Optimal. Leaf size=471 \[ -\frac{4807 a^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{442 b^{27/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{4807 a^{11/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{221 b^7 \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{24035 a^3 \sqrt{a x+b \sqrt [3]{x}}}{4641 b^5 x^{5/3}}-\frac{6555 a^2 \sqrt{a x+b \sqrt [3]{x}}}{1547 b^4 x^{7/3}}+\frac{4807 a^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{221 b^{27/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{4807 a^5 \sqrt{a x+b \sqrt [3]{x}}}{221 b^7 \sqrt [3]{x}}-\frac{4807 a^4 \sqrt{a x+b \sqrt [3]{x}}}{663 b^6 x}+\frac{437 a \sqrt{a x+b \sqrt [3]{x}}}{119 b^3 x^3}-\frac{23 \sqrt{a x+b \sqrt [3]{x}}}{7 b^2 x^{11/3}}+\frac{3}{b x^{10/3} \sqrt{a x+b \sqrt [3]{x}}} \]
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Rubi [A] time = 0.693825, antiderivative size = 471, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {2018, 2023, 2025, 2032, 329, 305, 220, 1196} \[ -\frac{4807 a^{11/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{221 b^7 \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{24035 a^3 \sqrt{a x+b \sqrt [3]{x}}}{4641 b^5 x^{5/3}}-\frac{6555 a^2 \sqrt{a x+b \sqrt [3]{x}}}{1547 b^4 x^{7/3}}-\frac{4807 a^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{442 b^{27/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{4807 a^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{221 b^{27/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{4807 a^5 \sqrt{a x+b \sqrt [3]{x}}}{221 b^7 \sqrt [3]{x}}-\frac{4807 a^4 \sqrt{a x+b \sqrt [3]{x}}}{663 b^6 x}+\frac{437 a \sqrt{a x+b \sqrt [3]{x}}}{119 b^3 x^3}-\frac{23 \sqrt{a x+b \sqrt [3]{x}}}{7 b^2 x^{11/3}}+\frac{3}{b x^{10/3} \sqrt{a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 2023
Rule 2025
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=3 \operatorname{Subst}\left (\int \frac{1}{x^{10} \left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3}{b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}+\frac{69 \operatorname{Subst}\left (\int \frac{1}{x^{11} \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac{3}{b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}-\frac{23 \sqrt{b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}-\frac{(437 a) \operatorname{Subst}\left (\int \frac{1}{x^9 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{14 b^2}\\ &=\frac{3}{b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}-\frac{23 \sqrt{b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac{437 a \sqrt{b \sqrt [3]{x}+a x}}{119 b^3 x^3}+\frac{\left (6555 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^7 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{238 b^3}\\ &=\frac{3}{b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}-\frac{23 \sqrt{b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac{437 a \sqrt{b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac{6555 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}-\frac{\left (72105 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^5 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{3094 b^4}\\ &=\frac{3}{b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}-\frac{23 \sqrt{b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac{437 a \sqrt{b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac{6555 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac{24035 a^3 \sqrt{b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}+\frac{\left (24035 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1326 b^5}\\ &=\frac{3}{b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}-\frac{23 \sqrt{b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac{437 a \sqrt{b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac{6555 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac{24035 a^3 \sqrt{b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac{4807 a^4 \sqrt{b \sqrt [3]{x}+a x}}{663 b^6 x}-\frac{\left (4807 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{442 b^6}\\ &=\frac{3}{b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}-\frac{23 \sqrt{b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac{437 a \sqrt{b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac{6555 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac{24035 a^3 \sqrt{b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac{4807 a^4 \sqrt{b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac{4807 a^5 \sqrt{b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}-\frac{\left (4807 a^6\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{442 b^7}\\ &=\frac{3}{b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}-\frac{23 \sqrt{b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac{437 a \sqrt{b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac{6555 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac{24035 a^3 \sqrt{b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac{4807 a^4 \sqrt{b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac{4807 a^5 \sqrt{b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}-\frac{\left (4807 a^6 \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\sqrt{b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{442 b^7 \sqrt{b \sqrt [3]{x}+a x}}\\ &=\frac{3}{b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}-\frac{23 \sqrt{b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac{437 a \sqrt{b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac{6555 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac{24035 a^3 \sqrt{b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac{4807 a^4 \sqrt{b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac{4807 a^5 \sqrt{b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}-\frac{\left (4807 a^6 \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{221 b^7 \sqrt{b \sqrt [3]{x}+a x}}\\ &=\frac{3}{b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}-\frac{23 \sqrt{b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac{437 a \sqrt{b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac{6555 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac{24035 a^3 \sqrt{b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac{4807 a^4 \sqrt{b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac{4807 a^5 \sqrt{b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}-\frac{\left (4807 a^{11/2} \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 )}{221 b^{13/2} \sqrt{b \sqrt [3]{x}+a x}}+\frac{\left (4807 a^{11/2} \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{a} x^2}{\sqrt{b}}}{\sqrt{b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{221 b^{13/2} \sqrt{b \sqrt [3]{x}+a x}}\\ &=\frac{3}{b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}-\frac{4807 a^{11/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{221 b^7 \left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right ) \sqrt{b \sqrt [3]{x}+a x}}-\frac{23 \sqrt{b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac{437 a \sqrt{b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac{6555 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac{24035 a^3 \sqrt{b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac{4807 a^4 \sqrt{b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac{4807 a^5 \sqrt{b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}+\frac{4807 a^{21/4} \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} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{221 b^{27/4} \sqrt{b \sqrt [3]{x}+a x}}-\frac{4807 a^{21/4} \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 )}{442 b^{27/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}
Mathematica [C] time = 0.0547918, size = 64, normalized size = 0.14 \[ -\frac{2 \sqrt{\frac{a x^{2/3}}{b}+1} \, _2F_1\left (-\frac{21}{4},\frac{3}{2};-\frac{17}{4};-\frac{a x^{2/3}}{b}\right )}{7 b x^{10/3} \sqrt{a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 413, normalized size = 0.9 \begin{align*}{\frac{1}{9282\,{x}^{7}{b}^{7}} \left ( -201894\,{a}^{5}b\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}}}}{x}^{{\frac{20}{3}}}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticE} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) +100947\,{a}^{5}b\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}}}}{x}^{{\frac{20}{3}}}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) +201894\,{x}^{{\frac{22}{3}}}\sqrt{b\sqrt [3]{x}+ax}{a}^{6}-19228\,{x}^{6}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{a}^{4}{b}^{2}-39452\,{x}^{{\frac{20}{3}}}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{a}^{5}b+174048\,{x}^{{\frac{20}{3}}}\sqrt{b\sqrt [3]{x}+ax}{a}^{5}b+3588\,{x}^{4}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }a{b}^{5}-5244\,{x}^{14/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{a}^{2}{b}^{4}+8740\,{x}^{16/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{a}^{3}{b}^{3}-2652\,{x}^{10/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{b}^{6} \right ) \left ( b+a{x}^{{\frac{2}{3}}} \right ) ^{-1}} \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{1}{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}} x^{4}}\,{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 (\frac{{\left (a^{4} x^{3} + 3 \, a^{2} b^{2} x^{\frac{5}{3}} - 2 \, a b^{3} x -{\left (2 \, a^{3} b x^{2} - b^{4}\right )} x^{\frac{1}{3}}\right )} \sqrt{a x + b x^{\frac{1}{3}}}}{a^{6} x^{9} + 2 \, a^{3} b^{3} x^{7} + b^{6} x^{5}}, 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 \frac{1}{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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