Optimal. Leaf size=70 \[ \frac{23}{2} a^2 b x^2+\frac{32 a^5}{b (a-b x)}+\frac{80 a^4 \log (a-b x)}{b}+49 a^3 x+\frac{7}{3} a b^2 x^3+\frac{b^3 x^4}{4} \]
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Rubi [A] time = 0.0509393, antiderivative size = 70, 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} \[ \frac{23}{2} a^2 b x^2+\frac{32 a^5}{b (a-b x)}+\frac{80 a^4 \log (a-b x)}{b}+49 a^3 x+\frac{7}{3} a b^2 x^3+\frac{b^3 x^4}{4} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^7}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac{(a+b x)^5}{(a-b x)^2} \, dx\\ &=\int \left (49 a^3+23 a^2 b x+7 a b^2 x^2+b^3 x^3+\frac{32 a^5}{(a-b x)^2}-\frac{80 a^4}{a-b x}\right ) \, dx\\ &=49 a^3 x+\frac{23}{2} a^2 b x^2+\frac{7}{3} a b^2 x^3+\frac{b^3 x^4}{4}+\frac{32 a^5}{b (a-b x)}+\frac{80 a^4 \log (a-b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0269898, size = 71, normalized size = 1.01 \[ \frac{23}{2} a^2 b x^2-\frac{32 a^5}{b (b x-a)}+\frac{80 a^4 \log (a-b x)}{b}+49 a^3 x+\frac{7}{3} a b^2 x^3+\frac{b^3 x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 67, normalized size = 1. \begin{align*}{\frac{{b}^{3}{x}^{4}}{4}}+{\frac{7\,a{b}^{2}{x}^{3}}{3}}+{\frac{23\,{a}^{2}b{x}^{2}}{2}}+49\,x{a}^{3}-32\,{\frac{{a}^{5}}{b \left ( bx-a \right ) }}+80\,{\frac{{a}^{4}\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 = 1.09474, size = 89, normalized size = 1.27 \begin{align*} \frac{1}{4} \, b^{3} x^{4} + \frac{7}{3} \, a b^{2} x^{3} + \frac{23}{2} \, a^{2} b x^{2} - \frac{32 \, a^{5}}{b^{2} x - a b} + 49 \, a^{3} x + \frac{80 \, a^{4} \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 = 1.70059, size = 192, normalized size = 2.74 \begin{align*} \frac{3 \, b^{5} x^{5} + 25 \, a b^{4} x^{4} + 110 \, a^{2} b^{3} x^{3} + 450 \, a^{3} b^{2} x^{2} - 588 \, a^{4} b x - 384 \, a^{5} + 960 \,{\left (a^{4} b x - a^{5}\right )} \log \left (b x - a\right )}{12 \,{\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 = 0.442633, size = 65, normalized size = 0.93 \begin{align*} - \frac{32 a^{5}}{- a b + b^{2} x} + \frac{80 a^{4} \log{\left (- a + b x \right )}}{b} + 49 a^{3} x + \frac{23 a^{2} b x^{2}}{2} + \frac{7 a b^{2} x^{3}}{3} + \frac{b^{3} x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22894, size = 105, normalized size = 1.5 \begin{align*} \frac{80 \, a^{4} \log \left ({\left | b x - a \right |}\right )}{b} - \frac{32 \, a^{5}}{{\left (b x - a\right )} b} + \frac{3 \, b^{11} x^{4} + 28 \, a b^{10} x^{3} + 138 \, a^{2} b^{9} x^{2} + 588 \, a^{3} b^{8} x}{12 \, b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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