Optimal. Leaf size=184 \[ -\frac {b d e \left (2 c^2 d^2-e^2\right ) x}{c^3}-\frac {b e^2 \left (10 c^2 d^2-e^2\right ) x^2}{10 c^3}-\frac {b d e^3 x^3}{3 c}-\frac {b e^4 x^4}{20 c}-\frac {b d \left (c^4 d^4-10 c^2 d^2 e^2+5 e^4\right ) \text {ArcTan}(c x)}{5 c^4 e}+\frac {(d+e x)^5 (a+b \text {ArcTan}(c x))}{5 e}-\frac {b \left (5 c^4 d^4-10 c^2 d^2 e^2+e^4\right ) \log \left (1+c^2 x^2\right )}{10 c^5} \]
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Rubi [A]
time = 0.10, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4972, 716, 649,
209, 266} \begin {gather*} \frac {(d+e x)^5 (a+b \text {ArcTan}(c x))}{5 e}-\frac {b d \text {ArcTan}(c x) \left (c^4 d^4-10 c^2 d^2 e^2+5 e^4\right )}{5 c^4 e}-\frac {b e^2 x^2 \left (10 c^2 d^2-e^2\right )}{10 c^3}-\frac {b d e x \left (2 c^2 d^2-e^2\right )}{c^3}-\frac {b \left (5 c^4 d^4-10 c^2 d^2 e^2+e^4\right ) \log \left (c^2 x^2+1\right )}{10 c^5}-\frac {b d e^3 x^3}{3 c}-\frac {b e^4 x^4}{20 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 649
Rule 716
Rule 4972
Rubi steps
\begin {align*} \int (d+e x)^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 e}-\frac {(b c) \int \frac {(d+e x)^5}{1+c^2 x^2} \, dx}{5 e}\\ &=\frac {(d+e x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 e}-\frac {(b c) \int \left (\frac {5 d e^2 \left (2 c^2 d^2-e^2\right )}{c^4}+\frac {e^3 \left (10 c^2 d^2-e^2\right ) x}{c^4}+\frac {5 d e^4 x^2}{c^2}+\frac {e^5 x^3}{c^2}+\frac {c^4 d^5-10 c^2 d^3 e^2+5 d e^4+e \left (5 c^4 d^4-10 c^2 d^2 e^2+e^4\right ) x}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx}{5 e}\\ &=-\frac {b d e \left (2 c^2 d^2-e^2\right ) x}{c^3}-\frac {b e^2 \left (10 c^2 d^2-e^2\right ) x^2}{10 c^3}-\frac {b d e^3 x^3}{3 c}-\frac {b e^4 x^4}{20 c}+\frac {(d+e x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 e}-\frac {b \int \frac {c^4 d^5-10 c^2 d^3 e^2+5 d e^4+e \left (5 c^4 d^4-10 c^2 d^2 e^2+e^4\right ) x}{1+c^2 x^2} \, dx}{5 c^3 e}\\ &=-\frac {b d e \left (2 c^2 d^2-e^2\right ) x}{c^3}-\frac {b e^2 \left (10 c^2 d^2-e^2\right ) x^2}{10 c^3}-\frac {b d e^3 x^3}{3 c}-\frac {b e^4 x^4}{20 c}+\frac {(d+e x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 e}-\frac {\left (b \left (5 c^4 d^4-10 c^2 d^2 e^2+e^4\right )\right ) \int \frac {x}{1+c^2 x^2} \, dx}{5 c^3}-\frac {\left (b d \left (c^4 d^4-10 c^2 d^2 e^2+5 e^4\right )\right ) \int \frac {1}{1+c^2 x^2} \, dx}{5 c^3 e}\\ &=-\frac {b d e \left (2 c^2 d^2-e^2\right ) x}{c^3}-\frac {b e^2 \left (10 c^2 d^2-e^2\right ) x^2}{10 c^3}-\frac {b d e^3 x^3}{3 c}-\frac {b e^4 x^4}{20 c}-\frac {b d \left (c^4 d^4-10 c^2 d^2 e^2+5 e^4\right ) \tan ^{-1}(c x)}{5 c^4 e}+\frac {(d+e x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 e}-\frac {b \left (5 c^4 d^4-10 c^2 d^2 e^2+e^4\right ) \log \left (1+c^2 x^2\right )}{10 c^5}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 255, normalized size = 1.39 \begin {gather*} \frac {(d+e x)^5 (a+b \text {ArcTan}(c x))-\frac {b \left (c^2 e^2 x \left (-6 e^2 (10 d+e x)+c^2 \left (120 d^3+60 d^2 e x+20 d e^2 x^2+3 e^3 x^3\right )\right )+6 \left (-10 c^2 d^2 e^2 \left (\sqrt {-c^2} d+e\right )+e^4 \left (5 \sqrt {-c^2} d+e\right )+c^4 d^4 \left (\sqrt {-c^2} d+5 e\right )\right ) \log \left (1-\sqrt {-c^2} x\right )-6 \left (c^4 d^4 \left (\sqrt {-c^2} d-5 e\right )-10 c^2 d^2 \left (\sqrt {-c^2} d-e\right ) e^2+\left (5 \sqrt {-c^2} d-e\right ) e^4\right ) \log \left (1+\sqrt {-c^2} x\right )\right )}{12 c^5}}{5 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 241, normalized size = 1.31
