Optimal. Leaf size=146 \[ \frac {b (1-p) \left (a+b \sqrt [3]{x}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{a^2 x^{2/3}}-\frac {\left (a+b \sqrt [3]{x}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{a x}-\frac {b^2 (1-2 p) (1-p) \left (a+b \sqrt [3]{x}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{a^3 \sqrt [3]{x}} \]
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Rubi [C] time = 0.10, antiderivative size = 162, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 3, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {1356, 266, 65} \[ \frac {2 b^3 (1-2 p) (1-p) p \left (\frac {b \sqrt [3]{x}}{a}+1\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (1,2 p+1;2 (p+1);\frac {\sqrt [3]{x} b}{a}+1\right )}{a^3 (2 p+1)}+\frac {3 b^3 \left (\frac {b \sqrt [3]{x}}{a}+1\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (4,2 p+1;2 (p+1);\frac {\sqrt [3]{x} b}{a}+1\right )}{a^3 (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 266
Rule 1356
Rubi steps
\begin {align*} \int \left (\frac {\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{x^2}-\frac {2 b^3 (1-2 p) (1-p) p \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{3 a^3 x}\right ) \, dx &=-\frac {\left (2 b^3 (1-2 p) (1-p) p\right ) \int \frac {\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{x} \, dx}{3 a^3}+\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-\frac {\left (2 b^3 (1-2 p) (1-p) p \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}{3 a^3}\\ &=\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 {\left (2 b^3 (1-2 p) (1-p) p \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 )}{a^3}\\ &=\frac {2 b^3 (1-2 p) (1-p) p \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 )}{a^3 (1+2 p)}+\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 [C] time = 0.08, size = 101, normalized size = 0.69 \[ \frac {b^3 \left (a+b \sqrt [3]{x}\right ) \left (\left (a+b \sqrt [3]{x}\right )^2\right )^p \left (2 p \left (2 p^2-3 p+1\right ) \, _2F_1\left (1,2 p+1;2 (p+1);\frac {\sqrt [3]{x} b}{a}+1\right )+3 \, _2F_1\left (4,2 p+1;2 (p+1);\frac {\sqrt [3]{x} b}{a}+1\right )\right )}{a^3 (2 a p+a)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.32, size = 82, normalized size = 0.56 \[ -\frac {{\left (a^{2} b p x^{\frac {1}{3}} + a^{3} + {\left (2 \, b^{3} p^{2} - 3 \, b^{3} p + b^{3}\right )} x + 2 \, {\left (a b^{2} p^{2} - a b^{2} p\right )} x^{\frac {2}{3}}\right )} {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p}}{a^{3} x} \]
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 \[ \int -\frac {2 \, {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p} b^{3} {\left (2 \, p - 1\right )} {\left (p - 1\right )} p}{3 \, a^{3} x} + \frac {{\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p}}{x^{2}}\,{d x} \]
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 \[ \int -\frac {2 \left (-2 p +1\right ) \left (-p +1\right ) b^{3} p \left (b^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+a^{2}\right )^{p}}{3 a^{3} x}+\frac {\left (b^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+a^{2}\right )^{p}}{x^{2}}\, dx \]
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 \[ \int -\frac {2 \, {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p} b^{3} {\left (2 \, p - 1\right )} {\left (p - 1\right )} p}{3 \, a^{3} x} + \frac {{\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.65, size = 69, normalized size = 0.47 \[ -\frac {{\left (a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}\right )}^p\,\left (\frac {b^3\,x\,\left (2\,p^2-3\,p+1\right )}{a^3}+\frac {b\,p\,x^{1/3}}{a}+\frac {2\,b^2\,p\,x^{2/3}\,\left (p-1\right )}{a^2}+1\right )}{x} \]
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 \[ - \frac {\int \left (- \frac {3 a^{3} \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}\right )^{p}}{x^{2}}\right )\, dx + \int \frac {2 b^{3} p \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}\right )^{p}}{x}\, dx + \int \left (- \frac {6 b^{3} p^{2} \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}\right )^{p}}{x}\right )\, dx + \int \frac {4 b^{3} p^{3} \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}\right )^{p}}{x}\, dx}{3 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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