Optimal. Leaf size=115 \[ -\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \tan ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \tan ^{-1}(c x)\right )+\frac{b \left (c^4 d^2-6 c^2 d e-3 e^2\right ) \log \left (c^2 x^2+1\right )}{6 c}-\frac{1}{3} b c d \log (x) \left (c^2 d-6 e\right )-\frac{b c d^2}{6 x^2} \]
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Rubi [A] time = 0.168281, antiderivative size = 115, normalized size of antiderivative = 1., 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 270
Rule 4976
Rule 12
Rule 1251
Rule 893
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*}
Mathematica [A] time = 0.115644, size = 119, normalized size = 1.03 \[ \frac{1}{6} \left (-\frac{2 a d^2}{x^3}-\frac{12 a d e}{x}+6 a e^2 x+\frac{b \left (c^4 d^2-6 c^2 d e-3 e^2\right ) \log \left (c^2 x^2+1\right )}{c}-2 b c d \log (x) \left (c^2 d-6 e\right )-\frac{2 b \tan ^{-1}(c x) \left (d^2+6 d e x^2-3 e^2 x^4\right )}{x^3}-\frac{b c d^2}{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 147, normalized size = 1.3 \begin{align*} ax{e}^{2}-2\,{\frac{aed}{x}}-{\frac{a{d}^{2}}{3\,{x}^{3}}}+b\arctan \left ( cx \right ) x{e}^{2}-2\,{\frac{b\arctan \left ( cx \right ) ed}{x}}-{\frac{b{d}^{2}\arctan \left ( cx \right ) }{3\,{x}^{3}}}+{\frac{{c}^{3}b\ln \left ({c}^{2}{x}^{2}+1 \right ){d}^{2}}{6}}-cb\ln \left ({c}^{2}{x}^{2}+1 \right ) ed-{\frac{b\ln \left ({c}^{2}{x}^{2}+1 \right ){e}^{2}}{2\,c}}-{\frac{{c}^{3}b{d}^{2}\ln \left ( cx \right ) }{3}}+2\,cb\ln \left ( cx \right ) de-{\frac{cb{d}^{2}}{6\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963767, size = 182, normalized size = 1.58 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38202, size = 311, normalized size = 2.7 \begin{align*} \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 \left (x\right ) - 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.52562, size = 180, normalized size = 1.57 \begin{align*} \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{\left (x \right )}}{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{\left (x \right )} - 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09679, size = 232, normalized size = 2.02 \begin{align*} \frac{b c^{4} d^{2} x^{3} \log \left (c^{2} x^{2} + 1\right ) - 2 \, b c^{4} d^{2} x^{3} \log \left (x\right ) - 6 \, b c^{2} d x^{3} e \log \left (c^{2} x^{2} + 1\right ) + 12 \, b c^{2} d x^{3} e \log \left (x\right ) + 6 \, b c x^{4} \arctan \left (c x\right ) e^{2} + 6 \, a c x^{4} e^{2} - 12 \, b c d x^{2} \arctan \left (c x\right ) e - b c^{2} d^{2} x - 12 \, a c d x^{2} e - 3 \, b x^{3} e^{2} \log \left (c^{2} x^{2} + 1\right ) - 2 \, b c d^{2} \arctan \left (c x\right ) - 2 \, a c d^{2}}{6 \, c x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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