Optimal. Leaf size=69 \[ -\frac {3 \left (\frac {b \sqrt [3]{x}}{a}+1\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (1,2 p+1;2 (p+1);\frac {\sqrt [3]{x} b}{a}+1\right )}{2 p+1} \]
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Rubi [A] time = 0.03, antiderivative size = 69, 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 = {1356, 266, 65} \[ -\frac {3 \left (\frac {b \sqrt [3]{x}}{a}+1\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (1,2 p+1;2 (p+1);\frac {\sqrt [3]{x} b}{a}+1\right )}{2 p+1} \]
Antiderivative was successfully verified.
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Rule 65
Rule 266
Rule 1356
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{x} \, 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} \, 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} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {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 (1,1+2 p;2 (1+p);1+\frac {b \sqrt [3]{x}}{a}\right )}{1+2 p}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 58, normalized size = 0.84 \[ -\frac {3 \left (a+b \sqrt [3]{x}\right ) \left (\left (a+b \sqrt [3]{x}\right )^2\right )^p \, _2F_1\left (1,2 p+1;2 p+2;\frac {\sqrt [3]{x} b}{a}+1\right )}{a (2 p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p}}{x}, x\right ) \]
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 \[ \int \frac {{\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \frac {\left (b^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+a^{2}\right )^{p}}{x}\, dx \]
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 \[ \int \frac {{\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}\right )}^p}{x} \,d x \]
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 \[ \int \frac {\left (\left (a + b \sqrt [3]{x}\right )^{2}\right )^{p}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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