Integrand size = 16, antiderivative size = 144 \[ \int (d+e x)^3 (a+b \arctan (c x)) \, dx=-\frac {b e \left (6 c^2 d^2-e^2\right ) x}{4 c^3}-\frac {b d e^2 x^2}{2 c}-\frac {b e^3 x^3}{12 c}-\frac {b \left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) \arctan (c x)}{4 c^4 e}+\frac {(d+e x)^4 (a+b \arctan (c x))}{4 e}-\frac {b d (c d-e) (c d+e) \log \left (1+c^2 x^2\right )}{2 c^3} \]
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Time = 0.08 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4972, 716, 649, 209, 266} \[ \int (d+e x)^3 (a+b \arctan (c x)) \, dx=\frac {(d+e x)^4 (a+b \arctan (c x))}{4 e}-\frac {b \arctan (c x) \left (c^4 d^4-6 c^2 d^2 e^2+e^4\right )}{4 c^4 e}-\frac {b e x \left (6 c^2 d^2-e^2\right )}{4 c^3}-\frac {b d (c d-e) (c d+e) \log \left (c^2 x^2+1\right )}{2 c^3}-\frac {b d e^2 x^2}{2 c}-\frac {b e^3 x^3}{12 c} \]
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Rule 209
Rule 266
Rule 649
Rule 716
Rule 4972
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^4 (a+b \arctan (c x))}{4 e}-\frac {(b c) \int \frac {(d+e x)^4}{1+c^2 x^2} \, dx}{4 e} \\ & = \frac {(d+e x)^4 (a+b \arctan (c x))}{4 e}-\frac {(b c) \int \left (\frac {e^2 \left (6 c^2 d^2-e^2\right )}{c^4}+\frac {4 d e^3 x}{c^2}+\frac {e^4 x^2}{c^2}+\frac {c^4 d^4-6 c^2 d^2 e^2+e^4+4 c^2 d (c d-e) e (c d+e) x}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx}{4 e} \\ & = -\frac {b e \left (6 c^2 d^2-e^2\right ) x}{4 c^3}-\frac {b d e^2 x^2}{2 c}-\frac {b e^3 x^3}{12 c}+\frac {(d+e x)^4 (a+b \arctan (c x))}{4 e}-\frac {b \int \frac {c^4 d^4-6 c^2 d^2 e^2+e^4+4 c^2 d (c d-e) e (c d+e) x}{1+c^2 x^2} \, dx}{4 c^3 e} \\ & = -\frac {b e \left (6 c^2 d^2-e^2\right ) x}{4 c^3}-\frac {b d e^2 x^2}{2 c}-\frac {b e^3 x^3}{12 c}+\frac {(d+e x)^4 (a+b \arctan (c x))}{4 e}-\frac {(b d (c d-e) (c d+e)) \int \frac {x}{1+c^2 x^2} \, dx}{c}-\frac {\left (b \left (c^4 d^4-6 c^2 d^2 e^2+e^4\right )\right ) \int \frac {1}{1+c^2 x^2} \, dx}{4 c^3 e} \\ & = -\frac {b e \left (6 c^2 d^2-e^2\right ) x}{4 c^3}-\frac {b d e^2 x^2}{2 c}-\frac {b e^3 x^3}{12 c}-\frac {b \left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) \arctan (c x)}{4 c^4 e}+\frac {(d+e x)^4 (a+b \arctan (c x))}{4 e}-\frac {b d (c d-e) (c d+e) \log \left (1+c^2 x^2\right )}{2 c^3} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.51 \[ \int (d+e x)^3 (a+b \arctan (c x)) \, dx=\frac {(d+e x)^4 (a+b \arctan (c x))-\frac {b c \left (2 \sqrt {-c^2} e^2 x \left (-3 e^2+c^2 \left (18 d^2+6 d e x+e^2 x^2\right )\right )-3 \left (c^4 d^4+e^3 \left (4 \sqrt {-c^2} d+e\right )-2 c^2 d^2 e \left (2 \sqrt {-c^2} d+3 e\right )\right ) \log \left (1-\sqrt {-c^2} x\right )+3 \left (c^4 d^4+2 c^2 d^2 \left (2 \sqrt {-c^2} d-3 e\right ) e+e^3 \left (-4 \sqrt {-c^2} d+e\right )\right ) \log \left (1+\sqrt {-c^2} x\right )\right )}{6 \left (-c^2\right )^{5/2}}}{4 e} \]
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Time = 1.48 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.33
method | result | size |
