Optimal. Leaf size=40 \[ \frac {b \left (a+b x^2\right )^9}{180 a^2 x^{18}}-\frac {\left (a+b x^2\right )^9}{20 a x^{20}} \]
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Rubi [A] time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac {b \left (a+b x^2\right )^9}{180 a^2 x^{18}}-\frac {\left (a+b x^2\right )^9}{20 a x^{20}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^8}{x^{21}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{11}} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^9}{20 a x^{20}}-\frac {b \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{10}} \, dx,x,x^2\right )}{20 a}\\ &=-\frac {\left (a+b x^2\right )^9}{20 a x^{20}}+\frac {b \left (a+b x^2\right )^9}{180 a^2 x^{18}}\\ \end {align*}
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Mathematica [B] time = 0.00, size = 106, normalized size = 2.65 \[ -\frac {a^8}{20 x^{20}}-\frac {4 a^7 b}{9 x^{18}}-\frac {7 a^6 b^2}{4 x^{16}}-\frac {4 a^5 b^3}{x^{14}}-\frac {35 a^4 b^4}{6 x^{12}}-\frac {28 a^3 b^5}{5 x^{10}}-\frac {7 a^2 b^6}{2 x^8}-\frac {4 a b^7}{3 x^6}-\frac {b^8}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 92, normalized size = 2.30 \[ -\frac {45 \, b^{8} x^{16} + 240 \, a b^{7} x^{14} + 630 \, a^{2} b^{6} x^{12} + 1008 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 720 \, a^{5} b^{3} x^{6} + 315 \, a^{6} b^{2} x^{4} + 80 \, a^{7} b x^{2} + 9 \, a^{8}}{180 \, x^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.00, size = 92, normalized size = 2.30 \[ -\frac {45 \, b^{8} x^{16} + 240 \, a b^{7} x^{14} + 630 \, a^{2} b^{6} x^{12} + 1008 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 720 \, a^{5} b^{3} x^{6} + 315 \, a^{6} b^{2} x^{4} + 80 \, a^{7} b x^{2} + 9 \, a^{8}}{180 \, x^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 91, normalized size = 2.28 \[ -\frac {b^{8}}{4 x^{4}}-\frac {4 a \,b^{7}}{3 x^{6}}-\frac {7 a^{2} b^{6}}{2 x^{8}}-\frac {28 a^{3} b^{5}}{5 x^{10}}-\frac {35 a^{4} b^{4}}{6 x^{12}}-\frac {4 a^{5} b^{3}}{x^{14}}-\frac {7 a^{6} b^{2}}{4 x^{16}}-\frac {4 a^{7} b}{9 x^{18}}-\frac {a^{8}}{20 x^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.35, size = 92, normalized size = 2.30 \[ -\frac {45 \, b^{8} x^{16} + 240 \, a b^{7} x^{14} + 630 \, a^{2} b^{6} x^{12} + 1008 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 720 \, a^{5} b^{3} x^{6} + 315 \, a^{6} b^{2} x^{4} + 80 \, a^{7} b x^{2} + 9 \, a^{8}}{180 \, x^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 92, normalized size = 2.30 \[ -\frac {\frac {a^8}{20}+\frac {4\,a^7\,b\,x^2}{9}+\frac {7\,a^6\,b^2\,x^4}{4}+4\,a^5\,b^3\,x^6+\frac {35\,a^4\,b^4\,x^8}{6}+\frac {28\,a^3\,b^5\,x^{10}}{5}+\frac {7\,a^2\,b^6\,x^{12}}{2}+\frac {4\,a\,b^7\,x^{14}}{3}+\frac {b^8\,x^{16}}{4}}{x^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.94, size = 99, normalized size = 2.48 \[ \frac {- 9 a^{8} - 80 a^{7} b x^{2} - 315 a^{6} b^{2} x^{4} - 720 a^{5} b^{3} x^{6} - 1050 a^{4} b^{4} x^{8} - 1008 a^{3} b^{5} x^{10} - 630 a^{2} b^{6} x^{12} - 240 a b^{7} x^{14} - 45 b^{8} x^{16}}{180 x^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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