Optimal. Leaf size=371 \[ \frac{8388608 b^{12} \left (a x+b x^{2/3}\right )^{3/2}}{152108775 a^{13} x}-\frac{4194304 b^{11} \left (a x+b x^{2/3}\right )^{3/2}}{50702925 a^{12} x^{2/3}}+\frac{1048576 b^{10} \left (a x+b x^{2/3}\right )^{3/2}}{10140585 a^{11} \sqrt [3]{x}}-\frac{524288 b^9 \left (a x+b x^{2/3}\right )^{3/2}}{4345965 a^{10}}+\frac{65536 b^8 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{482885 a^9}-\frac{360448 b^7 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{2414425 a^8}+\frac{90112 b^6 x \left (a x+b x^{2/3}\right )^{3/2}}{557175 a^7}-\frac{45056 b^5 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{260015 a^6}+\frac{2816 b^4 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{15295 a^5}-\frac{1408 b^3 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7245 a^4}+\frac{352 b^2 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{1725 a^3}-\frac{16 b x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a} \]
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Rubi [A] time = 0.627282, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2016, 2002, 2014} \[ \frac{8388608 b^{12} \left (a x+b x^{2/3}\right )^{3/2}}{152108775 a^{13} x}-\frac{4194304 b^{11} \left (a x+b x^{2/3}\right )^{3/2}}{50702925 a^{12} x^{2/3}}+\frac{1048576 b^{10} \left (a x+b x^{2/3}\right )^{3/2}}{10140585 a^{11} \sqrt [3]{x}}-\frac{524288 b^9 \left (a x+b x^{2/3}\right )^{3/2}}{4345965 a^{10}}+\frac{65536 b^8 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{482885 a^9}-\frac{360448 b^7 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{2414425 a^8}+\frac{90112 b^6 x \left (a x+b x^{2/3}\right )^{3/2}}{557175 a^7}-\frac{45056 b^5 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{260015 a^6}+\frac{2816 b^4 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{15295 a^5}-\frac{1408 b^3 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7245 a^4}+\frac{352 b^2 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{1725 a^3}-\frac{16 b x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a} \]
Antiderivative was successfully verified.
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Rule 2016
Rule 2002
Rule 2014
Rubi steps
\begin{align*} \int x^3 \sqrt{b x^{2/3}+a x} \, dx &=\frac{2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}-\frac{(8 b) \int x^{8/3} \sqrt{b x^{2/3}+a x} \, dx}{9 a}\\ &=-\frac{16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}+\frac{\left (176 b^2\right ) \int x^{7/3} \sqrt{b x^{2/3}+a x} \, dx}{225 a^2}\\ &=\frac{352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac{16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}-\frac{\left (704 b^3\right ) \int x^2 \sqrt{b x^{2/3}+a x} \, dx}{1035 a^3}\\ &=-\frac{1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac{352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac{16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}+\frac{\left (1408 b^4\right ) \int x^{5/3} \sqrt{b x^{2/3}+a x} \, dx}{2415 a^4}\\ &=\frac{2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac{1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac{352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac{16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}-\frac{\left (22528 b^5\right ) \int x^{4/3} \sqrt{b x^{2/3}+a x} \, dx}{45885 a^5}\\ &=-\frac{45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac{2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac{1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac{352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac{16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}+\frac{\left (45056 b^6\right ) \int x \sqrt{b x^{2/3}+a x} \, dx}{111435 a^6}\\ &=\frac{90112 b^6 x \left (b x^{2/3}+a x\right )^{3/2}}{557175 a^7}-\frac{45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac{2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac{1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac{352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac{16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}-\frac{\left (180224 b^7\right ) \int x^{2/3} \sqrt{b x^{2/3}+a x} \, dx}{557175 a^7}\\ &=-\frac{360448 b^7 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{2414425 a^8}+\frac{90112 b^6 x \left (b x^{2/3}+a x\right )^{3/2}}{557175 a^7}-\frac{45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac{2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac{1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac{352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac{16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}+\frac{\left (360448 b^8\right ) \int \sqrt [3]{x} \sqrt{b x^{2/3}+a x} \, dx}{1448655 a^8}\\ &=\frac{65536 b^8 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{482885 a^9}-\frac{360448 b^7 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{2414425 a^8}+\frac{90112 b^6 x \left (b x^{2/3}+a x\right )^{3/2}}{557175 a^7}-\frac{45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac{2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac{1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac{352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac{16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}-\frac{\left (262144 b^9\right ) \int \sqrt{b x^{2/3}+a x} \, dx}{1448655 a^9}\\ &=-\frac{524288 b^9 \left (b x^{2/3}+a x\right )^{3/2}}{4345965 a^{10}}+\frac{65536 b^8 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{482885 a^9}-\frac{360448 b^7 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{2414425 