Optimal. Leaf size=110 \[ -\frac{d \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac{b c \left (3 c^2 d-5 e\right )}{30 x^2}-\frac{1}{30} b c^3 \left (3 c^2 d-5 e\right ) \log \left (c^2 x^2+1\right )+\frac{1}{15} b c^3 \log (x) \left (3 c^2 d-5 e\right )-\frac{b c d}{20 x^4} \]
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Rubi [A] time = 0.128825, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {14, 4976, 12, 446, 77} \[ -\frac{d \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac{b c \left (3 c^2 d-5 e\right )}{30 x^2}-\frac{1}{30} b c^3 \left (3 c^2 d-5 e\right ) \log \left (c^2 x^2+1\right )+\frac{1}{15} b c^3 \log (x) \left (3 c^2 d-5 e\right )-\frac{b c d}{20 x^4} \]
Antiderivative was successfully verified.
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Rule 14
Rule 4976
Rule 12
Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right )}{x^6} \, dx &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac{-3 d-5 e x^2}{15 x^5 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{1}{15} (b c) \int \frac{-3 d-5 e x^2}{x^5 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{1}{30} (b c) \operatorname{Subst}\left (\int \frac{-3 d-5 e x}{x^3 \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{1}{30} (b c) \operatorname{Subst}\left (\int \left (-\frac{3 d}{x^3}+\frac{3 c^2 d-5 e}{x^2}+\frac{-3 c^4 d+5 c^2 e}{x}+\frac{3 c^6 d-5 c^4 e}{1+c^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac{b c d}{20 x^4}+\frac{b c \left (3 c^2 d-5 e\right )}{30 x^2}-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac{1}{15} b c^3 \left (3 c^2 d-5 e\right ) \log (x)-\frac{1}{30} b c^3 \left (3 c^2 d-5 e\right ) \log \left (1+c^2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0422192, size = 123, normalized size = 1.12 \[ -\frac{a d}{5 x^5}-\frac{a e}{3 x^3}+\frac{1}{10} b c d \left (\frac{c^2}{x^2}-c^4 \log \left (c^2 x^2+1\right )+2 c^4 \log (x)-\frac{1}{2 x^4}\right )+\frac{1}{6} b c e \left (c^2 \log \left (c^2 x^2+1\right )-2 c^2 \log (x)-\frac{1}{x^2}\right )-\frac{b d \tan ^{-1}(c x)}{5 x^5}-\frac{b e \tan ^{-1}(c x)}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 120, normalized size = 1.1 \begin{align*} -{\frac{ad}{5\,{x}^{5}}}-{\frac{ae}{3\,{x}^{3}}}-{\frac{\arctan \left ( cx \right ) bd}{5\,{x}^{5}}}-{\frac{b\arctan \left ( cx \right ) e}{3\,{x}^{3}}}-{\frac{{c}^{5}b\ln \left ({c}^{2}{x}^{2}+1 \right ) d}{10}}+{\frac{b{c}^{3}e\ln \left ({c}^{2}{x}^{2}+1 \right ) }{6}}+{\frac{{c}^{5}bd\ln \left ( cx \right ) }{5}}-{\frac{{c}^{3}b\ln \left ( cx \right ) e}{3}}+{\frac{b{c}^{3}d}{10\,{x}^{2}}}-{\frac{bce}{6\,{x}^{2}}}-{\frac{bcd}{20\,{x}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955741, size = 157, normalized size = 1.43 \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 + \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 e - \frac{a e}{3 \, x^{3}} - \frac{a d}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85409, size = 269, normalized size = 2.45 \begin{align*} -\frac{2 \,{\left (3 \, b c^{5} d - 5 \, b c^{3} e\right )} x^{5} \log \left (c^{2} x^{2} + 1\right ) - 4 \,{\left (3 \, b c^{5} d - 5 \, b c^{3} e\right )} x^{5} \log \left (x\right ) + 3 \, b c d x + 20 \, a e x^{2} - 2 \,{\left (3 \, b c^{3} d - 5 \, b c e\right )} x^{3} + 12 \, a d + 4 \,{\left (5 \, b e x^{2} + 3 \, b d\right )} \arctan \left (c x\right )}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.17923, size = 153, normalized size = 1.39 \begin{align*} \begin{cases} - \frac{a d}{5 x^{5}} - \frac{a e}{3 x^{3}} + \frac{b c^{5} d \log{\left (x \right )}}{5} - \frac{b c^{5} d \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{10} + \frac{b c^{3} d}{10 x^{2}} - \frac{b c^{3} e \log{\left (x \right )}}{3} + \frac{b c^{3} e \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{6} - \frac{b c d}{20 x^{4}} - \frac{b c e}{6 x^{2}} - \frac{b d \operatorname{atan}{\left (c x \right )}}{5 x^{5}} - \frac{b e \operatorname{atan}{\left (c x \right )}}{3 x^{3}} & \text{for}\: c \neq 0 \\a \left (- \frac{d}{5 x^{5}} - \frac{e}{3 x^{3}}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10962, size = 174, normalized size = 1.58 \begin{align*} -\frac{6 \, b c^{5} d x^{5} \log \left (c^{2} x^{2} + 1\right ) - 12 \, b c^{5} d x^{5} \log \left (x\right ) - 10 \, b c^{3} x^{5} e \log \left (c^{2} x^{2} + 1\right ) + 20 \, b c^{3} x^{5} e \log \left (x\right ) - 6 \, b c^{3} d x^{3} + 10 \, b c x^{3} e + 20 \, b x^{2} \arctan \left (c x\right ) e + 3 \, b c d x + 20 \, a x^{2} e + 12 \, b d \arctan \left (c x\right ) + 12 \, a d}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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