Optimal. Leaf size=298 \[ -\frac{884 b^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{100947 a^{21/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{1768 b^5 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{168245 a^4}+\frac{1768 b^4 x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{216315 a^3}-\frac{136 b^3 x^2 \sqrt{a x+b \sqrt [3]{x}}}{19665 a^2}+\frac{1768 b^6 \sqrt{a x+b \sqrt [3]{x}}}{100947 a^5}+\frac{8 b^2 x^{8/3} \sqrt{a x+b \sqrt [3]{x}}}{1311 a}+\frac{4}{69} b x^{10/3} \sqrt{a x+b \sqrt [3]{x}}+\frac{2}{9} x^3 \left (a x+b \sqrt [3]{x}\right )^{3/2} \]
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Rubi [A] time = 0.50061, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2018, 2021, 2024, 2011, 329, 220} \[ -\frac{1768 b^5 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{168245 a^4}+\frac{1768 b^4 x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{216315 a^3}-\frac{136 b^3 x^2 \sqrt{a x+b \sqrt [3]{x}}}{19665 a^2}-\frac{884 b^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{100947 a^{21/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{1768 b^6 \sqrt{a x+b \sqrt [3]{x}}}{100947 a^5}+\frac{8 b^2 x^{8/3} \sqrt{a x+b \sqrt [3]{x}}}{1311 a}+\frac{4}{69} b x^{10/3} \sqrt{a x+b \sqrt [3]{x}}+\frac{2}{9} x^3 \left (a x+b \sqrt [3]{x}\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 2021
Rule 2024
Rule 2011
Rule 329
Rule 220
Rubi steps
\begin{align*} \int x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx &=3 \operatorname{Subst}\left (\int x^8 \left (b x+a x^3\right )^{3/2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac{1}{3} (2 b) \operatorname{Subst}\left (\int x^9 \sqrt{b x+a x^3} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{4}{69} b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac{1}{69} \left (4 b^2\right ) \operatorname{Subst}\left (\int \frac{x^{10}}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{8 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a}+\frac{4}{69} b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac{\left (68 b^3\right ) \operatorname{Subst}\left (\int \frac{x^8}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1311 a}\\ &=-\frac{136 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{19665 a^2}+\frac{8 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a}+\frac{4}{69} b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac{\left (884 b^4\right ) \operatorname{Subst}\left (\int \frac{x^6}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{19665 a^2}\\ &=\frac{1768 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{216315 a^3}-\frac{136 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{19665 a^2}+\frac{8 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a}+\frac{4}{69} b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac{\left (884 b^5\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{24035 a^3}\\ &=-\frac{1768 b^5 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{168245 a^4}+\frac{1768 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{216315 a^3}-\frac{136 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{19665 a^2}+\frac{8 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a}+\frac{4}{69} b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac{\left (884 b^6\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{33649 a^4}\\ &=\frac{1768 b^6 \sqrt{b \sqrt [3]{x}+a x}}{100947 a^5}-\frac{1768 b^5 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{168245 a^4}+\frac{1768 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{216315 a^3}-\frac{136 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{19665 a^2}+\frac{8 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a}+\frac{4}{69} b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac{\left (884 b^7\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{100947 a^5}\\ &=\frac{1768 b^6 \sqrt{b \sqrt [3]{x}+a x}}{100947 a^5}-\frac{1768 b^5 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{168245 a^4}+\frac{1768 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{216315 a^3}-\frac{136 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{19665 a^2}+\frac{8 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a}+\frac{4}{69} b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac{\left (884 b^7 \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{100947 a^5 \sqrt{b \sqrt [3]{x}+a x}}\\ &=\frac{1768 b^6 \sqrt{b \sqrt [3]{x}+a x}}{100947 a^5}-\frac{1768 b^5 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{168245 a^4}+\frac{1768 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{216315 a^3}-\frac{136 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{19665 a^2}+\frac{8 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a}+\frac{4}{69} b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac{\left (1768 b^7 \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{100947 a^5 \sqrt{b \sqrt [3]{x}+a x}}\\ &=\frac{1768 b^6 \sqrt{b \sqrt [3]{x}+a x}}{100947 a^5}-\frac{1768 b^5 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{168245 a^4}+\frac{1768 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{216315 a^3}-\frac{136 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{19665 a^2}+\frac{8 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a}+\frac{4}{69} b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac{884 b^{27/4} \left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right ) \sqrt{\frac{b+a x^{2/3}}{\left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{100947 a^{21/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}
Mathematica [C] time = 0.137646, size = 142, normalized size = 0.48 \[ \frac{2 \sqrt{a x+b \sqrt [3]{x}} \left (\left (a x^{2/3}+b\right )^2 \sqrt{\frac{a x^{2/3}}{b}+1} \left (12155 a^2 b^2 x^{4/3}-17765 a^3 b x^2+24035 a^4 x^{8/3}-7293 a b^3 x^{2/3}+3315 b^4\right )-3315 b^6 \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{a x^{2/3}}{b}\right )\right )}{216315 a^5 \sqrt{\frac{a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 197, normalized size = 0.7 \begin{align*} -{\frac{2}{1514205\,{a}^{6}} \left ( -216755\,{x}^{11/3}{a}^{6}{b}^{2}-380380\,{x}^{13/3}{a}^{7}b+616\,{x}^{3}{a}^{5}{b}^{3}+6630\,{b}^{7}\sqrt{-ab}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) +1768\,{x}^{5/3}{a}^{3}{b}^{5}-952\,{x}^{7/3}{a}^{4}{b}^{4}-168245\,{x}^{5}{a}^{8}-5304\,x{a}^{2}{b}^{6}-13260\,\sqrt [3]{x}a{b}^{7} \right ){\frac{1}{\sqrt{\sqrt [3]{x} \left ( b+a{x}^{{\frac{2}{3}}} \right ) }}}} \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{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a x^{3} + b x^{\frac{7}{3}}\right )} \sqrt{a x + b x^{\frac{1}{3}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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