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 = 0.02, 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 = {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 191
Rule 1343
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 = 0.02, 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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fricas [A] time = 0.70, size = 79, normalized size = 1.52 \[ \frac {{\left (a x x^{\left (\frac {1}{2 \, p + 1}\right )} + b x\right )} \left (\frac {a^{2} x^{\frac {2}{2 \, p + 1}} + 2 \, a b x^{\left (\frac {1}{2 \, p + 1}\right )} + b^{2}}{x^{\frac {2}{2 \, p + 1}}}\right )^{p}}{a x^{\left (\frac {1}{2 \, p + 1}\right )}} \]
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 \[ \int {\left (a^{2} + \frac {b^{2}}{x^{\frac {2}{2 \, p + 1}}} + \frac {2 \, a b}{x^{\left (\frac {1}{2 \, p + 1}\right )}}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \left (2 a b \,x^{-\frac {1}{2 p +1}}+b^{2} x^{-\frac {2}{2 p +1}}+a^{2}\right )^{p}\, dx \]
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 \[ \int {\left (a^{2} + \frac {b^{2}}{x^{\frac {2}{2 \, p + 1}}} + \frac {2 \, a b}{x^{\left (\frac {1}{2 \, p + 1}\right )}}\right )}^{p}\,{d x} \]
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 \[ \int {\left (a^2+\frac {b^2}{x^{\frac {2}{2\,p+1}}}+\frac {2\,a\,b}{x^{\frac {1}{2\,p+1}}}\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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