Optimal. Leaf size=100 \[ -\frac {1}{a x \left (a+b x^2\right )^{7/2}}-\frac {8 b x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac {48 b x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac {64 b x}{35 a^4 \left (a+b x^2\right )^{3/2}}-\frac {128 b x}{35 a^5 \sqrt {a+b x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {277, 198, 197}
\begin {gather*} -\frac {128 b x}{35 a^5 \sqrt {a+b x^2}}-\frac {64 b x}{35 a^4 \left (a+b x^2\right )^{3/2}}-\frac {48 b x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac {8 b x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac {1}{a x \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 277
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx &=-\frac {1}{a x \left (a+b x^2\right )^{7/2}}-\frac {(8 b) \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx}{a}\\ &=-\frac {1}{a x \left (a+b x^2\right )^{7/2}}-\frac {8 b x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac {(48 b) \int \frac {1}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a^2}\\ &=-\frac {1}{a x \left (a+b x^2\right )^{7/2}}-\frac {8 b x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac {48 b x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac {(192 b) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^3}\\ &=-\frac {1}{a x \left (a+b x^2\right )^{7/2}}-\frac {8 b x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac {48 b x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac {64 b x}{35 a^4 \left (a+b x^2\right )^{3/2}}-\frac {(128 b) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{35 a^4}\\ &=-\frac {1}{a x \left (a+b x^2\right )^{7/2}}-\frac {8 b x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac {48 b x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac {64 b x}{35 a^4 \left (a+b x^2\right )^{3/2}}-\frac {128 b x}{35 a^5 \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 64, normalized size = 0.64 \begin {gather*} \frac {-35 a^4-280 a^3 b x^2-560 a^2 b^2 x^4-448 a b^3 x^6-128 b^4 x^8}{35 a^5 x \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 98, normalized size = 0.98
method | result | size |
gosper | \(-\frac {128 b^{4} x^{8}+448 a \,b^{3} x^{6}+560 a^{2} b^{2} x^{4}+280 a^{3} b \,x^{2}+35 a^{4}}{35 x \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{5}}\) | \(61\) |
trager | \(-\frac {128 b^{4} x^{8}+448 a \,b^{3} x^{6}+560 a^{2} b^{2} x^{4}+280 a^{3} b \,x^{2}+35 a^{4}}{35 x \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{5}}\) | \(61\) |
default | \(-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\) | \(98\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}}{a^{5} x}-\frac {\sqrt {b \,x^{2}+a}\, x \left (93 b^{3} x^{6}+308 a \,b^{2} x^{4}+350 a^{2} b \,x^{2}+140 a^{3}\right ) b}{35 \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right ) a^{5}}\) | \(109\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 82, normalized size = 0.82 \begin {gather*} -\frac {128 \, b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} - \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.45, size = 103, normalized size = 1.03 \begin {gather*} -\frac {{\left (128 \, b^{4} x^{8} + 448 \, a b^{3} x^{6} + 560 \, a^{2} b^{2} x^{4} + 280 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt {b x^{2} + a}}{35 \, {\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\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. 400 vs.
\(2 (97) = 194\).
time = 1.40, size = 400, normalized size = 4.00 \begin {gather*} - \frac {35 a^{4} b^{\frac {33}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac {280 a^{3} b^{\frac {35}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac {560 a^{2} b^{\frac {37}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac {448 a b^{\frac {39}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac {128 b^{\frac {41}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.62, size = 90, normalized size = 0.90 \begin {gather*} -\frac {{\left ({\left (x^{2} {\left (\frac {93 \, b^{4} x^{2}}{a^{5}} + \frac {308 \, b^{3}}{a^{4}}\right )} + \frac {350 \, b^{2}}{a^{3}}\right )} x^{2} + \frac {140 \, b}{a^{2}}\right )} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.71, size = 76, normalized size = 0.76 \begin {gather*} -\frac {\frac {1}{a^4}+\frac {128\,b\,x^2}{35\,a^5}}{x\,\sqrt {b\,x^2+a}}-\frac {29\,b\,x}{35\,a^4\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {13\,b\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {b\,x}{7\,a^2\,{\left (b\,x^2+a\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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