Optimal. Leaf size=246 \[ \frac {663 a^{15/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{154 b^{21/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {663 a^3 \sqrt {a x+b \sqrt [3]{x}}}{77 b^5 x^{2/3}}-\frac {1989 a^2 \sqrt {a x+b \sqrt [3]{x}}}{385 b^4 x^{4/3}}+\frac {221 a \sqrt {a x+b \sqrt [3]{x}}}{55 b^3 x^2}-\frac {17 \sqrt {a x+b \sqrt [3]{x}}}{5 b^2 x^{8/3}}+\frac {3}{b x^{7/3} \sqrt {a x+b \sqrt [3]{x}}} \]
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Rubi [A] time = 0.36, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2018, 2023, 2025, 2011, 329, 220} \[ \frac {663 a^3 \sqrt {a x+b \sqrt [3]{x}}}{77 b^5 x^{2/3}}-\frac {1989 a^2 \sqrt {a x+b \sqrt [3]{x}}}{385 b^4 x^{4/3}}+\frac {663 a^{15/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{154 b^{21/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {221 a \sqrt {a x+b \sqrt [3]{x}}}{55 b^3 x^2}-\frac {17 \sqrt {a x+b \sqrt [3]{x}}}{5 b^2 x^{8/3}}+\frac {3}{b x^{7/3} \sqrt {a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2011
Rule 2018
Rule 2023
Rule 2025
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=3 \operatorname {Subst}\left (\int \frac {1}{x^7 \left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}+\frac {51 \operatorname {Subst}\left (\int \frac {1}{x^8 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}-\frac {(221 a) \operatorname {Subst}\left (\int \frac {1}{x^6 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{10 b^2}\\ &=\frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}+\frac {\left (1989 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{110 b^3}\\ &=\frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}-\frac {\left (1989 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{154 b^4}\\ &=\frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}+\frac {663 a^3 \sqrt {b \sqrt [3]{x}+a x}}{77 b^5 x^{2/3}}+\frac {\left (663 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{154 b^5}\\ &=\frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}+\frac {663 a^3 \sqrt {b \sqrt [3]{x}+a x}}{77 b^5 x^{2/3}}+\frac {\left (663 a^4 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{154 b^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}+\frac {663 a^3 \sqrt {b \sqrt [3]{x}+a x}}{77 b^5 x^{2/3}}+\frac {\left (663 a^4 \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 )}{77 b^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}+\frac {663 a^3 \sqrt {b \sqrt [3]{x}+a x}}{77 b^5 x^{2/3}}+\frac {663 a^{15/4} \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 )}{154 b^{21/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 64, normalized size = 0.26 \[ -\frac {2 \sqrt {\frac {a x^{2/3}}{b}+1} \, _2F_1\left (-\frac {15}{4},\frac {3}{2};-\frac {11}{4};-\frac {a x^{2/3}}{b}\right )}{5 b x^{7/3} \sqrt {a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} x^{3} + 3 \, a^{2} b^{2} x^{\frac {5}{3}} - 2 \, a b^{3} x - {\left (2 \, a^{3} b x^{2} - b^{4}\right )} x^{\frac {1}{3}}\right )} \sqrt {a x + b x^{\frac {1}{3}}}}{a^{6} x^{8} + 2 \, a^{3} b^{3} x^{6} + b^{6} x^{4}}, x\right ) \]
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 \[ \int \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 261, normalized size = 1.06 \[ \frac {2310 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{4} x^{5}+4320 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{4} x^{5}+3315 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, \sqrt {-a b}\, a^{3} x^{\frac {14}{3}} \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+2652 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{3} b \,x^{\frac {13}{3}}-884 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{2} b^{2} x^{\frac {11}{3}}+476 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a \,b^{3} x^{3}-308 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, b^{4} x^{\frac {7}{3}}}{770 \left (a \,x^{\frac {2}{3}}+b \right ) b^{5} x^{5}} \]
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 \[ \int \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
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 \[ \int \frac {1}{x^3\,{\left (a\,x+b\,x^{1/3}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{3} \left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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