Optimal. Leaf size=115 \[ -\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \tan ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{3} b c d \log (x) \left (c^2 d-6 e\right )+\frac {b \left (c^4 d^2-6 c^2 d e-3 e^2\right ) \log \left (c^2 x^2+1\right )}{6 c}-\frac {b c d^2}{6 x^2} \]
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Rubi [A] time = 0.17, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {270, 4976, 12, 1251, 893} \[ -\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \tan ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \tan ^{-1}(c x)\right )+\frac {b \left (c^4 d^2-6 c^2 d e-3 e^2\right ) \log \left (c^2 x^2+1\right )}{6 c}-\frac {1}{3} b c d \log (x) \left (c^2 d-6 e\right )-\frac {b c d^2}{6 x^2} \]
Antiderivative was successfully verified.
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Rule 12
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^4} \, dx &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \tan ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac {-d^2-6 d e x^2+3 e^2 x^4}{3 x^3 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \tan ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{3} (b c) \int \frac {-d^2-6 d e x^2+3 e^2 x^4}{x^3 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \tan ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{6} (b c) \operatorname {Subst}\left (\int \frac {-d^2-6 d e x+3 e^2 x^2}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \tan ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{6} (b c) \operatorname {Subst}\left (\int \left (-\frac {d^2}{x^2}+\frac {d \left (c^2 d-6 e\right )}{x}+\frac {-c^4 d^2+6 c^2 d e+3 e^2}{1+c^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac {b c d^2}{6 x^2}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \tan ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{3} b c d \left (c^2 d-6 e\right ) \log (x)+\frac {b \left (c^4 d^2-6 c^2 d e-3 e^2\right ) \log \left (1+c^2 x^2\right )}{6 c}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 119, normalized size = 1.03 \[ \frac {1}{6} \left (-\frac {2 a d^2}{x^3}-\frac {12 a d e}{x}+6 a e^2 x-2 b c d \log (x) \left (c^2 d-6 e\right )+\frac {b \left (c^4 d^2-6 c^2 d e-3 e^2\right ) \log \left (c^2 x^2+1\right )}{c}-\frac {2 b \tan ^{-1}(c x) \left (d^2+6 d e x^2-3 e^2 x^4\right )}{x^3}-\frac {b c d^2}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 139, normalized size = 1.21 \[ \frac {6 \, a c e^{2} x^{4} - b c^{2} d^{2} x - 12 \, a c d e x^{2} + {\left (b c^{4} d^{2} - 6 \, b c^{2} d e - 3 \, b e^{2}\right )} x^{3} \log \left (c^{2} x^{2} + 1\right ) - 2 \, {\left (b c^{4} d^{2} - 6 \, b c^{2} d e\right )} x^{3} \log \relax (x) - 2 \, a c d^{2} + 2 \, {\left (3 \, b c e^{2} x^{4} - 6 \, b c d e x^{2} - b c d^{2}\right )} \arctan \left (c x\right )}{6 \, c x^{3}} \]
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 = 147, normalized size = 1.28 \[ a x \,e^{2}-\frac {a \,d^{2}}{3 x^{3}}-\frac {2 a e d}{x}+b \arctan \left (c x \right ) x \,e^{2}-\frac {b \arctan \left (c x \right ) d^{2}}{3 x^{3}}-\frac {2 b \arctan \left (c x \right ) e d}{x}-\frac {c^{3} b \,d^{2} \ln \left (c x \right )}{3}+2 c b \ln \left (c x \right ) d e -\frac {b c \,d^{2}}{6 x^{2}}+\frac {c^{3} b \ln \left (c^{2} x^{2}+1\right ) d^{2}}{6}-c b \ln \left (c^{2} x^{2}+1\right ) e d -\frac {b \ln \left (c^{2} x^{2}+1\right ) e^{2}}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 135, normalized size = 1.17 \[ \frac {1}{6} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b d^{2} - {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b d e + a e^{2} x + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b e^{2}}{2 \, c} - \frac {2 \, a d e}{x} - \frac {a d^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 142, normalized size = 1.23 \[ a\,e^2\,x-\frac {a\,d^2}{3\,x^3}+\frac {b\,c^3\,d^2\,\ln \left (c^2\,x^2+1\right )}{6}-\frac {b\,e^2\,\ln \left (c^2\,x^2+1\right )}{2\,c}-\frac {b\,c^3\,d^2\,\ln \relax (x)}{3}-\frac {2\,a\,d\,e}{x}+b\,e^2\,x\,\mathrm {atan}\left (c\,x\right )-\frac {b\,c\,d^2}{6\,x^2}-\frac {b\,d^2\,\mathrm {atan}\left (c\,x\right )}{3\,x^3}-b\,c\,d\,e\,\ln \left (c^2\,x^2+1\right )+2\,b\,c\,d\,e\,\ln \relax (x)-\frac {2\,b\,d\,e\,\mathrm {atan}\left (c\,x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.87, size = 180, normalized size = 1.57 \[ \begin {cases} - \frac {a d^{2}}{3 x^{3}} - \frac {2 a d e}{x} + a e^{2} x - \frac {b c^{3} d^{2} \log {\relax (x )}}{3} + \frac {b c^{3} d^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{6} - \frac {b c d^{2}}{6 x^{2}} + 2 b c d e \log {\relax (x )} - b c d e \log {\left (x^{2} + \frac {1}{c^{2}} \right )} - \frac {b d^{2} \operatorname {atan}{\left (c x \right )}}{3 x^{3}} - \frac {2 b d e \operatorname {atan}{\left (c x \right )}}{x} + b e^{2} x \operatorname {atan}{\left (c x \right )} - \frac {b e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c} & \text {for}\: c \neq 0 \\a \left (- \frac {d^{2}}{3 x^{3}} - \frac {2 d e}{x} + e^{2} x\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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