Optimal. Leaf size=10 \[ \frac {x}{(a-b x)^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {641, 34}
\begin {gather*} \frac {x}{(a-b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 34
Rule 641
Rubi steps
\begin {align*} \int \frac {(a+b x)^4}{\left (a^2-b^2 x^2\right )^3} \, dx &=\int \frac {a+b x}{(a-b x)^3} \, dx\\ &=\frac {x}{(a-b x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {x}{(a-b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(27\) vs.
\(2(10)=20\).
time = 0.44, size = 28, normalized size = 2.80
method | result | size |
gosper | \(\frac {x}{\left (-b x +a \right )^{2}}\) | \(11\) |
risch | \(\frac {x}{\left (-b x +a \right )^{2}}\) | \(11\) |
default | \(-\frac {1}{b \left (-b x +a \right )}+\frac {a}{b \left (-b x +a \right )^{2}}\) | \(28\) |
norman | \(\frac {b^{2} x^{3}+2 a b \,x^{2}+a^{2} x}{\left (-b^{2} x^{2}+a^{2}\right )^{2}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 20, normalized size = 2.00 \begin {gather*} \frac {x}{b^{2} x^{2} - 2 \, a b x + a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.70, size = 20, normalized size = 2.00 \begin {gather*} \frac {x}{b^{2} x^{2} - 2 \, a b x + a^{2}} \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. 17 vs.
\(2 (7) = 14\).
time = 0.10, size = 17, normalized size = 1.70 \begin {gather*} \frac {x}{a^{2} - 2 a b x + b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.89, size = 11, normalized size = 1.10 \begin {gather*} \frac {x}{{\left (b x - a\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.40, size = 10, normalized size = 1.00 \begin {gather*} \frac {x}{{\left (a-b\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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