Optimal. Leaf size=248 \[ -\frac{65536 b^7 \sqrt{a x+b x^{2/3}}}{2145 a^9 \sqrt [3]{x}}+\frac{32768 b^6 \sqrt{a x+b x^{2/3}}}{2145 a^8}-\frac{8192 b^5 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{715 a^7}+\frac{4096 b^4 x^{2/3} \sqrt{a x+b x^{2/3}}}{429 a^6}-\frac{3584 b^3 x \sqrt{a x+b x^{2/3}}}{429 a^5}+\frac{5376 b^2 x^{4/3} \sqrt{a x+b x^{2/3}}}{715 a^4}-\frac{448 b x^{5/3} \sqrt{a x+b x^{2/3}}}{65 a^3}+\frac{32 x^2 \sqrt{a x+b x^{2/3}}}{5 a^2}-\frac{6 x^3}{a \sqrt{a x+b x^{2/3}}} \]
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Rubi [A] time = 0.413968, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2015, 2016, 2002, 2014} \[ -\frac{65536 b^7 \sqrt{a x+b x^{2/3}}}{2145 a^9 \sqrt [3]{x}}+\frac{32768 b^6 \sqrt{a x+b x^{2/3}}}{2145 a^8}-\frac{8192 b^5 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{715 a^7}+\frac{4096 b^4 x^{2/3} \sqrt{a x+b x^{2/3}}}{429 a^6}-\frac{3584 b^3 x \sqrt{a x+b x^{2/3}}}{429 a^5}+\frac{5376 b^2 x^{4/3} \sqrt{a x+b x^{2/3}}}{715 a^4}-\frac{448 b x^{5/3} \sqrt{a x+b x^{2/3}}}{65 a^3}+\frac{32 x^2 \sqrt{a x+b x^{2/3}}}{5 a^2}-\frac{6 x^3}{a \sqrt{a x+b x^{2/3}}} \]
Antiderivative was successfully verified.
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Rule 2015
Rule 2016
Rule 2002
Rule 2014
Rubi steps
\begin{align*} \int \frac{x^3}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx &=-\frac{6 x^3}{a \sqrt{b x^{2/3}+a x}}+\frac{16 \int \frac{x^2}{\sqrt{b x^{2/3}+a x}} \, dx}{a}\\ &=-\frac{6 x^3}{a \sqrt{b x^{2/3}+a x}}+\frac{32 x^2 \sqrt{b x^{2/3}+a x}}{5 a^2}-\frac{(224 b) \int \frac{x^{5/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{15 a^2}\\ &=-\frac{6 x^3}{a \sqrt{b x^{2/3}+a x}}-\frac{448 b x^{5/3} \sqrt{b x^{2/3}+a x}}{65 a^3}+\frac{32 x^2 \sqrt{b x^{2/3}+a x}}{5 a^2}+\frac{\left (896 b^2\right ) \int \frac{x^{4/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{65 a^3}\\ &=-\frac{6 x^3}{a \sqrt{b x^{2/3}+a x}}+\frac{5376 b^2 x^{4/3} \sqrt{b x^{2/3}+a x}}{715 a^4}-\frac{448 b x^{5/3} \sqrt{b x^{2/3}+a x}}{65 a^3}+\frac{32 x^2 \sqrt{b x^{2/3}+a x}}{5 a^2}-\frac{\left (1792 b^3\right ) \int \frac{x}{\sqrt{b x^{2/3}+a x}} \, dx}{143 a^4}\\ &=-\frac{6 x^3}{a \sqrt{b x^{2/3}+a x}}-\frac{3584 b^3 x \sqrt{b x^{2/3}+a x}}{429 a^5}+\frac{5376 b^2 x^{4/3} \sqrt{b x^{2/3}+a x}}{715 a^4}-\frac{448 b x^{5/3} \sqrt{b x^{2/3}+a x}}{65 a^3}+\frac{32 x^2 \sqrt{b x^{2/3}+a x}}{5 a^2}+\frac{\left (14336 b^4\right ) \int \frac{x^{2/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{1287 a^5}\\ &=-\frac{6 x^3}{a \sqrt{b x^{2/3}+a x}}+\frac{4096 b^4 x^{2/3} \sqrt{b x^{2/3}+a x}}{429 a^6}-\frac{3584 b^3 x \sqrt{b x^{2/3}+a x}}{429 a^5}+\frac{5376 b^2 x^{4/3} \sqrt{b x^{2/3}+a x}}{715 a^4}-\frac{448 b x^{5/3} \sqrt{b x^{2/3}+a x}}{65 a^3}+\frac{32 x^2 \sqrt{b x^{2/3}+a x}}{5 a^2}-\frac{\left (4096 b^5\right ) \int \frac{\sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}} \, dx}{429 a^6}\\ &=-\frac{6 x^3}{a \sqrt{b x^{2/3}+a x}}-\frac{8192 b^5 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{715 a^7}+\frac{4096 b^4 