Optimal. Leaf size=296 \[ -\frac {3 \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 a^{3/4} b^{3/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {3 \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{a^{3/4} b^{3/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {3 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{\sqrt {a} b \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {3 x^{2/3}}{b \sqrt {a x+b \sqrt [3]{x}}} \]
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Rubi [A] time = 0.26, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {2006, 2018, 2032, 329, 305, 220, 1196} \[ -\frac {3 \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 a^{3/4} b^{3/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {3 \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{a^{3/4} b^{3/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {3 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{\sqrt {a} b \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {3 x^{2/3}}{b \sqrt {a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2006
Rule 2018
Rule 2032
Rubi steps
\begin {align*} \int \frac {1}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=\frac {3 x^{2/3}}{b \sqrt {b \sqrt [3]{x}+a x}}-\frac {\int \frac {1}{\sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}} \, dx}{2 b}\\ &=\frac {3 x^{2/3}}{b \sqrt {b \sqrt [3]{x}+a x}}-\frac {3 \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac {3 x^{2/3}}{b \sqrt {b \sqrt [3]{x}+a x}}-\frac {\left (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 )}{2 b \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3 x^{2/3}}{b \sqrt {b \sqrt [3]{x}+a x}}-\frac {\left (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 )}{b \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3 x^{2/3}}{b \sqrt {b \sqrt [3]{x}+a x}}-\frac {\left (3 \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 )}{\sqrt {a} \sqrt {b} \sqrt {b \sqrt [3]{x}+a x}}+\frac {\left (3 \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 )}{\sqrt {a} \sqrt {b} \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {3 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{\sqrt {a} b \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}+\frac {3 x^{2/3}}{b \sqrt {b \sqrt [3]{x}+a x}}+\frac {3 \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 )}{a^{3/4} b^{3/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {3 \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 )}{2 a^{3/4} b^{3/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 62, normalized size = 0.21 \[ \frac {2 x^{2/3} \sqrt {\frac {a x^{2/3}}{b}+1} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\frac {a x^{2/3}}{b}\right )}{b \sqrt {a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 7.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} x^{3} + 3 \, a^{2} b^{2} x^{\frac {5}{3}} - 2 \, a b^{3} x - {\left (2 \, a^{3} b x^{2} - b^{4}\right )} x^{\frac {1}{3}}\right )} \sqrt {a x + b x^{\frac {1}{3}}}}{a^{6} x^{5} + 2 \, a^{3} b^{3} x^{3} + b^{6} x}, x\right ) \]
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 \[ \int \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 242, normalized size = 0.82 \[ \frac {-3 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, b \EllipticE \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+\frac {3 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, b \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2}+3 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a \,x^{\frac {2}{3}}}{\left (a \,x^{\frac {2}{3}}+b \right ) a b \,x^{\frac {1}{3}}} \]
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 \[ \int \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.35, size = 40, normalized size = 0.14 \[ \frac {2\,x\,{\left (\frac {a\,x^{2/3}}{b}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{4},\frac {3}{2};\ \frac {7}{4};\ -\frac {a\,x^{2/3}}{b}\right )}{{\left (a\,x+b\,x^{1/3}\right )}^{3/2}} \]
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 \[ \int \frac {1}{\left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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