Optimal. Leaf size=109 \[ -\frac{32 b^3 \left (a x+b x^{2/3}\right )^{3/2}}{105 a^4 x}+\frac{16 b^2 \left (a x+b x^{2/3}\right )^{3/2}}{35 a^3 x^{2/3}}-\frac{4 b \left (a x+b x^{2/3}\right )^{3/2}}{7 a^2 \sqrt [3]{x}}+\frac{2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a} \]
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Rubi [A] time = 0.136737, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2002, 2016, 2014} \[ -\frac{32 b^3 \left (a x+b x^{2/3}\right )^{3/2}}{105 a^4 x}+\frac{16 b^2 \left (a x+b x^{2/3}\right )^{3/2}}{35 a^3 x^{2/3}}-\frac{4 b \left (a x+b x^{2/3}\right )^{3/2}}{7 a^2 \sqrt [3]{x}}+\frac{2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a} \]
Antiderivative was successfully verified.
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Rule 2002
Rule 2016
Rule 2014
Rubi steps
\begin{align*} \int \sqrt{b x^{2/3}+a x} \, dx &=\frac{2 \left (b x^{2/3}+a x\right )^{3/2}}{3 a}-\frac{(2 b) \int \frac{\sqrt{b x^{2/3}+a x}}{\sqrt [3]{x}} \, dx}{3 a}\\ &=\frac{2 \left (b x^{2/3}+a x\right )^{3/2}}{3 a}-\frac{4 b \left (b x^{2/3}+a x\right )^{3/2}}{7 a^2 \sqrt [3]{x}}+\frac{\left (8 b^2\right ) \int \frac{\sqrt{b x^{2/3}+a x}}{x^{2/3}} \, dx}{21 a^2}\\ &=\frac{2 \left (b x^{2/3}+a x\right )^{3/2}}{3 a}+\frac{16 b^2 \left (b x^{2/3}+a x\right )^{3/2}}{35 a^3 x^{2/3}}-\frac{4 b \left (b x^{2/3}+a x\right )^{3/2}}{7 a^2 \sqrt [3]{x}}-\frac{\left (16 b^3\right ) \int \frac{\sqrt{b x^{2/3}+a x}}{x} \, dx}{105 a^3}\\ &=\frac{2 \left (b x^{2/3}+a x\right )^{3/2}}{3 a}-\frac{32 b^3 \left (b x^{2/3}+a x\right )^{3/2}}{105 a^4 x}+\frac{16 b^2 \left (b x^{2/3}+a x\right )^{3/2}}{35 a^3 x^{2/3}}-\frac{4 b \left (b x^{2/3}+a x\right )^{3/2}}{7 a^2 \sqrt [3]{x}}\\ \end{align*}
Mathematica [A] time = 0.0395579, size = 70, normalized size = 0.64 \[ \frac{2 \left (a \sqrt [3]{x}+b\right ) \sqrt{a x+b x^{2/3}} \left (-30 a^2 b x^{2/3}+35 a^3 x+24 a b^2 \sqrt [3]{x}-16 b^3\right )}{105 a^4 \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 57, normalized size = 0.5 \begin{align*} -{\frac{2}{105\,{a}^{4}}\sqrt{b{x}^{{\frac{2}{3}}}+ax} \left ( b+a\sqrt [3]{x} \right ) \left ( 30\,{x}^{2/3}{a}^{2}b-24\,\sqrt [3]{x}a{b}^{2}-35\,x{a}^{3}+16\,{b}^{3} \right ){\frac{1}{\sqrt [3]{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a x + b x^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a x + b x^{\frac{2}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11866, size = 89, normalized size = 0.82 \begin{align*} \frac{32 \, b^{\frac{9}{2}}}{105 \, a^{4}} + \frac{2 \,{\left (35 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} - 135 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b + 189 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} b^{3}\right )}}{105 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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