Optimal. Leaf size=75 \[ \frac {3 b^3 \left (1+\frac {b \sqrt [3]{x}}{a}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (4,1+2 p;2 (1+p);1+\frac {b \sqrt [3]{x}}{a}\right )}{a^3 (1+2 p)} \]
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Rubi [A]
time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1370, 272, 67}
\begin {gather*} \frac {3 b^3 \left (\frac {b \sqrt [3]{x}}{a}+1\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (4,2 p+1;2 (p+1);\frac {\sqrt [3]{x} b}{a}+1\right )}{a^3 (2 p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 272
Rule 1370
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{x^2} \, dx &=\left (\left (1+\frac {b \sqrt [3]{x}}{a}\right )^{-2 p} \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p\right ) \int \frac {\left (1+\frac {b \sqrt [3]{x}}{a}\right )^{2 p}}{x^2} \, dx\\ &=\left (3 \left (1+\frac {b \sqrt [3]{x}}{a}\right )^{-2 p} \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x}{a}\right )^{2 p}}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3 b^3 \left (1+\frac {b \sqrt [3]{x}}{a}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (4,1+2 p;2 (1+p);1+\frac {b \sqrt [3]{x}}{a}\right )}{a^3 (1+2 p)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 61, normalized size = 0.81 \begin {gather*} \frac {3 b^3 \left (a+b \sqrt [3]{x}\right ) \left (\left (a+b \sqrt [3]{x}\right )^2\right )^p \, _2F_1\left (4,1+2 p;2+2 p;1+\frac {b \sqrt [3]{x}}{a}\right )}{a^4 (1+2 p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2} x^{\frac {2}{3}}\right )^{p}}{x^{2}}\, dx\]
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 (\left (a + b \sqrt [3]{x}\right )^{2}\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^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}\right )}^p}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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