Optimal. Leaf size=17 \[ -\frac {1}{2 b c^8 (a+b x)^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {21, 32}
\begin {gather*} -\frac {1}{2 b c^8 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin {align*} \int \frac {(a+b x)^5}{(a c+b c x)^8} \, dx &=\frac {\int \frac {1}{(a+b x)^3} \, dx}{c^8}\\ &=-\frac {1}{2 b c^8 (a+b x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{2 b c^8 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 16, normalized size = 0.94
method | result | size |
gosper | \(-\frac {1}{2 b \,c^{8} \left (b x +a \right )^{2}}\) | \(16\) |
default | \(-\frac {1}{2 b \,c^{8} \left (b x +a \right )^{2}}\) | \(16\) |
risch | \(-\frac {1}{2 b \,c^{8} \left (b x +a \right )^{2}}\) | \(16\) |
norman | \(\frac {-\frac {5 a^{3} b \,x^{2}}{c}-\frac {a^{5}}{2 b c}-\frac {b^{4} x^{5}}{2 c}-\frac {5 a \,b^{3} x^{4}}{2 c}-\frac {5 b^{2} a^{2} x^{3}}{c}-\frac {5 a^{4} x}{2 c}}{c^{7} \left (b x +a \right )^{7}}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs.
\(2 (15) = 30\).
time = 0.29, size = 33, normalized size = 1.94 \begin {gather*} -\frac {1}{2 \, {\left (b^{3} c^{8} x^{2} + 2 \, a b^{2} c^{8} x + a^{2} b c^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs.
\(2 (15) = 30\).
time = 0.43, size = 33, normalized size = 1.94 \begin {gather*} -\frac {1}{2 \, {\left (b^{3} c^{8} x^{2} + 2 \, a b^{2} c^{8} x + a^{2} b c^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs.
\(2 (15) = 30\).
time = 0.10, size = 36, normalized size = 2.12 \begin {gather*} - \frac {1}{2 a^{2} b c^{8} + 4 a b^{2} c^{8} x + 2 b^{3} c^{8} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.14, size = 15, normalized size = 0.88 \begin {gather*} -\frac {1}{2 \, {\left (b x + a\right )}^{2} b c^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 35, normalized size = 2.06 \begin {gather*} -\frac {1}{2\,a^2\,b\,c^8+4\,a\,b^2\,c^8\,x+2\,b^3\,c^8\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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