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 = 0.0240573, antiderivative size = 47, normalized size of antiderivative = 1., 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 = 0.0232191, 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]
________________________________________________________________________________________
Maple [A] time = 0.048, size = 42, normalized size = 0.9 \begin{align*}{c}^{2}x-{\frac{2\,ac+{b}^{2}}{3\,{x}^{3}}}-{\frac{2\,ab}{5\,{x}^{5}}}-{\frac{{a}^{2}}{7\,{x}^{7}}}-2\,{\frac{bc}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.983932, size = 59, normalized size = 1.26 \begin{align*} 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.45975, size = 113, normalized size = 2.4 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.873508, size = 44, normalized size = 0.94 \begin{align*} 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14199, size = 62, normalized size = 1.32 \begin{align*} 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]