Optimal. Leaf size=164 \[ -\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac {40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac {320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}+\frac {512 b^5 \left (a+b x^2\right )^{11/2}}{676039 a^6 x^{13}}-\frac {1024 b^6 \left (a+b x^2\right )^{11/2}}{7436429 a^7 x^{11}} \]
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Rubi [A]
time = 0.05, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270}
\begin {gather*} -\frac {1024 b^6 \left (a+b x^2\right )^{11/2}}{7436429 a^7 x^{11}}+\frac {512 b^5 \left (a+b x^2\right )^{11/2}}{676039 a^6 x^{13}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}+\frac {320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac {40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{9/2}}{x^{24}} \, dx &=-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}-\frac {(12 b) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{22}} \, dx}{23 a}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}+\frac {\left (40 b^2\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx}{161 a^2}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac {40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}-\frac {\left (320 b^3\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx}{3059 a^3}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac {40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac {320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}+\frac {\left (1920 b^4\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{16}} \, dx}{52003 a^4}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac {40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac {320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}-\frac {\left (512 b^5\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{14}} \, dx}{52003 a^5}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac {40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac {320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}+\frac {512 b^5 \left (a+b x^2\right )^{11/2}}{676039 a^6 x^{13}}+\frac {\left (1024 b^6\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{12}} \, dx}{676039 a^6}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac {40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac {320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}+\frac {512 b^5 \left (a+b x^2\right )^{11/2}}{676039 a^6 x^{13}}-\frac {1024 b^6 \left (a+b x^2\right )^{11/2}}{7436429 a^7 x^{11}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 86, normalized size = 0.52 \begin {gather*} \frac {\left (a+b x^2\right )^{11/2} \left (-323323 a^6+184756 a^5 b x^2-97240 a^4 b^2 x^4+45760 a^3 b^3 x^6-18304 a^2 b^4 x^8+5632 a b^5 x^{10}-1024 b^6 x^{12}\right )}{7436429 a^7 x^{23}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 26.68, size = 157, normalized size = 0.96
method | result | size |
gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (1024 b^{6} x^{12}-5632 a \,b^{5} x^{10}+18304 a^{2} b^{4} x^{8}-45760 a^{3} x^{6} b^{3}+97240 a^{4} b^{2} x^{4}-184756 a^{5} b \,x^{2}+323323 a^{6}\right )}{7436429 x^{23} a^{7}}\) | \(83\) |
trager | \(-\frac {\left (1024 b^{11} x^{22}-512 a \,b^{10} x^{20}+384 a^{2} b^{9} x^{18}-320 a^{3} b^{8} x^{16}+280 a^{4} b^{7} x^{14}-252 a^{5} b^{6} x^{12}+231 a^{6} b^{5} x^{10}+530959 a^{7} b^{4} x^{8}+1826110 a^{8} b^{3} x^{6}+2406690 b^{2} a^{9} x^{4}+1431859 a^{10} b \,x^{2}+323323 a^{11}\right ) \sqrt {b \,x^{2}+a}}{7436429 a^{7} x^{23}}\) | \(138\) |
risch | \(-\frac {\left (1024 b^{11} x^{22}-512 a \,b^{10} x^{20}+384 a^{2} b^{9} x^{18}-320 a^{3} b^{8} x^{16}+280 a^{4} b^{7} x^{14}-252 a^{5} b^{6} x^{12}+231 a^{6} b^{5} x^{10}+530959 a^{7} b^{4} x^{8}+1826110 a^{8} b^{3} x^{6}+2406690 b^{2} a^{9} x^{4}+1431859 a^{10} b \,x^{2}+323323 a^{11}\right ) \sqrt {b \,x^{2}+a}}{7436429 a^{7} x^{23}}\) | \(138\) |
