Optimal. Leaf size=82 \[ \frac {1}{2} a^2 c x^2+\frac {1}{4} a^2 e x^4+\frac {2}{5} a b c x^5+\frac {d \left (a+b x^3\right )^3}{9 b}+\frac {2}{7} a b e x^7+\frac {1}{8} b^2 c x^8+\frac {1}{10} b^2 e x^{10} \]
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Rubi [A] time = 0.05, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1582, 1850} \[ \frac {1}{2} a^2 c x^2+\frac {1}{4} a^2 e x^4+\frac {2}{5} a b c x^5+\frac {d \left (a+b x^3\right )^3}{9 b}+\frac {2}{7} a b e x^7+\frac {1}{8} b^2 c x^8+\frac {1}{10} b^2 e x^{10} \]
Antiderivative was successfully verified.
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Rule 1582
Rule 1850
Rubi steps
\begin {align*} \int x \left (c+d x+e x^2\right ) \left (a+b x^3\right )^2 \, dx &=\frac {d \left (a+b x^3\right )^3}{9 b}+\int \left (a+b x^3\right )^2 \left (-d x^2+x \left (c+d x+e x^2\right )\right ) \, dx\\ &=\frac {d \left (a+b x^3\right )^3}{9 b}+\int \left (a^2 c x+a^2 e x^3+2 a b c x^4+2 a b e x^6+b^2 c x^7+b^2 e x^9\right ) \, dx\\ &=\frac {1}{2} a^2 c x^2+\frac {1}{4} a^2 e x^4+\frac {2}{5} a b c x^5+\frac {2}{7} a b e x^7+\frac {1}{8} b^2 c x^8+\frac {1}{10} b^2 e x^{10}+\frac {d \left (a+b x^3\right )^3}{9 b}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 97, normalized size = 1.18 \[ \frac {1}{2} a^2 c x^2+\frac {1}{3} a^2 d x^3+\frac {1}{4} a^2 e x^4+\frac {2}{5} a b c x^5+\frac {1}{3} a b d x^6+\frac {2}{7} a b e x^7+\frac {1}{8} b^2 c x^8+\frac {1}{9} b^2 d x^9+\frac {1}{10} b^2 e x^{10} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 79, normalized size = 0.96 \[ \frac {1}{10} x^{10} e b^{2} + \frac {1}{9} x^{9} d b^{2} + \frac {1}{8} x^{8} c b^{2} + \frac {2}{7} x^{7} e b a + \frac {1}{3} x^{6} d b a + \frac {2}{5} x^{5} c b a + \frac {1}{4} x^{4} e a^{2} + \frac {1}{3} x^{3} d a^{2} + \frac {1}{2} x^{2} c a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 82, normalized size = 1.00 \[ \frac {1}{10} \, b^{2} x^{10} e + \frac {1}{9} \, b^{2} d x^{9} + \frac {1}{8} \, b^{2} c x^{8} + \frac {2}{7} \, a b x^{7} e + \frac {1}{3} \, a b d x^{6} + \frac {2}{5} \, a b c x^{5} + \frac {1}{4} \, a^{2} x^{4} e + \frac {1}{3} \, a^{2} d x^{3} + \frac {1}{2} \, a^{2} c x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 80, normalized size = 0.98 \[ \frac {1}{10} b^{2} e \,x^{10}+\frac {1}{9} b^{2} d \,x^{9}+\frac {1}{8} b^{2} c \,x^{8}+\frac {2}{7} a b e \,x^{7}+\frac {1}{3} a b d \,x^{6}+\frac {2}{5} a b c \,x^{5}+\frac {1}{4} a^{2} e \,x^{4}+\frac {1}{3} a^{2} d \,x^{3}+\frac {1}{2} a^{2} c \,x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 79, normalized size = 0.96 \[ \frac {1}{10} \, b^{2} e x^{10} + \frac {1}{9} \, b^{2} d x^{9} + \frac {1}{8} \, b^{2} c x^{8} + \frac {2}{7} \, a b e x^{7} + \frac {1}{3} \, a b d x^{6} + \frac {2}{5} \, a b c x^{5} + \frac {1}{4} \, a^{2} e x^{4} + \frac {1}{3} \, a^{2} d x^{3} + \frac {1}{2} \, a^{2} c x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 79, normalized size = 0.96 \[ \frac {e\,a^2\,x^4}{4}+\frac {d\,a^2\,x^3}{3}+\frac {c\,a^2\,x^2}{2}+\frac {2\,e\,a\,b\,x^7}{7}+\frac {d\,a\,b\,x^6}{3}+\frac {2\,c\,a\,b\,x^5}{5}+\frac {e\,b^2\,x^{10}}{10}+\frac {d\,b^2\,x^9}{9}+\frac {c\,b^2\,x^8}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 94, normalized size = 1.15 \[ \frac {a^{2} c x^{2}}{2} + \frac {a^{2} d x^{3}}{3} + \frac {a^{2} e x^{4}}{4} + \frac {2 a b c x^{5}}{5} + \frac {a b d x^{6}}{3} + \frac {2 a b e x^{7}}{7} + \frac {b^{2} c x^{8}}{8} + \frac {b^{2} d x^{9}}{9} + \frac {b^{2} e x^{10}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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