Optimal. Leaf size=44 \[ -\frac {(a+b x)^{m-1} \, _2F_1\left (2,m-1;m;\frac {a+b x}{2 a}\right )}{4 a^2 b (1-m)} \]
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Rubi [A] time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, 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} \[ -\frac {(a+b x)^{m-1} \, _2F_1\left (2,m-1;m;\frac {a+b x}{2 a}\right )}{4 a^2 b (1-m)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 627
Rubi steps
\begin {align*} \int \frac {(a+b x)^m}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac {(a+b x)^{-2+m}}{(a-b x)^2} \, dx\\ &=-\frac {(a+b x)^{-1+m} \, _2F_1\left (2,-1+m;m;\frac {a+b x}{2 a}\right )}{4 a^2 b (1-m)}\\ \end {align*}
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Mathematica [B] time = 0.16, size = 102, normalized size = 2.32 \[ \frac {(a+b x)^m \left (\frac {2 (a+b x) \, _2F_1\left (1,m+1;m+2;\frac {a+b x}{2 a}\right )}{m+1}+\frac {(a+b x) \, _2F_1\left (2,m+1;m+2;\frac {a+b x}{2 a}\right )}{m+1}+4 a \left (\frac {a}{(m-1) (a+b x)}+\frac {1}{m}\right )\right )}{16 a^4 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{m}}{b^{4} x^{4} - 2 \, a^{2} b^{2} x^{2} + a^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{m}}{{\left (b^{2} x^{2} - a^{2}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.79, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{m}}{\left (-b^{2} x^{2}+a^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{m}}{{\left (b^{2} x^{2} - a^{2}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,x\right )}^m}{{\left (a^2-b^2\,x^2\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x\right )^{m}}{\left (- a + b x\right )^{2} \left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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