Optimal. Leaf size=47 \[ -\frac {a^2}{7 x^7}-\frac {2 a c+b^2}{3 x^3}-\frac {2 a b}{5 x^5}-\frac {2 b c}{x}+c^2 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 47, 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} \[ -\frac {a^2}{7 x^7}-\frac {2 a c+b^2}{3 x^3}-\frac {2 a b}{5 x^5}-\frac {2 b c}{x}+c^2 x \]
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^8} \, dx &=\int \left (c^2+\frac {a^2}{x^8}+\frac {2 a b}{x^6}+\frac {b^2+2 a c}{x^4}+\frac {2 b c}{x^2}\right ) \, dx\\ &=-\frac {a^2}{7 x^7}-\frac {2 a b}{5 x^5}-\frac {b^2+2 a c}{3 x^3}-\frac {2 b c}{x}+c^2 x\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 49, normalized size = 1.04 \[ -\frac {a^2}{7 x^7}+\frac {-2 a c-b^2}{3 x^3}-\frac {2 a b}{5 x^5}-\frac {2 b c}{x}+c^2 x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.93, size = 46, normalized size = 0.98 \[ \frac {105 \, c^{2} x^{8} - 210 \, b c x^{6} - 35 \, {\left (b^{2} + 2 \, a c\right )} x^{4} - 42 \, a b x^{2} - 15 \, a^{2}}{105 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 46, normalized size = 0.98 \[ c^{2} x - \frac {210 \, b c x^{6} + 35 \, b^{2} x^{4} + 70 \, a c x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 42, normalized size = 0.89 \[ c^{2} x -\frac {2 b c}{x}-\frac {2 a b}{5 x^{5}}-\frac {2 a c +b^{2}}{3 x^{3}}-\frac {a^{2}}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.34, size = 44, normalized size = 0.94 \[ c^{2} x - \frac {210 \, b c x^{6} + 35 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.17, size = 45, normalized size = 0.96 \[ c^2\,x-\frac {\frac {a^2}{7}+x^4\,\left (\frac {b^2}{3}+\frac {2\,a\,c}{3}\right )+\frac {2\,a\,b\,x^2}{5}+2\,b\,c\,x^6}{x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.76, size = 46, normalized size = 0.98 \[ c^{2} x + \frac {- 15 a^{2} - 42 a b x^{2} - 210 b c x^{6} + x^{4} \left (- 70 a c - 35 b^{2}\right )}{105 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________