Optimal. Leaf size=72 \[ -\frac {a^3 \left (a+b x^2\right )^9}{18 b^4}+\frac {3 a^2 \left (a+b x^2\right )^{10}}{20 b^4}-\frac {3 a \left (a+b x^2\right )^{11}}{22 b^4}+\frac {\left (a+b x^2\right )^{12}}{24 b^4} \]
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Rubi [A]
time = 0.08, antiderivative size = 72, 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^3 \left (a+b x^2\right )^9}{18 b^4}+\frac {3 a^2 \left (a+b x^2\right )^{10}}{20 b^4}+\frac {\left (a+b x^2\right )^{12}}{24 b^4}-\frac {3 a \left (a+b x^2\right )^{11}}{22 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^7 \left (a+b x^2\right )^8 \, dx &=\frac {1}{2} \text {Subst}\left (\int x^3 (a+b x)^8 \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a^3 (a+b x)^8}{b^3}+\frac {3 a^2 (a+b x)^9}{b^3}-\frac {3 a (a+b x)^{10}}{b^3}+\frac {(a+b x)^{11}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^3 \left (a+b x^2\right )^9}{18 b^4}+\frac {3 a^2 \left (a+b x^2\right )^{10}}{20 b^4}-\frac {3 a \left (a+b x^2\right )^{11}}{22 b^4}+\frac {\left (a+b x^2\right )^{12}}{24 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 106, normalized size = 1.47 \begin {gather*} \frac {a^8 x^8}{8}+\frac {4}{5} a^7 b x^{10}+\frac {7}{3} a^6 b^2 x^{12}+4 a^5 b^3 x^{14}+\frac {35}{8} a^4 b^4 x^{16}+\frac {28}{9} a^3 b^5 x^{18}+\frac {7}{5} a^2 b^6 x^{20}+\frac {4}{11} a b^7 x^{22}+\frac {b^8 x^{24}}{24} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 91, normalized size = 1.26
method | result | size |
gosper | \(\frac {4}{11} a \,b^{7} x^{22}+\frac {1}{24} b^{8} x^{24}+\frac {1}{8} a^{8} x^{8}+\frac {4}{5} a^{7} b \,x^{10}+\frac {7}{3} a^{6} b^{2} x^{12}+4 a^{5} b^{3} x^{14}+\frac {35}{8} a^{4} b^{4} x^{16}+\frac {28}{9} a^{3} b^{5} x^{18}+\frac {7}{5} a^{2} b^{6} x^{20}\) | \(91\) |
default | \(\frac {4}{11} a \,b^{7} x^{22}+\frac {1}{24} b^{8} x^{24}+\frac {1}{8} a^{8} x^{8}+\frac {4}{5} a^{7} b \,x^{10}+\frac {7}{3} a^{6} b^{2} x^{12}+4 a^{5} b^{3} x^{14}+\frac {35}{8} a^{4} b^{4} x^{16}+\frac {28}{9} a^{3} b^{5} x^{18}+\frac {7}{5} a^{2} b^{6} x^{20}\) | \(91\) |
norman | \(\frac {4}{11} a \,b^{7} x^{22}+\frac {1}{24} b^{8} x^{24}+\frac {1}{8} a^{8} x^{8}+\frac {4}{5} a^{7} b \,x^{10}+\frac {7}{3} a^{6} b^{2} x^{12}+4 a^{5} b^{3} x^{14}+\frac {35}{8} a^{4} b^{4} x^{16}+\frac {28}{9} a^{3} b^{5} x^{18}+\frac {7}{5} a^{2} b^{6} x^{20}\) | \(91\) |
risch | \(\frac {4}{11} a \,b^{7} x^{22}+\frac {1}{24} b^{8} x^{24}+\frac {1}{8} a^{8} x^{8}+\frac {4}{5} a^{7} b \,x^{10}+\frac {7}{3} a^{6} b^{2} x^{12}+4 a^{5} b^{3} x^{14}+\frac {35}{8} a^{4} b^{4} x^{16}+\frac {28}{9} a^{3} b^{5} x^{18}+\frac {7}{5} a^{2} b^{6} x^{20}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 90, normalized size = 1.25 \begin {gather*} \frac {1}{24} \, b^{8} x^{24} + \frac {4}{11} \, a b^{7} x^{22} + \frac {7}{5} \, a^{2} b^{6} x^{20} + \frac {28}{9} \, a^{3} b^{5} x^{18} + \frac {35}{8} \, a^{4} b^{4} x^{16} + 4 \, a^{5} b^{3} x^{14} + \frac {7}{3} \, a^{6} b^{2} x^{12} + \frac {4}{5} \, a^{7} b x^{10} + \frac {1}{8} \, a^{8} x^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.40, size = 90, normalized size = 1.25 \begin {gather*} \frac {1}{24} \, b^{8} x^{24} + \frac {4}{11} \, a b^{7} x^{22} + \frac {7}{5} \, a^{2} b^{6} x^{20} + \frac {28}{9} \, a^{3} b^{5} x^{18} + \frac {35}{8} \, a^{4} b^{4} x^{16} + 4 \, a^{5} b^{3} x^{14} + \frac {7}{3} \, a^{6} b^{2} x^{12} + \frac {4}{5} \, a^{7} b x^{10} + \frac {1}{8} \, a^{8} x^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 105, normalized size = 1.46 \begin {gather*} \frac {a^{8} x^{8}}{8} + \frac {4 a^{7} b x^{10}}{5} + \frac {7 a^{6} b^{2} x^{12}}{3} + 4 a^{5} b^{3} x^{14} + \frac {35 a^{4} b^{4} x^{16}}{8} + \frac {28 a^{3} b^{5} x^{18}}{9} + \frac {7 a^{2} b^{6} x^{20}}{5} + \frac {4 a b^{7} x^{22}}{11} + \frac {b^{8} x^{24}}{24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.83, size = 90, normalized size = 1.25 \begin {gather*} \frac {1}{24} \, b^{8} x^{24} + \frac {4}{11} \, a b^{7} x^{22} + \frac {7}{5} \, a^{2} b^{6} x^{20} + \frac {28}{9} \, a^{3} b^{5} x^{18} + \frac {35}{8} \, a^{4} b^{4} x^{16} + 4 \, a^{5} b^{3} x^{14} + \frac {7}{3} \, a^{6} b^{2} x^{12} + \frac {4}{5} \, a^{7} b x^{10} + \frac {1}{8} \, a^{8} x^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 90, normalized size = 1.25 \begin {gather*} \frac {a^8\,x^8}{8}+\frac {4\,a^7\,b\,x^{10}}{5}+\frac {7\,a^6\,b^2\,x^{12}}{3}+4\,a^5\,b^3\,x^{14}+\frac {35\,a^4\,b^4\,x^{16}}{8}+\frac {28\,a^3\,b^5\,x^{18}}{9}+\frac {7\,a^2\,b^6\,x^{20}}{5}+\frac {4\,a\,b^7\,x^{22}}{11}+\frac {b^8\,x^{24}}{24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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