Optimal. Leaf size=55 \[ \frac{16 a^4}{b (a-b x)}+\frac{32 a^3 \log (a-b x)}{b}+17 a^2 x+3 a b x^2+\frac{b^2 x^3}{3} \]
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Rubi [A] time = 0.0410961, antiderivative size = 55, normalized size of antiderivative = 1., 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} \[ \frac{16 a^4}{b (a-b x)}+\frac{32 a^3 \log (a-b x)}{b}+17 a^2 x+3 a b x^2+\frac{b^2 x^3}{3} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^6}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac{(a+b x)^4}{(a-b x)^2} \, dx\\ &=\int \left (17 a^2+6 a b x+b^2 x^2+\frac{16 a^4}{(a-b x)^2}-\frac{32 a^3}{a-b x}\right ) \, dx\\ &=17 a^2 x+3 a b x^2+\frac{b^2 x^3}{3}+\frac{16 a^4}{b (a-b x)}+\frac{32 a^3 \log (a-b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0280465, size = 56, normalized size = 1.02 \[ -\frac{16 a^4}{b (b x-a)}+\frac{32 a^3 \log (a-b x)}{b}+17 a^2 x+3 a b x^2+\frac{b^2 x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 56, normalized size = 1. \begin{align*}{\frac{{b}^{2}{x}^{3}}{3}}+3\,ab{x}^{2}+17\,{a}^{2}x-16\,{\frac{{a}^{4}}{b \left ( bx-a \right ) }}+32\,{\frac{{a}^{3}\ln \left ( bx-a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11841, size = 74, normalized size = 1.35 \begin{align*} \frac{1}{3} \, b^{2} x^{3} + 3 \, a b x^{2} - \frac{16 \, a^{4}}{b^{2} x - a b} + 17 \, a^{2} x + \frac{32 \, a^{3} \log \left (b x - a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66251, size = 157, normalized size = 2.85 \begin{align*} \frac{b^{4} x^{4} + 8 \, a b^{3} x^{3} + 42 \, a^{2} b^{2} x^{2} - 51 \, a^{3} b x - 48 \, a^{4} + 96 \,{\left (a^{3} b x - a^{4}\right )} \log \left (b x - a\right )}{3 \,{\left (b^{2} x - a b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.397923, size = 49, normalized size = 0.89 \begin{align*} - \frac{16 a^{4}}{- a b + b^{2} x} + \frac{32 a^{3} \log{\left (- a + b x \right )}}{b} + 17 a^{2} x + 3 a b x^{2} + \frac{b^{2} x^{3}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21901, size = 89, normalized size = 1.62 \begin{align*} \frac{32 \, a^{3} \log \left ({\left | b x - a \right |}\right )}{b} - \frac{16 \, a^{4}}{{\left (b x - a\right )} b} + \frac{b^{8} x^{3} + 9 \, a b^{7} x^{2} + 51 \, a^{2} b^{6} x}{3 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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