Optimal. Leaf size=105 \[ a^3 c x+\frac {1}{2} a^3 d x^2+\frac {3}{4} a^2 b c x^4+\frac {3}{5} a^2 b d x^5+\frac {3}{7} a b^2 c x^7+\frac {3}{8} a b^2 d x^8+\frac {e \left (a+b x^3\right )^4}{12 b}+\frac {1}{10} b^3 c x^{10}+\frac {1}{11} b^3 d x^{11} \]
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Rubi [A] time = 0.10, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1582, 1850} \[ \frac {3}{4} a^2 b c x^4+\frac {3}{5} a^2 b d x^5+a^3 c x+\frac {1}{2} a^3 d x^2+\frac {3}{7} a b^2 c x^7+\frac {3}{8} a b^2 d x^8+\frac {e \left (a+b x^3\right )^4}{12 b}+\frac {1}{10} b^3 c x^{10}+\frac {1}{11} b^3 d x^{11} \]
Antiderivative was successfully verified.
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Rule 1582
Rule 1850
Rubi steps
\begin {align*} \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^3 \, dx &=\frac {e \left (a+b x^3\right )^4}{12 b}+\int (c+d x) \left (a+b x^3\right )^3 \, dx\\ &=\frac {e \left (a+b x^3\right )^4}{12 b}+\int \left (a^3 c+a^3 d x+3 a^2 b c x^3+3 a^2 b d x^4+3 a b^2 c x^6+3 a b^2 d x^7+b^3 c x^9+b^3 d x^{10}\right ) \, dx\\ &=a^3 c x+\frac {1}{2} a^3 d x^2+\frac {3}{4} a^2 b c x^4+\frac {3}{5} a^2 b d x^5+\frac {3}{7} a b^2 c x^7+\frac {3}{8} a b^2 d x^8+\frac {1}{10} b^3 c x^{10}+\frac {1}{11} b^3 d x^{11}+\frac {e \left (a+b x^3\right )^4}{12 b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 134, normalized size = 1.28 \[ a^3 c x+\frac {1}{2} a^3 d x^2+\frac {1}{3} a^3 e x^3+\frac {3}{4} a^2 b c x^4+\frac {3}{5} a^2 b d x^5+\frac {1}{2} a^2 b e x^6+\frac {3}{7} a b^2 c x^7+\frac {3}{8} a b^2 d x^8+\frac {1}{3} a b^2 e x^9+\frac {1}{10} b^3 c x^{10}+\frac {1}{11} b^3 d x^{11}+\frac {1}{12} b^3 e x^{12} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 112, normalized size = 1.07 \[ \frac {1}{12} x^{12} e b^{3} + \frac {1}{11} x^{11} d b^{3} + \frac {1}{10} x^{10} c b^{3} + \frac {1}{3} x^{9} e b^{2} a + \frac {3}{8} x^{8} d b^{2} a + \frac {3}{7} x^{7} c b^{2} a + \frac {1}{2} x^{6} e b a^{2} + \frac {3}{5} x^{5} d b a^{2} + \frac {3}{4} x^{4} c b a^{2} + \frac {1}{3} x^{3} e a^{3} + \frac {1}{2} x^{2} d a^{3} + x c a^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 116, normalized size = 1.10 \[ \frac {1}{12} \, b^{3} x^{12} e + \frac {1}{11} \, b^{3} d x^{11} + \frac {1}{10} \, b^{3} c x^{10} + \frac {1}{3} \, a b^{2} x^{9} e + \frac {3}{8} \, a b^{2} d x^{8} + \frac {3}{7} \, a b^{2} c x^{7} + \frac {1}{2} \, a^{2} b x^{6} e + \frac {3}{5} \, a^{2} b d x^{5} + \frac {3}{4} \, a^{2} b c x^{4} + \frac {1}{3} \, a^{3} x^{3} e + \frac {1}{2} \, a^{3} d x^{2} + a^{3} c x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 113, normalized size = 1.08 \[ \frac {1}{12} b^{3} e \,x^{12}+\frac {1}{11} b^{3} d \,x^{11}+\frac {1}{10} b^{3} c \,x^{10}+\frac {1}{3} a \,b^{2} e \,x^{9}+\frac {3}{8} a \,b^{2} d \,x^{8}+\frac {3}{7} a \,b^{2} c \,x^{7}+\frac {1}{2} a^{2} b e \,x^{6}+\frac {3}{5} a^{2} b d \,x^{5}+\frac {3}{4} a^{2} b c \,x^{4}+\frac {1}{3} a^{3} e \,x^{3}+\frac {1}{2} a^{3} d \,x^{2}+a^{3} c x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 112, normalized size = 1.07 \[ \frac {1}{12} \, b^{3} e x^{12} + \frac {1}{11} \, b^{3} d x^{11} + \frac {1}{10} \, b^{3} c x^{10} + \frac {1}{3} \, a b^{2} e x^{9} + \frac {3}{8} \, a b^{2} d x^{8} + \frac {3}{7} \, a b^{2} c x^{7} + \frac {1}{2} \, a^{2} b e x^{6} + \frac {3}{5} \, a^{2} b d x^{5} + \frac {3}{4} \, a^{2} b c x^{4} + \frac {1}{3} \, a^{3} e x^{3} + \frac {1}{2} \, a^{3} d x^{2} + a^{3} c x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 112, normalized size = 1.07 \[ \frac {e\,a^3\,x^3}{3}+\frac {d\,a^3\,x^2}{2}+c\,a^3\,x+\frac {e\,a^2\,b\,x^6}{2}+\frac {3\,d\,a^2\,b\,x^5}{5}+\frac {3\,c\,a^2\,b\,x^4}{4}+\frac {e\,a\,b^2\,x^9}{3}+\frac {3\,d\,a\,b^2\,x^8}{8}+\frac {3\,c\,a\,b^2\,x^7}{7}+\frac {e\,b^3\,x^{12}}{12}+\frac {d\,b^3\,x^{11}}{11}+\frac {c\,b^3\,x^{10}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 134, normalized size = 1.28 \[ a^{3} c x + \frac {a^{3} d x^{2}}{2} + \frac {a^{3} e x^{3}}{3} + \frac {3 a^{2} b c x^{4}}{4} + \frac {3 a^{2} b d x^{5}}{5} + \frac {a^{2} b e x^{6}}{2} + \frac {3 a b^{2} c x^{7}}{7} + \frac {3 a b^{2} d x^{8}}{8} + \frac {a b^{2} e x^{9}}{3} + \frac {b^{3} c x^{10}}{10} + \frac {b^{3} d x^{11}}{11} + \frac {b^{3} e x^{12}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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