Optimal. Leaf size=230 \[ \frac {b e \left (e (3+m)-2 c^2 d (5+m)\right ) x^{2+m}}{c^3 (2+m) (3+m) (5+m)}-\frac {b e^2 x^{4+m}}{c (4+m) (5+m)}+\frac {d^2 x^{1+m} (a+b \text {ArcTan}(c x))}{1+m}+\frac {2 d e x^{3+m} (a+b \text {ArcTan}(c x))}{3+m}+\frac {e^2 x^{5+m} (a+b \text {ArcTan}(c x))}{5+m}-\frac {b \left (e^2 \left (3+4 m+m^2\right )-2 c^2 d e \left (5+6 m+m^2\right )+c^4 d^2 \left (15+8 m+m^2\right )\right ) x^{2+m} \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};-c^2 x^2\right )}{c^3 (1+m) (2+m) (3+m) (5+m)} \]
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Rubi [A]
time = 0.22, antiderivative size = 226, normalized size of antiderivative = 0.98, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {276, 5096,
1275, 371} \begin {gather*} \frac {d^2 x^{m+1} (a+b \text {ArcTan}(c x))}{m+1}+\frac {2 d e x^{m+3} (a+b \text {ArcTan}(c x))}{m+3}+\frac {e^2 x^{m+5} (a+b \text {ArcTan}(c x))}{m+5}-\frac {b e x^{m+2} \left (\frac {2 c^2 d}{m+3}-\frac {e}{m+5}\right )}{c^3 (m+2)}-\frac {b x^{m+2} \left (c^4 d^2 \left (m^2+8 m+15\right )-2 c^2 d e \left (m^2+6 m+5\right )+e^2 \left (m^2+4 m+3\right )\right ) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};-c^2 x^2\right )}{c^3 (m+1) (m+2) (m+3) (m+5)}-\frac {b e^2 x^{m+4}}{c (m+4) (m+5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 371
Rule 1275
Rule 5096
Rubi steps
\begin {align*} \int x^m \left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac {d^2 x^{1+m} \left (a+b \tan ^{-1}(c x)\right )}{1+m}+\frac {2 d e x^{3+m} \left (a+b \tan ^{-1}(c x)\right )}{3+m}+\frac {e^2 x^{5+m} \left (a+b \tan ^{-1}(c x)\right )}{5+m}-(b c) \int \frac {x^{1+m} \left (\frac {d^2}{1+m}+\frac {2 d e x^2}{3+m}+\frac {e^2 x^4}{5+m}\right )}{1+c^2 x^2} \, dx\\ &=\frac {d^2 x^{1+m} \left (a+b \tan ^{-1}(c x)\right )}{1+m}+\frac {2 d e x^{3+m} \left (a+b \tan ^{-1}(c x)\right )}{3+m}+\frac {e^2 x^{5+m} \left (a+b \tan ^{-1}(c x)\right )}{5+m}-(b c) \int \left (\frac {e \left (\frac {2 c^2 d}{3+m}-\frac {e}{5+m}\right ) x^{1+m}}{c^4}+\frac {e^2 x^{3+m}}{c^2 (5+m)}+\frac {\left (15 c^4 d^2-10 c^2 d e+3 e^2+8 c^4 d^2 m-12 c^2 d e m+4 e^2 m+c^4 d^2 m^2-2 c^2 d e m^2+e^2 m^2\right ) x^{1+m}}{c^4 (1+m) (3+m) (5+m) \left (1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {b e \left (\frac {2 c^2 d}{3+m}-\frac {e}{5+m}\right ) x^{2+m}}{c^3 (2+m)}-\frac {b e^2 x^{4+m}}{c (4+m) (5+m)}+\frac {d^2 x^{1+m} \left (a+b \tan ^{-1}(c x)\right )}{1+m}+\frac {2 d e x^{3+m} \left (a+b \tan ^{-1}(c x)\right )}{3+m}+\frac {e^2 x^{5+m} \left (a+b \tan ^{-1}(c x)\right )}{5+m}-\frac {\left (b \left (e^2 \left (3+4 m+m^2\right )-2 c^2 d e \left (5+6 m+m^2\right )+c^4 d^2 \left (15+8 m+m^2\right )\right )\right ) \int \frac {x^{1+m}}{1+c^2 x^2} \, dx}{c^3 (1+m) (3+m) (5+m)}\\ &=-\frac {b e \left (\frac {2 c^2 d}{3+m}-\frac {e}{5+m}\right ) x^{2+m}}{c^3 (2+m)}-\frac {b e^2 x^{4+m}}{c (4+m) (5+m)}+\frac {d^2 x^{1+m} \left (a+b \tan ^{-1}(c x)\right )}{1+m}+\frac {2 d e x^{3+m} \left (a+b \tan ^{-1}(c x)\right )}{3+m}+\frac {e^2 x^{5+m} \left (a+b \tan ^{-1}(c x)\right )}{5+m}-\frac {b \left (e^2 \left (3+4 m+m^2\right )-2 c^2 d e \left (5+6 m+m^2\right )+c^4 d^2 \left (15+8 m+m^2\right )\right ) x^{2+m} \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};-c^2 x^2\right )}{c^3 (1+m) (2+m) (3+m) (5+m)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in
optimal.
time = 18.93, size = 8408, normalized size = 36.56 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.91, size = 0, normalized size = 0.00 \[\int x^{m} \left (e \,x^{2}+d \right )^{2} \left (a +b \arctan \left (c x \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{m} \left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^m\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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