Optimal. Leaf size=18 \[ \frac {a x}{c^4}+\frac {b x^2}{2 c^4} \]
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Rubi [A]
time = 0.00, antiderivative size = 18, 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 = {21}
\begin {gather*} \frac {a x}{c^4}+\frac {b x^2}{2 c^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rubi steps
\begin {align*} \int \frac {(a+b x)^5}{(a c+b c x)^4} \, dx &=\frac {\int (a+b x) \, dx}{c^4}\\ &=\frac {a x}{c^4}+\frac {b x^2}{2 c^4}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 16, normalized size = 0.89 \begin {gather*} \frac {a x+\frac {b x^2}{2}}{c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 15, normalized size = 0.83
method | result | size |
gosper | \(\frac {x \left (b x +2 a \right )}{2 c^{4}}\) | \(14\) |
default | \(\frac {\frac {1}{2} x^{2} b +a x}{c^{4}}\) | \(15\) |
risch | \(\frac {a x}{c^{4}}+\frac {b \,x^{2}}{2 c^{4}}\) | \(17\) |
norman | \(\frac {\frac {b^{4} x^{5}}{2 c}+\frac {5 a \,b^{3} x^{4}}{2 c}-\frac {9 a^{5}}{2 b c}-\frac {10 a^{3} b \,x^{2}}{c}-\frac {25 a^{4} x}{2 c}}{c^{3} \left (b x +a \right )^{3}}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 15, normalized size = 0.83 \begin {gather*} \frac {b x^{2} + 2 \, a x}{2 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 15, normalized size = 0.83 \begin {gather*} \frac {b x^{2} + 2 \, a x}{2 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 15, normalized size = 0.83 \begin {gather*} \frac {a x}{c^{4}} + \frac {b x^{2}}{2 c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.56, size = 15, normalized size = 0.83 \begin {gather*} \frac {b x^{2} + 2 \, a x}{2 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 13, normalized size = 0.72 \begin {gather*} \frac {x\,\left (2\,a+b\,x\right )}{2\,c^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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