Optimal. Leaf size=77 \[ a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{2} a b c x^4+\frac {2}{5} a b d x^5+\frac {e \left (a+b x^3\right )^3}{9 b}+\frac {1}{7} b^2 c x^7+\frac {1}{8} b^2 d x^8 \]
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Rubi [A] time = 0.06, antiderivative size = 77, 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} \[ a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{2} a b c x^4+\frac {2}{5} a b d x^5+\frac {e \left (a+b x^3\right )^3}{9 b}+\frac {1}{7} b^2 c x^7+\frac {1}{8} b^2 d x^8 \]
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 )^2 \, dx &=\frac {e \left (a+b x^3\right )^3}{9 b}+\int (c+d x) \left (a+b x^3\right )^2 \, dx\\ &=\frac {e \left (a+b x^3\right )^3}{9 b}+\int \left (a^2 c+a^2 d x+2 a b c x^3+2 a b d x^4+b^2 c x^6+b^2 d x^7\right ) \, dx\\ &=a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{2} a b c x^4+\frac {2}{5} a b d x^5+\frac {1}{7} b^2 c x^7+\frac {1}{8} b^2 d x^8+\frac {e \left (a+b x^3\right )^3}{9 b}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 92, normalized size = 1.19 \[ a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{3} a^2 e x^3+\frac {1}{2} a b c x^4+\frac {2}{5} a b d x^5+\frac {1}{3} a b e x^6+\frac {1}{7} b^2 c x^7+\frac {1}{8} b^2 d x^8+\frac {1}{9} b^2 e x^9 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 76, normalized size = 0.99 \[ \frac {1}{9} x^{9} e b^{2} + \frac {1}{8} x^{8} d b^{2} + \frac {1}{7} x^{7} c b^{2} + \frac {1}{3} x^{6} e b a + \frac {2}{5} x^{5} d b a + \frac {1}{2} x^{4} c b a + \frac {1}{3} x^{3} e a^{2} + \frac {1}{2} x^{2} d a^{2} + x c a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 79, normalized size = 1.03 \[ \frac {1}{9} \, b^{2} x^{9} e + \frac {1}{8} \, b^{2} d x^{8} + \frac {1}{7} \, b^{2} c x^{7} + \frac {1}{3} \, a b x^{6} e + \frac {2}{5} \, a b d x^{5} + \frac {1}{2} \, a b c x^{4} + \frac {1}{3} \, a^{2} x^{3} e + \frac {1}{2} \, a^{2} d x^{2} + a^{2} c x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 77, normalized size = 1.00 \[ \frac {1}{9} b^{2} e \,x^{9}+\frac {1}{8} b^{2} d \,x^{8}+\frac {1}{7} b^{2} c \,x^{7}+\frac {1}{3} a b e \,x^{6}+\frac {2}{5} a b d \,x^{5}+\frac {1}{2} a b c \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 76, normalized size = 0.99 \[ \frac {1}{9} \, b^{2} e x^{9} + \frac {1}{8} \, b^{2} d x^{8} + \frac {1}{7} \, b^{2} c x^{7} + \frac {1}{3} \, a b e x^{6} + \frac {2}{5} \, a b d x^{5} + \frac {1}{2} \, a b c x^{4} + \frac {1}{3} \, a^{2} e x^{3} + \frac {1}{2} \, a^{2} d x^{2} + a^{2} c x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 76, normalized size = 0.99 \[ \frac {e\,a^2\,x^3}{3}+\frac {d\,a^2\,x^2}{2}+c\,a^2\,x+\frac {e\,a\,b\,x^6}{3}+\frac {2\,d\,a\,b\,x^5}{5}+\frac {c\,a\,b\,x^4}{2}+\frac {e\,b^2\,x^9}{9}+\frac {d\,b^2\,x^8}{8}+\frac {c\,b^2\,x^7}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 88, normalized size = 1.14 \[ a^{2} c x + \frac {a^{2} d x^{2}}{2} + \frac {a^{2} e x^{3}}{3} + \frac {a b c x^{4}}{2} + \frac {2 a b d x^{5}}{5} + \frac {a b e x^{6}}{3} + \frac {b^{2} c x^{7}}{7} + \frac {b^{2} d x^{8}}{8} + \frac {b^{2} e x^{9}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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