Optimal. Leaf size=90 \[ \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{4 b^{3/2}}-\frac {3 a \sqrt {a x+b x^{2/3}}}{4 b x^{2/3}}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 x} \]
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Rubi [A] time = 0.14, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2020, 2025, 2029, 206} \begin {gather*} \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{4 b^{3/2}}-\frac {3 a \sqrt {a x+b x^{2/3}}}{4 b x^{2/3}}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2020
Rule 2025
Rule 2029
Rubi steps
\begin {align*} \int \frac {\sqrt {b x^{2/3}+a x}}{x^2} \, dx &=-\frac {3 \sqrt {b x^{2/3}+a x}}{2 x}+\frac {1}{4} a \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{2 x}-\frac {3 a \sqrt {b x^{2/3}+a x}}{4 b x^{2/3}}-\frac {a^2 \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{8 b}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{2 x}-\frac {3 a \sqrt {b x^{2/3}+a x}}{4 b x^{2/3}}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{4 b}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{2 x}-\frac {3 a \sqrt {b x^{2/3}+a x}}{4 b x^{2/3}}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{4 b^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 57, normalized size = 0.63 \begin {gather*} -\frac {2 a^2 \left (a \sqrt [3]{x}+b\right ) \sqrt {a x+b x^{2/3}} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {\sqrt [3]{x} a}{b}+1\right )}{b^3 \sqrt [3]{x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 76, normalized size = 0.84 \begin {gather*} \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{4 b^{3/2}}-\frac {3 \left (a \sqrt [3]{x}+2 b\right ) \sqrt {a x+b x^{2/3}}}{4 b x} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 72, normalized size = 0.80 \begin {gather*} -\frac {3 \, {\left (\frac {a^{3} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} + \frac {{\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{3} + \sqrt {a x^{\frac {1}{3}} + b} a^{3} b}{a^{2} b x^{\frac {2}{3}}}\right )}}{4 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 80, normalized size = 0.89 \begin {gather*} \frac {3 \sqrt {a x +b \,x^{\frac {2}{3}}}\, \left (a^{2} b \,x^{\frac {2}{3}} \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )-\sqrt {a \,x^{\frac {1}{3}}+b}\, b^{\frac {5}{2}}-\left (a \,x^{\frac {1}{3}}+b \right )^{\frac {3}{2}} b^{\frac {3}{2}}\right )}{4 \sqrt {a \,x^{\frac {1}{3}}+b}\, b^{\frac {5}{2}} x} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {a x + b x^{\frac {2}{3}}}}{x^{2}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {a\,x+b\,x^{2/3}}}{x^2} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {a x + b x^{\frac {2}{3}}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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