Optimal. Leaf size=77 \[ \frac {x^{12}}{1008 a^4 \left (a+b x^2\right )^6}+\frac {x^{12}}{168 a^3 \left (a+b x^2\right )^7}+\frac {x^{12}}{48 a^2 \left (a+b x^2\right )^8}+\frac {x^{12}}{18 a \left (a+b x^2\right )^9} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 37
Rule 45
Rule 266
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 = 0.02, size = 68, normalized size = 0.88 \[ -\frac {a^5+9 a^4 b x^2+36 a^3 b^2 x^4+84 a^2 b^3 x^6+126 a b^4 x^8+126 b^5 x^{10}}{1008 b^6 \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.56, size = 157, normalized size = 2.04 \[ -\frac {126 \, b^{5} x^{10} + 126 \, a b^{4} x^{8} + 84 \, a^{2} b^{3} x^{6} + 36 \, a^{3} b^{2} x^{4} + 9 \, a^{4} b x^{2} + a^{5}}{1008 \, {\left (b^{15} x^{18} + 9 \, a b^{14} x^{16} + 36 \, a^{2} b^{13} x^{14} + 84 \, a^{3} b^{12} x^{12} + 126 \, a^{4} b^{11} x^{10} + 126 \, a^{5} b^{10} x^{8} + 84 \, a^{6} b^{9} x^{6} + 36 \, a^{7} b^{8} x^{4} + 9 \, a^{8} b^{7} x^{2} + a^{9} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.63, size = 66, normalized size = 0.86 \[ -\frac {126 \, b^{5} x^{10} + 126 \, a b^{4} x^{8} + 84 \, a^{2} b^{3} x^{6} + 36 \, a^{3} b^{2} x^{4} + 9 \, a^{4} b x^{2} + a^{5}}{1008 \, {\left (b x^{2} + a\right )}^{9} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 99, normalized size = 1.29 \[ \frac {a^{5}}{18 \left (b \,x^{2}+a \right )^{9} b^{6}}-\frac {5 a^{4}}{16 \left (b \,x^{2}+a \right )^{8} b^{6}}+\frac {5 a^{3}}{7 \left (b \,x^{2}+a \right )^{7} b^{6}}-\frac {5 a^{2}}{6 \left (b \,x^{2}+a \right )^{6} b^{6}}+\frac {a}{2 \left (b \,x^{2}+a \right )^{5} b^{6}}-\frac {1}{8 \left (b \,x^{2}+a \right )^{4} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.43, size = 157, normalized size = 2.04 \[ -\frac {126 \, b^{5} x^{10} + 126 \, a b^{4} x^{8} + 84 \, a^{2} b^{3} x^{6} + 36 \, a^{3} b^{2} x^{4} + 9 \, a^{4} b x^{2} + a^{5}}{1008 \, {\left (b^{15} x^{18} + 9 \, a b^{14} x^{16} + 36 \, a^{2} b^{13} x^{14} + 84 \, a^{3} b^{12} x^{12} + 126 \, a^{4} b^{11} x^{10} + 126 \, a^{5} b^{10} x^{8} + 84 \, a^{6} b^{9} x^{6} + 36 \, a^{7} b^{8} x^{4} + 9 \, a^{8} b^{7} x^{2} + a^{9} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 159, normalized size = 2.06 \[ -\frac {a^5+9\,a^4\,b\,x^2+36\,a^3\,b^2\,x^4+84\,a^2\,b^3\,x^6+126\,a\,b^4\,x^8+126\,b^5\,x^{10}}{1008\,a^9\,b^6+9072\,a^8\,b^7\,x^2+36288\,a^7\,b^8\,x^4+84672\,a^6\,b^9\,x^6+127008\,a^5\,b^{10}\,x^8+127008\,a^4\,b^{11}\,x^{10}+84672\,a^3\,b^{12}\,x^{12}+36288\,a^2\,b^{13}\,x^{14}+9072\,a\,b^{14}\,x^{16}+1008\,b^{15}\,x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.18, size = 167, normalized size = 2.17 \[ \frac {- a^{5} - 9 a^{4} b x^{2} - 36 a^{3} b^{2} x^{4} - 84 a^{2} b^{3} x^{6} - 126 a b^{4} x^{8} - 126 b^{5} x^{10}}{1008 a^{9} b^{6} + 9072 a^{8} b^{7} x^{2} + 36288 a^{7} b^{8} x^{4} + 84672 a^{6} b^{9} x^{6} + 127008 a^{5} b^{10} x^{8} + 127008 a^{4} b^{11} x^{10} + 84672 a^{3} b^{12} x^{12} + 36288 a^{2} b^{13} x^{14} + 9072 a b^{14} x^{16} + 1008 b^{15} x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________