Optimal. Leaf size=129 \[ \frac {a^6 \left (a+b x^2\right )^9}{18 b^7}-\frac {3 a^5 \left (a+b x^2\right )^{10}}{10 b^7}+\frac {15 a^4 \left (a+b x^2\right )^{11}}{22 b^7}-\frac {5 a^3 \left (a+b x^2\right )^{12}}{6 b^7}+\frac {15 a^2 \left (a+b x^2\right )^{13}}{26 b^7}+\frac {\left (a+b x^2\right )^{15}}{30 b^7}-\frac {3 a \left (a+b x^2\right )^{14}}{14 b^7} \]
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Rubi [A] time = 0.21, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {15 a^2 \left (a+b x^2\right )^{13}}{26 b^7}-\frac {5 a^3 \left (a+b x^2\right )^{12}}{6 b^7}+\frac {15 a^4 \left (a+b x^2\right )^{11}}{22 b^7}-\frac {3 a^5 \left (a+b x^2\right )^{10}}{10 b^7}+\frac {a^6 \left (a+b x^2\right )^9}{18 b^7}+\frac {\left (a+b x^2\right )^{15}}{30 b^7}-\frac {3 a \left (a+b x^2\right )^{14}}{14 b^7} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{13} \left (a+b x^2\right )^8 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^6 (a+b x)^8 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^6 (a+b x)^8}{b^6}-\frac {6 a^5 (a+b x)^9}{b^6}+\frac {15 a^4 (a+b x)^{10}}{b^6}-\frac {20 a^3 (a+b x)^{11}}{b^6}+\frac {15 a^2 (a+b x)^{12}}{b^6}-\frac {6 a (a+b x)^{13}}{b^6}+\frac {(a+b x)^{14}}{b^6}\right ) \, dx,x,x^2\right )\\ &=\frac {a^6 \left (a+b x^2\right )^9}{18 b^7}-\frac {3 a^5 \left (a+b x^2\right )^{10}}{10 b^7}+\frac {15 a^4 \left (a+b x^2\right )^{11}}{22 b^7}-\frac {5 a^3 \left (a+b x^2\right )^{12}}{6 b^7}+\frac {15 a^2 \left (a+b x^2\right )^{13}}{26 b^7}-\frac {3 a \left (a+b x^2\right )^{14}}{14 b^7}+\frac {\left (a+b x^2\right )^{15}}{30 b^7}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 108, normalized size = 0.84 \[ \frac {a^8 x^{14}}{14}+\frac {1}{2} a^7 b x^{16}+\frac {14}{9} a^6 b^2 x^{18}+\frac {14}{5} a^5 b^3 x^{20}+\frac {35}{11} a^4 b^4 x^{22}+\frac {7}{3} a^3 b^5 x^{24}+\frac {14}{13} a^2 b^6 x^{26}+\frac {2}{7} a b^7 x^{28}+\frac {b^8 x^{30}}{30} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 90, normalized size = 0.70 \[ \frac {1}{30} x^{30} b^{8} + \frac {2}{7} x^{28} b^{7} a + \frac {14}{13} x^{26} b^{6} a^{2} + \frac {7}{3} x^{24} b^{5} a^{3} + \frac {35}{11} x^{22} b^{4} a^{4} + \frac {14}{5} x^{20} b^{3} a^{5} + \frac {14}{9} x^{18} b^{2} a^{6} + \frac {1}{2} x^{16} b a^{7} + \frac {1}{14} x^{14} a^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.06, size = 90, normalized size = 0.70 \[ \frac {1}{30} \, b^{8} x^{30} + \frac {2}{7} \, a b^{7} x^{28} + \frac {14}{13} \, a^{2} b^{6} x^{26} + \frac {7}{3} \, a^{3} b^{5} x^{24} + \frac {35}{11} \, a^{4} b^{4} x^{22} + \frac {14}{5} \, a^{5} b^{3} x^{20} + \frac {14}{9} \, a^{6} b^{2} x^{18} + \frac {1}{2} \, a^{7} b x^{16} + \frac {1}{14} \, a^{8} x^{14} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 91, normalized size = 0.71 \[ \frac {1}{30} b^{8} x^{30}+\frac {2}{7} a \,b^{7} x^{28}+\frac {14}{13} a^{2} b^{6} x^{26}+\frac {7}{3} a^{3} b^{5} x^{24}+\frac {35}{11} a^{4} b^{4} x^{22}+\frac {14}{5} a^{5} b^{3} x^{20}+\frac {14}{9} a^{6} b^{2} x^{18}+\frac {1}{2} a^{7} b \,x^{16}+\frac {1}{14} a^{8} x^{14} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 90, normalized size = 0.70 \[ \frac {1}{30} \, b^{8} x^{30} + \frac {2}{7} \, a b^{7} x^{28} + \frac {14}{13} \, a^{2} b^{6} x^{26} + \frac {7}{3} \, a^{3} b^{5} x^{24} + \frac {35}{11} \, a^{4} b^{4} x^{22} + \frac {14}{5} \, a^{5} b^{3} x^{20} + \frac {14}{9} \, a^{6} b^{2} x^{18} + \frac {1}{2} \, a^{7} b x^{16} + \frac {1}{14} \, a^{8} x^{14} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 90, normalized size = 0.70 \[ \frac {a^8\,x^{14}}{14}+\frac {a^7\,b\,x^{16}}{2}+\frac {14\,a^6\,b^2\,x^{18}}{9}+\frac {14\,a^5\,b^3\,x^{20}}{5}+\frac {35\,a^4\,b^4\,x^{22}}{11}+\frac {7\,a^3\,b^5\,x^{24}}{3}+\frac {14\,a^2\,b^6\,x^{26}}{13}+\frac {2\,a\,b^7\,x^{28}}{7}+\frac {b^8\,x^{30}}{30} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 105, normalized size = 0.81 \[ \frac {a^{8} x^{14}}{14} + \frac {a^{7} b x^{16}}{2} + \frac {14 a^{6} b^{2} x^{18}}{9} + \frac {14 a^{5} b^{3} x^{20}}{5} + \frac {35 a^{4} b^{4} x^{22}}{11} + \frac {7 a^{3} b^{5} x^{24}}{3} + \frac {14 a^{2} b^{6} x^{26}}{13} + \frac {2 a b^{7} x^{28}}{7} + \frac {b^{8} x^{30}}{30} \]
Verification of antiderivative is not currently implemented for this CAS.
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