Optimal. Leaf size=107 \[ -\frac {64 a^2 \sqrt {a x+b \sqrt {x}}}{5 b^4 \sqrt {x}}+\frac {32 a \sqrt {a x+b \sqrt {x}}}{5 b^3 x}-\frac {24 \sqrt {a x+b \sqrt {x}}}{5 b^2 x^{3/2}}+\frac {4}{b x \sqrt {a x+b \sqrt {x}}} \]
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Rubi [A] time = 0.16, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2015, 2016, 2014} \[ -\frac {64 a^2 \sqrt {a x+b \sqrt {x}}}{5 b^4 \sqrt {x}}-\frac {24 \sqrt {a x+b \sqrt {x}}}{5 b^2 x^{3/2}}+\frac {32 a \sqrt {a x+b \sqrt {x}}}{5 b^3 x}+\frac {4}{b x \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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Rule 2014
Rule 2015
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx &=\frac {4}{b x \sqrt {b \sqrt {x}+a x}}+\frac {6 \int \frac {1}{x^2 \sqrt {b \sqrt {x}+a x}} \, dx}{b}\\ &=\frac {4}{b x \sqrt {b \sqrt {x}+a x}}-\frac {24 \sqrt {b \sqrt {x}+a x}}{5 b^2 x^{3/2}}-\frac {(24 a) \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{5 b^2}\\ &=\frac {4}{b x \sqrt {b \sqrt {x}+a x}}-\frac {24 \sqrt {b \sqrt {x}+a x}}{5 b^2 x^{3/2}}+\frac {32 a \sqrt {b \sqrt {x}+a x}}{5 b^3 x}+\frac {\left (16 a^2\right ) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{5 b^3}\\ &=\frac {4}{b x \sqrt {b \sqrt {x}+a x}}-\frac {24 \sqrt {b \sqrt {x}+a x}}{5 b^2 x^{3/2}}+\frac {32 a \sqrt {b \sqrt {x}+a x}}{5 b^3 x}-\frac {64 a^2 \sqrt {b \sqrt {x}+a x}}{5 b^4 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 57, normalized size = 0.53 \[ -\frac {4 \left (16 a^3 x^{3/2}+8 a^2 b x-2 a b^2 \sqrt {x}+b^3\right )}{5 b^4 x \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.36, size = 79, normalized size = 0.74 \[ \frac {4 \, {\left (8 \, a^{3} b x^{2} - 3 \, a b^{3} x - {\left (16 \, a^{4} x^{2} - 10 \, a^{2} b^{2} x - b^{4}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{5 \, {\left (a^{2} b^{4} x^{3} - b^{6} x^{2}\right )}} \]
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 \[ \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 548, normalized size = 5.12 \[ \frac {2 \sqrt {a x +b \sqrt {x}}\, \left (-5 a^{5} b \,x^{\frac {9}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+5 a^{5} b \,x^{\frac {9}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-10 a^{4} b^{2} x^{4} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+10 a^{4} b^{2} x^{4} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-5 a^{3} b^{3} x^{\frac {7}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+5 a^{3} b^{3} x^{\frac {7}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+10 \sqrt {a x +b \sqrt {x}}\, a^{\frac {11}{2}} x^{\frac {9}{2}}+10 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {11}{2}} x^{\frac {9}{2}}+20 \sqrt {a x +b \sqrt {x}}\, a^{\frac {9}{2}} b \,x^{4}+20 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {9}{2}} b \,x^{4}+10 \sqrt {a x +b \sqrt {x}}\, a^{\frac {7}{2}} b^{2} x^{\frac {7}{2}}+10 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {7}{2}} b^{2} x^{\frac {7}{2}}-30 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{\frac {7}{2}}+10 \left (\left (a \sqrt {x}+b \right ) \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{\frac {7}{2}}-52 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} b \,x^{3}-16 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{2} x^{\frac {5}{2}}+4 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{3} x^{2}-2 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{4} x^{\frac {3}{2}}\right )}{5 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \left (a \sqrt {x}+b \right )^{2} \sqrt {a}\, b^{5} x^{\frac {7}{2}}} \]
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 \[ \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{x^{3/2}\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{\frac {3}{2}} \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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