Optimal. Leaf size=77 \[ \frac {\left (1+\frac {b \sqrt [3]{x}}{a}\right )^{-2 p} \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p x (d x)^m \, _2F_1\left (3 (1+m),-2 p;4+3 m;-\frac {b \sqrt [3]{x}}{a}\right )}{1+m} \]
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Rubi [A]
time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1370, 350, 348,
66} \begin {gather*} \frac {x (d x)^m \left (\frac {b \sqrt [3]{x}}{a}+1\right )^{-2 p} \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (3 (m+1),-2 p;3 m+4;-\frac {b \sqrt [3]{x}}{a}\right )}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 348
Rule 350
Rule 1370
Rubi steps
\begin {align*} \int \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p (d x)^m \, 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 \left (1+\frac {b \sqrt [3]{x}}{a}\right )^{2 p} (d x)^m \, 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 x^{-m} (d x)^m\right ) \int \left (1+\frac {b \sqrt [3]{x}}{a}\right )^{2 p} x^m \, 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 x^{-m} (d x)^m\right ) \text {Subst}\left (\int x^{-1+3 (1+m)} \left (1+\frac {b x}{a}\right )^{2 p} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {\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 x (d x)^m \, _2F_1\left (3 (1+m),-2 p;4+3 m;-\frac {b \sqrt [3]{x}}{a}\right )}{1+m}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 68, normalized size = 0.88 \begin {gather*} \frac {\left (\left (a+b \sqrt [3]{x}\right )^2\right )^p \left (1+\frac {b \sqrt [3]{x}}{a}\right )^{-2 p} x (d x)^m \, _2F_1\left (3 (1+m),-2 p;1+3 (1+m);-\frac {b \sqrt [3]{x}}{a}\right )}{1+m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2} x^{\frac {2}{3}}\right )^{p} \left (d x \right )^{m}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\left (d\,x\right )}^m\,{\left (a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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