Optimal. Leaf size=116 \[ -\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{230945 a^5 x^{11}}+\frac {64 b^3 \left (a+b x^2\right )^{11/2}}{20995 a^4 x^{13}}-\frac {16 b^2 \left (a+b x^2\right )^{11/2}}{1615 a^3 x^{15}}+\frac {8 b \left (a+b x^2\right )^{11/2}}{323 a^2 x^{17}}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}} \]
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Rubi [A] time = 0.04, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 264} \[ -\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{230945 a^5 x^{11}}+\frac {64 b^3 \left (a+b x^2\right )^{11/2}}{20995 a^4 x^{13}}-\frac {16 b^2 \left (a+b x^2\right )^{11/2}}{1615 a^3 x^{15}}+\frac {8 b \left (a+b x^2\right )^{11/2}}{323 a^2 x^{17}}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx &=-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}-\frac {(8 b) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx}{19 a}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}+\frac {8 b \left (a+b x^2\right )^{11/2}}{323 a^2 x^{17}}+\frac {\left (48 b^2\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{16}} \, dx}{323 a^2}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}+\frac {8 b \left (a+b x^2\right )^{11/2}}{323 a^2 x^{17}}-\frac {16 b^2 \left (a+b x^2\right )^{11/2}}{1615 a^3 x^{15}}-\frac {\left (64 b^3\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{14}} \, dx}{1615 a^3}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}+\frac {8 b \left (a+b x^2\right )^{11/2}}{323 a^2 x^{17}}-\frac {16 b^2 \left (a+b x^2\right )^{11/2}}{1615 a^3 x^{15}}+\frac {64 b^3 \left (a+b x^2\right )^{11/2}}{20995 a^4 x^{13}}+\frac {\left (128 b^4\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{12}} \, dx}{20995 a^4}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}+\frac {8 b \left (a+b x^2\right )^{11/2}}{323 a^2 x^{17}}-\frac {16 b^2 \left (a+b x^2\right )^{11/2}}{1615 a^3 x^{15}}+\frac {64 b^3 \left (a+b x^2\right )^{11/2}}{20995 a^4 x^{13}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{230945 a^5 x^{11}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 64, normalized size = 0.55 \[ -\frac {\left (a+b x^2\right )^{11/2} \left (12155 a^4-5720 a^3 b x^2+2288 a^2 b^2 x^4-704 a b^3 x^6+128 b^4 x^8\right )}{230945 a^5 x^{19}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.35, size = 115, normalized size = 0.99 \[ -\frac {{\left (128 \, b^{9} x^{18} - 64 \, a b^{8} x^{16} + 48 \, a^{2} b^{7} x^{14} - 40 \, a^{3} b^{6} x^{12} + 35 \, a^{4} b^{5} x^{10} + 23063 \, a^{5} b^{4} x^{8} + 75086 \, a^{6} b^{3} x^{6} + 95238 \, a^{7} b^{2} x^{4} + 55055 \, a^{8} b x^{2} + 12155 \, a^{9}\right )} \sqrt {b x^{2} + a}}{230945 \, a^{5} x^{19}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.17, size = 408, normalized size = 3.52 \[ \frac {256 \, {\left (92378 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{28} b^{\frac {19}{2}} + 554268 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{26} a b^{\frac {19}{2}} + 1939938 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{24} a^{2} b^{\frac {19}{2}} + 4018443 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} a^{3} b^{\frac {19}{2}} + 5866003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a^{4} b^{\frac {19}{2}} + 5773625 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{5} b^{\frac {19}{2}} + 4094025 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{6} b^{\frac {19}{2}} + 1889550 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{7} b^{\frac {19}{2}} + 581400 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{8} b^{\frac {19}{2}} + 80750 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{9} b^{\frac {19}{2}} + 3876 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{10} b^{\frac {19}{2}} - 969 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{11} b^{\frac {19}{2}} + 171 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{12} b^{\frac {19}{2}} - 19 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{13} b^{\frac {19}{2}} + a^{14} b^{\frac {19}{2}}\right )}}{230945 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{19}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 61, normalized size = 0.53 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (128 b^{4} x^{8}-704 a \,b^{3} x^{6}+2288 a^{2} b^{2} x^{4}-5720 a^{3} b \,x^{2}+12155 a^{4}\right )}{230945 a^{5} x^{19}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.57, size = 96, normalized size = 0.83 \[ -\frac {128 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{4}}{230945 \, a^{5} x^{11}} + \frac {64 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{20995 \, a^{4} x^{13}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{1615 \, a^{3} x^{15}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{323 \, a^{2} x^{17}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{19 \, a x^{19}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.84, size = 191, normalized size = 1.65 \[ \frac {8\,b^6\,\sqrt {b\,x^2+a}}{46189\,a^2\,x^7}-\frac {23063\,b^4\,\sqrt {b\,x^2+a}}{230945\,x^{11}}-\frac {6826\,a\,b^3\,\sqrt {b\,x^2+a}}{20995\,x^{13}}-\frac {77\,a^3\,b\,\sqrt {b\,x^2+a}}{323\,x^{17}}-\frac {7\,b^5\,\sqrt {b\,x^2+a}}{46189\,a\,x^9}-\frac {a^4\,\sqrt {b\,x^2+a}}{19\,x^{19}}-\frac {48\,b^7\,\sqrt {b\,x^2+a}}{230945\,a^3\,x^5}+\frac {64\,b^8\,\sqrt {b\,x^2+a}}{230945\,a^4\,x^3}-\frac {128\,b^9\,\sqrt {b\,x^2+a}}{230945\,a^5\,x}-\frac {666\,a^2\,b^2\,\sqrt {b\,x^2+a}}{1615\,x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.26, size = 1182, normalized size = 10.19 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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