Optimal. Leaf size=124 \[ d^2 x \left (a+b \tan ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {b e x^2 \left (10 c^2 d-3 e\right )}{30 c^3}-\frac {b \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) \log \left (c^2 x^2+1\right )}{30 c^5}-\frac {b e^2 x^4}{20 c} \]
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Rubi [A] time = 0.16, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {194, 4912, 1594, 1247, 698} \[ d^2 x \left (a+b \tan ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {b \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) \log \left (c^2 x^2+1\right )}{30 c^5}-\frac {b e x^2 \left (10 c^2 d-3 e\right )}{30 c^3}-\frac {b e^2 x^4}{20 c} \]
Antiderivative was successfully verified.
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Rule 194
Rule 698
Rule 1247
Rule 1594
Rule 4912
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \tan ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac {d^2 x+\frac {2}{3} d e x^3+\frac {e^2 x^5}{5}}{1+c^2 x^2} \, dx\\ &=d^2 x \left (a+b \tan ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac {x \left (d^2+\frac {2}{3} d e x^2+\frac {e^2 x^4}{5}\right )}{1+c^2 x^2} \, dx\\ &=d^2 x \left (a+b \tan ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {d^2+\frac {2 d e x}{3}+\frac {e^2 x^2}{5}}{1+c^2 x} \, dx,x,x^2\right )\\ &=d^2 x \left (a+b \tan ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{2} (b c) \operatorname {Subst}\left (\int \left (\frac {\left (10 c^2 d-3 e\right ) e}{15 c^4}+\frac {e^2 x}{5 c^2}+\frac {15 c^4 d^2-10 c^2 d e+3 e^2}{15 c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {b \left (10 c^2 d-3 e\right ) e x^2}{30 c^3}-\frac {b e^2 x^4}{20 c}+d^2 x \left (a+b \tan ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {b \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) \log \left (1+c^2 x^2\right )}{30 c^5}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 130, normalized size = 1.05 \[ \frac {c^2 x \left (4 a c^3 \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+b e x \left (6 e-c^2 \left (20 d+3 e x^2\right )\right )\right )+4 b c^5 x \tan ^{-1}(c x) \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-2 b \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) \log \left (c^2 x^2+1\right )}{60 c^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 150, normalized size = 1.21 \[ \frac {12 \, a c^{5} e^{2} x^{5} + 40 \, a c^{5} d e x^{3} - 3 \, b c^{4} e^{2} x^{4} + 60 \, a c^{5} d^{2} x - 2 \, {\left (10 \, b c^{4} d e - 3 \, b c^{2} e^{2}\right )} x^{2} + 4 \, {\left (3 \, b c^{5} e^{2} x^{5} + 10 \, b c^{5} d e x^{3} + 15 \, b c^{5} d^{2} x\right )} \arctan \left (c x\right ) - 2 \, {\left (15 \, b c^{4} d^{2} - 10 \, b c^{2} d e + 3 \, b e^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{60 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 151, normalized size = 1.22 \[ \frac {a \,x^{5} e^{2}}{5}+\frac {2 a \,x^{3} d e}{3}+a x \,d^{2}+\frac {b \arctan \left (c x \right ) x^{5} e^{2}}{5}+\frac {2 b \arctan \left (c x \right ) x^{3} d e}{3}+b \arctan \left (c x \right ) d^{2} x -\frac {b \,x^{2} d e}{3 c}-\frac {b \,e^{2} x^{4}}{20 c}+\frac {b \,x^{2} e^{2}}{10 c^{3}}-\frac {b \ln \left (c^{2} x^{2}+1\right ) d^{2}}{2 c}+\frac {b \ln \left (c^{2} x^{2}+1\right ) e d}{3 c^{3}}-\frac {b \ln \left (c^{2} x^{2}+1\right ) e^{2}}{10 c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 147, normalized size = 1.19 \[ \frac {1}{5} \, a e^{2} x^{5} + \frac {2}{3} \, a d e x^{3} + \frac {1}{3} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b d e + \frac {1}{20} \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b e^{2} + a d^{2} x + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d^{2}}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 150, normalized size = 1.21 \[ \frac {a\,e^2\,x^5}{5}+a\,d^2\,x-\frac {b\,d^2\,\ln \left (c^2\,x^2+1\right )}{2\,c}-\frac {b\,e^2\,\ln \left (c^2\,x^2+1\right )}{10\,c^5}-\frac {b\,e^2\,x^4}{20\,c}+\frac {b\,e^2\,x^2}{10\,c^3}+\frac {2\,a\,d\,e\,x^3}{3}+b\,d^2\,x\,\mathrm {atan}\left (c\,x\right )+\frac {b\,e^2\,x^5\,\mathrm {atan}\left (c\,x\right )}{5}+\frac {b\,d\,e\,\ln \left (c^2\,x^2+1\right )}{3\,c^3}-\frac {b\,d\,e\,x^2}{3\,c}+\frac {2\,b\,d\,e\,x^3\,\mathrm {atan}\left (c\,x\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.63, size = 194, normalized size = 1.56 \[ \begin {cases} a d^{2} x + \frac {2 a d e x^{3}}{3} + \frac {a e^{2} x^{5}}{5} + b d^{2} x \operatorname {atan}{\left (c x \right )} + \frac {2 b d e x^{3} \operatorname {atan}{\left (c x \right )}}{3} + \frac {b e^{2} x^{5} \operatorname {atan}{\left (c x \right )}}{5} - \frac {b d^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c} - \frac {b d e x^{2}}{3 c} - \frac {b e^{2} x^{4}}{20 c} + \frac {b d e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{3 c^{3}} + \frac {b e^{2} x^{2}}{10 c^{3}} - \frac {b e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{10 c^{5}} & \text {for}\: c \neq 0 \\a \left (d^{2} x + \frac {2 d e x^{3}}{3} + \frac {e^{2} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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