Optimal. Leaf size=47 \[ -\frac {a^2}{3 x^3}+x \left (2 a c+b^2\right )-\frac {2 a b}{x}+\frac {2}{3} b c x^3+\frac {c^2 x^5}{5} \]
[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} \[ -\frac {a^2}{3 x^3}+x \left (2 a c+b^2\right )-\frac {2 a b}{x}+\frac {2}{3} b c x^3+\frac {c^2 x^5}{5} \]
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^4} \, dx &=\int \left (b^2 \left (1+\frac {2 a c}{b^2}\right )+\frac {a^2}{x^4}+\frac {2 a b}{x^2}+2 b c x^2+c^2 x^4\right ) \, dx\\ &=-\frac {a^2}{3 x^3}-\frac {2 a b}{x}+\left (b^2+2 a c\right ) x+\frac {2}{3} b c x^3+\frac {c^2 x^5}{5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 47, normalized size = 1.00 \[ -\frac {a^2}{3 x^3}+x \left (2 a c+b^2\right )-\frac {2 a b}{x}+\frac {2}{3} b c x^3+\frac {c^2 x^5}{5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.93, size = 46, normalized size = 0.98 \[ \frac {3 \, c^{2} x^{8} + 10 \, b c x^{6} + 15 \, {\left (b^{2} + 2 \, a c\right )} x^{4} - 30 \, a b x^{2} - 5 \, a^{2}}{15 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 42, normalized size = 0.89 \[ \frac {1}{5} \, c^{2} x^{5} + \frac {2}{3} \, b c x^{3} + b^{2} x + 2 \, a c x - \frac {6 \, a b x^{2} + a^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 42, normalized size = 0.89 \[ \frac {c^{2} x^{5}}{5}+\frac {2 b c \,x^{3}}{3}+2 a c x +b^{2} x -\frac {2 a b}{x}-\frac {a^{2}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.37, size = 42, normalized size = 0.89 \[ \frac {1}{5} \, c^{2} x^{5} + \frac {2}{3} \, b c x^{3} + {\left (b^{2} + 2 \, a c\right )} x - \frac {6 \, a b x^{2} + a^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 44, normalized size = 0.94 \[ x\,\left (b^2+2\,a\,c\right )-\frac {\frac {a^2}{3}+2\,b\,a\,x^2}{x^3}+\frac {c^2\,x^5}{5}+\frac {2\,b\,c\,x^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.18, size = 46, normalized size = 0.98 \[ \frac {2 b c x^{3}}{3} + \frac {c^{2} x^{5}}{5} + x \left (2 a c + b^{2}\right ) + \frac {- a^{2} - 6 a b x^{2}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________