parts | \(\frac {a \left (e x +d \right )^{4}}{4 e}+\frac {b \left (\frac {c \,e^{3} \arctan \left (c x \right ) x^{4}}{4}+c \,e^{2} \arctan \left (c x \right ) x^{3} d +\frac {3 c e \arctan \left (c x \right ) x^{2} d^{2}}{2}+\arctan \left (c x \right ) c x \,d^{3}+\frac {c \arctan \left (c x \right ) d^{4}}{4 e}-\frac {6 c^{3} d^{2} e^{2} x +2 e^{3} c^{3} d \,x^{2}+\frac {e^{4} c^{3} x^{3}}{3}-c \,e^{4} x +\frac {\left (4 c^{3} d^{3} e -4 c d \,e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+\left (c^{4} d^{4}-6 c^{2} d^{2} e^{2}+e^{4}\right ) \arctan \left (c x \right )}{4 c^{3} e}\right )}{c}\) | \(191\) |
derivativedivides | \(\frac {\frac {a \left (c e x +c d \right )^{4}}{4 c^{3} e}+\frac {b \left (\frac {\arctan \left (c x \right ) c^{4} d^{4}}{4 e}+\arctan \left (c x \right ) c^{4} d^{3} x +\frac {3 e \arctan \left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \arctan \left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \arctan \left (c x \right ) c^{4} x^{4}}{4}-\frac {6 c^{3} d^{2} e^{2} x +2 e^{3} c^{3} d \,x^{2}+\frac {e^{4} c^{3} x^{3}}{3}-c \,e^{4} x +\frac {\left (4 c^{3} d^{3} e -4 c d \,e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+\left (c^{4} d^{4}-6 c^{2} d^{2} e^{2}+e^{4}\right ) \arctan \left (c x \right )}{4 e}\right )}{c^{3}}}{c}\) | \(208\) |
default | \(\frac {\frac {a \left (c e x +c d \right )^{4}}{4 c^{3} e}+\frac {b \left (\frac {\arctan \left (c x \right ) c^{4} d^{4}}{4 e}+\arctan \left (c x \right ) c^{4} d^{3} x +\frac {3 e \arctan \left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \arctan \left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \arctan \left (c x \right ) c^{4} x^{4}}{4}-\frac {6 c^{3} d^{2} e^{2} x +2 e^{3} c^{3} d \,x^{2}+\frac {e^{4} c^{3} x^{3}}{3}-c \,e^{4} x +\frac {\left (4 c^{3} d^{3} e -4 c d \,e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+\left (c^{4} d^{4}-6 c^{2} d^{2} e^{2}+e^{4}\right ) \arctan \left (c x \right )}{4 e}\right )}{c^{3}}}{c}\) | \(208\) |
parallelrisch | \(-\frac {-3 x^{4} \arctan \left (c x \right ) b \,c^{4} e^{3}-3 x^{4} a \,c^{4} e^{3}-12 x^{3} \arctan \left (c x \right ) b \,c^{4} d \,e^{2}-12 x^{3} a \,c^{4} d \,e^{2}-18 x^{2} \arctan \left (c x \right ) b \,c^{4} d^{2} e +x^{3} b \,c^{3} e^{3}-18 x^{2} a \,c^{4} d^{2} e -12 b \,d^{3} \arctan \left (c x \right ) x \,c^{4}+6 x^{2} b \,c^{3} d \,e^{2}-12 x a \,c^{4} d^{3}+6 \ln \left (c^{2} x^{2}+1\right ) b \,c^{3} d^{3}+18 x b \,c^{3} d^{2} e -18 \arctan \left (c x \right ) b \,c^{2} d^{2} e -6 \ln \left (c^{2} x^{2}+1\right ) b c d \,e^{2}-3 x b c \,e^{3}+3 \arctan \left (c x \right ) b \,e^{3}}{12 c^{4}}\) | \(223\) |
risch | \(\frac {3 i e b \,d^{2} x^{2} \ln \left (-i c x +1\right )}{4}+\frac {i e^{3} b \,x^{4} \ln \left (-i c x +1\right )}{8}+\frac {i b \,d^{3} x \ln \left (-i c x +1\right )}{2}-\frac {i \left (e x +d \right )^{4} b \ln \left (i c x +1\right )}{8 e}+\frac {i b \,d^{4} \ln \left (c^{2} x^{2}+1\right )}{16 e}-\frac {b \,d^{4} \arctan \left (c x \right )}{8 e}+\frac {e^{2} b d \ln \left (c^{2} x^{2}+1\right )}{2 c^{3}}+\frac {i e^{2} b d \,x^{3} \ln \left (-i c x +1\right )}{2}-\frac {b \,d^{3} \ln \left (c^{2} x^{2}+1\right )}{2 c}+\frac {3 e b \,d^{2} \arctan \left (c x \right )}{2 c^{2}}-\frac {b d \,e^{2} x^{2}}{2 c}-\frac {3 e b \,d^{2} x}{2 c}-\frac {e^{3} b \arctan \left (c x \right )}{4 c^{4}}+\frac {x^{4} e^{3} a}{4}-\frac {b \,e^{3} x^{3}}{12 c}+\frac {e^{3} b x}{4 c^{3}}+x^{3} e^{2} d a +\frac {3 x^{2} e \,d^{2} a}{2}+x \,d^{3} a\) | \(275\) |
