Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b \sqrt [3]{x}\right )^6}{2 b^3}+\frac{3 \left (a+b \sqrt [3]{x}\right )^8}{8 b^3}-\frac{6 a \left (a+b \sqrt [3]{x}\right )^7}{7 b^3} \]
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Rubi [A] time = 0.0392421, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac{a^2 \left (a+b \sqrt [3]{x}\right )^6}{2 b^3}+\frac{3 \left (a+b \sqrt [3]{x}\right )^8}{8 b^3}-\frac{6 a \left (a+b \sqrt [3]{x}\right )^7}{7 b^3} \]
Antiderivative was successfully verified.
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Rule 190
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt [3]{x}\right )^5 \, dx &=3 \operatorname{Subst}\left (\int x^2 (a+b x)^5 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^5}{b^2}-\frac{2 a (a+b x)^6}{b^2}+\frac{(a+b x)^7}{b^2}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{a^2 \left (a+b \sqrt [3]{x}\right )^6}{2 b^3}-\frac{6 a \left (a+b \sqrt [3]{x}\right )^7}{7 b^3}+\frac{3 \left (a+b \sqrt [3]{x}\right )^8}{8 b^3}\\ \end{align*}
Mathematica [A] time = 0.0227737, size = 41, normalized size = 0.69 \[ \frac{\left (a+b \sqrt [3]{x}\right )^6 \left (a^2-6 a b \sqrt [3]{x}+21 b^2 x^{2/3}\right )}{56 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 55, normalized size = 0.9 \begin{align*} x{a}^{5}+{\frac{3\,{b}^{5}}{8}{x}^{{\frac{8}{3}}}}+{\frac{15\,a{b}^{4}}{7}{x}^{{\frac{7}{3}}}}+5\,{a}^{2}{b}^{3}{x}^{2}+6\,{a}^{3}{b}^{2}{x}^{5/3}+{\frac{15\,{a}^{4}b}{4}{x}^{{\frac{4}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957892, size = 73, normalized size = 1.24 \begin{align*} \frac{3}{8} \, b^{5} x^{\frac{8}{3}} + \frac{15}{7} \, a b^{4} x^{\frac{7}{3}} + 5 \, a^{2} b^{3} x^{2} + 6 \, a^{3} b^{2} x^{\frac{5}{3}} + \frac{15}{4} \, a^{4} b x^{\frac{4}{3}} + a^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47549, size = 140, normalized size = 2.37 \begin{align*} 5 \, a^{2} b^{3} x^{2} + a^{5} x + \frac{3}{8} \,{\left (b^{5} x^{2} + 16 \, a^{3} b^{2} x\right )} x^{\frac{2}{3}} + \frac{15}{28} \,{\left (4 \, a b^{4} x^{2} + 7 \, a^{4} b x\right )} x^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.5631, size = 68, normalized size = 1.15 \begin{align*} a^{5} x + \frac{15 a^{4} b x^{\frac{4}{3}}}{4} + 6 a^{3} b^{2} x^{\frac{5}{3}} + 5 a^{2} b^{3} x^{2} + \frac{15 a b^{4} x^{\frac{7}{3}}}{7} + \frac{3 b^{5} x^{\frac{8}{3}}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15102, size = 73, normalized size = 1.24 \begin{align*} \frac{3}{8} \, b^{5} x^{\frac{8}{3}} + \frac{15}{7} \, a b^{4} x^{\frac{7}{3}} + 5 \, a^{2} b^{3} x^{2} + 6 \, a^{3} b^{2} x^{\frac{5}{3}} + \frac{15}{4} \, a^{4} b x^{\frac{4}{3}} + a^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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