Optimal. Leaf size=163 \[ \frac {5 a^{7/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{7 b^{9/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {10 a \sqrt {a x+b \sqrt [3]{x}}}{7 b^2 x^{2/3}}-\frac {6 \sqrt {a x+b \sqrt [3]{x}}}{7 b x^{4/3}} \]
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Rubi [A] time = 0.20, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2018, 2025, 2011, 329, 220} \[ \frac {5 a^{7/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{7 b^{9/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {10 a \sqrt {a x+b \sqrt [3]{x}}}{7 b^2 x^{2/3}}-\frac {6 \sqrt {a x+b \sqrt [3]{x}}}{7 b x^{4/3}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2011
Rule 2018
Rule 2025
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {b \sqrt [3]{x}+a x}} \, dx &=3 \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{7 b x^{4/3}}-\frac {(15 a) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{7 b}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{7 b x^{4/3}}+\frac {10 a \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{2/3}}+\frac {\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{7 b^2}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{7 b x^{4/3}}+\frac {10 a \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{2/3}}+\frac {\left (5 a^2 \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 )}{7 b^2 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{7 b x^{4/3}}+\frac {10 a \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{2/3}}+\frac {\left (10 a^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 )}{7 b^2 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{7 b x^{4/3}}+\frac {10 a \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{2/3}}+\frac {5 a^{7/4} \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 )}{7 b^{9/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 59, normalized size = 0.36 \[ -\frac {6 \sqrt {\frac {a x^{2/3}}{b}+1} \, _2F_1\left (-\frac {7}{4},\frac {1}{2};-\frac {3}{4};-\frac {a x^{2/3}}{b}\right )}{7 x \sqrt {a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} x^{2} - a b x^{\frac {4}{3}} + b^{2} x^{\frac {2}{3}}\right )} \sqrt {a x + b x^{\frac {1}{3}}}}{a^{3} x^{5} + b^{3} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 142, normalized size = 0.87 \[ \frac {10 a^{2} x^{\frac {5}{3}}+5 \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, a \,x^{\frac {4}{3}} \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+4 a b x -6 b^{2} x^{\frac {1}{3}}}{7 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, b^{2} x^{\frac {4}{3}}} \]
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 \[ \int \frac {1}{\sqrt {a x + b x^{\frac {1}{3}}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^2\,\sqrt {a\,x+b\,x^{1/3}}} \,d x \]
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 \[ \int \frac {1}{x^{2} \sqrt {a x + b \sqrt [3]{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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