Optimal. Leaf size=105 \[ -\frac{d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}+\frac{b c \left (2 c^2 d-3 e\right )}{36 x^3}-\frac{b c^3 \left (2 c^2 d-3 e\right )}{12 x}-\frac{1}{12} b c^4 \left (2 c^2 d-3 e\right ) \tan ^{-1}(c x)-\frac{b c d}{30 x^5} \]
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Rubi [A] time = 0.109091, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}+\frac{b c \left (2 c^2 d-3 e\right )}{36 x^3}-\frac{b c^3 \left (2 c^2 d-3 e\right )}{12 x}-\frac{1}{12} b c^4 \left (2 c^2 d-3 e\right ) \tan ^{-1}(c x)-\frac{b c d}{30 x^5} \]
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^7} \, dx &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-(b c) \int \frac{-2 d-3 e x^2}{12 x^6 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{1}{12} (b c) \int \frac{-2 d-3 e x^2}{x^6 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{b c d}{30 x^5}-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{1}{12} \left (b c \left (2 c^2 d-3 e\right )\right ) \int \frac{1}{x^4 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{b c d}{30 x^5}+\frac{b c \left (2 c^2 d-3 e\right )}{36 x^3}-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}+\frac{1}{12} \left (b c^3 \left (2 c^2 d-3 e\right )\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{b c d}{30 x^5}+\frac{b c \left (2 c^2 d-3 e\right )}{36 x^3}-\frac{b c^3 \left (2 c^2 d-3 e\right )}{12 x}-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{1}{12} \left (b c^5 \left (2 c^2 d-3 e\right )\right ) \int \frac{1}{1+c^2 x^2} \, dx\\ &=-\frac{b c d}{30 x^5}+\frac{b c \left (2 c^2 d-3 e\right )}{36 x^3}-\frac{b c^3 \left (2 c^2 d-3 e\right )}{12 x}-\frac{1}{12} b c^4 \left (2 c^2 d-3 e\right ) \tan ^{-1}(c x)-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}\\ \end{align*}
Mathematica [C] time = 0.0051924, size = 97, normalized size = 0.92 \[ -\frac{b c d \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},-c^2 x^2\right )}{30 x^5}-\frac{b c e \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},-c^2 x^2\right )}{12 x^3}-\frac{a d}{6 x^6}-\frac{a e}{4 x^4}-\frac{b d \tan ^{-1}(c x)}{6 x^6}-\frac{b e \tan ^{-1}(c x)}{4 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 106, normalized size = 1. \begin{align*} -{\frac{ae}{4\,{x}^{4}}}-{\frac{ad}{6\,{x}^{6}}}-{\frac{b\arctan \left ( cx \right ) e}{4\,{x}^{4}}}-{\frac{\arctan \left ( cx \right ) bd}{6\,{x}^{6}}}-{\frac{{c}^{6}b\arctan \left ( cx \right ) d}{6}}+{\frac{b{c}^{4}e\arctan \left ( cx \right ) }{4}}-{\frac{{c}^{5}bd}{6\,x}}+{\frac{b{c}^{3}e}{4\,x}}+{\frac{b{c}^{3}d}{18\,{x}^{3}}}-{\frac{bce}{12\,{x}^{3}}}-{\frac{bcd}{30\,{x}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45785, size = 139, normalized size = 1.32 \begin{align*} -\frac{1}{90} \,{\left ({\left (15 \, c^{5} \arctan \left (c x\right ) + \frac{15 \, c^{4} x^{4} - 5 \, c^{2} x^{2} + 3}{x^{5}}\right )} c + \frac{15 \, \arctan \left (c x\right )}{x^{6}}\right )} b d + \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 e - \frac{a e}{4 \, x^{4}} - \frac{a d}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76981, size = 238, normalized size = 2.27 \begin{align*} -\frac{15 \,{\left (2 \, b c^{5} d - 3 \, b c^{3} e\right )} x^{5} + 6 \, b c d x + 45 \, a e x^{2} - 5 \,{\left (2 \, b c^{3} d - 3 \, b c e\right )} x^{3} + 30 \, a d + 15 \,{\left ({\left (2 \, b c^{6} d - 3 \, b c^{4} e\right )} x^{6} + 3 \, b e x^{2} + 2 \, b d\right )} \arctan \left (c x\right )}{180 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.25256, size = 122, normalized size = 1.16 \begin{align*} - \frac{a d}{6 x^{6}} - \frac{a e}{4 x^{4}} - \frac{b c^{6} d \operatorname{atan}{\left (c x \right )}}{6} - \frac{b c^{5} d}{6 x} + \frac{b c^{4} e \operatorname{atan}{\left (c x \right )}}{4} + \frac{b c^{3} d}{18 x^{3}} + \frac{b c^{3} e}{4 x} - \frac{b c d}{30 x^{5}} - \frac{b c e}{12 x^{3}} - \frac{b d \operatorname{atan}{\left (c x \right )}}{6 x^{6}} - \frac{b e \operatorname{atan}{\left (c x \right )}}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19593, size = 174, normalized size = 1.66 \begin{align*} -\frac{30 \, b c^{6} d x^{6} \arctan \left (c x\right ) + 45 \, \pi b c^{4} x^{6} e \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 45 \, b c^{4} x^{6} \arctan \left (c x\right ) e + 30 \, b c^{5} d x^{5} - 45 \, b c^{3} x^{5} e - 10 \, b c^{3} d x^{3} + 15 \, b c x^{3} e + 45 \, b x^{2} \arctan \left (c x\right ) e + 6 \, b c d x + 45 \, a x^{2} e + 30 \, b d \arctan \left (c x\right ) + 30 \, a d}{180 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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