Optimal. Leaf size=66 \[ -8 a^3 x-\frac {2 a^2 (a+b x)^2}{b}-\frac {2 a (a+b x)^3}{3 b}-\frac {(a+b x)^4}{4 b}-\frac {16 a^4 \log (a-b x)}{b} \]
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Rubi [A]
time = 0.02, antiderivative size = 66, 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 {16 a^4 \log (a-b x)}{b}-8 a^3 x-\frac {2 a^2 (a+b x)^2}{b}-\frac {2 a (a+b x)^3}{3 b}-\frac {(a+b x)^4}{4 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}{a^2-b^2 x^2} \, dx &=\int \frac {(a+b x)^4}{a-b x} \, dx\\ &=\int \left (-8 a^3+\frac {16 a^4}{a-b x}-4 a^2 (a+b x)-2 a (a+b x)^2-(a+b x)^3\right ) \, dx\\ &=-8 a^3 x-\frac {2 a^2 (a+b x)^2}{b}-\frac {2 a (a+b x)^3}{3 b}-\frac {(a+b x)^4}{4 b}-\frac {16 a^4 \log (a-b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 54, normalized size = 0.82 \begin {gather*} -15 a^3 x-\frac {11}{2} a^2 b x^2-\frac {5}{3} a b^2 x^3-\frac {b^3 x^4}{4}-\frac {16 a^4 \log (a-b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 49, normalized size = 0.74
method | result | size |
default | \(-\frac {b^{3} x^{4}}{4}-\frac {5 a \,b^{2} x^{3}}{3}-\frac {11 a^{2} b \,x^{2}}{2}-15 a^{3} x -\frac {16 a^{4} \ln \left (-b x +a \right )}{b}\) | \(49\) |
norman | \(-\frac {b^{3} x^{4}}{4}-\frac {5 a \,b^{2} x^{3}}{3}-\frac {11 a^{2} b \,x^{2}}{2}-15 a^{3} x -\frac {16 a^{4} \ln \left (-b x +a \right )}{b}\) | \(49\) |
risch | \(-\frac {b^{3} x^{4}}{4}-\frac {5 a \,b^{2} x^{3}}{3}-\frac {11 a^{2} b \,x^{2}}{2}-15 a^{3} x -\frac {16 a^{4} \ln \left (-b x +a \right )}{b}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 49, normalized size = 0.74 \begin {gather*} -\frac {1}{4} \, b^{3} x^{4} - \frac {5}{3} \, a b^{2} x^{3} - \frac {11}{2} \, a^{2} b x^{2} - 15 \, a^{3} x - \frac {16 \, a^{4} \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 = 2.18, size = 54, normalized size = 0.82 \begin {gather*} -\frac {3 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 66 \, a^{2} b^{2} x^{2} + 180 \, a^{3} b x + 192 \, a^{4} \log \left (b x - a\right )}{12 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 53, normalized size = 0.80 \begin {gather*} - \frac {16 a^{4} \log {\left (- a + b x \right )}}{b} - 15 a^{3} x - \frac {11 a^{2} b x^{2}}{2} - \frac {5 a b^{2} x^{3}}{3} - \frac {b^{3} x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.21, size = 61, normalized size = 0.92 \begin {gather*} -\frac {16 \, a^{4} \log \left ({\left | b x - a \right |}\right )}{b} - \frac {3 \, b^{7} x^{4} + 20 \, a b^{6} x^{3} + 66 \, a^{2} b^{5} x^{2} + 180 \, a^{3} b^{4} x}{12 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 49, normalized size = 0.74 \begin {gather*} -15\,a^3\,x-\frac {b^3\,x^4}{4}-\frac {11\,a^2\,b\,x^2}{2}-\frac {5\,a\,b^2\,x^3}{3}-\frac {16\,a^4\,\ln \left (b\,x-a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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