Optimal. Leaf size=46 \[ -\frac{(a+b x)^{m-2} \, _2F_1\left (3,m-2;m-1;\frac{a+b x}{2 a}\right )}{8 a^3 b (2-m)} \]
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Rubi [A] time = 0.0151704, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {627, 68} \[ -\frac{(a+b x)^{m-2} \, _2F_1\left (3,m-2;m-1;\frac{a+b x}{2 a}\right )}{8 a^3 b (2-m)} \]
Antiderivative was successfully verified.
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Rule 627
Rule 68
Rubi steps
\begin{align*} \int \frac{(a+b x)^m}{\left (a^2-b^2 x^2\right )^3} \, dx &=\int \frac{(a+b x)^{-3+m}}{(a-b x)^3} \, dx\\ &=-\frac{(a+b x)^{-2+m} \, _2F_1\left (3,-2+m;-1+m;\frac{a+b x}{2 a}\right )}{8 a^3 b (2-m)}\\ \end{align*}
Mathematica [B] time = 0.185065, size = 153, normalized size = 3.33 \[ \frac{(a+b x)^m \left (\frac{8 a^3}{(m-2) (a+b x)^2}+\frac{12 a^2}{(m-1) (a+b x)}+\frac{6 (a+b x) \, _2F_1\left (1,m+1;m+2;\frac{a+b x}{2 a}\right )}{m+1}+\frac{3 (a+b x) \, _2F_1\left (2,m+1;m+2;\frac{a+b x}{2 a}\right )}{m+1}+\frac{(a+b x) \, _2F_1\left (3,m+1;m+2;\frac{a+b x}{2 a}\right )}{m+1}+\frac{12 a}{m}\right )}{64 a^6 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.536, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (b x + a\right )}^{m}}{{\left (b^{2} x^{2} - a^{2}\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (b x + a\right )}^{m}}{b^{6} x^{6} - 3 \, a^{2} b^{4} x^{4} + 3 \, a^{4} b^{2} x^{2} - a^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\left (a + b x\right )^{m}}{- a^{6} + 3 a^{4} b^{2} x^{2} - 3 a^{2} b^{4} x^{4} + b^{6} x^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (b x + a\right )}^{m}}{{\left (b^{2} x^{2} - a^{2}\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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