Optimal. Leaf size=83 \[ -\frac {32 a^5 \log (a-b x)}{b}-16 a^4 x-\frac {4 a^3 (a+b x)^2}{b}-\frac {4 a^2 (a+b x)^3}{3 b}-\frac {a (a+b x)^4}{2 b}-\frac {(a+b x)^5}{5 b} \]
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Rubi [A] time = 0.03, antiderivative size = 83, 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 = {627, 43} \[ -\frac {4 a^3 (a+b x)^2}{b}-\frac {4 a^2 (a+b x)^3}{3 b}-\frac {32 a^5 \log (a-b x)}{b}-16 a^4 x-\frac {a (a+b x)^4}{2 b}-\frac {(a+b x)^5}{5 b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 627
Rubi steps
\begin {align*} \int \frac {(a+b x)^6}{a^2-b^2 x^2} \, dx &=\int \frac {(a+b x)^5}{a-b x} \, dx\\ &=\int \left (-16 a^4+\frac {32 a^5}{a-b x}-8 a^3 (a+b x)-4 a^2 (a+b x)^2-2 a (a+b x)^3-(a+b x)^4\right ) \, dx\\ &=-16 a^4 x-\frac {4 a^3 (a+b x)^2}{b}-\frac {4 a^2 (a+b x)^3}{3 b}-\frac {a (a+b x)^4}{2 b}-\frac {(a+b x)^5}{5 b}-\frac {32 a^5 \log (a-b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 65, normalized size = 0.78 \[ -\frac {32 a^5 \log (a-b x)}{b}-31 a^4 x-13 a^3 b x^2-\frac {16}{3} a^2 b^2 x^3-\frac {3}{2} a b^3 x^4-\frac {b^4 x^5}{5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 65, normalized size = 0.78 \[ -\frac {6 \, b^{5} x^{5} + 45 \, a b^{4} x^{4} + 160 \, a^{2} b^{3} x^{3} + 390 \, a^{3} b^{2} x^{2} + 930 \, a^{4} b x + 960 \, a^{5} \log \left (b x - a\right )}{30 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 72, normalized size = 0.87 \[ -\frac {32 \, a^{5} \log \left ({\left | b x - a \right |}\right )}{b} - \frac {6 \, b^{9} x^{5} + 45 \, a b^{8} x^{4} + 160 \, a^{2} b^{7} x^{3} + 390 \, a^{3} b^{6} x^{2} + 930 \, a^{4} b^{5} x}{30 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 61, normalized size = 0.73 \[ -\frac {b^{4} x^{5}}{5}-\frac {3 a \,b^{3} x^{4}}{2}-\frac {16 a^{2} b^{2} x^{3}}{3}-13 a^{3} b \,x^{2}-\frac {32 a^{5} \ln \left (b x -a \right )}{b}-31 a^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 60, normalized size = 0.72 \[ -\frac {1}{5} \, b^{4} x^{5} - \frac {3}{2} \, a b^{3} x^{4} - \frac {16}{3} \, a^{2} b^{2} x^{3} - 13 \, a^{3} b x^{2} - 31 \, a^{4} x - \frac {32 \, a^{5} \log \left (b x - a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 60, normalized size = 0.72 \[ -31\,a^4\,x-\frac {b^4\,x^5}{5}-13\,a^3\,b\,x^2-\frac {3\,a\,b^3\,x^4}{2}-\frac {32\,a^5\,\ln \left (b\,x-a\right )}{b}-\frac {16\,a^2\,b^2\,x^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 65, normalized size = 0.78 \[ - \frac {32 a^{5} \log {\left (- a + b x \right )}}{b} - 31 a^{4} x - 13 a^{3} b x^{2} - \frac {16 a^{2} b^{2} x^{3}}{3} - \frac {3 a b^{3} x^{4}}{2} - \frac {b^{4} x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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