Optimal. Leaf size=107 \[ \frac {1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac {b \left (3 c^2 d-2 e\right ) \tan ^{-1}(c x)}{12 c^6}+\frac {b x \left (3 c^2 d-2 e\right )}{12 c^5}-\frac {b x^3 \left (3 c^2 d-2 e\right )}{36 c^3}-\frac {b e x^5}{30 c} \]
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Rubi [A] time = 0.11, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 4976, 459, 302, 203} \[ \frac {1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac {b x^3 \left (3 c^2 d-2 e\right )}{36 c^3}+\frac {b x \left (3 c^2 d-2 e\right )}{12 c^5}-\frac {b \left (3 c^2 d-2 e\right ) \tan ^{-1}(c x)}{12 c^6}-\frac {b e x^5}{30 c} \]
Antiderivative was successfully verified.
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Rule 14
Rule 203
Rule 302
Rule 459
Rule 4976
Rubi steps
\begin {align*} \int x^3 \left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac {1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac {x^4 \left (3 d+2 e x^2\right )}{12+12 c^2 x^2} \, dx\\ &=-\frac {b e x^5}{30 c}+\frac {1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )+\left (b c \left (-3 d+\frac {2 e}{c^2}\right )\right ) \int \frac {x^4}{12+12 c^2 x^2} \, dx\\ &=-\frac {b e x^5}{30 c}+\frac {1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )+\left (b c \left (-3 d+\frac {2 e}{c^2}\right )\right ) \int \left (-\frac {1}{12 c^4}+\frac {x^2}{12 c^2}+\frac {1}{c^4 \left (12+12 c^2 x^2\right )}\right ) \, dx\\ &=\frac {b \left (3 d-\frac {2 e}{c^2}\right ) x}{12 c^3}-\frac {b \left (3 d-\frac {2 e}{c^2}\right ) x^3}{36 c}-\frac {b e x^5}{30 c}+\frac {1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac {\left (b \left (-3 d+\frac {2 e}{c^2}\right )\right ) \int \frac {1}{12+12 c^2 x^2} \, dx}{c^3}\\ &=\frac {b \left (3 d-\frac {2 e}{c^2}\right ) x}{12 c^3}-\frac {b \left (3 d-\frac {2 e}{c^2}\right ) x^3}{36 c}-\frac {b e x^5}{30 c}-\frac {b \left (3 d-\frac {2 e}{c^2}\right ) \tan ^{-1}(c x)}{12 c^4}+\frac {1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 127, normalized size = 1.19 \[ \frac {1}{4} a d x^4+\frac {1}{6} a e x^6+\frac {b e \tan ^{-1}(c x)}{6 c^6}-\frac {b e x}{6 c^5}-\frac {b d \tan ^{-1}(c x)}{4 c^4}+\frac {b d x}{4 c^3}+\frac {b e x^3}{18 c^3}+\frac {1}{4} b d x^4 \tan ^{-1}(c x)-\frac {b d x^3}{12 c}+\frac {1}{6} b e x^6 \tan ^{-1}(c x)-\frac {b e x^5}{30 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 110, normalized size = 1.03 \[ \frac {30 \, a c^{6} e x^{6} + 45 \, a c^{6} d x^{4} - 6 \, b c^{5} e x^{5} - 5 \, {\left (3 \, b c^{5} d - 2 \, b c^{3} e\right )} x^{3} + 15 \, {\left (3 \, b c^{3} d - 2 \, b c e\right )} x + 15 \, {\left (2 \, b c^{6} e x^{6} + 3 \, b c^{6} d x^{4} - 3 \, b c^{2} d + 2 \, b e\right )} \arctan \left (c x\right )}{180 \, c^{6}} \]
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 = 106, normalized size = 0.99 \[ \frac {a e \,x^{6}}{6}+\frac {a \,x^{4} d}{4}+\frac {b \arctan \left (c x \right ) e \,x^{6}}{6}+\frac {b \arctan \left (c x \right ) x^{4} d}{4}-\frac {b e \,x^{5}}{30 c}-\frac {b d \,x^{3}}{12 c}+\frac {b \,x^{3} e}{18 c^{3}}+\frac {b d x}{4 c^{3}}-\frac {b e x}{6 c^{5}}-\frac {b d \arctan \left (c x \right )}{4 c^{4}}+\frac {b \arctan \left (c x \right ) e}{6 c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 108, normalized size = 1.01 \[ \frac {1}{6} \, a e x^{6} + \frac {1}{4} \, a d x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d + \frac {1}{90} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b e \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 105, normalized size = 0.98 \[ \frac {a\,d\,x^4}{4}+\frac {a\,e\,x^6}{6}+\frac {b\,d\,x}{4\,c^3}-\frac {b\,e\,x}{6\,c^5}-\frac {b\,d\,\mathrm {atan}\left (c\,x\right )}{4\,c^4}+\frac {b\,e\,\mathrm {atan}\left (c\,x\right )}{6\,c^6}+\frac {b\,d\,x^4\,\mathrm {atan}\left (c\,x\right )}{4}+\frac {b\,e\,x^6\,\mathrm {atan}\left (c\,x\right )}{6}-\frac {b\,d\,x^3}{12\,c}-\frac {b\,e\,x^5}{30\,c}+\frac {b\,e\,x^3}{18\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.00, size = 138, normalized size = 1.29 \[ \begin {cases} \frac {a d x^{4}}{4} + \frac {a e x^{6}}{6} + \frac {b d x^{4} \operatorname {atan}{\left (c x \right )}}{4} + \frac {b e x^{6} \operatorname {atan}{\left (c x \right )}}{6} - \frac {b d x^{3}}{12 c} - \frac {b e x^{5}}{30 c} + \frac {b d x}{4 c^{3}} + \frac {b e x^{3}}{18 c^{3}} - \frac {b d \operatorname {atan}{\left (c x \right )}}{4 c^{4}} - \frac {b e x}{6 c^{5}} + \frac {b e \operatorname {atan}{\left (c x \right )}}{6 c^{6}} & \text {for}\: c \neq 0 \\a \left (\frac {d x^{4}}{4} + \frac {e x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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