Optimal. Leaf size=240 \[ \frac {\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{10 e^2}-\frac {d \left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac {b \left (c^2 d-e\right )^4 \left (c^2 d+4 e\right ) \tan ^{-1}(c x)}{40 c^{10} e^2}-\frac {b e^2 x^7 \left (15 c^2 d-4 e\right )}{280 c^3}-\frac {b e x^5 \left (20 c^4 d^2-15 c^2 d e+4 e^2\right )}{200 c^5}+\frac {b x \left (10 c^6 d^3-20 c^4 d^2 e+15 c^2 d e^2-4 e^3\right )}{40 c^9}-\frac {b x^3 \left (10 c^6 d^3-20 c^4 d^2 e+15 c^2 d e^2-4 e^3\right )}{120 c^7}-\frac {b e^3 x^9}{90 c} \]
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Rubi [A] time = 0.46, antiderivative size = 285, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {266, 43, 4976, 12, 528, 388, 203} \[ \frac {\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{10 e^2}-\frac {d \left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}-\frac {b x \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) \left (d+e x^2\right )^2}{4200 c^5 e}+\frac {b x \left (750 c^4 d^2 e+5 c^6 d^3-1071 c^2 d e^2+420 e^3\right ) \left (d+e x^2\right )}{12600 c^7 e}+\frac {b x \left (-4977 c^4 d^2 e^2+1815 c^6 d^3 e+325 c^8 d^4+4305 c^2 d e^3-1260 e^4\right )}{12600 c^9 e}+\frac {b \left (c^2 d-e\right )^4 \left (c^2 d+4 e\right ) \tan ^{-1}(c x)}{40 c^{10} e^2}-\frac {b x \left (23 c^2 d-36 e\right ) \left (d+e x^2\right )^3}{2520 c^3 e}-\frac {b x \left (d+e x^2\right )^4}{90 c e} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 203
Rule 266
Rule 388
Rule 528
Rule 4976
Rubi steps
\begin {align*} \int x^3 \left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=-\frac {d \left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{10 e^2}-(b c) \int \frac {\left (d+e x^2\right )^4 \left (-d+4 e x^2\right )}{40 e^2 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d \left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{10 e^2}-\frac {(b c) \int \frac {\left (d+e x^2\right )^4 \left (-d+4 e x^2\right )}{1+c^2 x^2} \, dx}{40 e^2}\\ &=-\frac {b x \left (d+e x^2\right )^4}{90 c e}-\frac {d \left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{10 e^2}-\frac {b \int \frac {\left (d+e x^2\right )^3 \left (-d \left (9 c^2 d+4 e\right )+\left (23 c^2 d-36 e\right ) e x^2\right )}{1+c^2 x^2} \, dx}{360 c e^2}\\ &=-\frac {b \left (23 c^2 d-36 e\right ) x \left (d+e x^2\right )^3}{2520 c^3 e}-\frac {b x \left (d+e x^2\right )^4}{90 c e}-\frac {d \left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{10 e^2}-\frac {b \int \frac {\left (d+e x^2\right )^2 \left (-3 d \left (21 c^4 d^2+17 c^2 d e-12 e^2\right )+3 e \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) x^2\right )}{1+c^2 x^2} \, dx}{2520 c^3 e^2}\\ &=-\frac {b \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) x \left (d+e x^2\right )^2}{4200 c^5 e}-\frac {b \left (23 c^2 d-36 e\right ) x \left (d+e x^2\right )^3}{2520 c^3 e}-\frac {b x \left (d+e x^2\right )^4}{90 c e}-\frac {d \left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{10 