Optimal. Leaf size=313 \[ \frac{524288 b^{10} \sqrt{a x+b x^{2/3}}}{323323 a^{11} \sqrt [3]{x}}-\frac{262144 b^9 \sqrt{a x+b x^{2/3}}}{323323 a^{10}}+\frac{196608 b^8 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{323323 a^9}-\frac{163840 b^7 x^{2/3} \sqrt{a x+b x^{2/3}}}{323323 a^8}+\frac{20480 b^6 x \sqrt{a x+b x^{2/3}}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{a x+b x^{2/3}}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{a x+b x^{2/3}}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{a x+b x^{2/3}}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{a x+b x^{2/3}}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{a x+b x^{2/3}}}{133 a^2}+\frac{2 x^3 \sqrt{a x+b x^{2/3}}}{7 a} \]
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Rubi [A] time = 0.53092, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2016, 2002, 2014} \[ \frac{524288 b^{10} \sqrt{a x+b x^{2/3}}}{323323 a^{11} \sqrt [3]{x}}-\frac{262144 b^9 \sqrt{a x+b x^{2/3}}}{323323 a^{10}}+\frac{196608 b^8 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{323323 a^9}-\frac{163840 b^7 x^{2/3} \sqrt{a x+b x^{2/3}}}{323323 a^8}+\frac{20480 b^6 x \sqrt{a x+b x^{2/3}}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{a x+b x^{2/3}}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{a x+b x^{2/3}}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{a x+b x^{2/3}}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{a x+b x^{2/3}}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{a x+b x^{2/3}}}{133 a^2}+\frac{2 x^3 \sqrt{a x+b x^{2/3}}}{7 a} \]
Antiderivative was successfully verified.
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Rule 2016
Rule 2002
Rule 2014
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{b x^{2/3}+a x}} \, dx &=\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}-\frac{(20 b) \int \frac{x^{8/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{21 a}\\ &=-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}+\frac{\left (120 b^2\right ) \int \frac{x^{7/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{133 a^2}\\ &=\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}-\frac{\left (1920 b^3\right ) \int \frac{x^2}{\sqrt{b x^{2/3}+a x}} \, dx}{2261 a^3}\\ &=-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}+\frac{\left (256 b^4\right ) \int \frac{x^{5/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{323 a^4}\\ &=\frac{1536 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}-\frac{\left (3072 b^5\right ) \int \frac{x^{4/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{4199 a^5}\\ &=-\frac{18432 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}+\frac{\left (30720 b^6\right ) \int \frac{x}{\sqrt{b x^{2/3}+a x}} \, dx}{46189 a^6}\\ &=\frac{20480 b^6 x \sqrt{b x^{2/3}+a x}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}-\frac{\left (81920 b^7\right ) \int \frac{x^{2/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{138567 a^7}\\ &=-\frac{163840 b^7 x^{2/3} \sqrt{b x^{2/3}+a x}}{323323 a^8}+\frac{20480 b^6 x \sqrt{b x^{2/3}+a x}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}+\frac{\left (163840 b^8\right ) \int \frac{\sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}} \, dx}{323323 a^8}\\ &=\frac{196608 b^8 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{323323 a^9}-\frac{163840 b^7 x^{2/3} \sqrt{b x^{2/3}+a x}}{323323 a^8}+\frac{20480 b^6 x \sqrt{b x^{2/3}+a x}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}-\frac{\left (131072 b^9\right ) \int \frac{1}{\sqrt{b x^{2/3}+a x}} \, dx}{323323 a^9}\\ &=-\frac{262144 b^9 \sqrt{b x^{2/3}+a x}}{323323 a^{10}}+\frac{196608 b^8 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{323323 a^9}-\frac{163840 b^7 x^{2/3} \sqrt{b x^{2/3}+a x}}{323323 a^8}+\frac{20480 b^6 x \sqrt{b x^{2/3}+a x}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}+\frac{\left (262144 b^{10}\right ) \int \frac{1}{\sqrt [3]{x} \sqrt{b x^{2/3}+a x}} \, dx}{969969 a^{10}}\\ &=-\frac{262144 b^9 \sqrt{b x^{2/3}+a x}}{323323 a^{10}}+\frac{524288 b^{10} \sqrt{b x^{2/3}+a x}}{323323 a^{11} \sqrt [3]{x}}+\frac{196608 b^8 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{323323 a^9}-\frac{163840 b^7 x^{2/3} \sqrt{b x^{2/3}+a x}}{323323 a^8}+\frac{20480 b^6 x \sqrt{b x^{2/3}+a x}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}\\ \end{align*}
Mathematica [A] time = 0.115059, size = 148, normalized size = 0.47 \[ \frac{2 \sqrt{a x+b x^{2/3}} \left (51480 a^8 b^2 x^{8/3}-54912 a^7 b^3 x^{7/3}+59136 a^6 b^4 x^2-64512 a^5 b^5 x^{5/3}+71680 a^4 b^6 x^{4/3}+98304 a^2 b^8 x^{2/3}-81920 a^3 b^7 x-48620 a^9 b x^3+46189 a^{10} x^{10/3}-131072 a b^9 \sqrt [3]{x}+262144 b^{10}\right )}{323323 a^{11} \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 134, normalized size = 0.4 \begin{align*}{\frac{2}{323323\,{a}^{11}}\sqrt [3]{x} \left ( b+a\sqrt [3]{x} \right ) \left ( 46189\,{x}^{10/3}{a}^{10}-48620\,{x}^{3}{a}^{9}b+51480\,{x}^{8/3}{a}^{8}{b}^{2}-54912\,{x}^{7/3}{a}^{7}{b}^{3}+59136\,{x}^{2}{a}^{6}{b}^{4}-64512\,{x}^{5/3}{a}^{5}{b}^{5}+71680\,{x}^{4/3}{a}^{4}{b}^{6}-81920\,x{a}^{3}{b}^{7}+98304\,{x}^{2/3}{a}^{2}{b}^{8}-131072\,\sqrt [3]{x}a{b}^{9}+262144\,{b}^{10} \right ){\frac{1}{\sqrt{b{x}^{{\frac{2}{3}}}+ax}}}} \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^{3}}{\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 \frac{x^{3}}{\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 = 1.15723, size = 221, normalized size = 0.71 \begin{align*} -\frac{524288 \, b^{\frac{21}{2}}}{323323 \, a^{11}} + \frac{2 \,{\left (46189 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} - 510510 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} b + 2567565 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} b^{2} - 7759752 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} b^{3} + 15668730 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} b^{4} - 22221108 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} b^{5} + 22632610 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} b^{6} - 16628040 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b^{7} + 8729721 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} b^{8} - 3233230 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} b^{9} + 969969 \, \sqrt{a x^{\frac{1}{3}} + b} b^{10}\right )}}{323323 \, a^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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