Optimal. Leaf size=52 \[ \frac {x \left (a+b x^{\frac {1}{-1-2 p}}\right ) \left (a^2+2 a b x^{\frac {1}{-1-2 p}}+b^2 x^{-\frac {2}{1+2 p}}\right )^p}{a} \]
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Rubi [A]
time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1357, 197}
\begin {gather*} \frac {x \left (a+b x^{\frac {1}{-2 p-1}}\right ) \left (a^2+2 a b x^{\frac {1}{-2 p-1}}+b^2 x^{-\frac {2}{2 p+1}}\right )^p}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 1357
Rubi steps
\begin {align*} \int \left (a^2+b^2 x^{-\frac {2}{1+2 p}}+2 a b x^{-\frac {1}{1+2 p}}\right )^p \, dx &=\left (\left (a^2+b^2 x^{-\frac {2}{1+2 p}}+2 a b x^{-\frac {1}{1+2 p}}\right )^p \left (2 a b+2 b^2 x^{-\frac {1}{1+2 p}}\right )^{-2 p}\right ) \int \left (2 a b+2 b^2 x^{-\frac {1}{1+2 p}}\right )^{2 p} \, dx\\ &=\frac {x \left (a+b x^{\frac {1}{-1-2 p}}\right ) \left (a^2+2 a b x^{\frac {1}{-1-2 p}}+b^2 x^{-\frac {2}{1+2 p}}\right )^p}{a}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 58, normalized size = 1.12 \begin {gather*} \frac {x^{\frac {2 p}{1+2 p}} \left (b+a x^{\frac {1}{1+2 p}}\right ) \left (x^{-\frac {2}{1+2 p}} \left (b+a x^{\frac {1}{1+2 p}}\right )^2\right )^p}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (a^{2}+b^{2} x^{-\frac {2}{1+2 p}}+2 a b \,x^{-\frac {1}{1+2 p}}\right )^{p}\, dx\]
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 [A]
time = 0.37, size = 79, normalized size = 1.52 \begin {gather*} \frac {{\left (a x x^{\left (\frac {1}{2 \, p + 1}\right )} + b x\right )} \left (\frac {a^{2} x^{\frac {2}{2 \, p + 1}} + 2 \, a b x^{\left (\frac {1}{2 \, p + 1}\right )} + b^{2}}{x^{\frac {2}{2 \, p + 1}}}\right )^{p}}{a x^{\left (\frac {1}{2 \, p + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\left (a^2+\frac {b^2}{x^{\frac {2}{2\,p+1}}}+\frac {2\,a\,b}{x^{\frac {1}{2\,p+1}}}\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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