Optimal. Leaf size=139 \[ -\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {d e \left (a+b \tan ^{-1}(c x)\right )}{x^2}+a e^2 \log (x)+\frac {1}{4} b c^4 d^2 \tan ^{-1}(c x)+\frac {b c^3 d^2}{4 x}-b c^2 d e \tan ^{-1}(c x)-\frac {b c d^2}{12 x^3}-\frac {b c d e}{x}+\frac {1}{2} i b e^2 \text {Li}_2(-i c x)-\frac {1}{2} i b e^2 \text {Li}_2(i c x) \]
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Rubi [A] time = 0.17, antiderivative size = 139, normalized size of antiderivative = 1.00, 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 203
Rule 325
Rule 2391
Rule 4848
Rule 4852
Rule 4980
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*}
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Mathematica [C] time = 0.11, size = 130, normalized size = 0.94 \[ -\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {d e \left (a+b \tan ^{-1}(c x)\right )}{x^2}+a e^2 \log (x)-\frac {b c d^2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-c^2 x^2\right )}{12 x^3}-\frac {b c d e \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-c^2 x^2\right )}{x}+\frac {1}{2} i b e^2 \text {Li}_2(-i c x)-\frac {1}{2} i b e^2 \text {Li}_2(i c x) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} + {\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \arctan \left (c x\right )}{x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 190, normalized size = 1.37 \[ a \,e^{2} \ln \left (c x \right )-\frac {a \,d^{2}}{4 x^{4}}-\frac {a e d}{x^{2}}+b \arctan \left (c x \right ) e^{2} \ln \left (c x \right )-\frac {b \arctan \left (c x \right ) d^{2}}{4 x^{4}}-\frac {b \arctan \left (c x \right ) e d}{x^{2}}+\frac {i b \,e^{2} \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-\frac {i b \,e^{2} \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}+\frac {i b \,e^{2} \dilog \left (i c x +1\right )}{2}-\frac {i b \,e^{2} \dilog \left (-i c x +1\right )}{2}+\frac {b \,c^{3} d^{2}}{4 x}-\frac {b c d e}{x}-\frac {b c \,d^{2}}{12 x^{3}}+\frac {b \,c^{4} d^{2} \arctan \left (c x \right )}{4}-b \,c^{2} d e \arctan \left (c x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{12} \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b d^{2} - {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b d e + b e^{2} \int \frac {\arctan \left (c x\right )}{x}\,{d x} + a e^{2} \log \relax (x) - \frac {a d e}{x^{2}} - \frac {a d^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.76, size = 177, normalized size = 1.27 \[ \left \{\begin {array}{cl} a\,e^2\,\ln \relax (x)-\frac {\frac {a\,d^2}{4}+a\,e\,d\,x^2}{x^4} & \text {\ if\ \ }c=0\\ a\,e^2\,\ln \relax (x)-\frac {\frac {a\,d^2}{4}+a\,e\,d\,x^2}{x^4}-\frac {b\,d^2\,\left (\frac {\frac {c^2}{3}-c^4\,x^2}{x^3}-c^5\,\mathrm {atan}\left (c\,x\right )\right )}{4\,c}-2\,b\,d\,e\,\left (\frac {c^3\,\mathrm {atan}\left (c\,x\right )+\frac {c^2}{x}}{2\,c}+\frac {\mathrm {atan}\left (c\,x\right )}{2\,x^2}\right )-\frac {b\,d^2\,\mathrm {atan}\left (c\,x\right )}{4\,x^4}-\frac {b\,e^2\,{\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {b\,e^2\,{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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