Optimal. Leaf size=84 \[ \frac{b^3 \left (a+b x^2\right )^7}{1680 a^4 x^{14}}-\frac{b^2 \left (a+b x^2\right )^7}{240 a^3 x^{16}}+\frac{b \left (a+b x^2\right )^7}{60 a^2 x^{18}}-\frac{\left (a+b x^2\right )^7}{20 a x^{20}} \]
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Rubi [A] time = 0.056049, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {28, 266, 45, 37} \[ \frac{b^3 \left (a+b x^2\right )^7}{1680 a^4 x^{14}}-\frac{b^2 \left (a+b x^2\right )^7}{240 a^3 x^{16}}+\frac{b \left (a+b x^2\right )^7}{60 a^2 x^{18}}-\frac{\left (a+b x^2\right )^7}{20 a x^{20}} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{21}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^{21}} \, dx}{b^6}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^6}{x^{11}} \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac{\left (a+b x^2\right )^7}{20 a x^{20}}-\frac{3 \operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^6}{x^{10}} \, dx,x,x^2\right )}{20 a b^5}\\ &=-\frac{\left (a+b x^2\right )^7}{20 a x^{20}}+\frac{b \left (a+b x^2\right )^7}{60 a^2 x^{18}}+\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^6}{x^9} \, dx,x,x^2\right )}{30 a^2 b^4}\\ &=-\frac{\left (a+b x^2\right )^7}{20 a x^{20}}+\frac{b \left (a+b x^2\right )^7}{60 a^2 x^{18}}-\frac{b^2 \left (a+b x^2\right )^7}{240 a^3 x^{16}}-\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^6}{x^8} \, dx,x,x^2\right )}{240 a^3 b^3}\\ &=-\frac{\left (a+b x^2\right )^7}{20 a x^{20}}+\frac{b \left (a+b x^2\right )^7}{60 a^2 x^{18}}-\frac{b^2 \left (a+b x^2\right )^7}{240 a^3 x^{16}}+\frac{b^3 \left (a+b x^2\right )^7}{1680 a^4 x^{14}}\\ \end{align*}
Mathematica [A] time = 0.0078792, size = 82, normalized size = 0.98 \[ -\frac{15 a^4 b^2}{16 x^{16}}-\frac{10 a^3 b^3}{7 x^{14}}-\frac{5 a^2 b^4}{4 x^{12}}-\frac{a^5 b}{3 x^{18}}-\frac{a^6}{20 x^{20}}-\frac{3 a b^5}{5 x^{10}}-\frac{b^6}{8 x^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 69, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5}b}{3\,{x}^{18}}}-{\frac{15\,{a}^{4}{b}^{2}}{16\,{x}^{16}}}-{\frac{5\,{a}^{2}{b}^{4}}{4\,{x}^{12}}}-{\frac{3\,a{b}^{5}}{5\,{x}^{10}}}-{\frac{{a}^{6}}{20\,{x}^{20}}}-{\frac{10\,{a}^{3}{b}^{3}}{7\,{x}^{14}}}-{\frac{{b}^{6}}{8\,{x}^{8}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0085, size = 95, normalized size = 1.13 \begin{align*} -\frac{210 \, b^{6} x^{12} + 1008 \, a b^{5} x^{10} + 2100 \, a^{2} b^{4} x^{8} + 2400 \, a^{3} b^{3} x^{6} + 1575 \, a^{4} b^{2} x^{4} + 560 \, a^{5} b x^{2} + 84 \, a^{6}}{1680 \, x^{20}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69188, size = 174, normalized size = 2.07 \begin{align*} -\frac{210 \, b^{6} x^{12} + 1008 \, a b^{5} x^{10} + 2100 \, a^{2} b^{4} x^{8} + 2400 \, a^{3} b^{3} x^{6} + 1575 \, a^{4} b^{2} x^{4} + 560 \, a^{5} b x^{2} + 84 \, a^{6}}{1680 \, x^{20}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.902203, size = 75, normalized size = 0.89 \begin{align*} - \frac{84 a^{6} + 560 a^{5} b x^{2} + 1575 a^{4} b^{2} x^{4} + 2400 a^{3} b^{3} x^{6} + 2100 a^{2} b^{4} x^{8} + 1008 a b^{5} x^{10} + 210 b^{6} x^{12}}{1680 x^{20}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14007, size = 95, normalized size = 1.13 \begin{align*} -\frac{210 \, b^{6} x^{12} + 1008 \, a b^{5} x^{10} + 2100 \, a^{2} b^{4} x^{8} + 2400 \, a^{3} b^{3} x^{6} + 1575 \, a^{4} b^{2} x^{4} + 560 \, a^{5} b x^{2} + 84 \, a^{6}}{1680 \, x^{20}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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