Integrand size = 21, antiderivative size = 380 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=-\frac {a b d^2 x}{c}+\frac {a b d e x}{c^3}-\frac {a b e^2 x}{3 c^5}+\frac {b^2 d e x^2}{6 c^2}-\frac {4 b^2 e^2 x^2}{45 c^4}+\frac {b^2 e^2 x^4}{60 c^2}-\frac {b^2 d^2 x \arctan (c x)}{c}+\frac {b^2 d e x \arctan (c x)}{c^3}-\frac {b^2 e^2 x \arctan (c x)}{3 c^5}-\frac {b d e x^3 (a+b \arctan (c x))}{3 c}+\frac {b e^2 x^3 (a+b \arctan (c x))}{9 c^3}-\frac {b e^2 x^5 (a+b \arctan (c x))}{15 c}+\frac {d^2 (a+b \arctan (c x))^2}{2 c^2}-\frac {d e (a+b \arctan (c x))^2}{2 c^4}+\frac {e^2 (a+b \arctan (c x))^2}{6 c^6}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {1}{2} d e x^4 (a+b \arctan (c x))^2+\frac {1}{6} e^2 x^6 (a+b \arctan (c x))^2+\frac {b^2 d^2 \log \left (1+c^2 x^2\right )}{2 c^2}-\frac {2 b^2 d e \log \left (1+c^2 x^2\right )}{3 c^4}+\frac {23 b^2 e^2 \log \left (1+c^2 x^2\right )}{90 c^6} \]
[Out]
Time = 0.54 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {5100, 4946, 5036, 4930, 266, 5004, 272, 45} \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\frac {e^2 (a+b \arctan (c x))^2}{6 c^6}-\frac {d e (a+b \arctan (c x))^2}{2 c^4}+\frac {b e^2 x^3 (a+b \arctan (c x))}{9 c^3}+\frac {d^2 (a+b \arctan (c x))^2}{2 c^2}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {1}{2} d e x^4 (a+b \arctan (c x))^2-\frac {b d e x^3 (a+b \arctan (c x))}{3 c}+\frac {1}{6} e^2 x^6 (a+b \arctan (c x))^2-\frac {b e^2 x^5 (a+b \arctan (c x))}{15 c}-\frac {a b e^2 x}{3 c^5}+\frac {a b d e x}{c^3}-\frac {a b d^2 x}{c}-\frac {b^2 e^2 x \arctan (c x)}{3 c^5}+\frac {b^2 d e x \arctan (c x)}{c^3}-\frac {b^2 d^2 x \arctan (c x)}{c}-\frac {4 b^2 e^2 x^2}{45 c^4}+\frac {b^2 d^2 \log \left (c^2 x^2+1\right )}{2 c^2}+\frac {b^2 d e x^2}{6 c^2}+\frac {b^2 e^2 x^4}{60 c^2}+\frac {23 b^2 e^2 \log \left (c^2 x^2+1\right )}{90 c^6}-\frac {2 b^2 d e \log \left (c^2 x^2+1\right )}{3 c^4} \]
[In]
[Out]
Rule 45
Rule 266
Rule 272
Rule 4930
Rule 4946
Rule 5004
Rule 5036
Rule 5100
Rubi steps \begin{align*} \text {integral}& = \int \left (d^2 x (a+b \arctan (c x))^2+2 d e x^3 (a+b \arctan (c x))^2+e^2 x^5 (a+b \arctan (c x))^2\right ) \, dx \\ & = d^2 \int x (a+b \arctan (c x))^2 \, dx+(2 d e) \int x^3 (a+b \arctan (c x))^2 \, dx+e^2 \int x^5 (a+b \arctan (c x))^2 \, dx \\ & = \frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {1}{2} d e x^4 (a+b \arctan (c x))^2+\frac {1}{6} e^2 x^6 (a+b \arctan (c x))^2-\left (b c d^2\right ) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-(b c d e) \int \frac {x^4 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {1}{3} \left (b c e^2\right ) \int \frac {x^6 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = \frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {1}{2} d e x^4 (a+b \arctan (c x))^2+\frac {1}{6} e^2 x^6 (a+b \arctan (c x))^2-\frac {\left (b d^2\right ) \int (a+b \arctan (c x)) \, dx}{c}+\frac {\left (b d^2\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{c}-\frac {(b d e) \int x^2 (a+b \arctan (c x)) \, dx}{c}+\frac {(b d e) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{c}-\frac {\left (b e^2\right ) \int x^4 (a+b \arctan (c x)) \, dx}{3 c}+\frac {\left (b e^2\right ) \int \frac {x^4 (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{3 c} \\ & = -\frac {a b d^2 x}{c}-\frac {b d e x^3 (a+b \arctan (c x))}{3 c}-\frac {b e^2 x^5 (a+b \arctan (c x))}{15 c}+\frac {d^2 (a+b \arctan (c x))^2}{2 c^2}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {1}{2} d e x^4 (a+b \arctan (c x))^2+\frac {1}{6} e^2 x^6 (a+b \arctan (c x))^2-\frac {\left (b^2 d^2\right ) \int \arctan (c x) \, dx}{c}+\frac {1}{3} \left (b^2 d e\right ) \int \frac {x^3}{1+c^2 x^2} \, dx+\frac {(b d e) \int (a+b \arctan (c x)) \, dx}{c^3}-\frac {(b d e) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{c^3}+\frac {1}{15} \left (b^2 e^2\right ) \int \frac {x^5}{1+c^2 x^2} \, dx+\frac {\left (b e^2\right ) \int x^2 (a+b \arctan (c x)) \, dx}{3 c^3}-\frac {\left (b e^2\right ) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{3 c^3} \\ & = -\frac {a b d^2 x}{c}+\frac {a b d e x}{c^3}-\frac {b^2 d^2 x \arctan (c x)}{c}-\frac {b d e x^3 (a+b \arctan (c x))}{3 c}+\frac {b e^2 x^3 (a+b \arctan (c x))}{9 c^3}-\frac {b e^2 x^5 (a+b \arctan (c x))}{15 c}+\frac {d^2 (a+b \arctan (c x))^2}{2 c^2}-\frac {d e (a+b \arctan (c x))^2}{2 c^4}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {1}{2} d e x^4 (a+b \arctan (c x))^2+\frac {1}{6} e^2 x^6 (a+b \arctan (c x))^2+\left (b^2 d^2\right ) \int \frac {x}{1+c^2 x^2} \, dx+\frac {1}{6} \left (b^2 d e\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )+\frac {\left (b^2 d e\right ) \int \arctan (c x) \, dx}{c^3}+\frac {1}{30} \left (b^2 e^2\right ) \text {Subst}\left (\int \frac {x^2}{1+c^2 x} \, dx,x,x^2\right )-\frac {\left (b e^2\right ) \int (a+b \arctan (c x)) \, dx}{3 c^5}+\frac {\left (b e^2\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{3 c^5}-\frac {\left (b^2 e^2\right ) \int \frac {x^3}{1+c^2 x^2} \, dx}{9 c^2} \\ & = -\frac {a b d^2 x}{c}+\frac {a b d e x}{c^3}-\frac {a b e^2 x}{3 c^5}-\frac {b^2 d^2 x \arctan (c x)}{c}+\frac {b^2 d e x \arctan (c x)}{c^3}-\frac {b d e x^3 (a+b \arctan (c x))}{3 c}+\frac {b e^2 x^3 (a+b \arctan (c x))}{9 c^3}-\frac {b e^2 x^5 (a+b \arctan (c x))}{15 c}+\frac {d^2 (a+b \arctan (c x))^2}{2 c^2}-\frac {d e (a+b \arctan (c x))^2}{2 c^4}+\frac {e^2 (a+b \arctan (c x))^2}{6 c^6}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {1}{2} d e x^4 (a+b \arctan (c x))^2+\frac {1}{6} e^2 x^6 (a+b \arctan (c x))^2+\frac {b^2 d^2 \log \left (1+c^2 x^2\right )}{2 c^2}+\frac {1}{6} \left (b^2 d e\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {\left (b^2 d e\right ) \int \frac {x}{1+c^2 x^2} \, dx}{c^2}+\frac {1}{30} \left (b^2 e^2\right ) \text {Subst}\left (\int \left (-\frac {1}{c^4}+\frac {x}{c^2}+\frac {1}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {\left (b^2 e^2\right ) \int \arctan (c x) \, dx}{3 c^5}-\frac {\left (b^2 e^2\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )}{18 c^2} \\ & = -\frac {a b d^2 x}{c}+\frac {a b d e x}{c^3}-\frac {a b e^2 x}{3 c^5}+\frac {b^2 d e x^2}{6 c^2}-\frac {b^2 e^2 x^2}{30 c^4}+\frac {b^2 e^2 x^4}{60 c^2}-\frac {b^2 d^2 x \arctan (c x)}{c}+\frac {b^2 d e x \arctan (c x)}{c^3}-\frac {b^2 e^2 x \arctan (c x)}{3 c^5}-\frac {b d e x^3 (a+b \arctan (c x))}{3 c}+\frac {b e^2 x^3 (a+b \arctan (c x))}{9 c^3}-\frac {b e^2 x^5 (a+b \arctan (c x))}{15 c}+\frac {d^2 (a+b \arctan (c x))^2}{2 c^2}-\frac {d e (a+b \arctan (c x))^2}{2 c^4}+\frac {e^2 (a+b \arctan (c x))^2}{6 c^6}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {1}{2} d e x^4 (a+b \arctan (c x))^2+\frac {1}{6} e^2 x^6 (a+b \arctan (c x))^2+\frac {b^2 d^2 \log \left (1+c^2 x^2\right )}{2 c^2}-\frac {2 b^2 d e \log \left (1+c^2 x^2\right )}{3 c^4}+\frac {b^2 e^2 \log \left (1+c^2 x^2\right )}{30 c^6}+\frac {\left (b^2 e^2\right ) \int \frac {x}{1+c^2 x^2} \, dx}{3 c^4}-\frac {\left (b^2 e^2\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{18 c^2} \\ & = -\frac {a b d^2 x}{c}+\frac {a b d e x}{c^3}-\frac {a b e^2 x}{3 c^5}+\frac {b^2 d e x^2}{6 c^2}-\frac {4 b^2 e^2 x^2}{45 c^4}+\frac {b^2 e^2 x^4}{60 c^2}-\frac {b^2 d^2 x \arctan (c x)}{c}+\frac {b^2 d e x \arctan (c x)}{c^3}-\frac {b^2 e^2 x \arctan (c x)}{3 c^5}-\frac {b d e x^3 (a+b \arctan (c x))}{3 c}+\frac {b e^2 x^3 (a+b \arctan (c x))}{9 c^3}-\frac {b e^2 x^5 (a+b \arctan (c x))}{15 c}+\frac {d^2 (a+b \arctan (c x))^2}{2 c^2}-\frac {d e (a+b \arctan (c x))^2}{2 c^4}+\frac {e^2 (a+b \arctan (c x))^2}{6 c^6}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {1}{2} d e x^4 (a+b \arctan (c x))^2+\frac {1}{6} e^2 x^6 (a+b \arctan (c x))^2+\frac {b^2 d^2 \log \left (1+c^2 x^2\right )}{2 c^2}-\frac {2 b^2 d e \log \left (1+c^2 x^2\right )}{3 c^4}+\frac {23 b^2 e^2 \log \left (1+c^2 x^2\right )}{90 c^6} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.83 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\frac {c x \left (30 a^2 c^5 x \left (3 d^2+3 d e x^2+e^2 x^4\right )+b^2 c e x \left (-16 e+3 c^2 \left (10 d+e x^2\right )\right )-4 a b \left (15 e^2-5 c^2 e \left (9 d+e x^2\right )+3 c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )\right )\right )+4 b \left (-b c x \left (15 e^2-5 c^2 e \left (9 d+e x^2\right )+3 c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )\right )+15 a \left (3 c^4 d^2-3 c^2 d e+e^2+c^6 \left (3 d^2 x^2+3 d e x^4+e^2 x^6\right )\right )\right ) \arctan (c x)+30 b^2 \left (3 c^4 d^2-3 c^2 d e+e^2+c^6 \left (3 d^2 x^2+3 d e x^4+e^2 x^6\right )\right ) \arctan (c x)^2+2 b^2 \left (45 c^4 d^2-60 c^2 d e+23 e^2\right ) \log \left (1+c^2 x^2\right )}{180 c^6} \]
[In]
[Out]
Time = 0.43 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.29
method | result | size |
parts | \(\frac {a^{2} \left (e \,x^{2}+d \right )^{3}}{6 e}+\frac {b^{2} \arctan \left (c x \right )^{2} e^{2} x^{6}}{6}+\frac {b^{2} \arctan \left (c x \right )^{2} e \,x^{4} d}{2}+\frac {b^{2} \arctan \left (c x \right )^{2} x^{2} d^{2}}{2}-\frac {b^{2} e^{2} \arctan \left (c x \right ) x^{5}}{15 c}-\frac {b^{2} e \arctan \left (c x \right ) x^{3} d}{3 c}-\frac {b^{2} d^{2} x \arctan \left (c x \right )}{c}+\frac {b^{2} e^{2} \arctan \left (c x \right ) x^{3}}{9 c^{3}}+\frac {b^{2} d e x \arctan \left (c x \right )}{c^{3}}-\frac {b^{2} e^{2} x \arctan \left (c x \right )}{3 c^{5}}+\frac {b^{2} \arctan \left (c x \right )^{2} d^{2}}{2 c^{2}}-\frac {b^{2} e \arctan \left (c x \right )^{2} d}{2 c^{4}}+\frac {b^{2} e^{2} \arctan \left (c x \right )^{2}}{6 c^{6}}+\frac {b^{2} d e \,x^{2}}{6 c^{2}}+\frac {b^{2} e^{2} x^{4}}{60 c^{2}}-\frac {4 b^{2} e^{2} x^{2}}{45 c^{4}}+\frac {b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{2 c^{2}}-\frac {2 b^{2} d e \ln \left (c^{2} x^{2}+1\right )}{3 c^{4}}+\frac {23 b^{2} e^{2} \ln \left (c^{2} x^{2}+1\right )}{90 c^{6}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) c^{2} e^{2} x^{6}}{6}+\frac {\arctan \left (c x \right ) c^{2} e \,x^{4} d}{2}+\frac {\arctan \left (c x \right ) c^{2} x^{2} d^{2}}{2}+\frac {\arctan \left (c x \right ) c^{2} d^{3}}{6 e}-\frac {3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-3 c^{3} x d \,e^{2}-\frac {e^{3} c^{3} x^{3}}{3}+c x \,e^{3}+\left (c^{6} d^{3}-3 c^{4} d^{2} e +3 e^{2} d \,c^{2}-e^{3}\right ) \arctan \left (c x \right )}{6 c^{4} e}\right )}{c^{2}}\) | \(492\) |
derivativedivides | \(\frac {\frac {a^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b^{2} \arctan \left (c x \right )^{2} d^{2} c^{2} x^{2}}{2}+\frac {b^{2} c^{2} \arctan \left (c x \right )^{2} e d \,x^{4}}{2}+\frac {b^{2} c^{2} \arctan \left (c x \right )^{2} e^{2} x^{6}}{6}-b^{2} c \,d^{2} x \arctan \left (c x \right )-\frac {b^{2} c e \arctan \left (c x \right ) d \,x^{3}}{3}-\frac {b^{2} c \,e^{2} \arctan \left (c x \right ) x^{5}}{15}+\frac {b^{2} d e x \arctan \left (c x \right )}{c}+\frac {b^{2} e^{2} \arctan \left (c x \right ) x^{3}}{9 c}-\frac {b^{2} e^{2} x \arctan \left (c x \right )}{3 c^{3}}+\frac {b^{2} \arctan \left (c x \right )^{2} d^{2}}{2}-\frac {b^{2} e \arctan \left (c x \right )^{2} d}{2 c^{2}}+\frac {b^{2} e^{2} \arctan \left (c x \right )^{2}}{6 c^{4}}+\frac {b^{2} d e \,x^{2}}{6}+\frac {b^{2} e^{2} x^{4}}{60}-\frac {4 b^{2} e^{2} x^{2}}{45 c^{2}}+\frac {b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {2 b^{2} d e \ln \left (c^{2} x^{2}+1\right )}{3 c^{2}}+\frac {23 b^{2} e^{2} \ln \left (c^{2} x^{2}+1\right )}{90 c^{4}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) c^{6} d^{3}}{6 e}+\frac {\arctan \left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {\arctan \left (c x \right ) e \,c^{6} d \,x^{4}}{2}+\frac {\arctan \left (c x \right ) e^{2} c^{6} x^{6}}{6}-\frac {3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-3 c^{3} x d \,e^{2}-\frac {e^{3} c^{3} x^{3}}{3}+c x \,e^{3}+\left (c^{6} d^{3}-3 c^{4} d^{2} e +3 e^{2} d \,c^{2}-e^{3}\right ) \arctan \left (c x \right )}{6 e}\right )}{c^{4}}}{c^{2}}\) | \(494\) |
default | \(\frac {\frac {a^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b^{2} \arctan \left (c x \right )^{2} d^{2} c^{2} x^{2}}{2}+\frac {b^{2} c^{2} \arctan \left (c x \right )^{2} e d \,x^{4}}{2}+\frac {b^{2} c^{2} \arctan \left (c x \right )^{2} e^{2} x^{6}}{6}-b^{2} c \,d^{2} x \arctan \left (c x \right )-\frac {b^{2} c e \arctan \left (c x \right ) d \,x^{3}}{3}-\frac {b^{2} c \,e^{2} \arctan \left (c x \right ) x^{5}}{15}+\frac {b^{2} d e x \arctan \left (c x \right )}{c}+\frac {b^{2} e^{2} \arctan \left (c x \right ) x^{3}}{9 c}-\frac {b^{2} e^{2} x \arctan \left (c x \right )}{3 c^{3}}+\frac {b^{2} \arctan \left (c x \right )^{2} d^{2}}{2}-\frac {b^{2} e \arctan \left (c x \right )^{2} d}{2 c^{2}}+\frac {b^{2} e^{2} \arctan \left (c x \right )^{2}}{6 c^{4}}+\frac {b^{2} d e \,x^{2}}{6}+\frac {b^{2} e^{2} x^{4}}{60}-\frac {4 b^{2} e^{2} x^{2}}{45 c^{2}}+\frac {b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {2 b^{2} d e \ln \left (c^{2} x^{2}+1\right )}{3 c^{2}}+\frac {23 b^{2} e^{2} \ln \left (c^{2} x^{2}+1\right )}{90 c^{4}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) c^{6} d^{3}}{6 e}+\frac {\arctan \left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {\arctan \left (c x \right ) e \,c^{6} d \,x^{4}}{2}+\frac {\arctan \left (c x \right ) e^{2} c^{6} x^{6}}{6}-\frac {3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-3 c^{3} x d \,e^{2}-\frac {e^{3} c^{3} x^{3}}{3}+c x \,e^{3}+\left (c^{6} d^{3}-3 c^{4} d^{2} e +3 e^{2} d \,c^{2}-e^{3}\right ) \arctan \left (c x \right )}{6 e}\right )}{c^{4}}}{c^{2}}\) | \(494\) |
parallelrisch | \(\frac {46 e^{2} b^{2} \ln \left (c^{2} x^{2}+1\right )+30 \arctan \left (c x \right )^{2} b^{2} e^{2}+180 x^{4} \arctan \left (c x \right ) a b \,c^{6} d e +16 e^{2} b^{2}+180 a b \,c^{4} d^{2} \arctan \left (c x \right )-180 a b \,c^{2} d e \arctan \left (c x \right )+3 b^{2} c^{4} e^{2} x^{4}-16 b^{2} c^{2} e^{2} x^{2}-30 b^{2} c^{2} d e -60 x^{3} \arctan \left (c x \right ) b^{2} c^{5} d e +180 x \arctan \left (c x \right ) b^{2} c^{3} d e -120 b^{2} c^{2} d e \ln \left (c^{2} x^{2}+1\right )-12 a b \,c^{5} e^{2} x^{5}-180 a b \,c^{5} d^{2} x +20 a b \,c^{3} e^{2} x^{3}-60 a b c \,e^{2} x +30 b^{2} c^{4} d e \,x^{2}+30 c^{6} a^{2} e^{2} x^{6}+90 c^{6} a^{2} d^{2} x^{2}+60 x^{6} \arctan \left (c x \right ) a b \,c^{6} e^{2}+180 x^{2} \arctan \left (c x \right ) a b \,c^{6} d^{2}+90 x^{4} \arctan \left (c x \right )^{2} b^{2} c^{6} d e +30 x^{6} \arctan \left (c x \right )^{2} b^{2} c^{6} e^{2}+90 x^{2} \arctan \left (c x \right )^{2} b^{2} c^{6} d^{2}+90 \arctan \left (c x \right )^{2} b^{2} c^{4} d^{2}-90 c^{4} a^{2} d^{2}-60 a b \,c^{5} d e \,x^{3}+180 a b \,c^{3} d e x +60 a b \,e^{2} \arctan \left (c x \right )+90 b^{2} c^{4} d^{2} \ln \left (c^{2} x^{2}+1\right )+90 c^{6} a^{2} d e \,x^{4}-90 \arctan \left (c x \right )^{2} b^{2} c^{2} d e +20 x^{3} \arctan \left (c x \right ) b^{2} c^{3} e^{2}-12 x^{5} \arctan \left (c x \right ) b^{2} c^{5} e^{2}-180 x \arctan \left (c x \right ) b^{2} c^{5} d^{2}-60 x \arctan \left (c x \right ) b^{2} c \,e^{2}}{180 c^{6}}\) | \(537\) |
risch | \(\frac {x^{6} e^{2} a^{2}}{6}+\frac {x^{2} d^{2} a^{2}}{2}-\frac {4 b^{2} e^{2} x^{2}}{45 c^{4}}+\frac {b^{2} e^{2} x^{4}}{60 c^{2}}+\frac {23 b^{2} e^{2} \ln \left (c^{2} x^{2}+1\right )}{90 c^{6}}-\frac {a b \,e^{2} x}{3 c^{5}}+\frac {b^{2} d e \,x^{2}}{6 c^{2}}-\frac {2 b^{2} d e \ln \left (c^{2} x^{2}+1\right )}{3 c^{4}}+\frac {b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{2 c^{2}}-\frac {a b \,d^{2} x}{c}+\frac {a b d e x}{c^{3}}-\frac {b^{2} e^{2} x^{6} \ln \left (-i c x +1\right )^{2}}{24}-\frac {b^{2} d^{2} x^{2} \ln \left (-i c x +1\right )^{2}}{8}-\frac {b^{2} d^{2} \ln \left (-i c x +1\right )^{2}}{8 