Optimal. Leaf size=160 \[ 3 d^2 e x \left (a+b \tan ^{-1}(c x)\right )-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+d e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} e^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{b \left (15 c^4 d^2 e+5 c^6 d^3-5 c^2 d e^2+e^3\right ) \log \left (c^2 x^2+1\right )}{10 c^5}-\frac{b e^2 x^2 \left (5 c^2 d-e\right )}{10 c^3}+b c d^3 \log (x)-\frac{b e^3 x^4}{20 c} \]
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Rubi [A] time = 0.258139, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {270, 4976, 1799, 1620} \[ 3 d^2 e x \left (a+b \tan ^{-1}(c x)\right )-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+d e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} e^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{b \left (15 c^4 d^2 e+5 c^6 d^3-5 c^2 d e^2+e^3\right ) \log \left (c^2 x^2+1\right )}{10 c^5}-\frac{b e^2 x^2 \left (5 c^2 d-e\right )}{10 c^3}+b c d^3 \log (x)-\frac{b e^3 x^4}{20 c} \]
Antiderivative was successfully verified.
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Rule 270
Rule 4976
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^2} \, dx &=-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \tan ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} e^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac{-d^3+3 d^2 e x^2+d e^2 x^4+\frac{e^3 x^6}{5}}{x \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \tan ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} e^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{-d^3+3 d^2 e x+d e^2 x^2+\frac{e^3 x^3}{5}}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \tan ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} e^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} (b c) \operatorname{Subst}\left (\int \left (\frac{\left (5 c^2 d-e\right ) e^2}{5 c^4}-\frac{d^3}{x}+\frac{e^3 x}{5 c^2}+\frac{5 c^6 d^3+15 c^4 d^2 e-5 c^2 d e^2+e^3}{5 c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{b \left (5 c^2 d-e\right ) e^2 x^2}{10 c^3}-\frac{b e^3 x^4}{20 c}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \tan ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} e^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+b c d^3 \log (x)-\frac{b \left (5 c^6 d^3+15 c^4 d^2 e-5 c^2 d e^2+e^3\right ) \log \left (1+c^2 x^2\right )}{10 c^5}\\ \end{align*}
Mathematica [A] time = 0.149035, size = 169, normalized size = 1.06 \[ \frac{1}{20} \left (60 a d^2 e x-\frac{20 a d^3}{x}+20 a d e^2 x^3+4 a e^3 x^5-\frac{2 b \left (15 c^4 d^2 e+5 c^6 d^3-5 c^2 d e^2+e^3\right ) \log \left (c^2 x^2+1\right )}{c^5}+\frac{2 b e^2 x^2 \left (e-5 c^2 d\right )}{c^3}+\frac{4 b \tan ^{-1}(c x) \left (15 d^2 e x^2-5 d^3+5 d e^2 x^4+e^3 x^6\right )}{x}+20 b c d^3 \log (x)-\frac{b e^3 x^4}{c}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 211, normalized size = 1.3 \begin{align*}{\frac{a{x}^{5}{e}^{3}}{5}}+a{x}^{3}d{e}^{2}+3\,a{d}^{2}ex-{\frac{a{d}^{3}}{x}}+{\frac{b\arctan \left ( cx \right ){x}^{5}{e}^{3}}{5}}+b\arctan \left ( cx \right ){x}^{3}d{e}^{2}+3\,b\arctan \left ( cx \right ){d}^{2}ex-{\frac{b{d}^{3}\arctan \left ( cx \right ) }{x}}-{\frac{b{e}^{3}{x}^{4}}{20\,c}}-{\frac{b{x}^{2}d{e}^{2}}{2\,c}}+{\frac{b{e}^{3}{x}^{2}}{10\,{c}^{3}}}-{\frac{bc{d}^{3}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2}}-{\frac{3\,b\ln \left ({c}^{2}{x}^{2}+1 \right ){d}^{2}e}{2\,c}}+{\frac{b\ln \left ({c}^{2}{x}^{2}+1 \right ) d{e}^{2}}{2\,{c}^{3}}}-{\frac{b\ln \left ({c}^{2}{x}^{2}+1 \right ){e}^{3}}{10\,{c}^{5}}}+cb{d}^{3}\ln \left ( cx \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965819, size = 266, normalized size = 1.66 \begin{align*} \frac{1}{5} \, a e^{3} x^{5} + a d e^{2} x^{3} - \frac{1}{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^{3} + \frac{1}{2} \,{\left (2 \, x^{3} \arctan \left (c x\right ) - c{\left (\frac{x^{2}}{c^{2}} - \frac{\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b d e^{2} + \frac{1}{20} \,{\left (4 \, x^{5} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac{2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b e^{3} + 3 \, a d^{2} e x + \frac{3 \,{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d^{2} e}{2 \, c} - \frac{a d^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95102, size = 446, normalized size = 2.79 \begin{align*} \frac{4 \, a c^{5} e^{3} x^{6} + 20 \, a c^{5} d e^{2} x^{4} - b c^{4} e^{3} x^{5} + 20 \, b c^{6} d^{3} x \log \left (x\right ) + 60 \, a c^{5} d^{2} e x^{2} - 20 \, a c^{5} d^{3} - 2 \,{\left (5 \, b c^{4} d e^{2} - b c^{2} e^{3}\right )} x^{3} - 2 \,{\left (5 \, b c^{6} d^{3} + 15 \, b c^{4} d^{2} e - 5 \, b c^{2} d e^{2} + b e^{3}\right )} x \log \left (c^{2} x^{2} + 1\right ) + 4 \,{\left (b c^{5} e^{3} x^{6} + 5 \, b c^{5} d e^{2} x^{4} + 15 \, b c^{5} d^{2} e x^{2} - 5 \, b c^{5} d^{3}\right )} \arctan \left (c x\right )}{20 \, c^{5} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.54261, size = 258, normalized size = 1.61 \begin{align*} \begin{cases} - \frac{a d^{3}}{x} + 3 a d^{2} e x + a d e^{2} x^{3} + \frac{a e^{3} x^{5}}{5} + b c d^{3} \log{\left (x \right )} - \frac{b c d^{3} \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{2} - \frac{b d^{3} \operatorname{atan}{\left (c x \right )}}{x} + 3 b d^{2} e x \operatorname{atan}{\left (c x \right )} + b d e^{2} x^{3} \operatorname{atan}{\left (c x \right )} + \frac{b e^{3} x^{5} \operatorname{atan}{\left (c x \right )}}{5} - \frac{3 b d^{2} e \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{2 c} - \frac{b d e^{2} x^{2}}{2 c} - \frac{b e^{3} x^{4}}{20 c} + \frac{b d e^{2} \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{2 c^{3}} + \frac{b e^{3} x^{2}}{10 c^{3}} - \frac{b e^{3} \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{10 c^{5}} & \text{for}\: c \neq 0 \\a \left (- \frac{d^{3}}{x} + 3 d^{2} e x + d e^{2} x^{3} + \frac{e^{3} x^{5}}{5}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11454, size = 325, normalized size = 2.03 \begin{align*} \frac{4 \, b c^{5} x^{6} \arctan \left (c x\right ) e^{3} + 4 \, a c^{5} x^{6} e^{3} + 20 \, b c^{5} d x^{4} \arctan \left (c x\right ) e^{2} + 20 \, a c^{5} d x^{4} e^{2} + 60 \, b c^{5} d^{2} x^{2} \arctan \left (c x\right ) e - 10 \, b c^{6} d^{3} x \log \left (c^{2} x^{2} + 1\right ) + 20 \, b c^{6} d^{3} x \log \left (x\right ) - b c^{4} x^{5} e^{3} + 60 \, a c^{5} d^{2} x^{2} e - 20 \, b c^{5} d^{3} \arctan \left (c x\right ) - 10 \, b c^{4} d x^{3} e^{2} - 30 \, b c^{4} d^{2} x e \log \left (c^{2} x^{2} + 1\right ) - 20 \, a c^{5} d^{3} + 2 \, b c^{2} x^{3} e^{3} + 10 \, b c^{2} d x e^{2} \log \left (c^{2} x^{2} + 1\right ) - 2 \, b x e^{3} \log \left (c^{2} x^{2} + 1\right )}{20 \, c^{5} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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