Optimal. Leaf size=73 \[ -\frac {a^3}{b^4 \sqrt {a+\frac {b}{x^2}}}-\frac {3 a^2 \sqrt {a+\frac {b}{x^2}}}{b^4}+\frac {a \left (a+\frac {b}{x^2}\right )^{3/2}}{b^4}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{5 b^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 73, 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 = {272, 45}
\begin {gather*} -\frac {a^3}{b^4 \sqrt {a+\frac {b}{x^2}}}-\frac {3 a^2 \sqrt {a+\frac {b}{x^2}}}{b^4}+\frac {a \left (a+\frac {b}{x^2}\right )^{3/2}}{b^4}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{5 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^9} \, dx &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {x^3}{(a+b x)^{3/2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a^3}{b^3 (a+b x)^{3/2}}+\frac {3 a^2}{b^3 \sqrt {a+b x}}-\frac {3 a \sqrt {a+b x}}{b^3}+\frac {(a+b x)^{3/2}}{b^3}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {a^3}{b^4 \sqrt {a+\frac {b}{x^2}}}-\frac {3 a^2 \sqrt {a+\frac {b}{x^2}}}{b^4}+\frac {a \left (a+\frac {b}{x^2}\right )^{3/2}}{b^4}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{5 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 53, normalized size = 0.73 \begin {gather*} \frac {-b^3+2 a b^2 x^2-8 a^2 b x^4-16 a^3 x^6}{5 b^4 \sqrt {a+\frac {b}{x^2}} x^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 59, normalized size = 0.81
method | result | size |
gosper | \(-\frac {\left (a \,x^{2}+b \right ) \left (16 a^{3} x^{6}+8 a^{2} b \,x^{4}-2 a \,b^{2} x^{2}+b^{3}\right )}{5 x^{8} b^{4} \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}}}\) | \(59\) |
default | \(-\frac {\left (a \,x^{2}+b \right ) \left (16 a^{3} x^{6}+8 a^{2} b \,x^{4}-2 a \,b^{2} x^{2}+b^{3}\right )}{5 x^{8} b^{4} \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}}}\) | \(59\) |
trager | \(-\frac {\left (16 a^{3} x^{6}+8 a^{2} b \,x^{4}-2 a \,b^{2} x^{2}+b^{3}\right ) \sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{5 x^{4} b^{4} \left (a \,x^{2}+b \right )}\) | \(65\) |
risch | \(-\frac {\left (a \,x^{2}+b \right ) \left (11 a^{2} x^{4}-3 a b \,x^{2}+b^{2}\right )}{5 b^{4} x^{6} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}-\frac {a^{3}}{b^{4} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 63, normalized size = 0.86 \begin {gather*} -\frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}}}{5 \, b^{4}} + \frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a}{b^{4}} - \frac {3 \, \sqrt {a + \frac {b}{x^{2}}} a^{2}}{b^{4}} - \frac {a^{3}}{\sqrt {a + \frac {b}{x^{2}}} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 63, normalized size = 0.86 \begin {gather*} -\frac {{\left (16 \, a^{3} x^{6} + 8 \, a^{2} b x^{4} - 2 \, a b^{2} x^{2} + b^{3}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{5 \, {\left (a b^{4} x^{6} + b^{5} x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1844 vs.
\(2 (65) = 130\).
time = 1.84, size = 1844, normalized size = 25.26 \begin {gather*} - \frac {16 a^{\frac {21}{2}} b^{\frac {23}{2}} x^{16} \sqrt {\frac {a x^{2}}{b} + 1}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} - \frac {88 a^{\frac {19}{2}} b^{\frac {25}{2}} x^{14} \sqrt {\frac {a x^{2}}{b} + 1}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} - \frac {198 a^{\frac {17}{2}} b^{\frac {27}{2}} x^{12} \sqrt {\frac {a x^{2}}{b} + 1}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} - \frac {231 a^{\frac {15}{2}} b^{\frac {29}{2}} x^{10} \sqrt {\frac {a x^{2}}{b} + 1}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} - \frac {145 a^{\frac {13}{2}} b^{\frac {31}{2}} x^{8} \sqrt {\frac {a x^{2}}{b} + 1}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} - \frac {46 a^{\frac {11}{2}} b^{\frac {33}{2}} x^{6} \sqrt {\frac {a x^{2}}{b} + 1}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} - \frac {8 a^{\frac {9}{2}} b^{\frac {35}{2}} x^{4} \sqrt {\frac {a x^{2}}{b} + 1}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} - \frac {3 a^{\frac {7}{2}} b^{\frac {37}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} - \frac {a^{\frac {5}{2}} b^{\frac {39}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} + \frac {16 a^{11} b^{11} x^{17}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} + \frac {96 a^{10} b^{12} x^{15}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} + \frac {240 a^{9} b^{13} x^{13}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} + \frac {320 a^{8} b^{14} x^{11}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} + \frac {240 a^{7} b^{15} x^{9}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} + \frac {96 a^{6} b^{16} x^{7}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} + \frac {16 a^{5} b^{17} x^{5}}{5 a^{\frac {17}{2}} b^{15} x^{17} + 30 a^{\frac {15}{2}} b^{16} x^{15} + 75 a^{\frac {13}{2}} b^{17} x^{13} + 100 a^{\frac {11}{2}} b^{18} x^{11} + 75 a^{\frac {9}{2}} b^{19} x^{9} + 30 a^{\frac {7}{2}} b^{20} x^{7} + 5 a^{\frac {5}{2}} b^{21} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 169 vs.
\(2 (63) = 126\).
time = 1.54, size = 169, normalized size = 2.32 \begin {gather*} -\frac {a^{3} x}{\sqrt {a x^{2} + b} b^{4} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (5 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{8} a^{\frac {5}{2}} - 30 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{6} a^{\frac {5}{2}} b + 80 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{4} a^{\frac {5}{2}} b^{2} - 50 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} a^{\frac {5}{2}} b^{3} + 11 \, a^{\frac {5}{2}} b^{4}\right )}}{5 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b\right )}^{5} b^{3} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.41, size = 56, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {a+\frac {b}{x^2}}\,\left (16\,a^3\,x^6+8\,a^2\,b\,x^4-2\,a\,b^2\,x^2+b^3\right )}{5\,b^4\,x^4\,\left (a\,x^2+b\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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