Optimal. Leaf size=77 \[ \frac{x^{12}}{1008 a^4 \left (a+b x^2\right )^6}+\frac{x^{12}}{168 a^3 \left (a+b x^2\right )^7}+\frac{x^{12}}{48 a^2 \left (a+b x^2\right )^8}+\frac{x^{12}}{18 a \left (a+b x^2\right )^9} \]
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Rubi [A] time = 0.0389887, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac{x^{12}}{1008 a^4 \left (a+b x^2\right )^6}+\frac{x^{12}}{168 a^3 \left (a+b x^2\right )^7}+\frac{x^{12}}{48 a^2 \left (a+b x^2\right )^8}+\frac{x^{12}}{18 a \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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Rule 266
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^5}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{x^{12}}{18 a \left (a+b x^2\right )^9}+\frac{\operatorname{Subst}\left (\int \frac{x^5}{(a+b x)^9} \, dx,x,x^2\right )}{6 a}\\ &=\frac{x^{12}}{18 a \left (a+b x^2\right )^9}+\frac{x^{12}}{48 a^2 \left (a+b x^2\right )^8}+\frac{\operatorname{Subst}\left (\int \frac{x^5}{(a+b x)^8} \, dx,x,x^2\right )}{24 a^2}\\ &=\frac{x^{12}}{18 a \left (a+b x^2\right )^9}+\frac{x^{12}}{48 a^2 \left (a+b x^2\right )^8}+\frac{x^{12}}{168 a^3 \left (a+b x^2\right )^7}+\frac{\operatorname{Subst}\left (\int \frac{x^5}{(a+b x)^7} \, dx,x,x^2\right )}{168 a^3}\\ &=\frac{x^{12}}{18 a \left (a+b x^2\right )^9}+\frac{x^{12}}{48 a^2 \left (a+b x^2\right )^8}+\frac{x^{12}}{168 a^3 \left (a+b x^2\right )^7}+\frac{x^{12}}{1008 a^4 \left (a+b x^2\right )^6}\\ \end{align*}
Mathematica [A] time = 0.021162, size = 68, normalized size = 0.88 \[ -\frac{84 a^2 b^3 x^6+36 a^3 b^2 x^4+9 a^4 b x^2+a^5+126 a b^4 x^8+126 b^5 x^{10}}{1008 b^6 \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 99, normalized size = 1.3 \begin{align*} -{\frac{5\,{a}^{4}}{16\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{8}}}-{\frac{5\,{a}^{2}}{6\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{6}}}+{\frac{{a}^{5}}{18\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{a}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{1}{8\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{5\,{a}^{3}}{7\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.36448, size = 212, normalized size = 2.75 \begin{align*} -\frac{126 \, b^{5} x^{10} + 126 \, a b^{4} x^{8} + 84 \, a^{2} b^{3} x^{6} + 36 \, a^{3} b^{2} x^{4} + 9 \, a^{4} b x^{2} + a^{5}}{1008 \,{\left (b^{15} x^{18} + 9 \, a b^{14} x^{16} + 36 \, a^{2} b^{13} x^{14} + 84 \, a^{3} b^{12} x^{12} + 126 \, a^{4} b^{11} x^{10} + 126 \, a^{5} b^{10} x^{8} + 84 \, a^{6} b^{9} x^{6} + 36 \, a^{7} b^{8} x^{4} + 9 \, a^{8} b^{7} x^{2} + a^{9} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.24011, size = 350, normalized size = 4.55 \begin{align*} -\frac{126 \, b^{5} x^{10} + 126 \, a b^{4} x^{8} + 84 \, a^{2} b^{3} x^{6} + 36 \, a^{3} b^{2} x^{4} + 9 \, a^{4} b x^{2} + a^{5}}{1008 \,{\left (b^{15} x^{18} + 9 \, a b^{14} x^{16} + 36 \, a^{2} b^{13} x^{14} + 84 \, a^{3} b^{12} x^{12} + 126 \, a^{4} b^{11} x^{10} + 126 \, a^{5} b^{10} x^{8} + 84 \, a^{6} b^{9} x^{6} + 36 \, a^{7} b^{8} x^{4} + 9 \, a^{8} b^{7} x^{2} + a^{9} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.2542, size = 167, normalized size = 2.17 \begin{align*} - \frac{a^{5} + 9 a^{4} b x^{2} + 36 a^{3} b^{2} x^{4} + 84 a^{2} b^{3} x^{6} + 126 a b^{4} x^{8} + 126 b^{5} x^{10}}{1008 a^{9} b^{6} + 9072 a^{8} b^{7} x^{2} + 36288 a^{7} b^{8} x^{4} + 84672 a^{6} b^{9} x^{6} + 127008 a^{5} b^{10} x^{8} + 127008 a^{4} b^{11} x^{10} + 84672 a^{3} b^{12} x^{12} + 36288 a^{2} b^{13} x^{14} + 9072 a b^{14} x^{16} + 1008 b^{15} x^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.57667, size = 89, normalized size = 1.16 \begin{align*} -\frac{126 \, b^{5} x^{10} + 126 \, a b^{4} x^{8} + 84 \, a^{2} b^{3} x^{6} + 36 \, a^{3} b^{2} x^{4} + 9 \, a^{4} b x^{2} + a^{5}}{1008 \,{\left (b x^{2} + a\right )}^{9} b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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