Optimal. Leaf size=146 \[ -\frac {\left (a+b \sqrt [3]{x}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{a x}+\frac {b (1-p) \left (a+b \sqrt [3]{x}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{a^2 x^{2/3}}-\frac {b^2 (1-2 p) (1-p) \left (a+b \sqrt [3]{x}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{a^3 \sqrt [3]{x}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.07, 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 = {1370, 272, 67}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 272
Rule 1370
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 ) \text {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 ) \text {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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.25, size = 101, normalized size = 0.69 \begin {gather*} \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 (1-3 p+2 p^2\right ) \, _2F_1\left (1,1+2 p;2 (1+p);1+\frac {b \sqrt [3]{x}}{a}\right )+3 \, _2F_1\left (4,1+2 p;2 (1+p);1+\frac {b \sqrt [3]{x}}{a}\right )\right )}{a^3 (a+2 a p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2} x^{\frac {2}{3}}\right )^{p}}{x^{2}}-\frac {2 b^{3} \left (1-2 p \right ) \left (1-p \right ) p \left (a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2} x^{\frac {2}{3}}\right )^{p}}{3 a^{3} x}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.56, size = 82, normalized size = 0.56 \begin {gather*} -\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} \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*} - \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}} \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 [B]
time = 1.65, size = 69, normalized size = 0.47 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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