e^2}-\frac {b \int \frac {\left (d+e x^2\right ) \left (-3 d \left (105 c^6 d^3+110 c^4 d^2 e-195 c^2 d e^2+84 e^3\right )-3 e \left (5 c^6 d^3+750 c^4 d^2 e-1071 c^2 d e^2+420 e^3\right ) x^2\right )}{1+c^2 x^2} \, dx}{12600 c^5 e^2}\\ &=\frac {b \left (5 c^6 d^3+750 c^4 d^2 e-1071 c^2 d e^2+420 e^3\right ) x \left (d+e x^2\right )}{12600 c^7 e}-\frac {b \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) x \left (d+e x^2\right )^2}{4200 c^5 e}-\frac {b \left (23 c^2 d-36 e\right ) x \left (d+e x^2\right )^3}{2520 c^3 e}-\frac {b x \left (d+e x^2\right )^4}{90 c e}-\frac {d \left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{10 e^2}-\frac {b \int \frac {-3 d \left (315 c^8 d^4+325 c^6 d^3 e-1335 c^4 d^2 e^2+1323 c^2 d e^3-420 e^4\right )-3 e \left (325 c^8 d^4+1815 c^6 d^3 e-4977 c^4 d^2 e^2+4305 c^2 d e^3-1260 e^4\right ) x^2}{1+c^2 x^2} \, dx}{37800 c^7 e^2}\\ &=\frac {b \left (325 c^8 d^4+1815 c^6 d^3 e-4977 c^4 d^2 e^2+4305 c^2 d e^3-1260 e^4\right ) x}{12600 c^9 e}+\frac {b \left (5 c^6 d^3+750 c^4 d^2 e-1071 c^2 d e^2+420 e^3\right ) x \left (d+e x^2\right )}{12600 c^7 e}-\frac {b \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) x \left (d+e x^2\right )^2}{4200 c^5 e}-\frac {b \left (23 c^2 d-36 e\right ) x \left (d+e x^2\right )^3}{2520 c^3 e}-\frac {b x \left (d+e x^2\right )^4}{90 c e}-\frac {d \left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{10 e^2}+\frac {\left (b \left (c^2 d-e\right )^4 \left (c^2 d+4 e\right )\right ) \int \frac {1}{1+c^2 x^2} \, dx}{40 c^9 e^2}\\ &=\frac {b \left (325 c^8 d^4+1815 c^6 d^3 e-4977 c^4 d^2 e^2+4305 c^2 d e^3-1260 e^4\right ) x}{12600 c^9 e}+\frac {b \left (5 c^6 d^3+750 c^4 d^2 e-1071 c^2 d e^2+420 e^3\right ) x \left (d+e x^2\right )}{12600 c^7 e}-\frac {b \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) x \left (d+e x^2\right )^2}{4200 c^5 e}-\frac {b \left (23 c^2 d-36 e\right ) x \left (d+e x^2\right )^3}{2520 c^3 e}-\frac {b x \left (d+e x^2\right )^4}{90 c e}+\frac {b \left (c^2 d-e\right )^4 \left (c^2 d+4 e\right ) \tan ^{-1}(c x)}{40 c^{10} e^2}-\frac {d \left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{10 e^2}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 262, normalized size = 1.09 \[ \frac {c x \left (315 a c^9 x^3 \left (10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right )-b \left (5 c^8 \left (210 d^3 x^2+252 d^2 e x^4+135 d e^2 x^6+28 e^3 x^8\right )-15 c^6 \left (210 d^3+140 d^2 e x^2+63 d e^2 x^4+12 e^3 x^6\right )+63 c^4 e \left (100 d^2+25 d e x^2+4 e^2 x^4\right )-105 c^2 e^2 \left (45 d+4 e x^2\right )+1260 e^3\right )\right )+315 b \tan ^{-1}(c x) \left (c^{10} x^4 \left (10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right )-10 c^6 d^3+20 c^4 d^2 e-15 c^2 d e^2+4 e^3\right )}{12600 c^{10}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 304, normalized size = 1.27 \[ \frac {1260 \, a c^{10} e^{3} x^{10} + 4725 \, a c^{10} d e^{2} x^{8} - 140 \, b