Optimal. Leaf size=72 \[ -\frac {\left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (a^2-b^2 x^{2 n}\right ) \, _2F_1\left (1,1+p;2+p;1-\frac {b^2 x^{2 n}}{a^2}\right )}{2 a^2 n (1+p)} \]
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Rubi [A]
time = 0.10, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {273, 127, 272,
67} \begin {gather*} -\frac {\left (a^2-b^2 x^{2 n}\right ) \left (b x^n-a\right )^p \left (a+b x^n\right )^p \, _2F_1\left (1,p+1;p+2;1-\frac {b^2 x^{2 n}}{a^2}\right )}{2 a^2 n (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 127
Rule 272
Rule 273
Rubi steps
\begin {align*} \int \frac {\left (-a+b x^n\right )^p \left (a+b x^n\right )^p}{x} \, dx &=\frac {\text {Subst}\left (\int \frac {(-a+b x)^p (a+b x)^p}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\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 ) \text {Subst}\left (\int \frac {\left (-a^2+b^2 x^2\right )^p}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\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 ) \text {Subst}\left (\int \frac {\left (-a^2+b^2 x\right )^p}{x} \, dx,x,x^{2 n}\right )}{2 n}\\ &=-\frac {\left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (a^2-b^2 x^{2 n}\right ) \, _2F_1\left (1,1+p;2+p;1-\frac {b^2 x^{2 n}}{a^2}\right )}{2 a^2 n (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 73, normalized size = 1.01 \begin {gather*} \frac {\left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (-a^2+b^2 x^{2 n}\right ) \, _2F_1\left (1,1+p;2+p;1-\frac {b^2 x^{2 n}}{a^2}\right )}{2 a^2 n (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{n}-a \right )^{p} \left (a +b \,x^{n}\right )^{p}}{x}\, dx\]
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}\, 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} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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