method | result | size |
derivativedivides | \(\frac {\frac {\left (c e x +c d \right )^{5} a}{5 c^{4} e}+b \arctan \left (c x \right ) d^{4} c x +2 b c e \arctan \left (c x \right ) d^{3} x^{2}+2 b c \,e^{2} \arctan \left (c x \right ) d^{2} x^{3}+b c \,e^{3} \arctan \left (c x \right ) d \,x^{4}+\frac {b c \,e^{4} \arctan \left (c x \right ) x^{5}}{5}-2 b e \,d^{3} x -b \,e^{2} d^{2} x^{2}-\frac {b \,e^{3} d \,x^{3}}{3}-\frac {b \,e^{4} x^{4}}{20}+\frac {b \,e^{3} d x}{c^{2}}+\frac {b \,e^{4} x^{2}}{10 c^{2}}-\frac {b \ln \left (c^{2} x^{2}+1\right ) d^{4}}{2}+\frac {b \,e^{2} \ln \left (c^{2} x^{2}+1\right ) d^{2}}{c^{2}}-\frac {b \,e^{4} \ln \left (c^{2} x^{2}+1\right )}{10 c^{4}}+\frac {2 b e \arctan \left (c x \right ) d^{3}}{c}-\frac {b \,e^{3} \arctan \left (c x \right ) d}{c^{3}}}{c}\) | \(241\) |
default | \(\frac {\frac {\left (c e x +c d \right )^{5} a}{5 c^{4} e}+b \arctan \left (c x \right ) d^{4} c x +2 b c e \arctan \left (c x \right ) d^{3} x^{2}+2 b c \,e^{2} \arctan \left (c x \right ) d^{2} x^{3}+b c \,e^{3} \arctan \left (c x \right ) d \,x^{4}+\frac {b c \,e^{4} \arctan \left (c x \right ) x^{5}}{5}-2 b e \,d^{3} x -b \,e^{2} d^{2} x^{2}-\frac {b \,e^{3} d \,x^{3}}{3}-\frac {b \,e^{4} x^{4}}{20}+\frac {b \,e^{3} d x}{c^{2}}+\frac {b \,e^{4} x^{2}}{10 c^{2}}-\frac {b \ln \left (c^{2} x^{2}+1\right ) d^{4}}{2}+\frac {b \,e^{2} \ln \left (c^{2} x^{2}+1\right ) d^{2}}{c^{2}}-\frac {b \,e^{4} \ln \left (c^{2} x^{2}+1\right )}{10 c^{4}}+\frac {2 b e \arctan \left (c x \right ) d^{3}}{c}-\frac {b \,e^{3} \arctan \left (c x \right ) d}{c^{3}}}{c}\) | \(241\) |
risch | \(\frac {i e^{4} b \,x^{5} \ln \left (-i c x +1\right )}{10}+\frac {i b \,d^{5} \ln \left (c^{2} x^{2}+1\right )}{20 e}+i e b \,d^{3} x^{2} \ln \left (-i c x +1\right )+\frac {i b \,d^{4} x \ln \left (-i c x +1\right )}{2}+\frac {x^{5} e^{4} a}{5}-\frac {i \left (e x +d \right )^{5} b \ln \left (i c x +1\right )}{10 e}+x^{4} e^{3} d a +\frac {i e^{3} b d \,x^{4} \ln \left (-i c x +1\right )}{2}+2 x^{3} e^{2} d^{2} a -\frac {b \,e^{4} x^{4}}{20 c}+i e^{2} b \,d^{2} x^{3} \ln \left (-i c x +1\right )+2 x^{2} e \,d^{3} a -\frac {b d \,e^{3} x^{3}}{3 c}-\frac {b \,d^{5} \arctan \left (c x \right )}{10 e}+x a \,d^{4}-\frac {e^{2} b \,d^{2} x^{2}}{c}-\frac {b \,d^{4} \ln \left (c^{2} x^{2}+1\right )}{2 c}-\frac {2 e b \,d^{3} x}{c}+\frac {2 e b \,d^{3} \arctan \left (c x \right )}{c^{2}}+\frac {e^{4} b \,x^{2}}{10 c^{3}}+\frac {e^{2} b \,d^{2} \ln \left (c^{2} x^{2}+1\right )}{c^{3}}+\frac {e^{3} b d x}{c^{3}}-\frac {e^{3} b d \arctan \left (c x \right )}{c^{4}}-\frac {e^{4} b \ln \left (c^{2} x^{2}+1\right )}{10 c^{5}}\) | \(356\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 248, normalized size = 1.35 \begin {gather*} \frac {1}{5} \, a x^{5} e^{4} + a d x^{4} e^{3} + 2 \, a d^{2} x^{3} e^{2} + 2 \, a d^{3} x^{2} e + a d^{4} x + 2 \, {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} b d^{3} e + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d^{4}}{2 \, c} + {\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^{2} + \frac {1}{3} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d e^{3} + \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^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.38, size = 257, normalized size = 1.40 \begin {gather*} \frac {60 \, a c^{5} d^{4} x + 12 \, {\left (b c^{5} x^{5} e^{4} + 10 \, b c^{5} d^{2} x^{3} e^{2} + 5 \, b c^{5} d^{4} x + 5 \, {\left (b c^{5} d x^{4} - b c d\right )} e^{3} + 10 \, {\left (b c^{5} d^{3} x^{2} + b c^{3} d^{3}\right )} e\right )} \arctan \left (c x\right ) + 3 \, {\left (4 \, a c^{5} x^{5} - b c^{4} x^{4} + 2 \, b c^{2} x^{2}\right )} e^{4} + 20 \, {\left (3 \, a c^{5} d x^{4} - b c^{4} d x^{3} + 3 \, b c^{2} d x\right )} e^{3} + 60 \, {\left (2 \, a c^{5} d^{2} x^{3} - b c^{4} d^{2} x^{2}\right )} e^{2} + 120 \, {\left (a c^{5} d^{3} x^{2} - b c^{4} d^{3} x\right )} e - 6 \, {\left (5 \, b c^{4} d^{4} - 10 \, b c^{2} d^{2} e^{2} + b e^{4}\right )} \log \left (c^{2} x^{2} + 1\right )}{60 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 345 vs.
\(2 (170) = 340\).
time = 0.44, size = 345, normalized size = 1.88 \begin {gather*} \begin {cases} a d^{4} x + 2 a d^{3} e x^{2} + 2 a d^{2} e^{2} x^{3} + a d e^{3} x^{4} + \frac {a e^{4} x^{5}}{5} + b d^{4} x \operatorname {atan}{\left (c x \right )} + 2 b d^{3} e x^{2} \operatorname {atan}{\left (c x \right )} + 2 b d^{2} e^{2} x^{3} \operatorname {atan}{\left (c x \right )} + b d e^{3} x^{4} \operatorname {atan}{\left (c x \right )} + \frac {b e^{4} x^{5} \operatorname {atan}{\left (c x \right )}}{5} - \frac {b d^{4} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c} - \frac {2 b d^{3} e x}{c} - \frac {b d^{2} e^{2} x^{2}}{c} - \frac {b d e^{3} x^{3}}{3 c} - \frac {b e^{4} x^{4}}{20 c} + \frac {2 b d^{3} e \operatorname {atan}{\left (c x \right )}}{c^{2}} + \frac {b d^{2} e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{c^{3}} + \frac {b d e^{3} x}{c^{3}} + \frac {b e^{4} x^{2}}{10 c^{3}} - \frac {b d e^{3} \operatorname {atan}{\left (c x \right )}}{c^{4}} - \frac {b e^{4} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{10 c^{5}} & \text {for}\: c \neq 0 \\a \left (d^{4} x + 2 d^{3} e x^{2} + 2 d^{2} e^{2} x^{3} + d e^{3} x^{4} + \frac {e^{4} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.05, size = 273, normalized size = 1.48 \begin {gather*} \frac {a\,e^4\,x^5}{5}+a\,d^4\,x-\frac {b\,d^4\,\ln \left (c^2\,x^2+1\right )}{2\,c}-\frac {b\,e^4\,\ln \left (c^2\,x^2+1\right )}{10\,c^5}+2\,a\,d^2\,e^2\,x^3-\frac {b\,e^4\,x^4}{20\,c}+\frac {b\,e^4\,x^2}{10\,c^3}+b\,d^4\,x\,\mathrm {atan}\left (c\,x\right )+2\,a\,d^3\,e\,x^2+a\,d\,e^3\,x^4+\frac {b\,e^4\,x^5\,\mathrm {atan}\left (c\,x\right )}{5}-\frac {2\,b\,d^3\,e\,x}{c}+\frac {b\,d\,e^3\,x}{c^3}+\frac {2\,b\,d^3\,e\,\mathrm {atan}\left (c\,x\right )}{c^2}-\frac {b\,d\,e^3\,\mathrm {atan}\left (c\,x\right )}{c^4}+2\,b\,d^3\,e\,x^2\,\mathrm {atan}\left (c\,x\right )+b\,d\,e^3\,x^4\,\mathrm {atan}\left (c\,x\right )-\frac {b\,d\,e^3\,x^3}{3\,c}+2\,b\,d^2\,e^2\,x^3\,\mathrm {atan}\left (c\,x\right )+\frac {b\,d^2\,e^2\,\ln \left (c^2\,x^2+1\right )}{c^3}-\frac {b\,d^2\,e^2\,x^2}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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