Optimal. Leaf size=54 \[ -\frac {a^2}{11 x^{11}}-\frac {2 a c+b^2}{7 x^7}-\frac {2 a b}{9 x^9}-\frac {2 b c}{5 x^5}-\frac {c^2}{3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1108} \[ -\frac {a^2}{11 x^{11}}-\frac {2 a c+b^2}{7 x^7}-\frac {2 a b}{9 x^9}-\frac {2 b c}{5 x^5}-\frac {c^2}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1108
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^2}{x^{12}} \, dx &=\int \left (\frac {a^2}{x^{12}}+\frac {2 a b}{x^{10}}+\frac {b^2+2 a c}{x^8}+\frac {2 b c}{x^6}+\frac {c^2}{x^4}\right ) \, dx\\ &=-\frac {a^2}{11 x^{11}}-\frac {2 a b}{9 x^9}-\frac {b^2+2 a c}{7 x^7}-\frac {2 b c}{5 x^5}-\frac {c^2}{3 x^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 56, normalized size = 1.04 \[ -\frac {a^2}{11 x^{11}}+\frac {-2 a c-b^2}{7 x^7}-\frac {2 a b}{9 x^9}-\frac {2 b c}{5 x^5}-\frac {c^2}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.80, size = 46, normalized size = 0.85 \[ -\frac {1155 \, c^{2} x^{8} + 1386 \, b c x^{6} + 495 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 770 \, a b x^{2} + 315 \, a^{2}}{3465 \, x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 48, normalized size = 0.89 \[ -\frac {1155 \, c^{2} x^{8} + 1386 \, b c x^{6} + 495 \, b^{2} x^{4} + 990 \, a c x^{4} + 770 \, a b x^{2} + 315 \, a^{2}}{3465 \, x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 45, normalized size = 0.83 \[ -\frac {c^{2}}{3 x^{3}}-\frac {2 b c}{5 x^{5}}-\frac {2 a b}{9 x^{9}}-\frac {2 a c +b^{2}}{7 x^{7}}-\frac {a^{2}}{11 x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.38, size = 46, normalized size = 0.85 \[ -\frac {1155 \, c^{2} x^{8} + 1386 \, b c x^{6} + 495 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 770 \, a b x^{2} + 315 \, a^{2}}{3465 \, x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.16, size = 47, normalized size = 0.87 \[ -\frac {\frac {a^2}{11}+x^4\,\left (\frac {b^2}{7}+\frac {2\,a\,c}{7}\right )+\frac {c^2\,x^8}{3}+\frac {2\,a\,b\,x^2}{9}+\frac {2\,b\,c\,x^6}{5}}{x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.96, size = 49, normalized size = 0.91 \[ \frac {- 315 a^{2} - 770 a b x^{2} - 1386 b c x^{6} - 1155 c^{2} x^{8} + x^{4} \left (- 990 a c - 495 b^{2}\right )}{3465 x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________