Optimal. Leaf size=16 \[ \frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )^3}{a} \]
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Rubi [A] time = 0.0069336, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 37} \[ \frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )^3}{a} \]
Antiderivative was successfully verified.
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Rule 190
Rule 37
Rubi steps
\begin{align*} \int \left (a+\frac{b}{\sqrt [3]{x}}\right )^2 \, dx &=-\left (3 \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^4} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=\frac{\left (a+\frac{b}{\sqrt [3]{x}}\right )^3 x}{a}\\ \end{align*}
Mathematica [A] time = 0.0099199, size = 25, normalized size = 1.56 \[ a^2 x+3 a b x^{2/3}+3 b^2 \sqrt [3]{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 14, normalized size = 0.9 \begin{align*}{\frac{1}{a} \left ( b+a\sqrt [3]{x} \right ) ^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990099, size = 28, normalized size = 1.75 \begin{align*} a^{2} x + 3 \, a b x^{\frac{2}{3}} + 3 \, b^{2} x^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53712, size = 53, normalized size = 3.31 \begin{align*} a^{2} x + 3 \, a b x^{\frac{2}{3}} + 3 \, b^{2} x^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.15988, size = 24, normalized size = 1.5 \begin{align*} a^{2} x + 3 a b x^{\frac{2}{3}} + 3 b^{2} \sqrt [3]{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20793, size = 28, normalized size = 1.75 \begin{align*} a^{2} x + 3 \, a b x^{\frac{2}{3}} + 3 \, b^{2} x^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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