Optimal. Leaf size=82 \[ -\frac{d \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac{b c \left (c^2 d-2 e\right )}{4 x}+\frac{1}{4} b c^2 \left (c^2 d-2 e\right ) \tan ^{-1}(c x)-\frac{b c d}{12 x^3} \]
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Rubi [A] time = 0.0904979, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {14, 4976, 12, 453, 325, 203} \[ -\frac{d \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac{b c \left (c^2 d-2 e\right )}{4 x}+\frac{1}{4} b c^2 \left (c^2 d-2 e\right ) \tan ^{-1}(c x)-\frac{b c d}{12 x^3} \]
Antiderivative was successfully verified.
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Rule 14
Rule 4976
Rule 12
Rule 453
Rule 325
Rule 203
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-(b c) \int \frac{-d-2 e x^2}{4 x^4 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{1}{4} (b c) \int \frac{-d-2 e x^2}{x^4 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{b c d}{12 x^3}-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{1}{4} \left (b c \left (c^2 d-2 e\right )\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{b c d}{12 x^3}+\frac{b c \left (c^2 d-2 e\right )}{4 x}-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac{1}{4} \left (b c^3 \left (c^2 d-2 e\right )\right ) \int \frac{1}{1+c^2 x^2} \, dx\\ &=-\frac{b c d}{12 x^3}+\frac{b c \left (c^2 d-2 e\right )}{4 x}+\frac{1}{4} b c^2 \left (c^2 d-2 e\right ) \tan ^{-1}(c x)-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}\\ \end{align*}
Mathematica [C] time = 0.0052514, size = 97, normalized size = 1.18 \[ -\frac{b c d \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},-c^2 x^2\right )}{12 x^3}-\frac{b c e \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-c^2 x^2\right )}{2 x}-\frac{a d}{4 x^4}-\frac{a e}{2 x^2}-\frac{b d \tan ^{-1}(c x)}{4 x^4}-\frac{b e \tan ^{-1}(c x)}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 86, normalized size = 1.1 \begin{align*} -{\frac{ae}{2\,{x}^{2}}}-{\frac{ad}{4\,{x}^{4}}}-{\frac{b\arctan \left ( cx \right ) e}{2\,{x}^{2}}}-{\frac{\arctan \left ( cx \right ) bd}{4\,{x}^{4}}}+{\frac{{c}^{4}b\arctan \left ( cx \right ) d}{4}}-{\frac{b{c}^{2}e\arctan \left ( cx \right ) }{2}}+{\frac{b{c}^{3}d}{4\,x}}-{\frac{bce}{2\,x}}-{\frac{bcd}{12\,{x}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43652, size = 108, normalized size = 1.32 \begin{align*} \frac{1}{12} \,{\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac{3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac{3 \, \arctan \left (c x\right )}{x^{4}}\right )} b d - \frac{1}{2} \,{\left ({\left (c \arctan \left (c x\right ) + \frac{1}{x}\right )} c + \frac{\arctan \left (c x\right )}{x^{2}}\right )} b e - \frac{a e}{2 \, x^{2}} - \frac{a d}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68077, size = 177, normalized size = 2.16 \begin{align*} -\frac{b c d x + 6 \, a e x^{2} - 3 \,{\left (b c^{3} d - 2 \, b c e\right )} x^{3} + 3 \, a d - 3 \,{\left ({\left (b c^{4} d - 2 \, b c^{2} e\right )} x^{4} - 2 \, b e x^{2} - b d\right )} \arctan \left (c x\right )}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.49037, size = 99, normalized size = 1.21 \begin{align*} - \frac{a d}{4 x^{4}} - \frac{a e}{2 x^{2}} + \frac{b c^{4} d \operatorname{atan}{\left (c x \right )}}{4} + \frac{b c^{3} d}{4 x} - \frac{b c^{2} e \operatorname{atan}{\left (c x \right )}}{2} - \frac{b c d}{12 x^{3}} - \frac{b c e}{2 x} - \frac{b d \operatorname{atan}{\left (c x \right )}}{4 x^{4}} - \frac{b e \operatorname{atan}{\left (c x \right )}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16072, size = 143, normalized size = 1.74 \begin{align*} -\frac{3 \, \pi b c^{4} d x^{4} \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 3 \, b c^{4} d x^{4} \arctan \left (c x\right ) + 6 \, b c^{2} x^{4} \arctan \left (c x\right ) e - 3 \, b c^{3} d x^{3} + 6 \, b c x^{3} e + 6 \, b x^{2} \arctan \left (c x\right ) e + b c d x + 6 \, a x^{2} e + 3 \, b d \arctan \left (c x\right ) + 3 \, a d}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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