Optimal. Leaf size=188 \[ d^3 x \left (a+b \tan ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \tan ^{-1}(c x)\right )-\frac {b e^2 x^4 \left (21 c^2 d-5 e\right )}{140 c^3}-\frac {b e x^2 \left (35 c^4 d^2-21 c^2 d e+5 e^2\right )}{70 c^5}-\frac {b \left (35 c^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3\right ) \log \left (c^2 x^2+1\right )}{70 c^7}-\frac {b e^3 x^6}{42 c} \]
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Rubi [A] time = 0.15, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {194, 4912, 1810, 260} \[ d^2 e x^3 \left (a+b \tan ^{-1}(c x)\right )+d^3 x \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \tan ^{-1}(c x)\right )-\frac {b e x^2 \left (35 c^4 d^2-21 c^2 d e+5 e^2\right )}{70 c^5}-\frac {b \left (-35 c^4 d^2 e+35 c^6 d^3+21 c^2 d e^2-5 e^3\right ) \log \left (c^2 x^2+1\right )}{70 c^7}-\frac {b e^2 x^4 \left (21 c^2 d-5 e\right )}{140 c^3}-\frac {b e^3 x^6}{42 c} \]
Antiderivative was successfully verified.
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Rule 194
Rule 260
Rule 1810
Rule 4912
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=d^3 x \left (a+b \tan ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac {d^3 x+d^2 e x^3+\frac {3}{5} d e^2 x^5+\frac {e^3 x^7}{7}}{1+c^2 x^2} \, dx\\ &=d^3 x \left (a+b \tan ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \left (\frac {e \left (35 c^4 d^2-21 c^2 d e+5 e^2\right ) x}{35 c^6}+\frac {\left (21 c^2 d-5 e\right ) e^2 x^3}{35 c^4}+\frac {e^3 x^5}{7 c^2}+\frac {\left (35 c^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3\right ) x}{35 c^6 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {b e \left (35 c^4 d^2-21 c^2 d e+5 e^2\right ) x^2}{70 c^5}-\frac {b \left (21 c^2 d-5 e\right ) e^2 x^4}{140 c^3}-\frac {b e^3 x^6}{42 c}+d^3 x \left (a+b \tan ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \tan ^{-1}(c x)\right )-\frac {\left (b \left (35 c^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3\right )\right ) \int \frac {x}{1+c^2 x^2} \, dx}{35 c^5}\\ &=-\frac {b e \left (35 c^4 d^2-21 c^2 d e+5 e^2\right ) x^2}{70 c^5}-\frac {b \left (21 c^2 d-5 e\right ) e^2 x^4}{140 c^3}-\frac {b e^3 x^6}{42 c}+d^3 x \left (a+b \tan ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \tan ^{-1}(c x)\right )-\frac {b \left (35 c^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3\right ) \log \left (1+c^2 x^2\right )}{70 c^7}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 192, normalized size = 1.02 \[ \frac {c^2 x \left (12 a c^5 \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )-b e x \left (c^4 \left (210 d^2+63 d e x^2+10 e^2 x^4\right )-3 c^2 e \left (42 d+5 e x^2\right )+30 e^2\right )\right )+12 b c^7 x \tan ^{-1}(c x) \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )-6 b \left (35 c^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3\right ) \log \left (c^2 x^2+1\right )}{420 c^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 229, normalized size = 1.22 \[ \frac {60 \, a c^{7} e^{3} x^{7} + 252 \, a c^{7} d e^{2} x^{5} - 10 \, b c^{6} e^{3} x^{6} + 420 \, a c^{7} d^{2} e x^{3} + 420 \, a c^{7} d^{3} x - 3 \, {\left (21 \, b c^{6} d e^{2} - 5 \, b c^{4} e^{3}\right )} x^{4} - 6 \, {\left (35 \, b c^{6} d^{2} e - 21 \, b c^{4} d e^{2} + 5 \, b c^{2} e^{3}\right )} x^{2} + 12 \, {\left (5 \, b c^{7} e^{3} x^{7} + 21 \, b c^{7} d e^{2} x^{5} + 35 \, b c^{7} d^{2} e x^{3} + 35 \, b c^{7} d^{3} x\right )} \arctan \left (c x\right ) - 6 \, {\left (35 \, b c^{6} d^{3} - 