Optimal. Leaf size=160 \[ \frac{512 b^4 \sqrt{a x+b x^{2/3}}}{21 a^6 \sqrt [3]{x}}-\frac{256 b^3 \sqrt{a x+b x^{2/3}}}{21 a^5}+\frac{64 b^2 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{7 a^4}-\frac{160 b x^{2/3} \sqrt{a x+b x^{2/3}}}{21 a^3}+\frac{20 x \sqrt{a x+b x^{2/3}}}{3 a^2}-\frac{6 x^2}{a \sqrt{a x+b x^{2/3}}} \]
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Rubi [A] time = 0.242094, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2015, 2016, 2002, 2014} \[ \frac{512 b^4 \sqrt{a x+b x^{2/3}}}{21 a^6 \sqrt [3]{x}}-\frac{256 b^3 \sqrt{a x+b x^{2/3}}}{21 a^5}+\frac{64 b^2 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{7 a^4}-\frac{160 b x^{2/3} \sqrt{a x+b x^{2/3}}}{21 a^3}+\frac{20 x \sqrt{a x+b x^{2/3}}}{3 a^2}-\frac{6 x^2}{a \sqrt{a x+b x^{2/3}}} \]
Antiderivative was successfully verified.
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Rule 2015
Rule 2016
Rule 2002
Rule 2014
Rubi steps
\begin{align*} \int \frac{x^2}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx &=-\frac{6 x^2}{a \sqrt{b x^{2/3}+a x}}+\frac{10 \int \frac{x}{\sqrt{b x^{2/3}+a x}} \, dx}{a}\\ &=-\frac{6 x^2}{a \sqrt{b x^{2/3}+a x}}+\frac{20 x \sqrt{b x^{2/3}+a x}}{3 a^2}-\frac{(80 b) \int \frac{x^{2/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{9 a^2}\\ &=-\frac{6 x^2}{a \sqrt{b x^{2/3}+a x}}-\frac{160 b x^{2/3} \sqrt{b x^{2/3}+a x}}{21 a^3}+\frac{20 x \sqrt{b x^{2/3}+a x}}{3 a^2}+\frac{\left (160 b^2\right ) \int \frac{\sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}} \, dx}{21 a^3}\\ &=-\frac{6 x^2}{a \sqrt{b x^{2/3}+a x}}+\frac{64 b^2 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{7 a^4}-\frac{160 b x^{2/3} \sqrt{b x^{2/3}+a x}}{21 a^3}+\frac{20 x \sqrt{b x^{2/3}+a x}}{3 a^2}-\frac{\left (128 b^3\right ) \int \frac{1}{\sqrt{b x^{2/3}+a x}} \, dx}{21 a^4}\\ &=-\frac{6 x^2}{a \sqrt{b x^{2/3}+a x}}-\frac{256 b^3 \sqrt{b x^{2/3}+a x}}{21 a^5}+\frac{64 b^2 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{7 a^4}-\frac{160 b x^{2/3} \sqrt{b x^{2/3}+a x}}{21 a^3}+\frac{20 x \sqrt{b x^{2/3}+a x}}{3 a^2}+\frac{\left (256 b^4\right ) \int \frac{1}{\sqrt [3]{x} \sqrt{b x^{2/3}+a x}} \, dx}{63 a^5}\\ &=-\frac{6 x^2}{a \sqrt{b x^{2/3}+a x}}-\frac{256 b^3 \sqrt{b x^{2/3}+a x}}{21 a^5}+\frac{512 b^4 \sqrt{b x^{2/3}+a x}}{21 a^6 \sqrt [3]{x}}+\frac{64 b^2 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{7 a^4}-\frac{160 b x^{2/3} \sqrt{b x^{2/3}+a x}}{21 a^3}+\frac{20 x \sqrt{b x^{2/3}+a x}}{3 a^2}\\ \end{align*}
Mathematica [A] time = 0.0714595, size = 85, normalized size = 0.53 \[ \frac{32 a^3 b^2 x^{4/3}-64 a^2 b^3 x-20 a^4 b x^{5/3}+14 a^5 x^2+256 a b^4 x^{2/3}+512 b^5 \sqrt [3]{x}}{21 a^6 \sqrt{a x+b x^{2/3}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 77, normalized size = 0.5 \begin{align*}{\frac{2\,x}{21\,{a}^{6}} \left ( b+a\sqrt [3]{x} \right ) \left ( 7\,{x}^{5/3}{a}^{5}-10\,{x}^{4/3}{a}^{4}b+16\,x{a}^{3}{b}^{2}-32\,{x}^{2/3}{a}^{2}{b}^{3}+128\,\sqrt [3]{x}a{b}^{4}+256\,{b}^{5} \right ) \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{-{\frac{3}{2}}}} \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 \frac{x^{2}}{{\left (a x + b x^{\frac{2}{3}}\right )}^{\frac{3}{2}}}\,{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 \frac{x^{2}}{\left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14521, size = 151, normalized size = 0.94 \begin{align*} -\frac{512 \, b^{\frac{9}{2}}}{21 \, a^{6}} + \frac{6 \, b^{5}}{\sqrt{a x^{\frac{1}{3}} + b} a^{6}} + \frac{2 \,{\left (7 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{48} - 45 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{48} b + 126 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{48} b^{2} - 210 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{48} b^{3} + 315 \, \sqrt{a x^{\frac{1}{3}} + b} a^{48} b^{4}\right )}}{21 \, a^{54}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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