Optimal. Leaf size=77 \[ \frac {x (d x)^m \left (\frac {b \sqrt [3]{x}}{a}+1\right )^{-2 p} \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (3 (m+1),-2 p;3 m+4;-\frac {b \sqrt [3]{x}}{a}\right )}{m+1} \]
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Rubi [A] time = 0.04, 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 = {1356, 343, 341, 64} \[ \frac {x (d x)^m \left (\frac {b \sqrt [3]{x}}{a}+1\right )^{-2 p} \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (3 (m+1),-2 p;3 m+4;-\frac {b \sqrt [3]{x}}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 64
Rule 341
Rule 343
Rule 1356
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 ) \operatorname {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 = 0.03, size = 68, normalized size = 0.88 \[ \frac {x (d x)^m \left (\left (a+b \sqrt [3]{x}\right )^2\right )^p \left (\frac {b \sqrt [3]{x}}{a}+1\right )^{-2 p} \, _2F_1\left (3 (m+1),-2 p;3 (m+1)+1;-\frac {b \sqrt [3]{x}}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (b^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+a^{2}\right )^{p}\, dx \]
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 \[ \int {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p} \left (d x\right )^{m}\,{d x} \]
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 \[ \int {\left (d\,x\right )}^m\,{\left (a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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