Optimal. Leaf size=76 \[ \frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac {b \left (d^2-\frac {e^2}{c^2}\right ) \tan ^{-1}(c x)}{2 e}-\frac {b d \log \left (c^2 x^2+1\right )}{2 c}-\frac {b e x}{2 c} \]
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Rubi [A] time = 0.06, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4862, 702, 635, 203, 260} \[ \frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac {b \left (d^2-\frac {e^2}{c^2}\right ) \tan ^{-1}(c x)}{2 e}-\frac {b d \log \left (c^2 x^2+1\right )}{2 c}-\frac {b e x}{2 c} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 702
Rule 4862
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac {(b c) \int \frac {(d+e x)^2}{1+c^2 x^2} \, dx}{2 e}\\ &=\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac {(b c) \int \left (\frac {e^2}{c^2}+\frac {c^2 d^2-e^2+2 c^2 d e x}{c^2 \left (1+c^2 x^2\right )}\right ) \, dx}{2 e}\\ &=-\frac {b e x}{2 c}+\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac {b \int \frac {c^2 d^2-e^2+2 c^2 d e x}{1+c^2 x^2} \, dx}{2 c e}\\ &=-\frac {b e x}{2 c}+\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-(b c d) \int \frac {x}{1+c^2 x^2} \, dx-\frac {(b (c d-e) (c d+e)) \int \frac {1}{1+c^2 x^2} \, dx}{2 c e}\\ &=-\frac {b e x}{2 c}-\frac {b \left (d^2-\frac {e^2}{c^2}\right ) \tan ^{-1}(c x)}{2 e}+\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac {b d \log \left (1+c^2 x^2\right )}{2 c}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 77, normalized size = 1.01 \[ a d x+\frac {1}{2} a e x^2-\frac {b d \log \left (c^2 x^2+1\right )}{2 c}+\frac {b e \tan ^{-1}(c x)}{2 c^2}+b d x \tan ^{-1}(c x)+\frac {1}{2} b e x^2 \tan ^{-1}(c x)-\frac {b e x}{2 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.42, size = 71, normalized size = 0.93 \[ \frac {a c^{2} e x^{2} - b c d \log \left (c^{2} x^{2} + 1\right ) + {\left (2 \, a c^{2} d - b c e\right )} x + {\left (b c^{2} e x^{2} + 2 \, b c^{2} d x + b e\right )} \arctan \left (c x\right )}{2 \, c^{2}} \]
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 = 0.03, size = 68, normalized size = 0.89 \[ \frac {a \,x^{2} e}{2}+a d x +\frac {b \arctan \left (c x \right ) x^{2} e}{2}+b \arctan \left (c x \right ) x d -\frac {b e x}{2 c}-\frac {b d \ln \left (c^{2} x^{2}+1\right )}{2 c}+\frac {b e \arctan \left (c x \right )}{2 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 71, normalized size = 0.93 \[ \frac {1}{2} \, a e x^{2} + \frac {1}{2} \, {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} b e + a d x + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 67, normalized size = 0.88 \[ a\,d\,x+\frac {a\,e\,x^2}{2}+b\,d\,x\,\mathrm {atan}\left (c\,x\right )-\frac {b\,e\,x}{2\,c}+\frac {b\,e\,\mathrm {atan}\left (c\,x\right )}{2\,c^2}+\frac {b\,e\,x^2\,\mathrm {atan}\left (c\,x\right )}{2}-\frac {b\,d\,\ln \left (c^2\,x^2+1\right )}{2\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 87, normalized size = 1.14 \[ \begin {cases} a d x + \frac {a e x^{2}}{2} + b d x \operatorname {atan}{\left (c x \right )} + \frac {b e x^{2} \operatorname {atan}{\left (c x \right )}}{2} - \frac {b d \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c} - \frac {b e x}{2 c} + \frac {b e \operatorname {atan}{\left (c x \right )}}{2 c^{2}} & \text {for}\: c \neq 0 \\a \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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