Optimal. Leaf size=55 \[ \frac {1}{3} a^4 c^3 x^3-\frac {1}{2} a^3 b c^3 x^4+\frac {1}{3} a b^3 c^3 x^6-\frac {1}{7} b^4 c^3 x^7 \]
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Rubi [A] time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {75} \[ -\frac {1}{2} a^3 b c^3 x^4+\frac {1}{3} a^4 c^3 x^3+\frac {1}{3} a b^3 c^3 x^6-\frac {1}{7} b^4 c^3 x^7 \]
Antiderivative was successfully verified.
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Rule 75
Rubi steps
\begin {align*} \int x^2 (a+b x) (a c-b c x)^3 \, dx &=\int \left (a^4 c^3 x^2-2 a^3 b c^3 x^3+2 a b^3 c^3 x^5-b^4 c^3 x^6\right ) \, dx\\ &=\frac {1}{3} a^4 c^3 x^3-\frac {1}{2} a^3 b c^3 x^4+\frac {1}{3} a b^3 c^3 x^6-\frac {1}{7} b^4 c^3 x^7\\ \end {align*}
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Mathematica [A] time = 0.00, size = 47, normalized size = 0.85 \[ c^3 \left (\frac {a^4 x^3}{3}-\frac {1}{2} a^3 b x^4+\frac {1}{3} a b^3 x^6-\frac {1}{7} b^4 x^7\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 47, normalized size = 0.85 \[ -\frac {1}{7} x^{7} c^{3} b^{4} + \frac {1}{3} x^{6} c^{3} b^{3} a - \frac {1}{2} x^{4} c^{3} b a^{3} + \frac {1}{3} x^{3} c^{3} a^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 47, normalized size = 0.85 \[ -\frac {1}{7} \, b^{4} c^{3} x^{7} + \frac {1}{3} \, a b^{3} c^{3} x^{6} - \frac {1}{2} \, a^{3} b c^{3} x^{4} + \frac {1}{3} \, a^{4} c^{3} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 48, normalized size = 0.87 \[ -\frac {1}{7} b^{4} c^{3} x^{7}+\frac {1}{3} a \,b^{3} c^{3} x^{6}-\frac {1}{2} a^{3} b \,c^{3} x^{4}+\frac {1}{3} a^{4} c^{3} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 47, normalized size = 0.85 \[ -\frac {1}{7} \, b^{4} c^{3} x^{7} + \frac {1}{3} \, a b^{3} c^{3} x^{6} - \frac {1}{2} \, a^{3} b c^{3} x^{4} + \frac {1}{3} \, a^{4} c^{3} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 47, normalized size = 0.85 \[ \frac {a^4\,c^3\,x^3}{3}-\frac {a^3\,b\,c^3\,x^4}{2}+\frac {a\,b^3\,c^3\,x^6}{3}-\frac {b^4\,c^3\,x^7}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 49, normalized size = 0.89 \[ \frac {a^{4} c^{3} x^{3}}{3} - \frac {a^{3} b c^{3} x^{4}}{2} + \frac {a b^{3} c^{3} x^{6}}{3} - \frac {b^{4} c^{3} x^{7}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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