Optimal. Leaf size=17 \[ \frac {(a+b x)^5}{5 b c} \]
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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 {(a+b x)^5}{5 b c} \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} \, dx &=\frac {\int (a+b x)^4 \, dx}{c}\\ &=\frac {(a+b x)^5}{5 b c}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {(a+b x)^5}{5 b c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 16, normalized size = 0.94
method | result | size |
default | \(\frac {\left (b x +a \right )^{5}}{5 b c}\) | \(16\) |
gosper | \(\frac {x \left (b^{4} x^{4}+5 a \,b^{3} x^{3}+10 a^{2} b^{2} x^{2}+10 a^{3} b x +5 a^{4}\right )}{5 c}\) | \(47\) |
norman | \(\frac {a^{4} x}{c}+\frac {a \,b^{3} x^{4}}{c}+\frac {b^{4} x^{5}}{5 c}+\frac {2 a^{3} b \,x^{2}}{c}+\frac {2 b^{2} a^{2} x^{3}}{c}\) | \(58\) |
risch | \(\frac {b^{4} x^{5}}{5 c}+\frac {a \,b^{3} x^{4}}{c}+\frac {2 b^{2} a^{2} x^{3}}{c}+\frac {2 a^{3} b \,x^{2}}{c}+\frac {a^{4} x}{c}+\frac {a^{5}}{5 b c}\) | \(69\) |
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. 48 vs.
\(2 (15) = 30\).
time = 0.28, size = 48, normalized size = 2.82 \begin {gather*} \frac {b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 5 \, a^{4} x}{5 \, c} \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. 48 vs.
\(2 (15) = 30\).
time = 0.55, size = 48, normalized size = 2.82 \begin {gather*} \frac {b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 5 \, a^{4} x}{5 \, c} \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. 51 vs.
\(2 (10) = 20\).
time = 0.02, size = 51, normalized size = 3.00 \begin {gather*} \frac {a^{4} x}{c} + \frac {2 a^{3} b x^{2}}{c} + \frac {2 a^{2} b^{2} x^{3}}{c} + \frac {a b^{3} x^{4}}{c} + \frac {b^{4} x^{5}}{5 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (15) = 30\).
time = 2.23, size = 48, normalized size = 2.82 \begin {gather*} \frac {b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 5 \, a^{4} x}{5 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 57, normalized size = 3.35 \begin {gather*} \frac {a^4\,x}{c}+\frac {b^4\,x^5}{5\,c}+\frac {2\,a^3\,b\,x^2}{c}+\frac {a\,b^3\,x^4}{c}+\frac {2\,a^2\,b^2\,x^3}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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