Optimal. Leaf size=39 \[ \frac {x^{16}}{144 a^2 \left (a+b x^2\right )^8}+\frac {x^{16}}{18 a \left (a+b x^2\right )^9} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac {x^{16}}{144 a^2 \left (a+b x^2\right )^8}+\frac {x^{16}}{18 a \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{15}}{\left (a+b x^2\right )^{10}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^7}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac {x^{16}}{18 a \left (a+b x^2\right )^9}+\frac {\operatorname {Subst}\left (\int \frac {x^7}{(a+b x)^9} \, dx,x,x^2\right )}{18 a}\\ &=\frac {x^{16}}{18 a \left (a+b x^2\right )^9}+\frac {x^{16}}{144 a^2 \left (a+b x^2\right )^8}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 90, normalized size = 2.31 \[ -\frac {a^7+9 a^6 b x^2+36 a^5 b^2 x^4+84 a^4 b^3 x^6+126 a^3 b^4 x^8+126 a^2 b^5 x^{10}+84 a b^6 x^{12}+36 b^7 x^{14}}{144 b^8 \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 179, normalized size = 4.59 \[ -\frac {36 \, b^{7} x^{14} + 84 \, a b^{6} x^{12} + 126 \, a^{2} b^{5} x^{10} + 126 \, a^{3} b^{4} x^{8} + 84 \, a^{4} b^{3} x^{6} + 36 \, a^{5} b^{2} x^{4} + 9 \, a^{6} b x^{2} + a^{7}}{144 \, {\left (b^{17} x^{18} + 9 \, a b^{16} x^{16} + 36 \, a^{2} b^{15} x^{14} + 84 \, a^{3} b^{14} x^{12} + 126 \, a^{4} b^{13} x^{10} + 126 \, a^{5} b^{12} x^{8} + 84 \, a^{6} b^{11} x^{6} + 36 \, a^{7} b^{10} x^{4} + 9 \, a^{8} b^{9} x^{2} + a^{9} b^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.63, size = 88, normalized size = 2.26 \[ -\frac {36 \, b^{7} x^{14} + 84 \, a b^{6} x^{12} + 126 \, a^{2} b^{5} x^{10} + 126 \, a^{3} b^{4} x^{8} + 84 \, a^{4} b^{3} x^{6} + 36 \, a^{5} b^{2} x^{4} + 9 \, a^{6} b x^{2} + a^{7}}{144 \, {\left (b x^{2} + a\right )}^{9} b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 133, normalized size = 3.41 \[ \frac {a^{7}}{18 \left (b \,x^{2}+a \right )^{9} b^{8}}-\frac {7 a^{6}}{16 \left (b \,x^{2}+a \right )^{8} b^{8}}+\frac {3 a^{5}}{2 \left (b \,x^{2}+a \right )^{7} b^{8}}-\frac {35 a^{4}}{12 \left (b \,x^{2}+a \right )^{6} b^{8}}+\frac {7 a^{3}}{2 \left (b \,x^{2}+a \right )^{5} b^{8}}-\frac {21 a^{2}}{8 \left (b \,x^{2}+a \right )^{4} b^{8}}+\frac {7 a}{6 \left (b \,x^{2}+a \right )^{3} b^{8}}-\frac {1}{4 \left (b \,x^{2}+a \right )^{2} b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.47, size = 179, normalized size = 4.59 \[ -\frac {36 \, b^{7} x^{14} + 84 \, a b^{6} x^{12} + 126 \, a^{2} b^{5} x^{10} + 126 \, a^{3} b^{4} x^{8} + 84 \, a^{4} b^{3} x^{6} + 36 \, a^{5} b^{2} x^{4} + 9 \, a^{6} b x^{2} + a^{7}}{144 \, {\left (b^{17} x^{18} + 9 \, a b^{16} x^{16} + 36 \, a^{2} b^{15} x^{14} + 84 \, a^{3} b^{14} x^{12} + 126 \, a^{4} b^{13} x^{10} + 126 \, a^{5} b^{12} x^{8} + 84 \, a^{6} b^{11} x^{6} + 36 \, a^{7} b^{10} x^{4} + 9 \, a^{8} b^{9} x^{2} + a^{9} b^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.89, size = 181, normalized size = 4.64 \[ -\frac {a^7+9\,a^6\,b\,x^2+36\,a^5\,b^2\,x^4+84\,a^4\,b^3\,x^6+126\,a^3\,b^4\,x^8+126\,a^2\,b^5\,x^{10}+84\,a\,b^6\,x^{12}+36\,b^7\,x^{14}}{144\,a^9\,b^8+1296\,a^8\,b^9\,x^2+5184\,a^7\,b^{10}\,x^4+12096\,a^6\,b^{11}\,x^6+18144\,a^5\,b^{12}\,x^8+18144\,a^4\,b^{13}\,x^{10}+12096\,a^3\,b^{14}\,x^{12}+5184\,a^2\,b^{15}\,x^{14}+1296\,a\,b^{16}\,x^{16}+144\,b^{17}\,x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.42, size = 190, normalized size = 4.87 \[ \frac {- a^{7} - 9 a^{6} b x^{2} - 36 a^{5} b^{2} x^{4} - 84 a^{4} b^{3} x^{6} - 126 a^{3} b^{4} x^{8} - 126 a^{2} b^{5} x^{10} - 84 a b^{6} x^{12} - 36 b^{7} x^{14}}{144 a^{9} b^{8} + 1296 a^{8} b^{9} x^{2} + 5184 a^{7} b^{10} x^{4} + 12096 a^{6} b^{11} x^{6} + 18144 a^{5} b^{12} x^{8} + 18144 a^{4} b^{13} x^{10} + 12096 a^{3} b^{14} x^{12} + 5184 a^{2} b^{15} x^{14} + 1296 a b^{16} x^{16} + 144 b^{17} x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
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