Optimal. Leaf size=91 \[ \frac {a^4 \left (a+b x^2\right )^9}{18 b^5}-\frac {a^3 \left (a+b x^2\right )^{10}}{5 b^5}+\frac {3 a^2 \left (a+b x^2\right )^{11}}{11 b^5}-\frac {a \left (a+b x^2\right )^{12}}{6 b^5}+\frac {\left (a+b x^2\right )^{13}}{26 b^5} \]
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Rubi [A]
time = 0.10, antiderivative size = 91, 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 = {272, 45}
\begin {gather*} \frac {a^4 \left (a+b x^2\right )^9}{18 b^5}-\frac {a^3 \left (a+b x^2\right )^{10}}{5 b^5}+\frac {3 a^2 \left (a+b x^2\right )^{11}}{11 b^5}+\frac {\left (a+b x^2\right )^{13}}{26 b^5}-\frac {a \left (a+b x^2\right )^{12}}{6 b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^9 \left (a+b x^2\right )^8 \, dx &=\frac {1}{2} \text {Subst}\left (\int x^4 (a+b x)^8 \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^4 (a+b x)^8}{b^4}-\frac {4 a^3 (a+b x)^9}{b^4}+\frac {6 a^2 (a+b x)^{10}}{b^4}-\frac {4 a (a+b x)^{11}}{b^4}+\frac {(a+b x)^{12}}{b^4}\right ) \, dx,x,x^2\right )\\ &=\frac {a^4 \left (a+b x^2\right )^9}{18 b^5}-\frac {a^3 \left (a+b x^2\right )^{10}}{5 b^5}+\frac {3 a^2 \left (a+b x^2\right )^{11}}{11 b^5}-\frac {a \left (a+b x^2\right )^{12}}{6 b^5}+\frac {\left (a+b x^2\right )^{13}}{26 b^5}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 106, normalized size = 1.16 \begin {gather*} \frac {a^8 x^{10}}{10}+\frac {2}{3} a^7 b x^{12}+2 a^6 b^2 x^{14}+\frac {7}{2} a^5 b^3 x^{16}+\frac {35}{9} a^4 b^4 x^{18}+\frac {14}{5} a^3 b^5 x^{20}+\frac {14}{11} a^2 b^6 x^{22}+\frac {1}{3} a b^7 x^{24}+\frac {b^8 x^{26}}{26} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 91, normalized size = 1.00
method | result | size |
gosper | \(\frac {35}{9} a^{4} b^{4} x^{18}+\frac {14}{5} a^{3} b^{5} x^{20}+\frac {14}{11} a^{2} b^{6} x^{22}+\frac {1}{3} a \,b^{7} x^{24}+\frac {1}{26} b^{8} x^{26}+\frac {1}{10} a^{8} x^{10}+\frac {2}{3} a^{7} b \,x^{12}+2 a^{6} b^{2} x^{14}+\frac {7}{2} a^{5} b^{3} x^{16}\) | \(91\) |
default | \(\frac {35}{9} a^{4} b^{4} x^{18}+\frac {14}{5} a^{3} b^{5} x^{20}+\frac {14}{11} a^{2} b^{6} x^{22}+\frac {1}{3} a \,b^{7} x^{24}+\frac {1}{26} b^{8} x^{26}+\frac {1}{10} a^{8} x^{10}+\frac {2}{3} a^{7} b \,x^{12}+2 a^{6} b^{2} x^{14}+\frac {7}{2} a^{5} b^{3} x^{16}\) | \(91\) |
norman | \(\frac {35}{9} a^{4} b^{4} x^{18}+\frac {14}{5} a^{3} b^{5} x^{20}+\frac {14}{11} a^{2} b^{6} x^{22}+\frac {1}{3} a \,b^{7} x^{24}+\frac {1}{26} b^{8} x^{26}+\frac {1}{10} a^{8} x^{10}+\frac {2}{3} a^{7} b \,x^{12}+2 a^{6} b^{2} x^{14}+\frac {7}{2} a^{5} b^{3} x^{16}\) | \(91\) |
risch | \(\frac {35}{9} a^{4} b^{4} x^{18}+\frac {14}{5} a^{3} b^{5} x^{20}+\frac {14}{11} a^{2} b^{6} x^{22}+\frac {1}{3} a \,b^{7} x^{24}+\frac {1}{26} b^{8} x^{26}+\frac {1}{10} a^{8} x^{10}+\frac {2}{3} a^{7} b \,x^{12}+2 a^{6} b^{2} x^{14}+\frac {7}{2} a^{5} b^{3} x^{16}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 90, normalized size = 0.99 \begin {gather*} \frac {1}{26} \, b^{8} x^{26} + \frac {1}{3} \, a b^{7} x^{24} + \frac {14}{11} \, a^{2} b^{6} x^{22} + \frac {14}{5} \, a^{3} b^{5} x^{20} + \frac {35}{9} \, a^{4} b^{4} x^{18} + \frac {7}{2} \, a^{5} b^{3} x^{16} + 2 \, a^{6} b^{2} x^{14} + \frac {2}{3} \, a^{7} b x^{12} + \frac {1}{10} \, a^{8} x^{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.07, size = 90, normalized size = 0.99 \begin {gather*} \frac {1}{26} \, b^{8} x^{26} + \frac {1}{3} \, a b^{7} x^{24} + \frac {14}{11} \, a^{2} b^{6} x^{22} + \frac {14}{5} \, a^{3} b^{5} x^{20} + \frac {35}{9} \, a^{4} b^{4} x^{18} + \frac {7}{2} \, a^{5} b^{3} x^{16} + 2 \, a^{6} b^{2} x^{14} + \frac {2}{3} \, a^{7} b x^{12} + \frac {1}{10} \, a^{8} x^{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 104, normalized size = 1.14 \begin {gather*} \frac {a^{8} x^{10}}{10} + \frac {2 a^{7} b x^{12}}{3} + 2 a^{6} b^{2} x^{14} + \frac {7 a^{5} b^{3} x^{16}}{2} + \frac {35 a^{4} b^{4} x^{18}}{9} + \frac {14 a^{3} b^{5} x^{20}}{5} + \frac {14 a^{2} b^{6} x^{22}}{11} + \frac {a b^{7} x^{24}}{3} + \frac {b^{8} x^{26}}{26} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.34, size = 90, normalized size = 0.99 \begin {gather*} \frac {1}{26} \, b^{8} x^{26} + \frac {1}{3} \, a b^{7} x^{24} + \frac {14}{11} \, a^{2} b^{6} x^{22} + \frac {14}{5} \, a^{3} b^{5} x^{20} + \frac {35}{9} \, a^{4} b^{4} x^{18} + \frac {7}{2} \, a^{5} b^{3} x^{16} + 2 \, a^{6} b^{2} x^{14} + \frac {2}{3} \, a^{7} b x^{12} + \frac {1}{10} \, a^{8} x^{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.59, size = 90, normalized size = 0.99 \begin {gather*} \frac {a^8\,x^{10}}{10}+\frac {2\,a^7\,b\,x^{12}}{3}+2\,a^6\,b^2\,x^{14}+\frac {7\,a^5\,b^3\,x^{16}}{2}+\frac {35\,a^4\,b^4\,x^{18}}{9}+\frac {14\,a^3\,b^5\,x^{20}}{5}+\frac {14\,a^2\,b^6\,x^{22}}{11}+\frac {a\,b^7\,x^{24}}{3}+\frac {b^8\,x^{26}}{26} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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