Optimal. Leaf size=52 \[ \frac{x \left (a+b x^{\frac{1}{-2 p-1}}\right ) \left (a^2+2 a b x^{\frac{1}{-2 p-1}}+b^2 x^{-\frac{2}{2 p+1}}\right )^p}{a} \]
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Rubi [A] time = 0.0195689, antiderivative size = 52, normalized size of antiderivative = 1., 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 = {1343, 191} \[ \frac{x \left (a+b x^{\frac{1}{-2 p-1}}\right ) \left (a^2+2 a b x^{\frac{1}{-2 p-1}}+b^2 x^{-\frac{2}{2 p+1}}\right )^p}{a} \]
Antiderivative was successfully verified.
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Rule 1343
Rule 191
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*}
Mathematica [A] time = 0.0240551, size = 58, normalized size = 1.12 \[ \frac{x^{\frac{2 p}{2 p+1}} \left (a x^{\frac{1}{2 p+1}}+b\right ) \left (x^{-\frac{2}{2 p+1}} \left (a x^{\frac{1}{2 p+1}}+b\right )^2\right )^p}{a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.243, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}+{{b}^{2} \left ({x}^{2\, \left ( 1+2\,p \right ) ^{-1}} \right ) ^{-1}}+2\,{\frac{ab}{{x}^{ \left ( 1+2\,p \right ) ^{-1}}}} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} + \frac{b^{2}}{x^{\frac{2}{2 \, p + 1}}} + \frac{2 \, a b}{x^{\left (\frac{1}{2 \, p + 1}\right )}}\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65135, size = 163, normalized size = 3.13 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} + \frac{b^{2}}{x^{\frac{2}{2 \, p + 1}}} + \frac{2 \, a b}{x^{\left (\frac{1}{2 \, p + 1}\right )}}\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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