Optimal. Leaf size=139 \[ \frac{1}{2} i b e^2 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b e^2 \text{PolyLog}(2,i c x)-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{d e \left (a+b \tan ^{-1}(c x)\right )}{x^2}+a e^2 \log (x)+\frac{b c^3 d^2}{4 x}+\frac{1}{4} b c^4 d^2 \tan ^{-1}(c x)-b c^2 d e \tan ^{-1}(c x)-\frac{b c d^2}{12 x^3}-\frac{b c d e}{x} \]
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Rubi [A] time = 0.168279, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4980, 4852, 325, 203, 4848, 2391} \[ \frac{1}{2} i b e^2 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b e^2 \text{PolyLog}(2,i c x)-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{d e \left (a+b \tan ^{-1}(c x)\right )}{x^2}+a e^2 \log (x)+\frac{b c^3 d^2}{4 x}+\frac{1}{4} b c^4 d^2 \tan ^{-1}(c x)-b c^2 d e \tan ^{-1}(c x)-\frac{b c d^2}{12 x^3}-\frac{b c d e}{x} \]
Antiderivative was successfully verified.
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Rule 4980
Rule 4852
Rule 325
Rule 203
Rule 4848
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx &=\int \left (\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{x^5}+\frac{2 d e \left (a+b \tan ^{-1}(c x)\right )}{x^3}+\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d^2 \int \frac{a+b \tan ^{-1}(c x)}{x^5} \, dx+(2 d e) \int \frac{a+b \tan ^{-1}(c x)}{x^3} \, dx+e^2 \int \frac{a+b \tan ^{-1}(c x)}{x} \, dx\\ &=-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{d e \left (a+b \tan ^{-1}(c x)\right )}{x^2}+a e^2 \log (x)+\frac{1}{4} \left (b c d^2\right ) \int \frac{1}{x^4 \left (1+c^2 x^2\right )} \, dx+(b c d e) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx+\frac{1}{2} \left (i b e^2\right ) \int \frac{\log (1-i c x)}{x} \, dx-\frac{1}{2} \left (i b e^2\right ) \int \frac{\log (1+i c x)}{x} \, dx\\ &=-\frac{b c d^2}{12 x^3}-\frac{b c d e}{x}-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{d e \left (a+b \tan ^{-1}(c x)\right )}{x^2}+a e^2 \log (x)+\frac{1}{2} i b e^2 \text{Li}_2(-i c x)-\frac{1}{2} i b e^2 \text{Li}_2(i c x)-\frac{1}{4} \left (b c^3 d^2\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx-\left (b c^3 d e\right ) \int \frac{1}{1+c^2 x^2} \, dx\\ &=-\frac{b c d^2}{12 x^3}+\frac{b c^3 d^2}{4 x}-\frac{b c d e}{x}-b c^2 d e \tan ^{-1}(c x)-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{d e \left (a+b \tan ^{-1}(c x)\right )}{x^2}+a e^2 \log (x)+\frac{1}{2} i b e^2 \text{Li}_2(-i c x)-\frac{1}{2} i b e^2 \text{Li}_2(i c x)+\frac{1}{4} \left (b c^5 d^2\right ) \int \frac{1}{1+c^2 x^2} \, dx\\ &=-\frac{b c d^2}{12 x^3}+\frac{b c^3 d^2}{4 x}-\frac{b c d e}{x}+\frac{1}{4} b c^4 d^2 \tan ^{-1}(c x)-b c^2 d e \tan ^{-1}(c x)-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{d e \left (a+b \tan ^{-1}(c x)\right )}{x^2}+a e^2 \log (x)+\frac{1}{2} i b e^2 \text{Li}_2(-i c x)-\frac{1}{2} i b e^2 \text{Li}_2(i c x)\\ \end{align*}
Mathematica [C] time = 0.0945496, size = 130, normalized size = 0.94 \[ -\frac{b c d^2 \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},-c^2 x^2\right )}{12 x^3}-\frac{b c d e \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-c^2 x^2\right )}{x}+\frac{1}{2} i b e^2 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b e^2 \text{PolyLog}(2,i c x)-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{d e \left (a+b \tan ^{-1}(c x)\right )}{x^2}+a e^2 \log (x) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.059, size = 190, normalized size = 1.4 \begin{align*} -{\frac{aed}{{x}^{2}}}-{\frac{a{d}^{2}}{4\,{x}^{4}}}+a{e}^{2}\ln \left ( cx \right ) -{\frac{b\arctan \left ( cx \right ) ed}{{x}^{2}}}-{\frac{b{d}^{2}\arctan \left ( cx \right ) }{4\,{x}^{4}}}+b\arctan \left ( cx \right ){e}^{2}\ln \left ( cx \right ) +{\frac{i}{2}}b{e}^{2}\ln \left ( cx \right ) \ln \left ( 1+icx \right ) -{\frac{i}{2}}b{e}^{2}\ln \left ( cx \right ) \ln \left ( 1-icx \right ) +{\frac{i}{2}}b{e}^{2}{\it dilog} \left ( 1+icx \right ) -{\frac{i}{2}}b{e}^{2}{\it dilog} \left ( 1-icx \right ) +{\frac{b{c}^{4}{d}^{2}\arctan \left ( cx \right ) }{4}}-b{c}^{2}de\arctan \left ( cx \right ) +{\frac{b{c}^{3}{d}^{2}}{4\,x}}-{\frac{bcde}{x}}-{\frac{bc{d}^{2}}{12\,{x}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \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^{2} -{\left ({\left (c \arctan \left (c x\right ) + \frac{1}{x}\right )} c + \frac{\arctan \left (c x\right )}{x^{2}}\right )} b d e + b e^{2} \int \frac{\arctan \left (c x\right )}{x}\,{d x} + a e^{2} \log \left (x\right ) - \frac{a d e}{x^{2}} - \frac{a d^{2}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} +{\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \arctan \left (c x\right )}{x^{5}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{2}{\left (b \arctan \left (c x\right ) + a\right )}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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