c^{2}}-\frac {b^{2} e^{2} \ln \left (-i c x +1\right )^{2}}{24 c^{6}}-\frac {i b \left (30 x^{6} a \,c^{6} e^{2}+15 i b \,c^{6} e^{2} x^{6} \ln \left (-i c x +1\right )+90 x^{4} a \,c^{6} d e -6 b \,c^{5} e^{2} x^{5}+45 i b \,c^{6} d e \,x^{4} \ln \left (-i c x +1\right )+90 x^{2} a \,c^{6} d^{2}-30 b \,c^{5} d e \,x^{3}+45 i b \,c^{6} d^{2} x^{2} \ln \left (-i c x +1\right )-90 b \,c^{5} d^{2} x +10 b \,c^{3} e^{2} x^{3}+90 b \,c^{3} d e x +45 i \ln \left (-i c x +1\right ) b \,c^{4} d^{2}-30 b c \,e^{2} x -45 i \ln \left (-i c x +1\right ) b \,c^{2} d e +15 i \ln \left (-i c x +1\right ) b \,e^{2}\right ) \ln \left (i c x +1\right )}{180 c^{6}}-\frac {b^{2} d e \,x^{4} \ln \left (-i c x +1\right )^{2}}{8}+\frac {b^{2} d e \ln \left (-i c x +1\right )^{2}}{8 c^{4}}-\frac {i b^{2} e^{2} x^{5} \ln \left (-i c x +1\right )}{30 c}-\frac {i b^{2} d^{2} x \ln \left (-i c x +1\right )}{2 c}+\frac {i b^{2} e^{2} x^{3} \ln \left (-i c x +1\right )}{18 c^{3}}-\frac {i b^{2} e^{2} x \ln \left (-i c x +1\right )}{6 c^{5}}+\frac {i a b \,e^{2} x^{6} \ln \left (-i c x +1\right )}{6}-\frac {b^{2} \left (e^{2} c^{6} x^{6}+3 d \,c^{6} e \,x^{4}+3 d^{2} c^{6} x^{2}+3 c^{4} d^{2}-3 c^{2} d e +e^{2}\right ) \ln \left (i c x +1\right )^{2}}{24 c^{6}}+\frac {i a b d e \,x^{4} \ln \left (-i c x +1\right )}{2}-\frac {i b^{2} d e \,x^{3} \ln \left (-i c x +1\right )}{6 c}+\frac {i b^{2} d e x \ln \left (-i c x +1\right )}{2 c^{3}}-\frac {a b d e \,x^{3}}{3 c}-\frac {a b d e \arctan \left (c x \right )}{c^{4}}-\frac {a b \,e^{2} x^{5}}{15 c}+\frac {a b \,e^{2} x^{3}}{9 c^{3}}+\frac {a b \,d^{2} \arctan \left (c x \right )}{c^{2}}+\frac {a b \,e^{2} \arctan \left (c x \right )}{3 c^{6}}+\frac {x^{4} e d \,a^{2}}{2}+\frac {i a b \,d^{2} x^{2} \ln \left (-i c x +1\right )}{2}\) | \(849\) |
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Time = 0.28 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.09 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\frac {30 \, a^{2} c^{6} e^{2} x^{6} - 12 \, a b c^{5} e^{2} x^{5} + 3 \, {\left (30 \, a^{2} c^{6} d e + b^{2} c^{4} e^{2}\right )} x^{4} - 20 \, {\left (3 \, a b c^{5} d e - a b c^{3} e^{2}\right )} x^{3} + 2 \, {\left (45 \, a^{2} c^{6} d^{2} + 15 \, b^{2} c^{4} d e - 8 \, b^{2} c^{2} e^{2}\right )} x^{2} + 30 \, {\left (b^{2} c^{6} e^{2} x^{6} + 3 \, b^{2} c^{6} d e x^{4} + 3 \, b^{2} c^{6} d^{2} x^{2} + 3 \, b^{2} c^{4} d^{2} - 3 \, b^{2} c^{2} d e + b^{2} e^{2}\right )} \arctan \left (c x\right )^{2} - 60 \, {\left (3 \, a b c^{5} d^{2} - 3 \, a b c^{3} d e + a b c e^{2}\right )} x + 4 \, {\left (15 \, a b c^{6} e^{2} x^{6} + 45 \, a b c^{6} d e x^{4} - 3 \, b^{2} c^{5} e^{2} x^{5} + 45 \, a b c^{6} d^{2} x^{2} + 45 \, a b c^{4} d^{2} - 45 \, a b c^{2} d e + 15 \, a b e^{2} - 5 \, {\left (3 \, b^{2} c^{5} d e - b^{2} c^{3} e^{2}\right )} x^{3} - 15 \, {\left (3 \, b^{2} c^{5} d^{2} - 3 \, b^{2} c^{3} d e + b^{2} c e^{2}\right )} x\right )} \arctan \left (c x\right ) + 2 \, {\left (45 \, b^{2} c^{4} d^{2} - 60 \, b^{2} c^{2} d e + 23 \, b^{2} e^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{180 \, c^{6}} \]