c^{9} e^{3} x^{9} + 6300 \, a c^{10} d^{2} e x^{6} + 3150 \, a c^{10} d^{3} x^{4} - 45 \, {\left (15 \, b c^{9} d e^{2} - 4 \, b c^{7} e^{3}\right )} x^{7} - 63 \, {\left (20 \, b c^{9} d^{2} e - 15 \, b c^{7} d e^{2} + 4 \, b c^{5} e^{3}\right )} x^{5} - 105 \, {\left (10 \, b c^{9} d^{3} - 20 \, b c^{7} d^{2} e + 15 \, b c^{5} d e^{2} - 4 \, b c^{3} e^{3}\right )} x^{3} + 315 \, {\left (10 \, b c^{7} d^{3} - 20 \, b c^{5} d^{2} e + 15 \, b c^{3} d e^{2} - 4 \, b c e^{3}\right )} x + 315 \, {\left (4 \, b c^{10} e^{3} x^{10} + 15 \, b c^{10} d e^{2} x^{8} + 20 \, b c^{10} d^{2} e x^{6} + 10 \, b c^{10} d^{3} x^{4} - 10 \, b c^{6} d^{3} + 20 \, b c^{4} d^{2} e - 15 \, b c^{2} d e^{2} + 4 \, b e^{3}\right )} \arctan \left (c x\right )}{12600 \, c^{10}} \]
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 = 315, normalized size = 1.31 \[ \frac {b \arctan \left (c x \right ) e^{3}}{10 c^{10}}+\frac {b \,x^{3} e^{3}}{30 c^{7}}+\frac {b \arctan \left (c x \right ) x^{4} d^{3}}{4}+\frac {b \arctan \left (c x \right ) e^{3} x^{10}}{10}-\frac {b \,e^{3} x^{9}}{90 c}+\frac {b \,d^{3} x}{4 c^{3}}-\frac {b \,d^{3} x^{3}}{12 c}-\frac {b \,d^{3} \arctan \left (c x \right )}{4 c^{4}}-\frac {b \,e^{3} x}{10 c^{9}}+\frac {3 a d \,e^{2} x^{8}}{8}+\frac {a \,d^{2} e \,x^{6}}{2}+\frac {b \,x^{7} e^{3}}{70 c^{3}}-\frac {b \,x^{5} e^{3}}{50 c^{5}}-\frac {b \,d^{2} e x}{2 c^{5}}+\frac {b \arctan \left (c x \right ) d^{2} e}{2 c^{6}}-\frac {3 b \arctan \left (c x \right ) d \,e^{2}}{8 c^{8}}+\frac {3 b d \,e^{2} x}{8 c^{7}}+\frac {b \,x^{3} d^{2} e}{6 c^{3}}-\frac {b \,d^{2} e \,x^{5}}{10 c}+\frac {3 b \,x^{5} d \,e^{2}}{40 c^{3}}-\frac {3 b d \,e^{2} x^{7}}{56 c}-\frac {b \,x^{3} d \,e^{2}}{8 c^{5}}+\frac {3 b \arctan \left (c x \right ) d \,e^{2} x^{8}}{8}+\frac {b \arctan \left (c x \right ) d^{2} e \,x^{6}}{2}+\frac {a \,e^{3} x^{10}}{10}+\frac {a \,x^{4} d^{3}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 268, normalized size = 1.12 \[ \frac {1}{10} \, a e^{3} x^{10} + \frac {3}{8} \, a d e^{2} x^{8} + \frac {1}{2} \, a d^{2} e x^{6} + \frac {1}{4} \, a d^{3} x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d^{3} + \frac {1}{30} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b d^{2} e + \frac {1}{280} \, {\left (105 \, x^{8} \arctan \left (c x\right ) - c {\left (\frac {15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac {105 \, \arctan \left (c x\right )}{c^{9}}\right )}\right )} b d e^{2} + \frac {1}{3150} \, {\left (315 \, x^{10} \arctan \left (c x\right ) - c {\left (\frac {35 \, c^{8} x^{9} - 45 \, c^{6} x^{7} + 63 \, c^{4} x^{5} - 105 \, c^{2} x^{3} + 315 \, x}{c^{10}} - \frac {315 \, \arctan \left (c x\right )}{c^{11}}\right )}\right )} b e^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 599, normalized size = 2.50 \[ x^3\,\left (\frac {\frac {\frac {b\,e^3}{10\,c^3}-\frac {3\,b\,d\,e^2}{8\,c}}{c^2}+\frac {b\,d^2\,e}{2\,c}}{3\,c^2}-\frac {b\,d^3}{12\,c}\right )-x^8\,\left (\frac {a\,e^3}{8\,c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{8\,c^2}\right )+x^6\,\left (\frac {\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}}{6\,c^2}+\frac {a\,d\,e\,\left (d\,c^2+e\right )}{2\,c^2}\right )+x^7\,\left (\frac {b\,e^3}{70\,c^3}-\frac {3\,b\,d\,e^2}{56\,c}\right )+\mathrm {atan}\left (c\,x\right )\,\left (\frac {b\,d^3\,x^4}{4}+\frac {b\,d^2\,e\,x^6}{2}+\frac {3\,b\,d\,e^2\,x^8}{8}+\frac {b\,e^3\,x^{10}}{10}\right )-x^5\,\left (\frac {\frac {b\,e^3}{10\,c^3}-\frac {3\,b\,d\,e^2}{8\,c}}{5\,c^2}+\frac {b\,d^2\,e}{10\,c}\right )+x^2\,\left (\frac {\frac {\frac {\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}}{c^2}+\frac {3\,a\,d\,e\,\left (d\,c^2+e\right )}{c^2}}{c^2}-\frac {a\,d^2\,\left (d\,c^2+3\,e\right )}{c^2}}{2\,c^2}+\frac {a\,d^3}{2\,c^2}\right )-x^4\,\left (\frac {\frac {\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}}{c^2}+\frac {3\,a\,d\,e\,\left (d\,c^2+e\right )}{c^2}}{4\,c^2}-\frac {a\,d^2\,\left (d\,c^2+3\,e\right )}{4\,c^2}\right )+\frac {a\,e^3\,x^{10}}{10}-\frac {x\,\left (\frac {\frac {\frac {b\,e^3}{10\,c^3}-\frac {3\,b\,d\,e^2}{8\,c}}{c^2}+\frac {b\,d^2\,e}{2\,c}}{c^2}-\frac {b\,d^3}{4\,c}\right )}{c^2}-\frac {b\,e^3\,x^9}{90\,c}+\frac {b\,\mathrm {atan}\left (\frac {b\,c\,x\,\left (-10\,c^6\,d^3+20\,c^4\,d^2\,e-15\,c^2\,d\,e^2+4\,e^3\right )}{-10\,b\,c^6\,d^3+20\,b\,c^4\,d^2\,e-15\,b\,c^2\,d\,e^2+4\,b\,e^3}\right )\,\left (-10\,c^6\,d^3+20\,c^4\,d^2\,e-15\,c^2\,d\,e^2+4\,e^3\right )}{40\,c^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.75, size = 411, normalized size = 1.71 \[ \begin {cases} \frac {a d^{3} x^{4}}{4} + \frac {a d^{2} e x^{6}}{2} + \frac {3 a d e^{2} x^{8}}{8} + \frac {a e^{3} x^{10}}{10} + \frac {b d^{3} x^{4} \operatorname {atan}{\left (c x \right )}}{4} + \frac {b d^{2} e x^{6} \operatorname {atan}{\left (c x \right )}}{2} + \frac {3 b d e^{2} x^{8} \operatorname {atan}{\left (c x \right )}}{8} + \frac {b e^{3} x^{10} \operatorname {atan}{\left (c x \right )}}{10} - \frac {b d^{3} x^{3}}{12 c} - \frac {b d^{2} e x^{5}}{10 c} - \frac {3 b d e^{2} x^{7}}{56 c} - \frac {b e^{3} x^{9}}{90 c} + \frac {b d^{3} x}{4 c^{3}} + \frac {b d^{2} e x^{3}}{6 c^{3}} + \frac {3 b d e^{2} x^{5}}{40 c^{3}} + \frac {b e^{3} x^{7}}{70 c^{3}} - \frac {b d^{3} \operatorname {atan}{\left (c x \right )}}{4 c^{4}} - \frac {b d^{2} e x}{2 c^{5}} - \frac {b d e^{2} x^{3}}{8 c^{5}} - \frac {b e^{3} x^{5}}{50 c^{5}} + \frac {b d^{2} e \operatorname {atan}{\left (c x \right )}}{2 c^{6}} + \frac {3 b d e^{2} x}{8 c^{7}} + \frac {b e^{3} x^{3}}{30 c^{7}} - \frac {3 b d e^{2} \operatorname {atan}{\left (c x \right )}}{8 c^{8}} - \frac {b e^{3} x}{10 c^{9}} + \frac {b e^{3} \operatorname {atan}{\left (c x \right )}}{10 c^{10}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{3} x^{4}}{4} + \frac {d^{2} e x^{6}}{2} + \frac {3 d e^{2} x^{8}}{8} + \frac {e^{3} x^{10}}{10}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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