Optimal. Leaf size=177 \[ -\frac{d^2 e \left (a+b \tan ^{-1}(c x)\right )}{x^3}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{3 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{x}+e^3 x \left (a+b \tan ^{-1}(c x)\right )-\frac{b \left (-5 c^4 d^2 e+c^6 d^3+15 c^2 d e^2+5 e^3\right ) \log \left (c^2 x^2+1\right )}{10 c}+\frac{1}{5} b c d \log (x) \left (c^4 d^2-5 c^2 d e+15 e^2\right )+\frac{b c d^2 \left (c^2 d-5 e\right )}{10 x^2}-\frac{b c d^3}{20 x^4} \]
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Rubi [A] time = 0.285452, antiderivative size = 177, 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, 1799, 1620} \[ -\frac{d^2 e \left (a+b \tan ^{-1}(c x)\right )}{x^3}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{3 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{x}+e^3 x \left (a+b \tan ^{-1}(c x)\right )-\frac{b \left (-5 c^4 d^2 e+c^6 d^3+15 c^2 d e^2+5 e^3\right ) \log \left (c^2 x^2+1\right )}{10 c}+\frac{1}{5} b c d \log (x) \left (c^4 d^2-5 c^2 d e+15 e^2\right )+\frac{b c d^2 \left (c^2 d-5 e\right )}{10 x^2}-\frac{b c d^3}{20 x^4} \]
Antiderivative was successfully verified.
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Rule 270
Rule 4976
Rule 12
Rule 1799
Rule 1620
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right )}{x^6} \, dx &=-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{d^2 e \left (a+b \tan ^{-1}(c x)\right )}{x^3}-\frac{3 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{x}+e^3 x \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac{-d^3-5 d^2 e x^2-15 d e^2 x^4+5 e^3 x^6}{5 x^5 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{d^2 e \left (a+b \tan ^{-1}(c x)\right )}{x^3}-\frac{3 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{x}+e^3 x \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{5} (b c) \int \frac{-d^3-5 d^2 e x^2-15 d e^2 x^4+5 e^3 x^6}{x^5 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{d^2 e \left (a+b \tan ^{-1}(c x)\right )}{x^3}-\frac{3 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{x}+e^3 x \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{10} (b c) \operatorname{Subst}\left (\int \frac{-d^3-5 d^2 e x-15 d e^2 x^2+5 e^3 x^3}{x^3 \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{d^2 e \left (a+b \tan ^{-1}(c x)\right )}{x^3}-\frac{3 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{x}+e^3 x \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{10} (b c) \operatorname{Subst}\left (\int \left (-\frac{d^3}{x^3}+\frac{d^2 \left (c^2 d-5 e\right )}{x^2}-\frac{d \left (c^4 d^2-5 c^2 d e+15 e^2\right )}{x}+\frac{c^6 d^3-5 c^4 d^2 e+15 c^2 d e^2+5 e^3}{1+c^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac{b c d^3}{20 x^4}+\frac{b c d^2 \left (c^2 d-5 e\right )}{10 x^2}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{d^2 e \left (a+b \tan ^{-1}(c x)\right )}{x^3}-\frac{3 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{x}+e^3 x \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} b c d \left (c^4 d^2-5 c^2 d e+15 e^2\right ) \log (x)-\frac{b \left (c^6 d^3-5 c^4 d^2 e+15 c^2 d e^2+5 e^3\right ) \log \left (1+c^2 x^2\right )}{10 c}\\ \end{align*}
Mathematica [A] time = 0.172457, size = 184, normalized size = 1.04 \[ \frac{1}{20} \left (-\frac{20 a d^2 e}{x^3}-\frac{4 a d^3}{x^5}-\frac{60 a d e^2}{x}+20 a e^3 x-\frac{2 b \left (-5 c^4 d^2 e+c^6 d^3+15 c^2 d e^2+5 e^3\right ) \log \left (c^2 x^2+1\right )}{c}+4 b c d \log (x) \left (c^4 d^2-5 c^2 d e+15 e^2\right )+\frac{2 b c d^2 \left (c^2 d-5 e\right )}{x^2}-\frac{4 b \tan ^{-1}(c x) \left (5 d^2 e x^2+d^3+15 d e^2 x^4-5 e^3 x^6\right )}{x^5}-\frac{b c d^3}{x^4}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 236, normalized size = 1.3 \begin{align*} a{e}^{3}x-3\,{\frac{ad{e}^{2}}{x}}-{\frac{a{d}^{3}}{5\,{x}^{5}}}-{\frac{a{d}^{2}e}{{x}^{3}}}+b\arctan \left ( cx \right ){e}^{3}x-3\,{\frac{\arctan \left ( cx \right ) bd{e}^{2}}{x}}-{\frac{b{d}^{3}\arctan \left ( cx \right ) }{5\,{x}^{5}}}-{\frac{b{d}^{2}\arctan \left ( cx \right ) e}{{x}^{3}}}-{\frac{{c}^{5}b\ln \left ({c}^{2}{x}^{2}+1 \right ){d}^{3}}{10}}+{\frac{{c}^{3}b\ln \left ({c}^{2}{x}^{2}+1 \right ){d}^{2}e}{2}}-{\frac{3\,cb\ln \left ({c}^{2}{x}^{2}+1 \right ) d{e}^{2}}{2}}-{\frac{b\ln \left ({c}^{2}{x}^{2}+1 \right ){e}^{3}}{2\,c}}+{\frac{{c}^{3}b{d}^{3}}{10\,{x}^{2}}}-{\frac{cb{d}^{2}e}{2\,{x}^{2}}}-{\frac{cb{d}^{3}}{20\,{x}^{4}}}+{\frac{{c}^{5}b{d}^{3}\ln \left ( cx \right ) }{5}}-{c}^{3}b\ln \left ( cx \right ){d}^{2}e+3\,cb\ln \left ( cx \right ) d{e}^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.998314, size = 281, normalized size = 1.59 \begin{align*} -\frac{1}{20} \,{\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac{2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac{4 \, \arctan \left (c x\right )}{x^{5}}\right )} b d^{3} + \frac{1}{2} \,{\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} e - \frac{3}{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^{2} + a e^{3} x + \frac{{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b e^{3}}{2 \, c} - \frac{3 \, a d e^{2}}{x} - \frac{a d^{2} e}{x^{3}} - \frac{a d^{3}}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78263, size = 473, normalized size = 2.67 \begin{align*} \frac{20 \, a c e^{3} x^{6} - 60 \, a c d e^{2} x^{4} - b c^{2} d^{3} x - 20 \, a c d^{2} e x^{2} - 2 \,{\left (b c^{6} d^{3} - 5 \, b c^{4} d^{2} e + 15 \, b c^{2} d e^{2} + 5 \, b e^{3}\right )} x^{5} \log \left (c^{2} x^{2} + 1\right ) + 4 \,{\left (b c^{6} d^{3} - 5 \, b c^{4} d^{2} e + 15 \, b c^{2} d e^{2}\right )} x^{5} \log \left (x\right ) - 4 \, a c d^{3} + 2 \,{\left (b c^{4} d^{3} - 5 \, b c^{2} d^{2} e\right )} x^{3} + 4 \,{\left (5 \, b c e^{3} x^{6} - 15 \, b c d e^{2} x^{4} - 5 \, b c d^{2} e x^{2} - b c d^{3}\right )} \arctan \left (c x\right )}{20 \, c x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.91528, size = 289, normalized size = 1.63 \begin{align*} \begin{cases} - \frac{a d^{3}}{5 x^{5}} - \frac{a d^{2} e}{x^{3}} - \frac{3 a d e^{2}}{x} + a e^{3} x + \frac{b c^{5} d^{3} \log{\left (x \right )}}{5} - \frac{b c^{5} d^{3} \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{10} + \frac{b c^{3} d^{3}}{10 x^{2}} - b c^{3} d^{2} e \log{\left (x \right )} + \frac{b c^{3} d^{2} e \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{2} - \frac{b c d^{3}}{20 x^{4}} - \frac{b c d^{2} e}{2 x^{2}} + 3 b c d e^{2} \log{\left (x \right )} - \frac{3 b c d e^{2} \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{2} - \frac{b d^{3} \operatorname{atan}{\left (c x \right )}}{5 x^{5}} - \frac{b d^{2} e \operatorname{atan}{\left (c x \right )}}{x^{3}} - \frac{3 b d e^{2} \operatorname{atan}{\left (c x \right )}}{x} + b e^{3} x \operatorname{atan}{\left (c x \right )} - \frac{b e^{3} \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{2 c} & \text{for}\: c \neq 0 \\a \left (- \frac{d^{3}}{5 x^{5}} - \frac{d^{2} e}{x^{3}} - \frac{3 d e^{2}}{x} + e^{3} 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 = 1.10351, size = 359, normalized size = 2.03 \begin{align*} -\frac{2 \, b c^{6} d^{3} x^{5} \log \left (c^{2} x^{2} + 1\right ) - 4 \, b c^{6} d^{3} x^{5} \log \left (x\right ) - 10 \, b c^{4} d^{2} x^{5} e \log \left (c^{2} x^{2} + 1\right ) + 20 \, b c^{4} d^{2} x^{5} e \log \left (x\right ) - 2 \, b c^{4} d^{3} x^{3} + 30 \, b c^{2} d x^{5} e^{2} \log \left (c^{2} x^{2} + 1\right ) - 60 \, b c^{2} d x^{5} e^{2} \log \left (x\right ) - 20 \, b c x^{6} \arctan \left (c x\right ) e^{3} - 20 \, a c x^{6} e^{3} + 60 \, b c d x^{4} \arctan \left (c x\right ) e^{2} + 10 \, b c^{2} d^{2} x^{3} e + 60 \, a c d x^{4} e^{2} + 20 \, b c d^{2} x^{2} \arctan \left (c x\right ) e + 10 \, b x^{5} e^{3} \log \left (c^{2} x^{2} + 1\right ) + b c^{2} d^{3} x + 20 \, a c d^{2} x^{2} e + 4 \, b c d^{3} \arctan \left (c x\right ) + 4 \, a c d^{3}}{20 \, c x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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