Optimal. Leaf size=43 \[ \frac {2 a^2}{b (a-b x)^2}-\frac {4 a}{b (a-b x)}-\frac {\log (a-b x)}{b} \]
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Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {641, 45}
\begin {gather*} \frac {2 a^2}{b (a-b x)^2}-\frac {4 a}{b (a-b x)}-\frac {\log (a-b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 641
Rubi steps
\begin {align*} \int \frac {(a+b x)^5}{\left (a^2-b^2 x^2\right )^3} \, dx &=\int \frac {(a+b x)^2}{(a-b x)^3} \, dx\\ &=\int \left (\frac {4 a^2}{(a-b x)^3}-\frac {4 a}{(a-b x)^2}+\frac {1}{a-b x}\right ) \, dx\\ &=\frac {2 a^2}{b (a-b x)^2}-\frac {4 a}{b (a-b x)}-\frac {\log (a-b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 30, normalized size = 0.70 \begin {gather*} -\frac {\frac {2 a (a-2 b x)}{(a-b x)^2}+\log (a-b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 44, normalized size = 1.02
method | result | size |
risch | \(\frac {4 a x -\frac {2 a^{2}}{b}}{\left (-b x +a \right )^{2}}-\frac {\ln \left (-b x +a \right )}{b}\) | \(36\) |
default | \(\frac {2 a^{2}}{b \left (-b x +a \right )^{2}}-\frac {4 a}{b \left (-b x +a \right )}-\frac {\ln \left (-b x +a \right )}{b}\) | \(44\) |
norman | \(\frac {4 a \,b^{2} x^{3}-\frac {2 a^{4}}{b}+6 a^{2} b \,x^{2}}{\left (-b^{2} x^{2}+a^{2}\right )^{2}}-\frac {\ln \left (-b x +a \right )}{b}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 49, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (2 \, a b x - a^{2}\right )}}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} - \frac {\log \left (b x - a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.69, size = 60, normalized size = 1.40 \begin {gather*} \frac {4 \, a b x - 2 \, a^{2} - {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x - a\right )}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 41, normalized size = 0.95 \begin {gather*} - \frac {2 a^{2} - 4 a b x}{a^{2} b - 2 a b^{2} x + b^{3} x^{2}} - \frac {\log {\left (- a + b x \right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.14, size = 40, normalized size = 0.93 \begin {gather*} -\frac {\log \left ({\left | b x - a \right |}\right )}{b} + \frac {2 \, {\left (2 \, a b x - a^{2}\right )}}{{\left (b x - a\right )}^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.40, size = 46, normalized size = 1.07 \begin {gather*} \frac {4\,a\,x-\frac {2\,a^2}{b}}{a^2-2\,a\,b\,x+b^2\,x^2}-\frac {\ln \left (b\,x-a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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