Optimal. Leaf size=84 \[ -\frac {32 a^2 \sqrt {a x+b \sqrt {x}}}{15 b^3 \sqrt {x}}+\frac {16 a \sqrt {a x+b \sqrt {x}}}{15 b^2 x}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2016, 2014} \begin {gather*} -\frac {32 a^2 \sqrt {a x+b \sqrt {x}}}{15 b^3 \sqrt {x}}+\frac {16 a \sqrt {a x+b \sqrt {x}}}{15 b^2 x}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {b \sqrt {x}+a x}} \, dx &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{5 b x^{3/2}}-\frac {(4 a) \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{5 b}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{5 b x^{3/2}}+\frac {16 a \sqrt {b \sqrt {x}+a x}}{15 b^2 x}+\frac {\left (8 a^2\right ) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{15 b^2}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{5 b x^{3/2}}+\frac {16 a \sqrt {b \sqrt {x}+a x}}{15 b^2 x}-\frac {32 a^2 \sqrt {b \sqrt {x}+a x}}{15 b^3 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 48, normalized size = 0.57 \begin {gather*} -\frac {4 \sqrt {a x+b \sqrt {x}} \left (8 a^2 x-4 a b \sqrt {x}+3 b^2\right )}{15 b^3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 48, normalized size = 0.57 \begin {gather*} -\frac {4 \sqrt {a x+b \sqrt {x}} \left (8 a^2 x-4 a b \sqrt {x}+3 b^2\right )}{15 b^3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 42, normalized size = 0.50 \begin {gather*} \frac {4 \, {\left (4 \, a b x - {\left (8 \, a^{2} x + 3 \, b^{2}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{15 \, b^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 84, normalized size = 1.00 \begin {gather*} \frac {4 \, {\left (20 \, a {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{2} + 15 \, \sqrt {a} b {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + 3 \, b^{2}\right )}}{15 \, {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 218, normalized size = 2.60 \begin {gather*} \frac {\sqrt {a x +b \sqrt {x}}\, \left (-15 a^{3} b \,x^{\frac {7}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+15 a^{3} b \,x^{\frac {7}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+30 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {7}{2}} x^{\frac {7}{2}}+30 \sqrt {a x +b \sqrt {x}}\, a^{\frac {7}{2}} x^{\frac {7}{2}}-60 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} x^{\frac {5}{2}}+28 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b \,x^{2}-12 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{2} x^{\frac {3}{2}}\right )}{15 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}\, b^{4} x^{\frac {7}{2}}} \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 {1}{\sqrt {a x + b \sqrt {x}} 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 {1}{x^2\,\sqrt {a\,x+b\,\sqrt {x}}} \,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 {1}{x^{2} \sqrt {a x + b \sqrt {x}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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