Optimal. Leaf size=79 \[ \frac {32 a \sqrt {a x+b \sqrt {x}}}{3 b^3 \sqrt {x}}-\frac {16 \sqrt {a x+b \sqrt {x}}}{3 b^2 x}+\frac {4}{b \sqrt {x} \sqrt {a x+b \sqrt {x}}} \]
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Rubi [A] time = 0.12, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2015, 2016, 2014} \[ \frac {32 a \sqrt {a x+b \sqrt {x}}}{3 b^3 \sqrt {x}}-\frac {16 \sqrt {a x+b \sqrt {x}}}{3 b^2 x}+\frac {4}{b \sqrt {x} \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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Rule 2014
Rule 2015
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x \left (b \sqrt {x}+a x\right )^{3/2}} \, dx &=\frac {4}{b \sqrt {x} \sqrt {b \sqrt {x}+a x}}+\frac {4 \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{b}\\ &=\frac {4}{b \sqrt {x} \sqrt {b \sqrt {x}+a x}}-\frac {16 \sqrt {b \sqrt {x}+a x}}{3 b^2 x}-\frac {(8 a) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{3 b^2}\\ &=\frac {4}{b \sqrt {x} \sqrt {b \sqrt {x}+a x}}-\frac {16 \sqrt {b \sqrt {x}+a x}}{3 b^2 x}+\frac {32 a \sqrt {b \sqrt {x}+a x}}{3 b^3 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 48, normalized size = 0.61 \[ \frac {4 \left (8 a^2 x+4 a b \sqrt {x}-b^2\right )}{3 b^3 \sqrt {x} \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 63, normalized size = 0.80 \[ -\frac {4 \, {\left (4 \, a^{2} b x - b^{3} - {\left (8 \, a^{3} x - 5 \, a b^{2}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{3 \, {\left (a^{2} b^{3} x^{2} - b^{5} x\right )}} \]
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 \[ \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 524, normalized size = 6.63 \[ \frac {\sqrt {a x +b \sqrt {x}}\, \left (3 a^{4} b \,x^{\frac {7}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-3 a^{4} b \,x^{\frac {7}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+6 a^{3} b^{2} x^{3} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-6 a^{3} b^{2} x^{3} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+3 a^{2} b^{3} x^{\frac {5}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-3 a^{2} b^{3} x^{\frac {5}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-6 \sqrt {a x +b \sqrt {x}}\, a^{\frac {9}{2}} x^{\frac {7}{2}}-6 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {9}{2}} x^{\frac {7}{2}}-12 \sqrt {a x +b \sqrt {x}}\, a^{\frac {7}{2}} b \,x^{3}-12 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {7}{2}} b \,x^{3}-6 \sqrt {a x +b \sqrt {x}}\, a^{\frac {5}{2}} b^{2} x^{\frac {5}{2}}-6 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {5}{2}} b^{2} x^{\frac {5}{2}}+24 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} x^{\frac {5}{2}}-12 \left (\left (a \sqrt {x}+b \right ) \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} x^{\frac {5}{2}}+44 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b \,x^{2}+16 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{2} x^{\frac {3}{2}}-4 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{3} x \right )}{3 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \left (a \sqrt {x}+b \right )^{2} \sqrt {a}\, b^{4} x^{\frac {5}{2}}} \]
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 \[ \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x}\,{d x} \]
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 \[ \int \frac {1}{x\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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