a^8}+\frac{90112 b^6 x \left (b x^{2/3}+a x\right )^{3/2}}{557175 a^7}-\frac{45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac{2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac{1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac{352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac{16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}+\frac{\left (524288 b^{10}\right ) \int \frac{\sqrt{b x^{2/3}+a x}}{\sqrt [3]{x}} \, dx}{4345965 a^{10}}\\ &=-\frac{524288 b^9 \left (b x^{2/3}+a x\right )^{3/2}}{4345965 a^{10}}+\frac{1048576 b^{10} \left (b x^{2/3}+a x\right )^{3/2}}{10140585 a^{11} \sqrt [3]{x}}+\frac{65536 b^8 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{482885 a^9}-\frac{360448 b^7 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{2414425 a^8}+\frac{90112 b^6 x \left (b x^{2/3}+a x\right )^{3/2}}{557175 a^7}-\frac{45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac{2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac{1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac{352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac{16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}-\frac{\left (2097152 b^{11}\right ) \int \frac{\sqrt{b x^{2/3}+a x}}{x^{2/3}} \, dx}{30421755 a^{11}}\\ &=-\frac{524288 b^9 \left (b x^{2/3}+a x\right )^{3/2}}{4345965 a^{10}}-\frac{4194304 b^{11} \left (b x^{2/3}+a x\right )^{3/2}}{50702925 a^{12} x^{2/3}}+\frac{1048576 b^{10} \left (b x^{2/3}+a x\right )^{3/2}}{10140585 a^{11} \sqrt [3]{x}}+\frac{65536 b^8 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{482885 a^9}-\frac{360448 b^7 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{2414425 a^8}+\frac{90112 b^6 x \left (b x^{2/3}+a x\right )^{3/2}}{557175 a^7}-\frac{45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac{2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac{1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac{352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac{16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}+\frac{\left (4194304 b^{12}\right ) \int \frac{\sqrt{b x^{2/3}+a x}}{x} \, dx}{152108775 a^{12}}\\ &=-\frac{524288 b^9 \left (b x^{2/3}+a x\right )^{3/2}}{4345965 a^{10}}+\frac{8388608 b^{12} \left (b x^{2/3}+a x\right )^{3/2}}{152108775 a^{13} x}-\frac{4194304 b^{11} \left (b x^{2/3}+a x\right )^{3/2}}{50702925 a^{12} x^{2/3}}+\frac{1048576 b^{10} \left (b x^{2/3}+a x\right )^{3/2}}{10140585 a^{11} \sqrt [3]{x}}+\frac{65536 b^8 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{482885 a^9}-\frac{360448 b^7 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{2414425 a^8}+\frac{90112 b^6 x \left (b x^{2/3}+a x\right )^{3/2}}{557175 a^7}-\frac{45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac{2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac{1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac{352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac{16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}\\ \end{align*}
Mathematica [A] time = 0.135226, size = 181, normalized size = 0.49 \[ \frac{2 \left (a \sqrt [3]{x}+b\right ) \sqrt{a x+b x^{2/3}} \left (15519504 a^{10} b^2 x^{10/3}-14780480 a^9 b^3 x^3+14002560 a^8 b^4 x^{8/3}-13178880 a^7 b^5 x^{7/3}+12300288 a^6 b^6 x^2-11354112 a^5 b^7 x^{5/3}+10321920 a^4 b^8 x^{4/3}+7864320 a^2 b^{10} x^{2/3}-9175040 a^3 b^9 x-16224936 a^{11} b x^{11/3}+16900975 a^{12} x^4-6291456 a b^{11} \sqrt [3]{x}+4194304 b^{12}\right )}{152108775 a^{13} \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 156, normalized size = 0.4 \begin{align*} -{\frac{2}{152108775\,{a}^{13}}\sqrt{b{x}^{{\frac{2}{3}}}+ax} \left ( b+a\sqrt [3]{x} \right ) \left ( 16224936\,{x}^{11/3}{a}^{11}b-15519504\,{x}^{10/3}{a}^{10}{b}^{2}-14002560\,{x}^{8/3}{a}^{8}{b}^{4}+13178880\,{x}^{7/3}{a}^{7}{b}^{5}+11354112\,{x}^{5/3}{a}^{5}{b}^{7}-10321920\,{x}^{4/3}{a}^{4}{b}^{8}-16900975\,{x}^{4}{a}^{12}+14780480\,{x}^{3}{a}^{9}{b}^{3}-7864320\,{x}^{2/3}{a}^{2}{b}^{10}-12300288\,{x}^{2}{a}^{6}{b}^{6}+6291456\,\sqrt [3]{x}a{b}^{11}+9175040\,x{a}^{3}{b}^{9}-4194304\,{b}^{12} \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}}} x^{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 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 = 1.13299, size = 259, normalized size = 0.7 \begin{align*} -\frac{8388608 \, b^{\frac{27}{2}}}{152108775 \, a^{13}} + \frac{2 \,{\left (16900975 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{27}{2}} - 219036636 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{25}{2}} b + 1309458150 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{23}{2}} b^{2} - 4780561500 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} b^{3} + 11888501625 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} b^{4} - 21259438200 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} b^{5} + 28109701620 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} b^{6} - 27800803800 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} b^{7} + 20534684625 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} b^{8} - 11154643500 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} b^{9} + 4302505350 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b^{10} - 1095183180 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} b^{11} + 152108775 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} b^{12}\right )}}{152108775 \, a^{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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