Optimal. Leaf size=137 \[ \frac {512 a^3 \sqrt {a x+b \sqrt {x}}}{35 b^5 \sqrt {x}}-\frac {256 a^2 \sqrt {a x+b \sqrt {x}}}{35 b^4 x}+\frac {192 a \sqrt {a x+b \sqrt {x}}}{35 b^3 x^{3/2}}-\frac {32 \sqrt {a x+b \sqrt {x}}}{7 b^2 x^2}+\frac {4}{b x^{3/2} \sqrt {a x+b \sqrt {x}}} \]
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Rubi [A] time = 0.20, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2015, 2016, 2014} \[ \frac {512 a^3 \sqrt {a x+b \sqrt {x}}}{35 b^5 \sqrt {x}}-\frac {256 a^2 \sqrt {a x+b \sqrt {x}}}{35 b^4 x}+\frac {192 a \sqrt {a x+b \sqrt {x}}}{35 b^3 x^{3/2}}-\frac {32 \sqrt {a x+b \sqrt {x}}}{7 b^2 x^2}+\frac {4}{b x^{3/2} \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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Rule 2014
Rule 2015
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (b \sqrt {x}+a x\right )^{3/2}} \, dx &=\frac {4}{b x^{3/2} \sqrt {b \sqrt {x}+a x}}+\frac {8 \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx}{b}\\ &=\frac {4}{b x^{3/2} \sqrt {b \sqrt {x}+a x}}-\frac {32 \sqrt {b \sqrt {x}+a x}}{7 b^2 x^2}-\frac {(48 a) \int \frac {1}{x^2 \sqrt {b \sqrt {x}+a x}} \, dx}{7 b^2}\\ &=\frac {4}{b x^{3/2} \sqrt {b \sqrt {x}+a x}}-\frac {32 \sqrt {b \sqrt {x}+a x}}{7 b^2 x^2}+\frac {192 a \sqrt {b \sqrt {x}+a x}}{35 b^3 x^{3/2}}+\frac {\left (192 a^2\right ) \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{35 b^3}\\ &=\frac {4}{b x^{3/2} \sqrt {b \sqrt {x}+a x}}-\frac {32 \sqrt {b \sqrt {x}+a x}}{7 b^2 x^2}+\frac {192 a \sqrt {b \sqrt {x}+a x}}{35 b^3 x^{3/2}}-\frac {256 a^2 \sqrt {b \sqrt {x}+a x}}{35 b^4 x}-\frac {\left (128 a^3\right ) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{35 b^4}\\ &=\frac {4}{b x^{3/2} \sqrt {b \sqrt {x}+a x}}-\frac {32 \sqrt {b \sqrt {x}+a x}}{7 b^2 x^2}+\frac {192 a \sqrt {b \sqrt {x}+a x}}{35 b^3 x^{3/2}}-\frac {256 a^2 \sqrt {b \sqrt {x}+a x}}{35 b^4 x}+\frac {512 a^3 \sqrt {b \sqrt {x}+a x}}{35 b^5 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 72, normalized size = 0.53 \[ \frac {4 \left (128 a^4 x^2+64 a^3 b x^{3/2}-16 a^2 b^2 x+8 a b^3 \sqrt {x}-5 b^4\right )}{35 b^5 x^{3/2} \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 87, normalized size = 0.64 \[ -\frac {4 \, {\left (64 \, a^{4} b x^{2} - 24 \, a^{2} b^{3} x - 5 \, b^{5} - {\left (128 \, a^{5} x^{2} - 80 \, a^{3} b^{2} x - 13 \, a b^{4}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{35 \, {\left (a^{2} b^{5} x^{3} - b^{7} x^{2}\right )}} \]
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 \[ \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 570, normalized size = 4.16 \[ -\frac {\sqrt {a x +b \sqrt {x}}\, \left (-105 a^{6} b \,x^{\frac {11}{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 )+105 a^{6} b \,x^{\frac {11}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-210 a^{5} b^{2} x^{5} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+210 a^{5} b^{2} x^{5} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-105 a^{4} b^{3} 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 )+105 a^{4} b^{3} 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 )+210 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {13}{2}} x^{\frac {11}{2}}+210 \sqrt {a x +b \sqrt {x}}\, a^{\frac {13}{2}} x^{\frac {11}{2}}+420 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {11}{2}} b \,x^{5}+420 \sqrt {a x +b \sqrt {x}}\, a^{\frac {11}{2}} b \,x^{5}+210 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {9}{2}} b^{2} x^{\frac {9}{2}}+210 \sqrt {a x +b \sqrt {x}}\, a^{\frac {9}{2}} b^{2} x^{\frac {9}{2}}-560 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {11}{2}} x^{\frac {9}{2}}+140 \left (\left (a \sqrt {x}+b \right ) \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {11}{2}} x^{\frac {9}{2}}-932 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {9}{2}} b \,x^{4}-256 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2} x^{\frac {7}{2}}+64 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3} x^{3}-32 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{4} x^{\frac {5}{2}}+20 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{5} x^{2}\right )}{35 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \left (a \sqrt {x}+b \right )^{2} \sqrt {a}\, b^{6} x^{\frac {9}{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 \[ \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{2} \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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