Optimal. Leaf size=108 \[ -\frac {a^8}{28 x^{28}}-\frac {4 a^7 b}{13 x^{26}}-\frac {7 a^6 b^2}{6 x^{24}}-\frac {28 a^5 b^3}{11 x^{22}}-\frac {7 a^4 b^4}{2 x^{20}}-\frac {28 a^3 b^5}{9 x^{18}}-\frac {7 a^2 b^6}{4 x^{16}}-\frac {4 a b^7}{7 x^{14}}-\frac {b^8}{12 x^{12}} \]
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Rubi [A] time = 0.05, antiderivative size = 108, 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} \[ -\frac {7 a^6 b^2}{6 x^{24}}-\frac {28 a^5 b^3}{11 x^{22}}-\frac {7 a^4 b^4}{2 x^{20}}-\frac {28 a^3 b^5}{9 x^{18}}-\frac {7 a^2 b^6}{4 x^{16}}-\frac {4 a^7 b}{13 x^{26}}-\frac {a^8}{28 x^{28}}-\frac {4 a b^7}{7 x^{14}}-\frac {b^8}{12 x^{12}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^8}{x^{29}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{15}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^8}{x^{15}}+\frac {8 a^7 b}{x^{14}}+\frac {28 a^6 b^2}{x^{13}}+\frac {56 a^5 b^3}{x^{12}}+\frac {70 a^4 b^4}{x^{11}}+\frac {56 a^3 b^5}{x^{10}}+\frac {28 a^2 b^6}{x^9}+\frac {8 a b^7}{x^8}+\frac {b^8}{x^7}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^8}{28 x^{28}}-\frac {4 a^7 b}{13 x^{26}}-\frac {7 a^6 b^2}{6 x^{24}}-\frac {28 a^5 b^3}{11 x^{22}}-\frac {7 a^4 b^4}{2 x^{20}}-\frac {28 a^3 b^5}{9 x^{18}}-\frac {7 a^2 b^6}{4 x^{16}}-\frac {4 a b^7}{7 x^{14}}-\frac {b^8}{12 x^{12}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 108, normalized size = 1.00 \[ -\frac {a^8}{28 x^{28}}-\frac {4 a^7 b}{13 x^{26}}-\frac {7 a^6 b^2}{6 x^{24}}-\frac {28 a^5 b^3}{11 x^{22}}-\frac {7 a^4 b^4}{2 x^{20}}-\frac {28 a^3 b^5}{9 x^{18}}-\frac {7 a^2 b^6}{4 x^{16}}-\frac {4 a b^7}{7 x^{14}}-\frac {b^8}{12 x^{12}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 92, normalized size = 0.85 \[ -\frac {3003 \, b^{8} x^{16} + 20592 \, a b^{7} x^{14} + 63063 \, a^{2} b^{6} x^{12} + 112112 \, a^{3} b^{5} x^{10} + 126126 \, a^{4} b^{4} x^{8} + 91728 \, a^{5} b^{3} x^{6} + 42042 \, a^{6} b^{2} x^{4} + 11088 \, a^{7} b x^{2} + 1287 \, a^{8}}{36036 \, x^{28}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 92, normalized size = 0.85 \[ -\frac {3003 \, b^{8} x^{16} + 20592 \, a b^{7} x^{14} + 63063 \, a^{2} b^{6} x^{12} + 112112 \, a^{3} b^{5} x^{10} + 126126 \, a^{4} b^{4} x^{8} + 91728 \, a^{5} b^{3} x^{6} + 42042 \, a^{6} b^{2} x^{4} + 11088 \, a^{7} b x^{2} + 1287 \, a^{8}}{36036 \, x^{28}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 91, normalized size = 0.84 \[ -\frac {b^{8}}{12 x^{12}}-\frac {4 a \,b^{7}}{7 x^{14}}-\frac {7 a^{2} b^{6}}{4 x^{16}}-\frac {28 a^{3} b^{5}}{9 x^{18}}-\frac {7 a^{4} b^{4}}{2 x^{20}}-\frac {28 a^{5} b^{3}}{11 x^{22}}-\frac {7 a^{6} b^{2}}{6 x^{24}}-\frac {4 a^{7} b}{13 x^{26}}-\frac {a^{8}}{28 x^{28}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 92, normalized size = 0.85 \[ -\frac {3003 \, b^{8} x^{16} + 20592 \, a b^{7} x^{14} + 63063 \, a^{2} b^{6} x^{12} + 112112 \, a^{3} b^{5} x^{10} + 126126 \, a^{4} b^{4} x^{8} + 91728 \, a^{5} b^{3} x^{6} + 42042 \, a^{6} b^{2} x^{4} + 11088 \, a^{7} b x^{2} + 1287 \, a^{8}}{36036 \, x^{28}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.89, size = 92, normalized size = 0.85 \[ -\frac {\frac {a^8}{28}+\frac {4\,a^7\,b\,x^2}{13}+\frac {7\,a^6\,b^2\,x^4}{6}+\frac {28\,a^5\,b^3\,x^6}{11}+\frac {7\,a^4\,b^4\,x^8}{2}+\frac {28\,a^3\,b^5\,x^{10}}{9}+\frac {7\,a^2\,b^6\,x^{12}}{4}+\frac {4\,a\,b^7\,x^{14}}{7}+\frac {b^8\,x^{16}}{12}}{x^{28}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.20, size = 99, normalized size = 0.92 \[ \frac {- 1287 a^{8} - 11088 a^{7} b x^{2} - 42042 a^{6} b^{2} x^{4} - 91728 a^{5} b^{3} x^{6} - 126126 a^{4} b^{4} x^{8} - 112112 a^{3} b^{5} x^{10} - 63063 a^{2} b^{6} x^{12} - 20592 a b^{7} x^{14} - 3003 b^{8} x^{16}}{36036 x^{28}} \]
Verification of antiderivative is not currently implemented for this CAS.
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