x\,\left (6\,b^5\,c^7\,d^2\,e^8-162\,b^5\,c^9\,d^8\,e^2\right )+\mathrm {root}\left (46656\,a^6\,c^6+46656\,a^5\,b\,c^5\,e^2+19440\,a^4\,b^2\,c^4\,e^4+4320\,a^3\,b^3\,c^3\,e^6-11664\,a^3\,b^3\,c^5\,d^6+20412\,a^2\,b^4\,c^4\,d^6\,e^2+540\,a^2\,b^4\,c^2\,e^8-972\,a\,b^5\,c^3\,d^6\,e^4+36\,a\,b^5\,c\,e^{10}-54\,b^6\,c^2\,d^6\,e^6+729\,b^6\,c^4\,d^{12}+b^6\,e^{12},a,k\right )\,\left (x\,\left (486\,b^4\,c^{10}\,d^8+90\,b^4\,c^8\,d^2\,e^6\right )-\mathrm {root}\left (46656\,a^6\,c^6+46656\,a^5\,b\,c^5\,e^2+19440\,a^4\,b^2\,c^4\,e^4+4320\,a^3\,b^3\,c^3\,e^6-11664\,a^3\,b^3\,c^5\,d^6+20412\,a^2\,b^4\,c^4\,d^6\,e^2+540\,a^2\,b^4\,c^2\,e^8-972\,a\,b^5\,c^3\,d^6\,e^4+36\,a\,b^5\,c\,e^{10}-54\,b^6\,c^2\,d^6\,e^6+729\,b^6\,c^4\,d^{12}+b^6\,e^{12},a,k\right )\,\left (\mathrm {root}\left (46656\,a^6\,c^6+46656\,a^5\,b\,c^5\,e^2+19440\,a^4\,b^2\,c^4\,e^4+4320\,a^3\,b^3\,c^3\,e^6-11664\,a^3\,b^3\,c^5\,d^6+20412\,a^2\,b^4\,c^4\,d^6\,e^2+540\,a^2\,b^4\,c^2\,e^8-972\,a\,b^5\,c^3\,d^6\,e^4+36\,a\,b^5\,c\,e^{10}-54\,b^6\,c^2\,d^6\,e^6+729\,b^6\,c^4\,d^{12}+b^6\,e^{12},a,k\right )\,\left (3888\,b^2\,c^{10}\,d^3\,e+\mathrm {root}\left (46656\,a^6\,c^6+46656\,a^5\,b\,c^5\,e^2+19440\,a^4\,b^2\,c^4\,e^4+4320\,a^3\,b^3\,c^3\,e^6-11664\,a^3\,b^3\,c^5\,d^6+20412\,a^2\,b^4\,c^4\,d^6\,e^2+540\,a^2\,b^4\,c^2\,e^8-972\,a\,b^5\,c^3\,d^6\,e^4+36\,a\,b^5\,c\,e^{10}-54\,b^6\,c^2\,d^6\,e^6+729\,b^6\,c^4\,d^{12}+b^6\,e^{12},a,k\right )\,b\,c^{11}\,d^2\,x\,3888+648\,b^2\,c^{10}\,d^2\,e^2\,x\right )+972\,b^3\,c^9\,d^3\,e^3-324\,b^3\,c^9\,d^2\,e^4\,x\right )\right )-243\,b^5\,c^9\,d^9\,e+9\,b^5\,c^7\,d^3\,e^7\right )\,\mathrm {root}\left (46656\,a^6\,c^6+46656\,a^5\,b\,c^5\,e^2+19440\,a^4\,b^2\,c^4\,e^4+4320\,a^3\,b^3\,c^3\,e^6-11664\,a^3\,b^3\,c^5\,d^6+20412\,a^2\,b^4\,c^4\,d^6\,e^2+540\,a^2\,b^4\,c^2\,e^8-972\,a\,b^5\,c^3\,d^6\,e^4+36\,a\,b^5\,c\,e^{10}-54\,b^6\,c^2\,d^6\,e^6+729\,b^6\,c^4\,d^{12}+b^6\,e^{12},a,k\right )\right )+\frac {a\,e^2\,x^3}{3}+a\,d^2\,x+a\,d\,e\,x^2 \]
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