Optimal. Leaf size=383 \[ \frac {77 a^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{10 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {77 a^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {77 a^{5/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 b^4 \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}-\frac {77 a^2 \sqrt {a x+b \sqrt [3]{x}}}{5 b^4 \sqrt [3]{x}}+\frac {77 a \sqrt {a x+b \sqrt [3]{x}}}{15 b^3 x}-\frac {11 \sqrt {a x+b \sqrt [3]{x}}}{3 b^2 x^{5/3}}+\frac {3}{b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}} \]
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Rubi [A] time = 0.48, antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \frac {77 a^{5/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 b^4 \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {77 a^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{10 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {77 a^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {77 a^2 \sqrt {a x+b \sqrt [3]{x}}}{5 b^4 \sqrt [3]{x}}-\frac {11 \sqrt {a x+b \sqrt [3]{x}}}{3 b^2 x^{5/3}}+\frac {77 a \sqrt {a x+b \sqrt [3]{x}}}{15 b^3 x}+\frac {3}{b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2018
Rule 2023
Rule 2025
Rule 2032
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=3 \operatorname {Subst}\left (\int \frac {1}{x^4 \left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}+\frac {33 \operatorname {Subst}\left (\int \frac {1}{x^5 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}-\frac {(77 a) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{6 b^2}\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}+\frac {77 a \sqrt {b \sqrt [3]{x}+a x}}{15 b^3 x}+\frac {\left (77 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{10 b^3}\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}+\frac {77 a \sqrt {b \sqrt [3]{x}+a x}}{15 b^3 x}-\frac {77 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b^4 \sqrt [3]{x}}+\frac {\left (77 a^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{10 b^4}\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}+\frac {77 a \sqrt {b \sqrt [3]{x}+a x}}{15 b^3 x}-\frac {77 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b^4 \sqrt [3]{x}}+\frac {\left (77 a^3 \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 )}{10 b^4 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}+\frac {77 a \sqrt {b \sqrt [3]{x}+a x}}{15 b^3 x}-\frac {77 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b^4 \sqrt [3]{x}}+\frac {\left (77 a^3 \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 )}{5 b^4 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}+\frac {77 a \sqrt {b \sqrt [3]{x}+a x}}{15 b^3 x}-\frac {77 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b^4 \sqrt [3]{x}}+\frac {\left (77 a^{5/2} \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 )}{5 b^{7/2} \sqrt {b \sqrt [3]{x}+a x}}-\frac {\left (77 a^{5/2} \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 )}{5 b^{7/2} \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}+\frac {77 a^{5/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{5 b^4 \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}+\frac {77 a \sqrt {b \sqrt [3]{x}+a x}}{15 b^3 x}-\frac {77 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b^4 \sqrt [3]{x}}-\frac {77 a^{9/4} \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 )}{5 b^{15/4} \sqrt {b \sqrt [3]{x}+a x}}+\frac {77 a^{9/4} \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 )}{10 b^{15/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 64, normalized size = 0.17 \[ -\frac {2 \sqrt {\frac {a x^{2/3}}{b}+1} \, _2F_1\left (-\frac {9}{4},\frac {3}{2};-\frac {5}{4};-\frac {a x^{2/3}}{b}\right )}{3 b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 7.17, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} x^{3} + 3 \, a^{2} b^{2} x^{\frac {5}{3}} - 2 \, a b^{3} x - {\left (2 \, a^{3} b x^{2} - b^{4}\right )} x^{\frac {1}{3}}\right )} \sqrt {a x + b x^{\frac {1}{3}}}}{a^{6} x^{7} + 2 \, a^{3} b^{3} x^{5} + b^{6} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 339, normalized size = 0.89 \[ -\frac {-462 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{2} b \,x^{\frac {8}{3}} \EllipticE \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+231 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{2} b \,x^{\frac {8}{3}} \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+462 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{3} x^{\frac {10}{3}}+372 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{2} b \,x^{\frac {8}{3}}-64 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{2} b \,x^{\frac {8}{3}}-44 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a \,b^{2} x^{2}+20 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, b^{3} x^{\frac {4}{3}}}{30 \left (a \,x^{\frac {2}{3}}+b \right ) b^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^2\,{\left (a\,x+b\,x^{1/3}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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