Optimal. Leaf size=67 \[ -\frac {b e^2 (c+d x)^2}{6 d}+\frac {e^2 (c+d x)^3 (a+b \text {ArcTan}(c+d x))}{3 d}+\frac {b e^2 \log \left (1+(c+d x)^2\right )}{6 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5151, 12, 4946,
272, 45} \begin {gather*} \frac {e^2 (c+d x)^3 (a+b \text {ArcTan}(c+d x))}{3 d}-\frac {b e^2 (c+d x)^2}{6 d}+\frac {b e^2 \log \left ((c+d x)^2+1\right )}{6 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 272
Rule 4946
Rule 5151
Rubi steps
\begin {align*} \int (c e+d e x)^2 \left (a+b \tan ^{-1}(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int e^2 x^2 \left (a+b \tan ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int x^2 \left (a+b \tan ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \tan ^{-1}(c+d x)\right )}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {x^3}{1+x^2} \, dx,x,c+d x\right )}{3 d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \tan ^{-1}(c+d x)\right )}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {x}{1+x} \, dx,x,(c+d x)^2\right )}{6 d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \tan ^{-1}(c+d x)\right )}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \left (1+\frac {1}{-1-x}\right ) \, dx,x,(c+d x)^2\right )}{6 d}\\ &=-\frac {b e^2 (c+d x)^2}{6 d}+\frac {e^2 (c+d x)^3 \left (a+b \tan ^{-1}(c+d x)\right )}{3 d}+\frac {b e^2 \log \left (1+(c+d x)^2\right )}{6 d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 54, normalized size = 0.81 \begin {gather*} \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {ArcTan}(c+d x))-\frac {1}{6} b \left ((c+d x)^2-\log \left (1+(c+d x)^2\right )\right )\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 67, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {e^{2} \left (d x +c \right )^{3} a}{3}+\frac {e^{2} b \left (d x +c \right )^{3} \arctan \left (d x +c \right )}{3}-\frac {e^{2} \left (d x +c \right )^{2} b}{6}+\frac {e^{2} b \ln \left (1+\left (d x +c \right )^{2}\right )}{6}}{d}\) | \(67\) |
default | \(\frac {\frac {e^{2} \left (d x +c \right )^{3} a}{3}+\frac {e^{2} b \left (d x +c \right )^{3} \arctan \left (d x +c \right )}{3}-\frac {e^{2} \left (d x +c \right )^{2} b}{6}+\frac {e^{2} b \ln \left (1+\left (d x +c \right )^{2}\right )}{6}}{d}\) | \(67\) |
risch | \(-\frac {i e^{2} \left (d x +c \right )^{3} b \ln \left (1+i \left (d x +c \right )\right )}{6 d}+\frac {i e^{2} d^{2} b \,x^{3} \ln \left (1-i \left (d x +c \right )\right )}{6}+\frac {i e^{2} d b c \,x^{2} \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {i e^{2} b \,c^{2} x \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {e^{2} d^{2} a \,x^{3}}{3}+\frac {i e^{2} b \,c^{3} \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{12 d}+\frac {e^{2} b \,c^{3} \arctan \left (d x +c \right )}{6 d}+e^{2} d a c \,x^{2}+e^{2} a \,c^{2} x -\frac {e^{2} d b \,x^{2}}{6}-\frac {e^{2} b c x}{3}+\frac {e^{2} b \ln \left (-d^{2} x^{2}-2 c d x -c^{2}-1\right )}{6 d}\) | \(227\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 232 vs.
\(2 (58) = 116\).
time = 0.47, size = 232, normalized size = 3.46 \begin {gather*} \frac {1}{3} \, a d^{2} x^{3} e^{2} + a c d x^{2} e^{2} + {\left (x^{2} \arctan \left (d x + c\right ) - d {\left (\frac {x}{d^{2}} + \frac {{\left (c^{2} - 1\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{3}} - \frac {c \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{3}}\right )}\right )} b c d e^{2} + \frac {1}{6} \, {\left (2 \, x^{3} \arctan \left (d x + c\right ) - d {\left (\frac {d x^{2} - 4 \, c x}{d^{3}} - \frac {2 \, {\left (c^{3} - 3 \, c\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{4}} + \frac {{\left (3 \, c^{2} - 1\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{4}}\right )}\right )} b d^{2} e^{2} + a c^{2} x e^{2} + \frac {{\left (2 \, {\left (d x + c\right )} \arctan \left (d x + c\right ) - \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} b c^{2} e^{2}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.76, size = 113, normalized size = 1.69 \begin {gather*} \frac {2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \arctan \left (d x + c\right ) e^{2} + b e^{2} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) + {\left (2 \, a d^{3} x^{3} + {\left (6 \, a c - b\right )} d^{2} x^{2} + 2 \, {\left (3 \, a c^{2} - b c\right )} d x\right )} e^{2}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.21, size = 182, normalized size = 2.72 \begin {gather*} \begin {cases} a c^{2} e^{2} x + a c d e^{2} x^{2} + \frac {a d^{2} e^{2} x^{3}}{3} + \frac {b c^{3} e^{2} \operatorname {atan}{\left (c + d x \right )}}{3 d} + b c^{2} e^{2} x \operatorname {atan}{\left (c + d x \right )} + b c d e^{2} x^{2} \operatorname {atan}{\left (c + d x \right )} - \frac {b c e^{2} x}{3} + \frac {b d^{2} e^{2} x^{3} \operatorname {atan}{\left (c + d x \right )}}{3} - \frac {b d e^{2} x^{2}}{6} + \frac {b e^{2} \log {\left (\frac {c}{d} + x - \frac {i}{d} \right )}}{3 d} - \frac {i b e^{2} \operatorname {atan}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\c^{2} e^{2} x \left (a + b \operatorname {atan}{\left (c \right )}\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 = 1.17, size = 144, normalized size = 2.15 \begin {gather*} \frac {a\,d^2\,e^2\,x^3}{3}-\frac {b\,c\,e^2\,x}{3}+\frac {b\,e^2\,\ln \left (c^2+2\,c\,d\,x+d^2\,x^2+1\right )}{6\,d}+a\,c^2\,e^2\,x-\frac {b\,d\,e^2\,x^2}{6}+b\,c^2\,e^2\,x\,\mathrm {atan}\left (c+d\,x\right )+a\,c\,d\,e^2\,x^2+\frac {b\,c^3\,e^2\,\mathrm {atan}\left (c+d\,x\right )}{3\,d}+\frac {b\,d^2\,e^2\,x^3\,\mathrm {atan}\left (c+d\,x\right )}{3}+b\,c\,d\,e^2\,x^2\,\mathrm {atan}\left (c+d\,x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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