Optimal. Leaf size=59 \[ -\frac {2 a^2 c^4 (a-b x)^5}{5 b^2}-\frac {c^4 (a-b x)^7}{7 b^2}+\frac {a c^4 (a-b x)^6}{2 b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {75} \begin {gather*} -\frac {2 a^2 c^4 (a-b x)^5}{5 b^2}-\frac {c^4 (a-b x)^7}{7 b^2}+\frac {a c^4 (a-b x)^6}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 75
Rubi steps
\begin {align*} \int x (a+b x) (a c-b c x)^4 \, dx &=\int \left (\frac {2 a^2 (a c-b c x)^4}{b}-\frac {3 a (a c-b c x)^5}{b c}+\frac {(a c-b c x)^6}{b c^2}\right ) \, dx\\ &=-\frac {2 a^2 c^4 (a-b x)^5}{5 b^2}+\frac {a c^4 (a-b x)^6}{2 b^2}-\frac {c^4 (a-b x)^7}{7 b^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 85, normalized size = 1.44 \begin {gather*} \frac {1}{2} a^5 c^4 x^2-a^4 b c^4 x^3+\frac {1}{2} a^3 b^2 c^4 x^4+\frac {2}{5} a^2 b^3 c^4 x^5-\frac {1}{2} a b^4 c^4 x^6+\frac {1}{7} b^5 c^4 x^7 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x (a+b x) (a c-b c x)^4 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.13, size = 75, normalized size = 1.27 \begin {gather*} \frac {1}{7} x^{7} c^{4} b^{5} - \frac {1}{2} x^{6} c^{4} b^{4} a + \frac {2}{5} x^{5} c^{4} b^{3} a^{2} + \frac {1}{2} x^{4} c^{4} b^{2} a^{3} - x^{3} c^{4} b a^{4} + \frac {1}{2} x^{2} c^{4} a^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 75, normalized size = 1.27 \begin {gather*} \frac {1}{7} \, b^{5} c^{4} x^{7} - \frac {1}{2} \, a b^{4} c^{4} x^{6} + \frac {2}{5} \, a^{2} b^{3} c^{4} x^{5} + \frac {1}{2} \, a^{3} b^{2} c^{4} x^{4} - a^{4} b c^{4} x^{3} + \frac {1}{2} \, a^{5} c^{4} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 76, normalized size = 1.29 \begin {gather*} \frac {1}{7} b^{5} c^{4} x^{7}-\frac {1}{2} a \,b^{4} c^{4} x^{6}+\frac {2}{5} a^{2} b^{3} c^{4} x^{5}+\frac {1}{2} a^{3} b^{2} c^{4} x^{4}-a^{4} b \,c^{4} x^{3}+\frac {1}{2} a^{5} c^{4} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 75, normalized size = 1.27 \begin {gather*} \frac {1}{7} \, b^{5} c^{4} x^{7} - \frac {1}{2} \, a b^{4} c^{4} x^{6} + \frac {2}{5} \, a^{2} b^{3} c^{4} x^{5} + \frac {1}{2} \, a^{3} b^{2} c^{4} x^{4} - a^{4} b c^{4} x^{3} + \frac {1}{2} \, a^{5} c^{4} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 75, normalized size = 1.27 \begin {gather*} \frac {a^5\,c^4\,x^2}{2}-a^4\,b\,c^4\,x^3+\frac {a^3\,b^2\,c^4\,x^4}{2}+\frac {2\,a^2\,b^3\,c^4\,x^5}{5}-\frac {a\,b^4\,c^4\,x^6}{2}+\frac {b^5\,c^4\,x^7}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 80, normalized size = 1.36 \begin {gather*} \frac {a^{5} c^{4} x^{2}}{2} - a^{4} b c^{4} x^{3} + \frac {a^{3} b^{2} c^{4} x^{4}}{2} + \frac {2 a^{2} b^{3} c^{4} x^{5}}{5} - \frac {a b^{4} c^{4} x^{6}}{2} + \frac {b^{5} c^{4} x^{7}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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