Optimal. Leaf size=239 \[ \frac {663 b^{15/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{154 a^{21/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {663 b^3 \sqrt {a x+b \sqrt [3]{x}}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{55 a^3}+\frac {17 x^2 \sqrt {a x+b \sqrt [3]{x}}}{5 a^2}-\frac {3 x^3}{a \sqrt {a x+b \sqrt [3]{x}}} \]
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Rubi [A] time = 0.38, antiderivative size = 239, 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, 2022, 2024, 2011, 329, 220} \[ \frac {1989 b^2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{385 a^4}+\frac {663 b^{15/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{154 a^{21/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {663 b^3 \sqrt {a x+b \sqrt [3]{x}}}{77 a^5}-\frac {221 b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{55 a^3}+\frac {17 x^2 \sqrt {a x+b \sqrt [3]{x}}}{5 a^2}-\frac {3 x^3}{a \sqrt {a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2011
Rule 2018
Rule 2022
Rule 2024
Rubi steps
\begin {align*} \int \frac {x^3}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^{11}}{\left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {51 \operatorname {Subst}\left (\int \frac {x^8}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 a}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}-\frac {(221 b) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{10 a^2}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {\left (1989 b^2\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{110 a^3}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}-\frac {\left (1989 b^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{154 a^4}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {663 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {\left (663 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{154 a^5}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {663 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {\left (663 b^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 a^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {663 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {\left (663 b^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 a^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {663 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {663 b^{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 a^{21/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.13, size = 124, normalized size = 0.52 \[ \frac {\sqrt {a x+b \sqrt [3]{x}} \left (154 a^4 x^{8/3}-238 a^3 b x^2+442 a^2 b^2 x^{4/3}+3315 b^4 \sqrt {\frac {a x^{2/3}}{b}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {a x^{2/3}}{b}\right )-1326 a b^3 x^{2/3}-3315 b^4\right )}{385 a^5 \left (a x^{2/3}+b\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} x^{5} + 3 \, a^{2} b^{2} x^{\frac {11}{3}} - 2 \, a b^{3} x^{3} - {\left (2 \, a^{3} b x^{4} - b^{4} x^{2}\right )} x^{\frac {1}{3}}\right )} \sqrt {a x + b x^{\frac {1}{3}}}}{a^{6} x^{4} + 2 \, a^{3} b^{3} x^{2} + b^{6}}, x\right ) \]
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 \[ \int \frac {x^{3}}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 260, normalized size = 1.09 \[ -\frac {-308 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{5} x^{3}+476 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{4} b \,x^{\frac {7}{3}}-884 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{3} b^{2} x^{\frac {5}{3}}+2652 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{2} b^{3} x +4320 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a \,b^{4} x^{\frac {1}{3}}+2310 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a \,b^{4} x^{\frac {1}{3}}-3315 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, b^{4} \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{770 \left (a \,x^{\frac {2}{3}}+b \right ) a^{6} x^{\frac {1}{3}}} \]
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 \[ \int \frac {x^{3}}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^3}{{\left (a\,x+b\,x^{1/3}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3}}{\left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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