Optimal. Leaf size=57 \[ -\frac {2 a^2 \sqrt {a+\frac {b}{x}}}{b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{3/2}}{3 b^3} \]
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Rubi [A] time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac {2 a^2 \sqrt {a+\frac {b}{x}}}{b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+\frac {b}{x}} x^4} \, dx &=-\operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {a^2}{b^2 \sqrt {a+b x}}-\frac {2 a \sqrt {a+b x}}{b^2}+\frac {(a+b x)^{3/2}}{b^2}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 a^2 \sqrt {a+\frac {b}{x}}}{b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{3/2}}{3 b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 40, normalized size = 0.70 \[ -\frac {2 \sqrt {a+\frac {b}{x}} \left (8 a^2 x^2-4 a b x+3 b^2\right )}{15 b^3 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 38, normalized size = 0.67 \[ -\frac {2 \, {\left (8 \, a^{2} x^{2} - 4 \, a b x + 3 \, b^{2}\right )} \sqrt {\frac {a x + b}{x}}}{15 \, b^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 67, normalized size = 1.18 \[ -\frac {2 \, {\left (15 \, a^{2} \sqrt {\frac {a x + b}{x}} - \frac {10 \, {\left (a x + b\right )} a \sqrt {\frac {a x + b}{x}}}{x} + \frac {3 \, {\left (a x + b\right )}^{2} \sqrt {\frac {a x + b}{x}}}{x^{2}}\right )}}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 44, normalized size = 0.77 \[ -\frac {2 \left (a x +b \right ) \left (8 a^{2} x^{2}-4 a b x +3 b^{2}\right )}{15 \sqrt {\frac {a x +b}{x}}\, b^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 47, normalized size = 0.82 \[ -\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}}}{5 \, b^{3}} + \frac {4 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a}{3 \, b^{3}} - \frac {2 \, \sqrt {a + \frac {b}{x}} a^{2}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 54, normalized size = 0.95 \[ -\frac {6\,b^2\,\sqrt {a+\frac {b}{x}}+16\,a^2\,x^2\,\sqrt {a+\frac {b}{x}}-8\,a\,b\,x\,\sqrt {a+\frac {b}{x}}}{15\,b^3\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.25, size = 813, normalized size = 14.26 \[ - \frac {16 a^{\frac {15}{2}} b^{\frac {9}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} - \frac {40 a^{\frac {13}{2}} b^{\frac {11}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} - \frac {30 a^{\frac {11}{2}} b^{\frac {13}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} - \frac {10 a^{\frac {9}{2}} b^{\frac {15}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} - \frac {10 a^{\frac {7}{2}} b^{\frac {17}{2}} x \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} - \frac {6 a^{\frac {5}{2}} b^{\frac {19}{2}} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} + \frac {16 a^{8} b^{4} x^{\frac {11}{2}}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} + \frac {48 a^{7} b^{5} x^{\frac {9}{2}}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} + \frac {48 a^{6} b^{6} x^{\frac {7}{2}}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} + \frac {16 a^{5} b^{7} x^{\frac {5}{2}}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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