35 \, b c^{4} d^{2} e + 21 \, b c^{2} d e^{2} - 5 \, b e^{3}\right )} \log \left (c^{2} x^{2} + 1\right )}{420 \, c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 239, normalized size = 1.27 \[ \frac {a \,x^{7} e^{3}}{7}+\frac {3 a \,x^{5} d \,e^{2}}{5}+a \,x^{3} d^{2} e +a \,d^{3} x +\frac {b \arctan \left (c x \right ) x^{7} e^{3}}{7}+\frac {3 b \arctan \left (c x \right ) x^{5} d \,e^{2}}{5}+b \arctan \left (c x \right ) x^{3} d^{2} e +b \arctan \left (c x \right ) d^{3} x -\frac {b \,x^{2} d^{2} e}{2 c}-\frac {3 b \,x^{4} d \,e^{2}}{20 c}-\frac {b \,e^{3} x^{6}}{42 c}+\frac {3 b \,x^{2} d \,e^{2}}{10 c^{3}}+\frac {b \,e^{3} x^{4}}{28 c^{3}}-\frac {b \,x^{2} e^{3}}{14 c^{5}}-\frac {b \ln \left (c^{2} x^{2}+1\right ) d^{3}}{2 c}+\frac {b \ln \left (c^{2} x^{2}+1\right ) d^{2} e}{2 c^{3}}-\frac {3 b \ln \left (c^{2} x^{2}+1\right ) d \,e^{2}}{10 c^{5}}+\frac {b \ln \left (c^{2} x^{2}+1\right ) e^{3}}{14 c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 222, normalized size = 1.18 \[ \frac {1}{7} \, a e^{3} x^{7} + \frac {3}{5} \, a d e^{2} x^{5} + a d^{2} e x^{3} + \frac {1}{2} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b d^{2} e + \frac {3}{20} \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b d e^{2} + \frac {1}{84} \, {\left (12 \, x^{7} \arctan \left (c x\right ) - c {\left (\frac {2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} - \frac {6 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )}\right )} b e^{3} + a d^{3} x + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d^{3}}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 238, normalized size = 1.27 \[ \frac {a\,e^3\,x^7}{7}+a\,d^3\,x-\frac {b\,d^3\,\ln \left (c^2\,x^2+1\right )}{2\,c}+\frac {b\,e^3\,\ln \left (c^2\,x^2+1\right )}{14\,c^7}-\frac {b\,e^3\,x^6}{42\,c}+\frac {b\,e^3\,x^4}{28\,c^3}-\frac {b\,e^3\,x^2}{14\,c^5}+b\,d^3\,x\,\mathrm {atan}\left (c\,x\right )+a\,d^2\,e\,x^3+\frac {3\,a\,d\,e^2\,x^5}{5}+\frac {b\,e^3\,x^7\,\mathrm {atan}\left (c\,x\right )}{7}+b\,d^2\,e\,x^3\,\mathrm {atan}\left (c\,x\right )+\frac {3\,b\,d\,e^2\,x^5\,\mathrm {atan}\left (c\,x\right )}{5}+\frac {b\,d^2\,e\,\ln \left (c^2\,x^2+1\right )}{2\,c^3}-\frac {3\,b\,d\,e^2\,\ln \left (c^2\,x^2+1\right )}{10\,c^5}-\frac {b\,d^2\,e\,x^2}{2\,c}-\frac {3\,b\,d\,e^2\,x^4}{20\,c}+\frac {3\,b\,d\,e^2\,x^2}{10\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.10, size = 306, normalized size = 1.63 \[ \begin {cases} a d^{3} x + a d^{2} e x^{3} + \frac {3 a d e^{2} x^{5}}{5} + \frac {a e^{3} x^{7}}{7} + b d^{3} x \operatorname {atan}{\left (c x \right )} + b d^{2} e x^{3} \operatorname {atan}{\left (c x \right )} + \frac {3 b d e^{2} x^{5} \operatorname {atan}{\left (c x \right )}}{5} + \frac {b e^{3} x^{7} \operatorname {atan}{\left (c x \right )}}{7} - \frac {b d^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c} - \frac {b d^{2} e x^{2}}{2 c} - \frac {3 b d e^{2} x^{4}}{20 c} - \frac {b e^{3} x^{6}}{42 c} + \frac {b d^{2} e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c^{3}} + \frac {3 b d e^{2} x^{2}}{10 c^{3}} + \frac {b e^{3} x^{4}}{28 c^{3}} - \frac {3 b d e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{10 c^{5}} - \frac {b e^{3} x^{2}}{14 c^{5}} + \frac {b e^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{14 c^{7}} & \text {for}\: c \neq 0 \\a \left (d^{3} x + d^{2} e x^{3} + \frac {3 d e^{2} x^{5}}{5} + \frac {e^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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