Optimal. Leaf size=17 \[ \frac {(a+b x)^3}{3 b c^3} \]
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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)^3}{3 b c^3} \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)^3} \, dx &=\frac {\int (a+b x)^2 \, dx}{c^3}\\ &=\frac {(a+b x)^3}{3 b c^3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {(a+b x)^3}{3 b c^3} \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 | \(\frac {\left (b x +a \right )^{3}}{3 b \,c^{3}}\) | \(16\) |
gosper | \(\frac {x \left (x^{2} b^{2}+3 a b x +3 a^{2}\right )}{3 c^{3}}\) | \(25\) |
risch | \(\frac {b^{2} x^{3}}{3 c^{3}}+\frac {b a \,x^{2}}{c^{3}}+\frac {a^{2} x}{c^{3}}+\frac {a^{3}}{3 b \,c^{3}}\) | \(41\) |
norman | \(\frac {\frac {b^{4} x^{5}}{3 c}+\frac {5 a \,b^{3} x^{4}}{3 c}-\frac {3 a^{5}}{b c}+\frac {10 b^{2} a^{2} x^{3}}{3 c}-\frac {5 a^{4} x}{c}}{c^{2} \left (b x +a \right )^{2}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 26, normalized size = 1.53 \begin {gather*} \frac {b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x}{3 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 26, normalized size = 1.53 \begin {gather*} \frac {b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x}{3 \, c^{3}} \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. 29 vs.
\(2 (12) = 24\).
time = 0.03, size = 29, normalized size = 1.71 \begin {gather*} \frac {a^{2} x}{c^{3}} + \frac {a b x^{2}}{c^{3}} + \frac {b^{2} x^{3}}{3 c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.92, size = 26, normalized size = 1.53 \begin {gather*} \frac {b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x}{3 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 24, normalized size = 1.41 \begin {gather*} \frac {x\,\left (3\,a^2+3\,a\,b\,x+b^2\,x^2\right )}{3\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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