Optimal. Leaf size=59 \[ -\frac {a^2 c^5 (a-b x)^6}{3 b^2}+\frac {3 a c^5 (a-b x)^7}{7 b^2}-\frac {c^5 (a-b x)^8}{8 b^2} \]
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Rubi [A]
time = 0.02, 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 = {76}
\begin {gather*} -\frac {a^2 c^5 (a-b x)^6}{3 b^2}-\frac {c^5 (a-b x)^8}{8 b^2}+\frac {3 a c^5 (a-b x)^7}{7 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int x (a+b x) (a c-b c x)^5 \, dx &=\int \left (\frac {2 a^2 (a c-b c x)^5}{b}-\frac {3 a (a c-b c x)^6}{b c}+\frac {(a c-b c x)^7}{b c^2}\right ) \, dx\\ &=-\frac {a^2 c^5 (a-b x)^6}{3 b^2}+\frac {3 a c^5 (a-b x)^7}{7 b^2}-\frac {c^5 (a-b x)^8}{8 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 73, normalized size = 1.24 \begin {gather*} c^5 \left (\frac {a^6 x^2}{2}-\frac {4}{3} a^5 b x^3+\frac {5}{4} a^4 b^2 x^4-\frac {5}{6} a^2 b^4 x^6+\frac {4}{7} a b^5 x^7-\frac {b^6 x^8}{8}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 76, normalized size = 1.29
method | result | size |
gosper | \(\frac {x^{2} \left (-21 b^{6} x^{6}+96 a \,b^{5} x^{5}-140 a^{2} b^{4} x^{4}+210 a^{4} b^{2} x^{2}-224 a^{5} b x +84 a^{6}\right ) c^{5}}{168}\) | \(61\) |
default | \(-\frac {1}{8} b^{6} c^{5} x^{8}+\frac {4}{7} a \,b^{5} c^{5} x^{7}-\frac {5}{6} a^{2} c^{5} b^{4} x^{6}+\frac {5}{4} a^{4} c^{5} b^{2} x^{4}-\frac {4}{3} a^{5} c^{5} b \,x^{3}+\frac {1}{2} a^{6} c^{5} x^{2}\) | \(76\) |
norman | \(-\frac {1}{8} b^{6} c^{5} x^{8}+\frac {4}{7} a \,b^{5} c^{5} x^{7}-\frac {5}{6} a^{2} c^{5} b^{4} x^{6}+\frac {5}{4} a^{4} c^{5} b^{2} x^{4}-\frac {4}{3} a^{5} c^{5} b \,x^{3}+\frac {1}{2} a^{6} c^{5} x^{2}\) | \(76\) |
risch | \(-\frac {1}{8} b^{6} c^{5} x^{8}+\frac {4}{7} a \,b^{5} c^{5} x^{7}-\frac {5}{6} a^{2} c^{5} b^{4} x^{6}+\frac {5}{4} a^{4} c^{5} b^{2} x^{4}-\frac {4}{3} a^{5} c^{5} b \,x^{3}+\frac {1}{2} a^{6} c^{5} x^{2}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 75, normalized size = 1.27 \begin {gather*} -\frac {1}{8} \, b^{6} c^{5} x^{8} + \frac {4}{7} \, a b^{5} c^{5} x^{7} - \frac {5}{6} \, a^{2} b^{4} c^{5} x^{6} + \frac {5}{4} \, a^{4} b^{2} c^{5} x^{4} - \frac {4}{3} \, a^{5} b c^{5} x^{3} + \frac {1}{2} \, a^{6} c^{5} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.33, size = 75, normalized size = 1.27 \begin {gather*} -\frac {1}{8} \, b^{6} c^{5} x^{8} + \frac {4}{7} \, a b^{5} c^{5} x^{7} - \frac {5}{6} \, a^{2} b^{4} c^{5} x^{6} + \frac {5}{4} \, a^{4} b^{2} c^{5} x^{4} - \frac {4}{3} \, a^{5} b c^{5} x^{3} + \frac {1}{2} \, a^{6} c^{5} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 87, normalized size = 1.47 \begin {gather*} \frac {a^{6} c^{5} x^{2}}{2} - \frac {4 a^{5} b c^{5} x^{3}}{3} + \frac {5 a^{4} b^{2} c^{5} x^{4}}{4} - \frac {5 a^{2} b^{4} c^{5} x^{6}}{6} + \frac {4 a b^{5} c^{5} x^{7}}{7} - \frac {b^{6} c^{5} x^{8}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.63, size = 75, normalized size = 1.27 \begin {gather*} -\frac {1}{8} \, b^{6} c^{5} x^{8} + \frac {4}{7} \, a b^{5} c^{5} x^{7} - \frac {5}{6} \, a^{2} b^{4} c^{5} x^{6} + \frac {5}{4} \, a^{4} b^{2} c^{5} x^{4} - \frac {4}{3} \, a^{5} b c^{5} x^{3} + \frac {1}{2} \, a^{6} c^{5} 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^6\,c^5\,x^2}{2}-\frac {4\,a^5\,b\,c^5\,x^3}{3}+\frac {5\,a^4\,b^2\,c^5\,x^4}{4}-\frac {5\,a^2\,b^4\,c^5\,x^6}{6}+\frac {4\,a\,b^5\,c^5\,x^7}{7}-\frac {b^6\,c^5\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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