Optimal. Leaf size=109 \[ -\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {b \left (3 c^4 d^2+6 c^2 d e-e^2\right ) \log \left (c^2 x^2+1\right )}{6 c^3}+b c d^2 \log (x)-\frac {b e^2 x^2}{6 c} \]
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Rubi [A] time = 0.16, antiderivative size = 109, normalized size of antiderivative = 1.00, 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 = {270, 4976, 1251, 893} \[ -\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {b \left (3 c^4 d^2+6 c^2 d e-e^2\right ) \log \left (c^2 x^2+1\right )}{6 c^3}+b c d^2 \log (x)-\frac {b e^2 x^2}{6 c} \]
Antiderivative was successfully verified.
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Rule 270
Rule 893
Rule 1251
Rule 4976
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac {-d^2+2 d e x^2+\frac {e^2 x^4}{3}}{x \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {-d^2+2 d e x+\frac {e^2 x^2}{3}}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{2} (b c) \operatorname {Subst}\left (\int \left (\frac {e^2}{3 c^2}-\frac {d^2}{x}+\frac {3 c^4 d^2+6 c^2 d e-e^2}{3 c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {b e^2 x^2}{6 c}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+b c d^2 \log (x)-\frac {b \left (3 c^4 d^2+6 c^2 d e-e^2\right ) \log \left (1+c^2 x^2\right )}{6 c^3}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 114, normalized size = 1.05 \[ \frac {1}{6} \left (-\frac {6 a d^2}{x}+12 a d e x+2 a e^2 x^3+\frac {b \left (-3 c^4 d^2-6 c^2 d e+e^2\right ) \log \left (c^2 x^2+1\right )}{c^3}+\frac {2 b \tan ^{-1}(c x) \left (-3 d^2+6 d e x^2+e^2 x^4\right )}{x}+6 b c d^2 \log (x)-\frac {b e^2 x^2}{c}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 140, normalized size = 1.28 \[ \frac {2 \, a c^{3} e^{2} x^{4} + 6 \, b c^{4} d^{2} x \log \relax (x) + 12 \, a c^{3} d e x^{2} - b c^{2} e^{2} x^{3} - 6 \, a c^{3} d^{2} - {\left (3 \, b c^{4} d^{2} + 6 \, b c^{2} d e - b e^{2}\right )} x \log \left (c^{2} x^{2} + 1\right ) + 2 \, {\left (b c^{3} e^{2} x^{4} + 6 \, b c^{3} d e x^{2} - 3 \, b c^{3} d^{2}\right )} \arctan \left (c x\right )}{6 \, c^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 138, normalized size = 1.27 \[ \frac {a \,e^{2} x^{3}}{3}+2 a e d x -\frac {a \,d^{2}}{x}+\frac {b \arctan \left (c x \right ) x^{3} e^{2}}{3}+2 b \arctan \left (c x \right ) e d x -\frac {b \arctan \left (c x \right ) d^{2}}{x}-\frac {b \,e^{2} x^{2}}{6 c}+c b \,d^{2} \ln \left (c x \right )-\frac {c b \ln \left (c^{2} x^{2}+1\right ) d^{2}}{2}-\frac {b \ln \left (c^{2} x^{2}+1\right ) e d}{c}+\frac {b \ln \left (c^{2} x^{2}+1\right ) e^{2}}{6 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 130, normalized size = 1.19 \[ \frac {1}{3} \, a e^{2} x^{3} - \frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b d^{2} + \frac {1}{6} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b e^{2} + 2 \, a d e x + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d e}{c} - \frac {a d^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.70, size = 135, normalized size = 1.24 \[ \frac {a\,e^2\,x^3}{3}-\frac {a\,d^2}{x}+2\,a\,d\,e\,x+\frac {b\,e^2\,\ln \left (c^2\,x^2+1\right )}{6\,c^3}-\frac {b\,e^2\,x^2}{6\,c}-\frac {b\,c\,d^2\,\ln \left (c^2\,x^2+1\right )}{2}+b\,c\,d^2\,\ln \relax (x)-\frac {b\,d^2\,\mathrm {atan}\left (c\,x\right )}{x}+\frac {b\,e^2\,x^3\,\mathrm {atan}\left (c\,x\right )}{3}-\frac {b\,d\,e\,\ln \left (c^2\,x^2+1\right )}{c}+2\,b\,d\,e\,x\,\mathrm {atan}\left (c\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.83, size = 165, normalized size = 1.51 \[ \begin {cases} - \frac {a d^{2}}{x} + 2 a d e x + \frac {a e^{2} x^{3}}{3} + b c d^{2} \log {\relax (x )} - \frac {b c d^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2} - \frac {b d^{2} \operatorname {atan}{\left (c x \right )}}{x} + 2 b d e x \operatorname {atan}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {atan}{\left (c x \right )}}{3} - \frac {b d e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{c} - \frac {b e^{2} x^{2}}{6 c} + \frac {b e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{6 c^{3}} & \text {for}\: c \neq 0 \\a \left (- \frac {d^{2}}{x} + 2 d e x + \frac {e^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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