Optimal. Leaf size=94 \[ \frac {1}{3} d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac {b \left (5 c^2 d-3 e\right ) \log \left (c^2 x^2+1\right )}{30 c^5}-\frac {b x^2 \left (5 c^2 d-3 e\right )}{30 c^3}-\frac {b e x^4}{20 c} \]
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Rubi [A] time = 0.14, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {14, 4976, 446, 77} \[ \frac {1}{3} d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {b x^2 \left (5 c^2 d-3 e\right )}{30 c^3}+\frac {b \left (5 c^2 d-3 e\right ) \log \left (c^2 x^2+1\right )}{30 c^5}-\frac {b e x^4}{20 c} \]
Antiderivative was successfully verified.
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Rule 14
Rule 77
Rule 446
Rule 4976
Rubi steps
\begin {align*} \int x^2 \left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac {x^3 \left (5 d+3 e x^2\right )}{15+15 c^2 x^2} \, dx\\ &=\frac {1}{3} d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {x (5 d+3 e x)}{15+15 c^2 x} \, dx,x,x^2\right )\\ &=\frac {1}{3} d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{2} (b c) \operatorname {Subst}\left (\int \left (\frac {5 c^2 d-3 e}{15 c^4}+\frac {e x}{5 c^2}+\frac {-5 c^2 d+3 e}{15 c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {b \left (5 c^2 d-3 e\right ) x^2}{30 c^3}-\frac {b e x^4}{20 c}+\frac {1}{3} d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac {b \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{30 c^5}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 119, normalized size = 1.27 \[ \frac {1}{3} a d x^3+\frac {1}{5} a e x^5+\frac {b e x^2}{10 c^3}-\frac {b e \log \left (c^2 x^2+1\right )}{10 c^5}+\frac {b d \log \left (c^2 x^2+1\right )}{6 c^3}+\frac {1}{3} b d x^3 \tan ^{-1}(c x)-\frac {b d x^2}{6 c}+\frac {1}{5} b e x^5 \tan ^{-1}(c x)-\frac {b e x^4}{20 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 107, normalized size = 1.14 \[ \frac {12 \, a c^{5} e x^{5} + 20 \, a c^{5} d x^{3} - 3 \, b c^{4} e x^{4} - 2 \, {\left (5 \, b c^{4} d - 3 \, b c^{2} e\right )} x^{2} + 4 \, {\left (3 \, b c^{5} e x^{5} + 5 \, b c^{5} d x^{3}\right )} \arctan \left (c x\right ) + 2 \, {\left (5 \, b c^{2} d - 3 \, b e\right )} \log \left (c^{2} x^{2} + 1\right )}{60 \, c^{5}} \]
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 = 0.05, size = 102, normalized size = 1.09 \[ \frac {a e \,x^{5}}{5}+\frac {a d \,x^{3}}{3}+\frac {b e \,x^{5} \arctan \left (c x \right )}{5}+\frac {b \arctan \left (c x \right ) d \,x^{3}}{3}-\frac {b d \,x^{2}}{6 c}-\frac {b e \,x^{4}}{20 c}+\frac {b e \,x^{2}}{10 c^{3}}+\frac {b d \ln \left (c^{2} x^{2}+1\right )}{6 c^{3}}-\frac {b e \ln \left (c^{2} x^{2}+1\right )}{10 c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 105, normalized size = 1.12 \[ \frac {1}{5} \, a e x^{5} + \frac {1}{3} \, a d x^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b d + \frac {1}{20} \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b e \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 101, normalized size = 1.07 \[ \frac {a\,d\,x^3}{3}+\frac {a\,e\,x^5}{5}+\frac {b\,d\,x^3\,\mathrm {atan}\left (c\,x\right )}{3}+\frac {b\,e\,x^5\,\mathrm {atan}\left (c\,x\right )}{5}+\frac {b\,d\,\ln \left (c^2\,x^2+1\right )}{6\,c^3}-\frac {b\,e\,\ln \left (c^2\,x^2+1\right )}{10\,c^5}-\frac {b\,d\,x^2}{6\,c}-\frac {b\,e\,x^4}{20\,c}+\frac {b\,e\,x^2}{10\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.38, size = 128, normalized size = 1.36 \[ \begin {cases} \frac {a d x^{3}}{3} + \frac {a e x^{5}}{5} + \frac {b d x^{3} \operatorname {atan}{\left (c x \right )}}{3} + \frac {b e x^{5} \operatorname {atan}{\left (c x \right )}}{5} - \frac {b d x^{2}}{6 c} - \frac {b e x^{4}}{20 c} + \frac {b d \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{6 c^{3}} + \frac {b e x^{2}}{10 c^{3}} - \frac {b e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{10 c^{5}} & \text {for}\: c \neq 0 \\a \left (\frac {d x^{3}}{3} + \frac {e x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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