x^{2/3} \sqrt{b x^{2/3}+a x}}{429 a^6}-\frac{3584 b^3 x \sqrt{b x^{2/3}+a x}}{429 a^5}+\frac{5376 b^2 x^{4/3} \sqrt{b x^{2/3}+a x}}{715 a^4}-\frac{448 b x^{5/3} \sqrt{b x^{2/3}+a x}}{65 a^3}+\frac{32 x^2 \sqrt{b x^{2/3}+a x}}{5 a^2}+\frac{\left (16384 b^6\right ) \int \frac{1}{\sqrt{b x^{2/3}+a x}} \, dx}{2145 a^7}\\ &=-\frac{6 x^3}{a \sqrt{b x^{2/3}+a x}}+\frac{32768 b^6 \sqrt{b x^{2/3}+a x}}{2145 a^8}-\frac{8192 b^5 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{715 a^7}+\frac{4096 b^4 x^{2/3} \sqrt{b x^{2/3}+a x}}{429 a^6}-\frac{3584 b^3 x \sqrt{b x^{2/3}+a x}}{429 a^5}+\frac{5376 b^2 x^{4/3} \sqrt{b x^{2/3}+a x}}{715 a^4}-\frac{448 b x^{5/3} \sqrt{b x^{2/3}+a x}}{65 a^3}+\frac{32 x^2 \sqrt{b x^{2/3}+a x}}{5 a^2}-\frac{\left (32768 b^7\right ) \int \frac{1}{\sqrt [3]{x} \sqrt{b x^{2/3}+a x}} \, dx}{6435 a^8}\\ &=-\frac{6 x^3}{a \sqrt{b x^{2/3}+a x}}+\frac{32768 b^6 \sqrt{b x^{2/3}+a x}}{2145 a^8}-\frac{65536 b^7 \sqrt{b x^{2/3}+a x}}{2145 a^9 \sqrt [3]{x}}-\frac{8192 b^5 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{715 a^7}+\frac{4096 b^4 x^{2/3} \sqrt{b x^{2/3}+a x}}{429 a^6}-\frac{3584 b^3 x \sqrt{b x^{2/3}+a x}}{429 a^5}+\frac{5376 b^2 x^{4/3} \sqrt{b x^{2/3}+a x}}{715 a^4}-\frac{448 b x^{5/3} \sqrt{b x^{2/3}+a x}}{65 a^3}+\frac{32 x^2 \sqrt{b x^{2/3}+a x}}{5 a^2}\\ \end{align*}
Mathematica [A] time = 0.0885618, size = 122, normalized size = 0.49 \[ \frac{2 \left (672 a^6 b^2 x^{7/3}-896 a^5 b^3 x^2+1280 a^4 b^4 x^{5/3}-2048 a^3 b^5 x^{4/3}+4096 a^2 b^6 x-528 a^7 b x^{8/3}+429 a^8 x^3-16384 a b^7 x^{2/3}-32768 b^8 \sqrt [3]{x}\right )}{2145 a^9 \sqrt{a x+b x^{2/3}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 110, normalized size = 0.4 \begin{align*}{\frac{2\,x}{2145\,{a}^{9}} \left ( b+a\sqrt [3]{x} \right ) \left ( 429\,{x}^{8/3}{a}^{8}-528\,{x}^{7/3}{a}^{7}b+672\,{x}^{2}{a}^{6}{b}^{2}-896\,{x}^{5/3}{a}^{5}{b}^{3}+1280\,{x}^{4/3}{a}^{4}{b}^{4}-2048\,x{a}^{3}{b}^{5}+4096\,{x}^{2/3}{a}^{2}{b}^{6}-16384\,\sqrt [3]{x}a{b}^{7}-32768\,{b}^{8} \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^{3}}{{\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^{3}}{\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 = 1.21474, size = 220, normalized size = 0.89 \begin{align*} \frac{65536 \, b^{\frac{15}{2}}}{2145 \, a^{9}} - \frac{6 \, b^{8}}{\sqrt{a x^{\frac{1}{3}} + b} a^{9}} + \frac{2 \,{\left (429 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{126} - 3960 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{126} b + 16380 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{126} b^{2} - 40040 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{126} b^{3} + 64350 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{126} b^{4} - 72072 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{126} b^{5} + 60060 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{126} b^{6} - 51480 \, \sqrt{a x^{\frac{1}{3}} + b} a^{126} b^{7}\right )}}{2145 \, a^{135}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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