Optimal. Leaf size=76 \[ -\frac {\left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (-\frac {1}{2 n},-p;1-\frac {1}{2 n};\frac {b^2 x^{2 n}}{a^2}\right )}{x} \]
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Rubi [A]
time = 0.05, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {373, 372, 371}
\begin {gather*} -\frac {\left (b x^n-a\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (-\frac {1}{2 n},-p;1-\frac {1}{2 n};\frac {b^2 x^{2 n}}{a^2}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rule 373
Rubi steps
\begin {align*} \int \frac {\left (-a+b x^n\right )^p \left (a+b x^n\right )^p}{x^2} \, dx &=\left (\left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (-a^2+b^2 x^{2 n}\right )^{-p}\right ) \int \frac {\left (-a^2+b^2 x^{2 n}\right )^p}{x^2} \, dx\\ &=\left (\left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p}\right ) \int \frac {\left (1-\frac {b^2 x^{2 n}}{a^2}\right )^p}{x^2} \, dx\\ &=-\frac {\left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (-\frac {1}{2 n},-p;1-\frac {1}{2 n};\frac {b^2 x^{2 n}}{a^2}\right )}{x}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 78, normalized size = 1.03 \begin {gather*} -\frac {\left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (-\frac {1}{2 n},-p;1-\frac {1}{2 n};\frac {b^2 x^{2 n}}{a^2}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{n}-a \right )^{p} \left (a +b \,x^{n}\right )^{p}}{x^{2}}\, dx\]
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 \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^{n}\right )^{p} \left (a + b x^{n}\right )^{p}}{x^{2}}\, 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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x^n\right )}^p\,{\left (b\,x^n-a\right )}^p}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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