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Time = 0.26 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.36 \[ \int (d+e x)^3 (a+b \arctan (c x)) \, dx=\frac {3 \, a c^{4} e^{3} x^{4} + {\left (12 \, a c^{4} d e^{2} - b c^{3} e^{3}\right )} x^{3} + 6 \, {\left (3 \, a c^{4} d^{2} e - b c^{3} d e^{2}\right )} x^{2} + 3 \, {\left (4 \, a c^{4} d^{3} - 6 \, b c^{3} d^{2} e + b c e^{3}\right )} x + 3 \, {\left (b c^{4} e^{3} x^{4} + 4 \, b c^{4} d e^{2} x^{3} + 6 \, b c^{4} d^{2} e x^{2} + 4 \, b c^{4} d^{3} x + 6 \, b c^{2} d^{2} e - b e^{3}\right )} \arctan \left (c x\right ) - 6 \, {\left (b c^{3} d^{3} - b c d e^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{12 \, c^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (129) = 258\).
Time = 0.37 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.82 \[ \int (d+e x)^3 (a+b \arctan (c x)) \, dx=\begin {cases} a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + a d e^{2} x^{3} + \frac {a e^{3} x^{4}}{4} + b d^{3} x \operatorname {atan}{\left (c x \right )} + \frac {3 b d^{2} e x^{2} \operatorname {atan}{\left (c x \right )}}{2} + b d e^{2} x^{3} \operatorname {atan}{\left (c x \right )} + \frac {b e^{3} x^{4} \operatorname {atan}{\left (c x \right )}}{4} - \frac {b d^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c} - \frac {3 b d^{2} e x}{2 c} - \frac {b d e^{2} x^{2}}{2 c} - \frac {b e^{3} x^{3}}{12 c} + \frac {3 b d^{2} e \operatorname {atan}{\left (c x \right )}}{2 c^{2}} + \frac {b d e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c^{3}} + \frac {b e^{3} x}{4 c^{3}} - \frac {b e^{3} \operatorname {atan}{\left (c x \right )}}{4 c^{4}} & \text {for}\: c \neq 0 \\a \left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.29 \[ \int (d+e x)^3 (a+b \arctan (c x)) \, dx=\frac {1}{4} \, a e^{3} x^{4} + a d e^{2} x^{3} + \frac {3}{2} \, a d^{2} e x^{2} + \frac {3}{2} \, {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} b d^{2} e + \frac {1}{2} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b d e^{2} + \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b e^{3} + a d^{3} x + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d^{3}}{2 \, c} \]
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\[ \int (d+e x)^3 (a+b \arctan (c x)) \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )} \,d x } \]
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Time = 0.87 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.37 \[ \int (d+e x)^3 (a+b \arctan (c x)) \, dx=\frac {a\,e^3\,x^4}{4}+a\,d^3\,x-\frac {b\,d^3\,\ln \left (c^2\,x^2+1\right )}{2\,c}-\frac {b\,e^3\,x^3}{12\,c}+b\,d^3\,x\,\mathrm {atan}\left (c\,x\right )+\frac {3\,a\,d^2\,e\,x^2}{2}+a\,d\,e^2\,x^3+\frac {b\,e^3\,x}{4\,c^3}-\frac {b\,e^3\,\mathrm {atan}\left (c\,x\right )}{4\,c^4}+\frac {b\,e^3\,x^4\,\mathrm {atan}\left (c\,x\right )}{4}-\frac {3\,b\,d^2\,e\,x}{2\,c}+\frac {3\,b\,d^2\,e\,\mathrm {atan}\left (c\,x\right )}{2\,c^2}+\frac {3\,b\,d^2\,e\,x^2\,\mathrm {atan}\left (c\,x\right )}{2}+b\,d\,e^2\,x^3\,\mathrm {atan}\left (c\,x\right )+\frac {b\,d\,e^2\,\ln \left (c^2\,x^2+1\right )}{2\,c^3}-\frac {b\,d\,e^2\,x^2}{2\,c} \]
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