Optimal. Leaf size=204 \[ \frac {143 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{15/2} b^{5/2}}+\frac {143 x}{65536 a^7 b^2 \left (a+b x^2\right )}+\frac {143 x}{98304 a^6 b^2 \left (a+b x^2\right )^2}+\frac {143 x}{122880 a^5 b^2 \left (a+b x^2\right )^3}+\frac {143 x}{143360 a^4 b^2 \left (a+b x^2\right )^4}+\frac {143 x}{161280 a^3 b^2 \left (a+b x^2\right )^5}+\frac {13 x}{16128 a^2 b^2 \left (a+b x^2\right )^6}+\frac {x}{1344 a b^2 \left (a+b x^2\right )^7}-\frac {x}{96 b^2 \left (a+b x^2\right )^8}-\frac {x^3}{18 b \left (a+b x^2\right )^9} \]
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Rubi [A] time = 0.11, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {288, 199, 205} \[ \frac {143 x}{65536 a^7 b^2 \left (a+b x^2\right )}+\frac {143 x}{98304 a^6 b^2 \left (a+b x^2\right )^2}+\frac {143 x}{122880 a^5 b^2 \left (a+b x^2\right )^3}+\frac {143 x}{143360 a^4 b^2 \left (a+b x^2\right )^4}+\frac {143 x}{161280 a^3 b^2 \left (a+b x^2\right )^5}+\frac {13 x}{16128 a^2 b^2 \left (a+b x^2\right )^6}+\frac {143 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{15/2} b^{5/2}}+\frac {x}{1344 a b^2 \left (a+b x^2\right )^7}-\frac {x}{96 b^2 \left (a+b x^2\right )^8}-\frac {x^3}{18 b \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 288
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a+b x^2\right )^{10}} \, dx &=-\frac {x^3}{18 b \left (a+b x^2\right )^9}+\frac {\int \frac {x^2}{\left (a+b x^2\right )^9} \, dx}{6 b}\\ &=-\frac {x^3}{18 b \left (a+b x^2\right )^9}-\frac {x}{96 b^2 \left (a+b x^2\right )^8}+\frac {\int \frac {1}{\left (a+b x^2\right )^8} \, dx}{96 b^2}\\ &=-\frac {x^3}{18 b \left (a+b x^2\right )^9}-\frac {x}{96 b^2 \left (a+b x^2\right )^8}+\frac {x}{1344 a b^2 \left (a+b x^2\right )^7}+\frac {13 \int \frac {1}{\left (a+b x^2\right )^7} \, dx}{1344 a b^2}\\ &=-\frac {x^3}{18 b \left (a+b x^2\right )^9}-\frac {x}{96 b^2 \left (a+b x^2\right )^8}+\frac {x}{1344 a b^2 \left (a+b x^2\right )^7}+\frac {13 x}{16128 a^2 b^2 \left (a+b x^2\right )^6}+\frac {143 \int \frac {1}{\left (a+b x^2\right )^6} \, dx}{16128 a^2 b^2}\\ &=-\frac {x^3}{18 b \left (a+b x^2\right )^9}-\frac {x}{96 b^2 \left (a+b x^2\right )^8}+\frac {x}{1344 a b^2 \left (a+b x^2\right )^7}+\frac {13 x}{16128 a^2 b^2 \left (a+b x^2\right )^6}+\frac {143 x}{161280 a^3 b^2 \left (a+b x^2\right )^5}+\frac {143 \int \frac {1}{\left (a+b x^2\right )^5} \, dx}{17920 a^3 b^2}\\ &=-\frac {x^3}{18 b \left (a+b x^2\right )^9}-\frac {x}{96 b^2 \left (a+b x^2\right )^8}+\frac {x}{1344 a b^2 \left (a+b x^2\right )^7}+\frac {13 x}{16128 a^2 b^2 \left (a+b x^2\right )^6}+\frac {143 x}{161280 a^3 b^2 \left (a+b x^2\right )^5}+\frac {143 x}{143360 a^4 b^2 \left (a+b x^2\right )^4}+\frac {143 \int \frac {1}{\left (a+b x^2\right )^4} \, dx}{20480 a^4 b^2}\\ &=-\frac {x^3}{18 b \left (a+b x^2\right )^9}-\frac {x}{96 b^2 \left (a+b x^2\right )^8}+\frac {x}{1344 a b^2 \left (a+b x^2\right )^7}+\frac {13 x}{16128 a^2 b^2 \left (a+b x^2\right )^6}+\frac {143 x}{161280 a^3 b^2 \left (a+b x^2\right )^5}+\frac {143 x}{143360 a^4 b^2 \left (a+b x^2\right )^4}+\frac {143 x}{122880 a^5 b^2 \left (a+b x^2\right )^3}+\frac {143 \int \frac {1}{\left (a+b x^2\right )^3} \, dx}{24576 a^5 b^2}\\ &=-\frac {x^3}{18 b \left (a+b x^2\right )^9}-\frac {x}{96 b^2 \left (a+b x^2\right )^8}+\frac {x}{1344 a b^2 \left (a+b x^2\right )^7}+\frac {13 x}{16128 a^2 b^2 \left (a+b x^2\right )^6}+\frac {143 x}{161280 a^3 b^2 \left (a+b x^2\right )^5}+\frac {143 x}{143360 a^4 b^2 \left (a+b x^2\right )^4}+\frac {143 x}{122880 a^5 b^2 \left (a+b x^2\right )^3}+\frac {143 x}{98304 a^6 b^2 \left (a+b x^2\right )^2}+\frac {143 \int \frac {1}{\left (a+b x^2\right )^2} \, dx}{32768 a^6 b^2}\\ &=-\frac {x^3}{18 b \left (a+b x^2\right )^9}-\frac {x}{96 b^2 \left (a+b x^2\right )^8}+\frac {x}{1344 a b^2 \left (a+b x^2\right )^7}+\frac {13 x}{16128 a^2 b^2 \left (a+b x^2\right )^6}+\frac {143 x}{161280 a^3 b^2 \left (a+b x^2\right )^5}+\frac {143 x}{143360 a^4 b^2 \left (a+b x^2\right )^4}+\frac {143 x}{122880 a^5 b^2 \left (a+b x^2\right )^3}+\frac {143 x}{98304 a^6 b^2 \left (a+b x^2\right )^2}+\frac {143 x}{65536 a^7 b^2 \left (a+b x^2\right )}+\frac {143 \int \frac {1}{a+b x^2} \, dx}{65536 a^7 b^2}\\ &=-\frac {x^3}{18 b \left (a+b x^2\right )^9}-\frac {x}{96 b^2 \left (a+b x^2\right )^8}+\frac {x}{1344 a b^2 \left (a+b x^2\right )^7}+\frac {13 x}{16128 a^2 b^2 \left (a+b x^2\right )^6}+\frac {143 x}{161280 a^3 b^2 \left (a+b x^2\right )^5}+\frac {143 x}{143360 a^4 b^2 \left (a+b x^2\right )^4}+\frac {143 x}{122880 a^5 b^2 \left (a+b x^2\right )^3}+\frac {143 x}{98304 a^6 b^2 \left (a+b x^2\right )^2}+\frac {143 x}{65536 a^7 b^2 \left (a+b x^2\right )}+\frac {143 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{15/2} b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 138, normalized size = 0.68 \[ \frac {\frac {\sqrt {a} \sqrt {b} x \left (-45045 a^8-390390 a^7 b x^2+2633274 a^6 b^2 x^4+4349826 a^5 b^3 x^6+4685824 a^4 b^4 x^8+3317886 a^3 b^5 x^{10}+1495494 a^2 b^6 x^{12}+390390 a b^7 x^{14}+45045 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+45045 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{20643840 a^{15/2} b^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 654, normalized size = 3.21 \[ \left [\frac {90090 \, a b^{9} x^{17} + 780780 \, a^{2} b^{8} x^{15} + 2990988 \, a^{3} b^{7} x^{13} + 6635772 \, a^{4} b^{6} x^{11} + 9371648 \, a^{5} b^{5} x^{9} + 8699652 \, a^{6} b^{4} x^{7} + 5266548 \, a^{7} b^{3} x^{5} - 780780 \, a^{8} b^{2} x^{3} - 90090 \, a^{9} b x - 45045 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{41287680 \, {\left (a^{8} b^{12} x^{18} + 9 \, a^{9} b^{11} x^{16} + 36 \, a^{10} b^{10} x^{14} + 84 \, a^{11} b^{9} x^{12} + 126 \, a^{12} b^{8} x^{10} + 126 \, a^{13} b^{7} x^{8} + 84 \, a^{14} b^{6} x^{6} + 36 \, a^{15} b^{5} x^{4} + 9 \, a^{16} b^{4} x^{2} + a^{17} b^{3}\right )}}, \frac {45045 \, a b^{9} x^{17} + 390390 \, a^{2} b^{8} x^{15} + 1495494 \, a^{3} b^{7} x^{13} + 3317886 \, a^{4} b^{6} x^{11} + 4685824 \, a^{5} b^{5} x^{9} + 4349826 \, a^{6} b^{4} x^{7} + 2633274 \, a^{7} b^{3} x^{5} - 390390 \, a^{8} b^{2} x^{3} - 45045 \, a^{9} b x + 45045 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{20643840 \, {\left (a^{8} b^{12} x^{18} + 9 \, a^{9} b^{11} x^{16} + 36 \, a^{10} b^{10} x^{14} + 84 \, a^{11} b^{9} x^{12} + 126 \, a^{12} b^{8} x^{10} + 126 \, a^{13} b^{7} x^{8} + 84 \, a^{14} b^{6} x^{6} + 36 \, a^{15} b^{5} x^{4} + 9 \, a^{16} b^{4} x^{2} + a^{17} b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 128, normalized size = 0.63 \[ \frac {143 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{7} b^{2}} + \frac {45045 \, b^{8} x^{17} + 390390 \, a b^{7} x^{15} + 1495494 \, a^{2} b^{6} x^{13} + 3317886 \, a^{3} b^{5} x^{11} + 4685824 \, a^{4} b^{4} x^{9} + 4349826 \, a^{5} b^{3} x^{7} + 2633274 \, a^{6} b^{2} x^{5} - 390390 \, a^{7} b x^{3} - 45045 \, a^{8} x}{20643840 \, {\left (b x^{2} + a\right )}^{9} a^{7} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 122, normalized size = 0.60 \[ \frac {143 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}\, a^{7} b^{2}}+\frac {\frac {143 b^{6} x^{17}}{65536 a^{7}}+\frac {1859 b^{5} x^{15}}{98304 a^{6}}+\frac {11869 b^{4} x^{13}}{163840 a^{5}}+\frac {184327 b^{3} x^{11}}{1146880 a^{4}}+\frac {143 b^{2} x^{9}}{630 a^{3}}+\frac {241657 b \,x^{7}}{1146880 a^{2}}+\frac {20899 x^{5}}{163840 a}-\frac {1859 x^{3}}{98304 b}-\frac {143 a x}{65536 b^{2}}}{\left (b \,x^{2}+a \right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.19, size = 221, normalized size = 1.08 \[ \frac {45045 \, b^{8} x^{17} + 390390 \, a b^{7} x^{15} + 1495494 \, a^{2} b^{6} x^{13} + 3317886 \, a^{3} b^{5} x^{11} + 4685824 \, a^{4} b^{4} x^{9} + 4349826 \, a^{5} b^{3} x^{7} + 2633274 \, a^{6} b^{2} x^{5} - 390390 \, a^{7} b x^{3} - 45045 \, a^{8} x}{20643840 \, {\left (a^{7} b^{11} x^{18} + 9 \, a^{8} b^{10} x^{16} + 36 \, a^{9} b^{9} x^{14} + 84 \, a^{10} b^{8} x^{12} + 126 \, a^{11} b^{7} x^{10} + 126 \, a^{12} b^{6} x^{8} + 84 \, a^{13} b^{5} x^{6} + 36 \, a^{14} b^{4} x^{4} + 9 \, a^{15} b^{3} x^{2} + a^{16} b^{2}\right )}} + \frac {143 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{7} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.74, size = 204, normalized size = 1.00 \[ \frac {\frac {20899\,x^5}{163840\,a}-\frac {1859\,x^3}{98304\,b}+\frac {241657\,b\,x^7}{1146880\,a^2}+\frac {143\,b^2\,x^9}{630\,a^3}+\frac {184327\,b^3\,x^{11}}{1146880\,a^4}+\frac {11869\,b^4\,x^{13}}{163840\,a^5}+\frac {1859\,b^5\,x^{15}}{98304\,a^6}+\frac {143\,b^6\,x^{17}}{65536\,a^7}-\frac {143\,a\,x}{65536\,b^2}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}}+\frac {143\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{15/2}\,b^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.04, size = 291, normalized size = 1.43 \[ - \frac {143 \sqrt {- \frac {1}{a^{15} b^{5}}} \log {\left (- a^{8} b^{2} \sqrt {- \frac {1}{a^{15} b^{5}}} + x \right )}}{131072} + \frac {143 \sqrt {- \frac {1}{a^{15} b^{5}}} \log {\left (a^{8} b^{2} \sqrt {- \frac {1}{a^{15} b^{5}}} + x \right )}}{131072} + \frac {- 45045 a^{8} x - 390390 a^{7} b x^{3} + 2633274 a^{6} b^{2} x^{5} + 4349826 a^{5} b^{3} x^{7} + 4685824 a^{4} b^{4} x^{9} + 3317886 a^{3} b^{5} x^{11} + 1495494 a^{2} b^{6} x^{13} + 390390 a b^{7} x^{15} + 45045 b^{8} x^{17}}{20643840 a^{16} b^{2} + 185794560 a^{15} b^{3} x^{2} + 743178240 a^{14} b^{4} x^{4} + 1734082560 a^{13} b^{5} x^{6} + 2601123840 a^{12} b^{6} x^{8} + 2601123840 a^{11} b^{7} x^{10} + 1734082560 a^{10} b^{8} x^{12} + 743178240 a^{9} b^{9} x^{14} + 185794560 a^{8} b^{10} x^{16} + 20643840 a^{7} b^{11} x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
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