Optimal. Leaf size=46 \[ -\frac {(a+b x)^{-2+m} \, _2F_1\left (3,-2+m;-1+m;\frac {a+b x}{2 a}\right )}{8 a^3 b (2-m)} \]
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Rubi [A]
time = 0.01, antiderivative size = 46, 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 = {641, 70}
\begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 641
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*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(153\) vs. \(2(46)=92\).
time = 0.24, size = 153, normalized size = 3.33 \begin {gather*} \frac {(a+b x)^m \left (\frac {12 a}{m}+\frac {8 a^3}{(-2+m) (a+b x)^2}+\frac {12 a^2}{(-1+m) (a+b x)}+\frac {6 (a+b x) \, _2F_1\left (1,1+m;2+m;\frac {a+b x}{2 a}\right )}{1+m}+\frac {3 (a+b x) \, _2F_1\left (2,1+m;2+m;\frac {a+b x}{2 a}\right )}{1+m}+\frac {(a+b x) \, _2F_1\left (3,1+m;2+m;\frac {a+b x}{2 a}\right )}{1+m}\right )}{64 a^6 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{m}}{\left (-b^{2} x^{2}+a^{2}\right )^{3}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - \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 {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^m}{{\left (a^2-b^2\,x^2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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