Optimal. Leaf size=86 \[ -\frac{1}{16 a^4 b (a+b x)}-\frac{1}{16 a^3 b (a+b x)^2}-\frac{1}{12 a^2 b (a+b x)^3}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{16 a^5 b}-\frac{1}{8 a b (a+b x)^4} \]
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Rubi [A] time = 0.0530616, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {627, 44, 208} \[ -\frac{1}{16 a^4 b (a+b x)}-\frac{1}{16 a^3 b (a+b x)^2}-\frac{1}{12 a^2 b (a+b x)^3}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{16 a^5 b}-\frac{1}{8 a b (a+b x)^4} \]
Antiderivative was successfully verified.
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Rule 627
Rule 44
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^4 \left (a^2-b^2 x^2\right )} \, dx &=\int \frac{1}{(a-b x) (a+b x)^5} \, dx\\ &=\int \left (\frac{1}{2 a (a+b x)^5}+\frac{1}{4 a^2 (a+b x)^4}+\frac{1}{8 a^3 (a+b x)^3}+\frac{1}{16 a^4 (a+b x)^2}+\frac{1}{16 a^4 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=-\frac{1}{8 a b (a+b x)^4}-\frac{1}{12 a^2 b (a+b x)^3}-\frac{1}{16 a^3 b (a+b x)^2}-\frac{1}{16 a^4 b (a+b x)}+\frac{\int \frac{1}{a^2-b^2 x^2} \, dx}{16 a^4}\\ &=-\frac{1}{8 a b (a+b x)^4}-\frac{1}{12 a^2 b (a+b x)^3}-\frac{1}{16 a^3 b (a+b x)^2}-\frac{1}{16 a^4 b (a+b x)}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{16 a^5 b}\\ \end{align*}
Mathematica [A] time = 0.0243488, size = 82, normalized size = 0.95 \[ \frac{-2 a \left (19 a^2 b x+16 a^3+12 a b^2 x^2+3 b^3 x^3\right )-3 (a+b x)^4 \log (a-b x)+3 (a+b x)^4 \log (a+b x)}{96 a^5 b (a+b x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 92, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( bx+a \right ) }{32\,{a}^{5}b}}-{\frac{1}{16\,{a}^{4}b \left ( bx+a \right ) }}-{\frac{1}{16\,{a}^{3}b \left ( bx+a \right ) ^{2}}}-{\frac{1}{12\,b{a}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{1}{8\,ab \left ( bx+a \right ) ^{4}}}-{\frac{\ln \left ( bx-a \right ) }{32\,{a}^{5}b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03286, size = 151, normalized size = 1.76 \begin{align*} -\frac{3 \, b^{3} x^{3} + 12 \, a b^{2} x^{2} + 19 \, a^{2} b x + 16 \, a^{3}}{48 \,{\left (a^{4} b^{5} x^{4} + 4 \, a^{5} b^{4} x^{3} + 6 \, a^{6} b^{3} x^{2} + 4 \, a^{7} b^{2} x + a^{8} b\right )}} + \frac{\log \left (b x + a\right )}{32 \, a^{5} b} - \frac{\log \left (b x - a\right )}{32 \, a^{5} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.7232, size = 375, normalized size = 4.36 \begin{align*} -\frac{6 \, a b^{3} x^{3} + 24 \, a^{2} b^{2} x^{2} + 38 \, a^{3} b x + 32 \, a^{4} - 3 \,{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right ) + 3 \,{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (b x - a\right )}{96 \,{\left (a^{5} b^{5} x^{4} + 4 \, a^{6} b^{4} x^{3} + 6 \, a^{7} b^{3} x^{2} + 4 \, a^{8} b^{2} x + a^{9} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.708482, size = 107, normalized size = 1.24 \begin{align*} - \frac{16 a^{3} + 19 a^{2} b x + 12 a b^{2} x^{2} + 3 b^{3} x^{3}}{48 a^{8} b + 192 a^{7} b^{2} x + 288 a^{6} b^{3} x^{2} + 192 a^{5} b^{4} x^{3} + 48 a^{4} b^{5} x^{4}} - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{32} - \frac{\log{\left (\frac{a}{b} + x \right )}}{32}}{a^{5} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.223, size = 109, normalized size = 1.27 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right )}{32 \, a^{5} b} - \frac{\log \left ({\left | b x - a \right |}\right )}{32 \, a^{5} b} - \frac{3 \, a b^{3} x^{3} + 12 \, a^{2} b^{2} x^{2} + 19 \, a^{3} b x + 16 \, a^{4}}{48 \,{\left (b x + a\right )}^{4} a^{5} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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