Optimal. Leaf size=53 \[ \frac {a^2 \left (a+b x^2\right )^9}{18 b^3}-\frac {a \left (a+b x^2\right )^{10}}{10 b^3}+\frac {\left (a+b x^2\right )^{11}}{22 b^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 53, 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^2 \left (a+b x^2\right )^9}{18 b^3}+\frac {\left (a+b x^2\right )^{11}}{22 b^3}-\frac {a \left (a+b x^2\right )^{10}}{10 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^5 \left (a+b x^2\right )^8 \, dx &=\frac {1}{2} \text {Subst}\left (\int x^2 (a+b x)^8 \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2 (a+b x)^8}{b^2}-\frac {2 a (a+b x)^9}{b^2}+\frac {(a+b x)^{10}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 \left (a+b x^2\right )^9}{18 b^3}-\frac {a \left (a+b x^2\right )^{10}}{10 b^3}+\frac {\left (a+b x^2\right )^{11}}{22 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 103, normalized size = 1.94 \begin {gather*} \frac {a^8 x^6}{6}+a^7 b x^8+\frac {14}{5} a^6 b^2 x^{10}+\frac {14}{3} a^5 b^3 x^{12}+5 a^4 b^4 x^{14}+\frac {7}{2} a^3 b^5 x^{16}+\frac {14}{9} a^2 b^6 x^{18}+\frac {2}{5} a b^7 x^{20}+\frac {b^8 x^{22}}{22} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 90, normalized size = 1.70
method | result | size |
gosper | \(\frac {1}{6} a^{8} x^{6}+a^{7} b \,x^{8}+\frac {14}{5} a^{6} b^{2} x^{10}+\frac {14}{3} a^{5} b^{3} x^{12}+5 a^{4} b^{4} x^{14}+\frac {7}{2} a^{3} b^{5} x^{16}+\frac {14}{9} a^{2} b^{6} x^{18}+\frac {2}{5} a \,b^{7} x^{20}+\frac {1}{22} b^{8} x^{22}\) | \(90\) |
default | \(\frac {1}{6} a^{8} x^{6}+a^{7} b \,x^{8}+\frac {14}{5} a^{6} b^{2} x^{10}+\frac {14}{3} a^{5} b^{3} x^{12}+5 a^{4} b^{4} x^{14}+\frac {7}{2} a^{3} b^{5} x^{16}+\frac {14}{9} a^{2} b^{6} x^{18}+\frac {2}{5} a \,b^{7} x^{20}+\frac {1}{22} b^{8} x^{22}\) | \(90\) |
norman | \(\frac {1}{6} a^{8} x^{6}+a^{7} b \,x^{8}+\frac {14}{5} a^{6} b^{2} x^{10}+\frac {14}{3} a^{5} b^{3} x^{12}+5 a^{4} b^{4} x^{14}+\frac {7}{2} a^{3} b^{5} x^{16}+\frac {14}{9} a^{2} b^{6} x^{18}+\frac {2}{5} a \,b^{7} x^{20}+\frac {1}{22} b^{8} x^{22}\) | \(90\) |
risch | \(\frac {1}{6} a^{8} x^{6}+a^{7} b \,x^{8}+\frac {14}{5} a^{6} b^{2} x^{10}+\frac {14}{3} a^{5} b^{3} x^{12}+5 a^{4} b^{4} x^{14}+\frac {7}{2} a^{3} b^{5} x^{16}+\frac {14}{9} a^{2} b^{6} x^{18}+\frac {2}{5} a \,b^{7} x^{20}+\frac {1}{22} b^{8} x^{22}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 89, normalized size = 1.68 \begin {gather*} \frac {1}{22} \, b^{8} x^{22} + \frac {2}{5} \, a b^{7} x^{20} + \frac {14}{9} \, a^{2} b^{6} x^{18} + \frac {7}{2} \, a^{3} b^{5} x^{16} + 5 \, a^{4} b^{4} x^{14} + \frac {14}{3} \, a^{5} b^{3} x^{12} + \frac {14}{5} \, a^{6} b^{2} x^{10} + a^{7} b x^{8} + \frac {1}{6} \, a^{8} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.02, size = 89, normalized size = 1.68 \begin {gather*} \frac {1}{22} \, b^{8} x^{22} + \frac {2}{5} \, a b^{7} x^{20} + \frac {14}{9} \, a^{2} b^{6} x^{18} + \frac {7}{2} \, a^{3} b^{5} x^{16} + 5 \, a^{4} b^{4} x^{14} + \frac {14}{3} \, a^{5} b^{3} x^{12} + \frac {14}{5} \, a^{6} b^{2} x^{10} + a^{7} b x^{8} + \frac {1}{6} \, a^{8} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs.
\(2 (44) = 88\).
time = 0.02, size = 102, normalized size = 1.92 \begin {gather*} \frac {a^{8} x^{6}}{6} + a^{7} b x^{8} + \frac {14 a^{6} b^{2} x^{10}}{5} + \frac {14 a^{5} b^{3} x^{12}}{3} + 5 a^{4} b^{4} x^{14} + \frac {7 a^{3} b^{5} x^{16}}{2} + \frac {14 a^{2} b^{6} x^{18}}{9} + \frac {2 a b^{7} x^{20}}{5} + \frac {b^{8} x^{22}}{22} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.92, size = 89, normalized size = 1.68 \begin {gather*} \frac {1}{22} \, b^{8} x^{22} + \frac {2}{5} \, a b^{7} x^{20} + \frac {14}{9} \, a^{2} b^{6} x^{18} + \frac {7}{2} \, a^{3} b^{5} x^{16} + 5 \, a^{4} b^{4} x^{14} + \frac {14}{3} \, a^{5} b^{3} x^{12} + \frac {14}{5} \, a^{6} b^{2} x^{10} + a^{7} b x^{8} + \frac {1}{6} \, a^{8} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 89, normalized size = 1.68 \begin {gather*} \frac {a^8\,x^6}{6}+a^7\,b\,x^8+\frac {14\,a^6\,b^2\,x^{10}}{5}+\frac {14\,a^5\,b^3\,x^{12}}{3}+5\,a^4\,b^4\,x^{14}+\frac {7\,a^3\,b^5\,x^{16}}{2}+\frac {14\,a^2\,b^6\,x^{18}}{9}+\frac {2\,a\,b^7\,x^{20}}{5}+\frac {b^8\,x^{22}}{22} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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