Optimal. Leaf size=68 \[ d x \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {b \left (3 c^2 d-e\right ) \log \left (c^2 x^2+1\right )}{6 c^3}-\frac {b e x^2}{6 c} \]
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Rubi [A] time = 0.07, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4912, 1593, 444, 43} \[ d x \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {b \left (3 c^2 d-e\right ) \log \left (c^2 x^2+1\right )}{6 c^3}-\frac {b e x^2}{6 c} \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rule 1593
Rule 4912
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \, dx &=d x \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac {d x+\frac {e x^3}{3}}{1+c^2 x^2} \, dx\\ &=d x \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac {x \left (d+\frac {e x^2}{3}\right )}{1+c^2 x^2} \, dx\\ &=d x \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {d+\frac {e x}{3}}{1+c^2 x} \, dx,x,x^2\right )\\ &=d x \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{2} (b c) \operatorname {Subst}\left (\int \left (\frac {e}{3 c^2}+\frac {3 c^2 d-e}{3 c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {b e x^2}{6 c}+d x \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {b \left (3 c^2 d-e\right ) \log \left (1+c^2 x^2\right )}{6 c^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 85, normalized size = 1.25 \[ a d x+\frac {1}{3} a e x^3-\frac {b d \log \left (c^2 x^2+1\right )}{2 c}+\frac {b e \log \left (c^2 x^2+1\right )}{6 c^3}+b d x \tan ^{-1}(c x)+\frac {1}{3} b e x^3 \tan ^{-1}(c x)-\frac {b e x^2}{6 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 82, normalized size = 1.21 \[ \frac {2 \, a c^{3} e x^{3} + 6 \, a c^{3} d x - b c^{2} e x^{2} + 2 \, {\left (b c^{3} e x^{3} + 3 \, b c^{3} d x\right )} \arctan \left (c x\right ) - {\left (3 \, b c^{2} d - b e\right )} \log \left (c^{2} x^{2} + 1\right )}{6 \, c^{3}} \]
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 = 76, normalized size = 1.12 \[ \frac {a e \,x^{3}}{3}+a d x +\frac {b e \,x^{3} \arctan \left (c x \right )}{3}+b \arctan \left (c x \right ) d x -\frac {b e \,x^{2}}{6 c}-\frac {b d \ln \left (c^{2} x^{2}+1\right )}{2 c}+\frac {b e \ln \left (c^{2} x^{2}+1\right )}{6 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 80, normalized size = 1.18 \[ \frac {1}{3} \, a e x^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b e + a d x + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 75, normalized size = 1.10 \[ a\,d\,x+\frac {a\,e\,x^3}{3}+b\,d\,x\,\mathrm {atan}\left (c\,x\right )+\frac {b\,e\,x^3\,\mathrm {atan}\left (c\,x\right )}{3}-\frac {b\,d\,\ln \left (c^2\,x^2+1\right )}{2\,c}+\frac {b\,e\,\ln \left (c^2\,x^2+1\right )}{6\,c^3}-\frac {b\,e\,x^2}{6\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.71, size = 94, normalized size = 1.38 \[ \begin {cases} a d x + \frac {a e x^{3}}{3} + b d x \operatorname {atan}{\left (c x \right )} + \frac {b e x^{3} \operatorname {atan}{\left (c x \right )}}{3} - \frac {b d \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c} - \frac {b e x^{2}}{6 c} + \frac {b e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{6 c^{3}} & \text {for}\: c \neq 0 \\a \left (d x + \frac {e x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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