Optimal. Leaf size=103 \[ -\frac {b d e x}{c}-\frac {b e^2 x^2}{6 c}-\frac {b d \left (d^2-\frac {3 e^2}{c^2}\right ) \text {ArcTan}(c x)}{3 e}+\frac {(d+e x)^3 (a+b \text {ArcTan}(c x))}{3 e}-\frac {b \left (3 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{6 c^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4972, 716, 649,
209, 266} \begin {gather*} \frac {(d+e x)^3 (a+b \text {ArcTan}(c x))}{3 e}-\frac {b d \text {ArcTan}(c x) \left (d^2-\frac {3 e^2}{c^2}\right )}{3 e}-\frac {b \left (3 c^2 d^2-e^2\right ) \log \left (c^2 x^2+1\right )}{6 c^3}-\frac {b d e x}{c}-\frac {b e^2 x^2}{6 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)^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 e}-\frac {(b c) \int \frac {(d+e x)^3}{1+c^2 x^2} \, dx}{3 e}\\ &=\frac {(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 e}-\frac {(b c) \int \left (\frac {3 d e^2}{c^2}+\frac {e^3 x}{c^2}+\frac {c^2 d^3-3 d e^2+e \left (3 c^2 d^2-e^2\right ) x}{c^2 \left (1+c^2 x^2\right )}\right ) \, dx}{3 e}\\ &=-\frac {b d e x}{c}-\frac {b e^2 x^2}{6 c}+\frac {(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 e}-\frac {b \int \frac {c^2 d^3-3 d e^2+e \left (3 c^2 d^2-e^2\right ) x}{1+c^2 x^2} \, dx}{3 c e}\\ &=-\frac {b d e x}{c}-\frac {b e^2 x^2}{6 c}+\frac {(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 e}-\frac {1}{3} \left (b d \left (\frac {c d^2}{e}-\frac {3 e}{c}\right )\right ) \int \frac {1}{1+c^2 x^2} \, dx-\frac {\left (b \left (3 c^2 d^2-e^2\right )\right ) \int \frac {x}{1+c^2 x^2} \, dx}{3 c}\\ &=-\frac {b d e x}{c}-\frac {b e^2 x^2}{6 c}-\frac {b d \left (d^2-\frac {3 e^2}{c^2}\right ) \tan ^{-1}(c x)}{3 e}+\frac {(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 e}-\frac {b \left (3 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{6 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 163, normalized size = 1.58 \begin {gather*} \frac {(d+e x)^3 (a+b \text {ArcTan}(c x))-\frac {b \left (c^2 e^2 x (6 d+e x)+\left (-e^2 \left (3 \sqrt {-c^2} d+e\right )+c^2 d^2 \left (\sqrt {-c^2} d+3 e\right )\right ) \log \left (1-\sqrt {-c^2} x\right )-\left (c^2 d^2 \left (\sqrt {-c^2} d-3 e\right )+e^2 \left (-3 \sqrt {-c^2} d+e\right )\right ) \log \left (1+\sqrt {-c^2} x\right )\right )}{2 c^3}}{3 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 123, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {\frac {\left (c e x +c d \right )^{3} a}{3 c^{2} e}+b \arctan \left (c x \right ) d^{2} c x +b c e \arctan \left (c x \right ) d \,x^{2}+\frac {b c \,e^{2} \arctan \left (c x \right ) x^{3}}{3}-b e d x -\frac {b \,e^{2} x^{2}}{6}-\frac {b \ln \left (c^{2} x^{2}+1\right ) d^{2}}{2}+\frac {b \,e^{2} \ln \left (c^{2} x^{2}+1\right )}{6 c^{2}}+\frac {b e \arctan \left (c x \right ) d}{c}}{c}\) | \(123\) |
default | \(\frac {\frac {\left (c e x +c d \right )^{3} a}{3 c^{2} e}+b \arctan \left (c x \right ) d^{2} c x +b c e \arctan \left (c x \right ) d \,x^{2}+\frac {b c \,e^{2} \arctan \left (c x \right ) x^{3}}{3}-b e d x -\frac {b \,e^{2} x^{2}}{6}-\frac {b \ln \left (c^{2} x^{2}+1\right ) d^{2}}{2}+\frac {b \,e^{2} \ln \left (c^{2} x^{2}+1\right )}{6 c^{2}}+\frac {b e \arctan \left (c x \right ) d}{c}}{c}\) | \(123\) |
risch | \(-\frac {i \left (e x +d \right )^{3} b \ln \left (i c x +1\right )}{6 e}+\frac {i e b d \,x^{2} \ln \left (-i c x +1\right )}{2}+\frac {i b \,d^{2} x \ln \left (-i c x +1\right )}{2}-\frac {b \,d^{2} \ln \left (c^{2} x^{2}+1\right )}{2 c}+\frac {e b d \arctan \left (c x \right )}{c^{2}}+\frac {i b \,d^{3} \ln \left (c^{2} x^{2}+1\right )}{12 e}+\frac {e^{2} b \ln \left (c^{2} x^{2}+1\right )}{6 c^{3}}+x^{2} e d a +x \,d^{2} a -\frac {b d e x}{c}+\frac {i e^{2} b \,x^{3} \ln \left (-i c x +1\right )}{6}-\frac {b \,d^{3} \arctan \left (c x \right )}{6 e}+\frac {x^{3} e^{2} a}{3}-\frac {b \,e^{2} x^{2}}{6 c}\) | \(200\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 126, normalized size = 1.22 \begin {gather*} \frac {1}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x + {\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 d e + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d^{2}}{2 \, c} + \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^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.54, size = 132, normalized size = 1.28 \begin {gather*} \frac {6 \, a c^{3} d^{2} x + 2 \, {\left (b c^{3} x^{3} e^{2} + 3 \, b c^{3} d^{2} x + 3 \, {\left (b c^{3} d x^{2} + b c d\right )} e\right )} \arctan \left (c x\right ) + {\left (2 \, a c^{3} x^{3} - b c^{2} x^{2}\right )} e^{2} + 6 \, {\left (a c^{3} d x^{2} - b c^{2} d x\right )} e - {\left (3 \, b c^{2} d^{2} - b e^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{6 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 160, normalized size = 1.55 \begin {gather*} \begin {cases} a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} + b d^{2} x \operatorname {atan}{\left (c x \right )} + b d e x^{2} \operatorname {atan}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {atan}{\left (c x \right )}}{3} - \frac {b d^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c} - \frac {b d e x}{c} - \frac {b e^{2} x^{2}}{6 c} + \frac {b d e \operatorname {atan}{\left (c x \right )}}{c^{2}} + \frac {b e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{6 c^{3}} & \text {for}\: c \neq 0 \\a \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\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 = 0.35, size = 127, normalized size = 1.23 \begin {gather*} \frac {a\,e^2\,x^3}{3}+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 )}{6\,c^3}-\frac {b\,e^2\,x^2}{6\,c}+a\,d\,e\,x^2+b\,d^2\,x\,\mathrm {atan}\left (c\,x\right )+\frac {b\,e^2\,x^3\,\mathrm {atan}\left (c\,x\right )}{3}-\frac {b\,d\,e\,x}{c}+\frac {b\,d\,e\,\mathrm {atan}\left (c\,x\right )}{c^2}+b\,d\,e\,x^2\,\mathrm {atan}\left (c\,x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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