default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{23 a \,x^{23}}-\frac {12 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{21 a \,x^{21}}-\frac {10 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{19 a \,x^{19}}-\frac {8 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{17 a \,x^{17}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{15 a \,x^{15}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{13 a \,x^{13}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{143 a^{2} x^{11}}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\) | \(157\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 136, normalized size = 0.83 \begin {gather*} -\frac {1024 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{6}}{7436429 \, a^{7} x^{11}} + \frac {512 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{5}}{676039 \, a^{6} x^{13}} - \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{4}}{52003 \, a^{5} x^{15}} + \frac {320 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{52003 \, a^{4} x^{17}} - \frac {40 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{3059 \, a^{3} x^{19}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{161 \, a^{2} x^{21}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{23 \, a x^{23}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.61, size = 137, normalized size = 0.84 \begin {gather*} -\frac {{\left (1024 \, b^{11} x^{22} - 512 \, a b^{10} x^{20} + 384 \, a^{2} b^{9} x^{18} - 320 \, a^{3} b^{8} x^{16} + 280 \, a^{4} b^{7} x^{14} - 252 \, a^{5} b^{6} x^{12} + 231 \, a^{6} b^{5} x^{10} + 530959 \, a^{7} b^{4} x^{8} + 1826110 \, a^{8} b^{3} x^{6} + 2406690 \, a^{9} b^{2} x^{4} + 1431859 \, a^{10} b x^{2} + 323323 \, a^{11}\right )} \sqrt {b x^{2} + a}}{7436429 \, a^{7} x^{23}} \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. 1950 vs.
\(2 (156) = 312\).
time = 4.82, size = 1950, normalized size = 11.89 \begin {gather*} \text {Too large to display} \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. 462 vs.
\(2 (136) = 272\).
time = 0.61, size = 462, normalized size = 2.82 \begin {gather*} \frac {2048 \, {\left (4249388 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{32} b^{\frac {23}{2}} + 28683369 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{30} a b^{\frac {23}{2}} + 100922965 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{28} a^{2} b^{\frac {23}{2}} + 215656441 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{26} a^{3} b^{\frac {23}{2}} + 313006057 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{24} a^{4} b^{\frac {23}{2}} + 311653979 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} a^{5} b^{\frac {23}{2}} + 216800507 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a^{6} b^{\frac {23}{2}} + 100105775 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{7} b^{\frac {23}{2}} + 29173683 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{8} b^{\frac {23}{2}} + 4004231 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{9} b^{\frac {23}{2}} + 100947 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{10} b^{\frac {23}{2}} - 33649 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{11} b^{\frac {23}{2}} + 8855 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{12} b^{\frac {23}{2}} - 1771 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{13} b^{\frac {23}{2}} + 253 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{14} b^{\frac {23}{2}} - 23 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{15} b^{\frac {23}{2}} + a^{16} b^{\frac {23}{2}}\right )}}{7436429 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{23}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.47, size = 231, normalized size = 1.41 \begin {gather*} \frac {36\,b^6\,\sqrt {b\,x^2+a}}{1062347\,a^2\,x^{11}}-\frac {3713\,b^4\,\sqrt {b\,x^2+a}}{52003\,x^{15}}-\frac {12770\,a\,b^3\,\sqrt {b\,x^2+a}}{52003\,x^{17}}-\frac {31\,a^3\,b\,\sqrt {b\,x^2+a}}{161\,x^{21}}-\frac {3\,b^5\,\sqrt {b\,x^2+a}}{96577\,a\,x^{13}}-\frac {a^4\,\sqrt {b\,x^2+a}}{23\,x^{23}}-\frac {40\,b^7\,\sqrt {b\,x^2+a}}{1062347\,a^3\,x^9}+\frac {320\,b^8\,\sqrt {b\,x^2+a}}{7436429\,a^4\,x^7}-\frac {384\,b^9\,\sqrt {b\,x^2+a}}{7436429\,a^5\,x^5}+\frac {512\,b^{10}\,\sqrt {b\,x^2+a}}{7436429\,a^6\,x^3}-\frac {1024\,b^{11}\,\sqrt {b\,x^2+a}}{7436429\,a^7\,x}-\frac {990\,a^2\,b^2\,\sqrt {b\,x^2+a}}{3059\,x^{19}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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