Optimal. Leaf size=76 \[ -\frac {b e x}{2 c}-\frac {b \left (d^2-\frac {e^2}{c^2}\right ) \text {ArcTan}(c x)}{2 e}+\frac {(d+e x)^2 (a+b \text {ArcTan}(c x))}{2 e}-\frac {b d \log \left (1+c^2 x^2\right )}{2 c} \]
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Rubi [A]
time = 0.04, 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 = {4972, 716, 649,
209, 266} \begin {gather*} \frac {(d+e x)^2 (a+b \text {ArcTan}(c x))}{2 e}-\frac {b \text {ArcTan}(c x) \left (d^2-\frac {e^2}{c^2}\right )}{2 e}-\frac {b d \log \left (c^2 x^2+1\right )}{2 c}-\frac {b e x}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 649
Rule 716
Rule 4972
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 \begin {gather*} a d x-\frac {b e x}{2 c}+\frac {1}{2} a e x^2+\frac {b e \text {ArcTan}(c x)}{2 c^2}+b d x \text {ArcTan}(c x)+\frac {1}{2} b e x^2 \text {ArcTan}(c x)-\frac {b d \log \left (1+c^2 x^2\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 78, normalized size = 1.03
method | result | size |
derivativedivides | \(\frac {\frac {a \left (d \,c^{2} x +\frac {1}{2} e \,c^{2} x^{2}\right )}{c}+b \arctan \left (c x \right ) d c x +\frac {b c \arctan \left (c x \right ) e \,x^{2}}{2}-\frac {b e x}{2}-\frac {b d \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {b e \arctan \left (c x \right )}{2 c}}{c}\) | \(78\) |
default | \(\frac {\frac {a \left (d \,c^{2} x +\frac {1}{2} e \,c^{2} x^{2}\right )}{c}+b \arctan \left (c x \right ) d c x +\frac {b c \arctan \left (c x \right ) e \,x^{2}}{2}-\frac {b e x}{2}-\frac {b d \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {b e \arctan \left (c x \right )}{2 c}}{c}\) | \(78\) |
risch | \(-\frac {i b \left (e \,x^{2}+2 d x \right ) \ln \left (i c x +1\right )}{4}+\frac {i b e \,x^{2} \ln \left (-i c x +1\right )}{4}+\frac {i b d x \ln \left (-i c x +1\right )}{2}+\frac {a e \,x^{2}}{2}+a d x -\frac {b d \ln \left (c^{2} x^{2}+1\right )}{2 c}-\frac {b e x}{2 c}+\frac {\arctan \left (c x \right ) b e}{2 c^{2}}\) | \(101\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 73, normalized size = 0.96 \begin {gather*} \frac {1}{2} \, a x^{2} e + a d x + \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 + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.19, size = 73, normalized size = 0.96 \begin {gather*} \frac {2 \, a c^{2} d x - b c d \log \left (c^{2} x^{2} + 1\right ) + {\left (2 \, b c^{2} d x + {\left (b c^{2} x^{2} + b\right )} e\right )} \arctan \left (c x\right ) + {\left (a c^{2} x^{2} - b c x\right )} e}{2 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 87, normalized size = 1.14 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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