Optimal. Leaf size=75 \[ \frac{3 b^3 \left (\frac{b \sqrt [3]{x}}{a}+1\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (4,2 p+1;2 (p+1);\frac{\sqrt [3]{x} b}{a}+1\right )}{a^3 (2 p+1)} \]
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Rubi [A] time = 0.0364101, antiderivative size = 75, normalized size of antiderivative = 1., 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 = {1356, 266, 65} \[ \frac{3 b^3 \left (\frac{b \sqrt [3]{x}}{a}+1\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (4,2 p+1;2 (p+1);\frac{\sqrt [3]{x} b}{a}+1\right )}{a^3 (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 1356
Rule 266
Rule 65
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 ) \operatorname{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*}
Mathematica [A] time = 0.0151505, size = 61, normalized size = 0.81 \[ \frac{3 b^3 \left (a+b \sqrt [3]{x}\right ) \left (\left (a+b \sqrt [3]{x}\right )^2\right )^p \, _2F_1\left (4,2 p+1;2 p+2;\frac{\sqrt [3]{x} b}{a}+1\right )}{a^4 (2 p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.007, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ({a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}} \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 \frac{{\left (b^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + a^{2}\right )}^{p}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + a^{2}\right )}^{p}}{x^{2}}, x\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 \frac{{\left (b^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + a^{2}\right )}^{p}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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