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Time = 0.54 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.51 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\begin {cases} \frac {a^{2} d^{2} x^{2}}{2} + \frac {a^{2} d e x^{4}}{2} + \frac {a^{2} e^{2} x^{6}}{6} + a b d^{2} x^{2} \operatorname {atan}{\left (c x \right )} + a b d e x^{4} \operatorname {atan}{\left (c x \right )} + \frac {a b e^{2} x^{6} \operatorname {atan}{\left (c x \right )}}{3} - \frac {a b d^{2} x}{c} - \frac {a b d e x^{3}}{3 c} - \frac {a b e^{2} x^{5}}{15 c} + \frac {a b d^{2} \operatorname {atan}{\left (c x \right )}}{c^{2}} + \frac {a b d e x}{c^{3}} + \frac {a b e^{2} x^{3}}{9 c^{3}} - \frac {a b d e \operatorname {atan}{\left (c x \right )}}{c^{4}} - \frac {a b e^{2} x}{3 c^{5}} + \frac {a b e^{2} \operatorname {atan}{\left (c x \right )}}{3 c^{6}} + \frac {b^{2} d^{2} x^{2} \operatorname {atan}^{2}{\left (c x \right )}}{2} + \frac {b^{2} d e x^{4} \operatorname {atan}^{2}{\left (c x \right )}}{2} + \frac {b^{2} e^{2} x^{6} \operatorname {atan}^{2}{\left (c x \right )}}{6} - \frac {b^{2} d^{2} x \operatorname {atan}{\left (c x \right )}}{c} - \frac {b^{2} d e x^{3} \operatorname {atan}{\left (c x \right )}}{3 c} - \frac {b^{2} e^{2} x^{5} \operatorname {atan}{\left (c x \right )}}{15 c} + \frac {b^{2} d^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c^{2}} + \frac {b^{2} d^{2} \operatorname {atan}^{2}{\left (c x \right )}}{2 c^{2}} + \frac {b^{2} d e x^{2}}{6 c^{2}} + \frac {b^{2} e^{2} x^{4}}{60 c^{2}} + \frac {b^{2} d e x \operatorname {atan}{\left (c x \right )}}{c^{3}} + \frac {b^{2} e^{2} x^{3} \operatorname {atan}{\left (c x \right )}}{9 c^{3}} - \frac {2 b^{2} d e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{3 c^{4}} - \frac {b^{2} d e \operatorname {atan}^{2}{\left (c x \right )}}{2 c^{4}} - \frac {4 b^{2} e^{2} x^{2}}{45 c^{4}} - \frac {b^{2} e^{2} x \operatorname {atan}{\left (c x \right )}}{3 c^{5}} + \frac {23 b^{2} e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{90 c^{6}} + \frac {b^{2} e^{2} \operatorname {atan}^{2}{\left (c x \right )}}{6 c^{6}} & \text {for}\: c \neq 0 \\a^{2} \left (\frac {d^{2} x^{2}}{2} + \frac {d e x^{4}}{2} + \frac {e^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.31 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.14 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\frac {1}{6} \, b^{2} e^{2} x^{6} \arctan \left (c x\right )^{2} + \frac {1}{6} \, a^{2} e^{2} x^{6} + \frac {1}{2} \, b^{2} d e x^{4} \arctan \left (c x\right )^{2} + \frac {1}{2} \, a^{2} d e x^{4} + \frac {1}{2} \, b^{2} d^{2} x^{2} \arctan \left (c x\right )^{2} + \frac {1}{2} \, a^{2} d^{2} x^{2} + {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} a b d^{2} - \frac {1}{2} \, {\left (2 \, c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )} \arctan \left (c x\right ) + \frac {\arctan \left (c x\right )^{2} - \log \left (c^{2} x^{2} + 1\right )}{c^{2}}\right )} b^{2} d^{2} + \frac {1}{3} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} a b d e - \frac {1}{6} \, {\left (2 \, c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )} \arctan \left (c x\right ) - \frac {c^{2} x^{2} + 3 \, \arctan \left (c x\right )^{2} - 4 \, \log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )} b^{2} d e + \frac {1}{45} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} a b e^{2} - \frac {1}{180} \, {\left (4 \, c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )} \arctan \left (c x\right ) - \frac {3 \, c^{4} x^{4} - 16 \, c^{2} x^{2} - 30 \, \arctan \left (c x\right )^{2} + 46 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )} b^{2} e^{2} \]
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\[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \]
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Time = 5.64 (sec) , antiderivative size = 780, normalized size of antiderivative = 2.05 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\frac {a^2\,d^2\,x^2}{2}+\frac {a^2\,e^2\,x^6}{6}+\frac {b^2\,d^2\,\ln \left (c^2\,x^2+1\right )}{2\,c^2}+\frac {23\,b^2\,e^2\,\ln \left (c^2\,x^2+1\right )}{90\,c^6}+\frac {b^2\,e^2\,x^4}{60\,c^2}-\frac {4\,b^2\,e^2\,x^2}{45\,c^4}+\frac {b^2\,d^2\,{\mathrm {atan}\left (c\,x\right )}^2}{2\,c^2}+\frac {b^2\,e^2\,{\mathrm {atan}\left (c\,x\right )}^2}{6\,c^6}+\frac {b^2\,d^2\,x^2\,{\mathrm {atan}\left (c\,x\right )}^2}{2}+\frac {b^2\,e^2\,x^6\,{\mathrm {atan}\left (c\,x\right )}^2}{6}+\frac {a^2\,d\,e\,x^4}{2}-\frac {b^2\,e^2\,x^5\,\mathrm {atan}\left (c\,x\right )}{15\,c}+\frac {b^2\,e^2\,x^3\,\mathrm {atan}\left (c\,x\right )}{9\,c^3}-\frac {a\,b\,d^2\,x}{c}-\frac {a\,b\,e^2\,x}{3\,c^5}+a\,b\,d^2\,x^2\,\mathrm {atan}\left (c\,x\right )+\frac {a\,b\,e^2\,x^6\,\mathrm {atan}\left (c\,x\right )}{3}-\frac {2\,b^2\,d\,e\,\ln \left (c^2\,x^2+1\right )}{3\,c^4}-\frac {a\,b\,e^2\,x^5}{15\,c}+\frac {a\,b\,e^2\,x^3}{9\,c^3}+\frac {b^2\,d\,e\,x^2}{6\,c^2}-\frac {b^2\,d\,e\,{\mathrm {atan}\left (c\,x\right )}^2}{2\,c^4}-\frac {b^2\,d^2\,x\,\mathrm {atan}\left (c\,x\right )}{c}-\frac {b^2\,e^2\,x\,\mathrm {atan}\left (c\,x\right )}{3\,c^5}+\frac {b^2\,d\,e\,x^4\,{\mathrm {atan}\left (c\,x\right )}^2}{2}+\frac {a\,b\,d^2\,\mathrm {atan}\left (\frac {b\,c\,e^2\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}+\frac {3\,b\,c^5\,d^2\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}-\frac {3\,b\,c^3\,d\,e\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}\right )}{c^2}+\frac {a\,b\,e^2\,\mathrm {atan}\left (\frac {b\,c\,e^2\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}+\frac {3\,b\,c^5\,d^2\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}-\frac {3\,b\,c^3\,d\,e\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}\right )}{3\,c^6}-\frac {b^2\,d\,e\,x^3\,\mathrm {atan}\left (c\,x\right )}{3\,c}+\frac {a\,b\,d\,e\,x}{c^3}+a\,b\,d\,e\,x^4\,\mathrm {atan}\left (c\,x\right )-\frac {a\,b\,d\,e\,x^3}{3\,c}+\frac {b^2\,d\,e\,x\,\mathrm {atan}\left (c\,x\right )}{c^3}-\frac {a\,b\,d\,e\,\mathrm {atan}\left (\frac {b\,c\,e^2\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}+\frac {3\,b\,c^5\,d^2\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}-\frac {3\,b\,c^3\,d\,e\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}\right )}{c^4} \]
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