Optimal. Leaf size=58 \[ \frac {x^{14}}{504 a^3 \left (a+b x^2\right )^7}+\frac {x^{14}}{72 a^2 \left (a+b x^2\right )^8}+\frac {x^{14}}{18 a \left (a+b x^2\right )^9} \]
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Rubi [A] time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac {x^{14}}{504 a^3 \left (a+b x^2\right )^7}+\frac {x^{14}}{72 a^2 \left (a+b x^2\right )^8}+\frac {x^{14}}{18 a \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{13}}{\left (a+b x^2\right )^{10}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^6}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac {x^{14}}{18 a \left (a+b x^2\right )^9}+\frac {\operatorname {Subst}\left (\int \frac {x^6}{(a+b x)^9} \, dx,x,x^2\right )}{9 a}\\ &=\frac {x^{14}}{18 a \left (a+b x^2\right )^9}+\frac {x^{14}}{72 a^2 \left (a+b x^2\right )^8}+\frac {\operatorname {Subst}\left (\int \frac {x^6}{(a+b x)^8} \, dx,x,x^2\right )}{72 a^2}\\ &=\frac {x^{14}}{18 a \left (a+b x^2\right )^9}+\frac {x^{14}}{72 a^2 \left (a+b x^2\right )^8}+\frac {x^{14}}{504 a^3 \left (a+b x^2\right )^7}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 79, normalized size = 1.36 \[ -\frac {a^6+9 a^5 b x^2+36 a^4 b^2 x^4+84 a^3 b^3 x^6+126 a^2 b^4 x^8+126 a b^5 x^{10}+84 b^6 x^{12}}{504 b^7 \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 168, normalized size = 2.90 \[ -\frac {84 \, b^{6} x^{12} + 126 \, a b^{5} x^{10} + 126 \, a^{2} b^{4} x^{8} + 84 \, a^{3} b^{3} x^{6} + 36 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{504 \, {\left (b^{16} x^{18} + 9 \, a b^{15} x^{16} + 36 \, a^{2} b^{14} x^{14} + 84 \, a^{3} b^{13} x^{12} + 126 \, a^{4} b^{12} x^{10} + 126 \, a^{5} b^{11} x^{8} + 84 \, a^{6} b^{10} x^{6} + 36 \, a^{7} b^{9} x^{4} + 9 \, a^{8} b^{8} x^{2} + a^{9} b^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 77, normalized size = 1.33 \[ -\frac {84 \, b^{6} x^{12} + 126 \, a b^{5} x^{10} + 126 \, a^{2} b^{4} x^{8} + 84 \, a^{3} b^{3} x^{6} + 36 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{504 \, {\left (b x^{2} + a\right )}^{9} b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 116, normalized size = 2.00 \[ -\frac {a^{6}}{18 \left (b \,x^{2}+a \right )^{9} b^{7}}+\frac {3 a^{5}}{8 \left (b \,x^{2}+a \right )^{8} b^{7}}-\frac {15 a^{4}}{14 \left (b \,x^{2}+a \right )^{7} b^{7}}+\frac {5 a^{3}}{3 \left (b \,x^{2}+a \right )^{6} b^{7}}-\frac {3 a^{2}}{2 \left (b \,x^{2}+a \right )^{5} b^{7}}+\frac {3 a}{4 \left (b \,x^{2}+a \right )^{4} b^{7}}-\frac {1}{6 \left (b \,x^{2}+a \right )^{3} b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.46, size = 168, normalized size = 2.90 \[ -\frac {84 \, b^{6} x^{12} + 126 \, a b^{5} x^{10} + 126 \, a^{2} b^{4} x^{8} + 84 \, a^{3} b^{3} x^{6} + 36 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{504 \, {\left (b^{16} x^{18} + 9 \, a b^{15} x^{16} + 36 \, a^{2} b^{14} x^{14} + 84 \, a^{3} b^{13} x^{12} + 126 \, a^{4} b^{12} x^{10} + 126 \, a^{5} b^{11} x^{8} + 84 \, a^{6} b^{10} x^{6} + 36 \, a^{7} b^{9} x^{4} + 9 \, a^{8} b^{8} x^{2} + a^{9} b^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 170, normalized size = 2.93 \[ -\frac {a^6+9\,a^5\,b\,x^2+36\,a^4\,b^2\,x^4+84\,a^3\,b^3\,x^6+126\,a^2\,b^4\,x^8+126\,a\,b^5\,x^{10}+84\,b^6\,x^{12}}{504\,a^9\,b^7+4536\,a^8\,b^8\,x^2+18144\,a^7\,b^9\,x^4+42336\,a^6\,b^{10}\,x^6+63504\,a^5\,b^{11}\,x^8+63504\,a^4\,b^{12}\,x^{10}+42336\,a^3\,b^{13}\,x^{12}+18144\,a^2\,b^{14}\,x^{14}+4536\,a\,b^{15}\,x^{16}+504\,b^{16}\,x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.33, size = 178, normalized size = 3.07 \[ \frac {- a^{6} - 9 a^{5} b x^{2} - 36 a^{4} b^{2} x^{4} - 84 a^{3} b^{3} x^{6} - 126 a^{2} b^{4} x^{8} - 126 a b^{5} x^{10} - 84 b^{6} x^{12}}{504 a^{9} b^{7} + 4536 a^{8} b^{8} x^{2} + 18144 a^{7} b^{9} x^{4} + 42336 a^{6} b^{10} x^{6} + 63504 a^{5} b^{11} x^{8} + 63504 a^{4} b^{12} x^{10} + 42336 a^{3} b^{13} x^{12} + 18144 a^{2} b^{14} x^{14} + 4536 a b^{15} x^{16} + 504 b^{16} x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
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