Optimal. Leaf size=50 \[ \frac{3}{2} a^2 b^2 x^4+2 a^3 b x^2+a^4 \log (x)+\frac{2}{3} a b^3 x^6+\frac{b^4 x^8}{8} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0337603, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 266, 43} \[ \frac{3}{2} a^2 b^2 x^4+2 a^3 b x^2+a^4 \log (x)+\frac{2}{3} a b^3 x^6+\frac{b^4 x^8}{8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^4}{x} \, dx}{b^4}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^4}{x} \, dx,x,x^2\right )}{2 b^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a^3 b^5+\frac{a^4 b^4}{x}+6 a^2 b^6 x+4 a b^7 x^2+b^8 x^3\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=2 a^3 b x^2+\frac{3}{2} a^2 b^2 x^4+\frac{2}{3} a b^3 x^6+\frac{b^4 x^8}{8}+a^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0041856, size = 50, normalized size = 1. \[ \frac{3}{2} a^2 b^2 x^4+2 a^3 b x^2+a^4 \log (x)+\frac{2}{3} a b^3 x^6+\frac{b^4 x^8}{8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.041, size = 45, normalized size = 0.9 \begin{align*} 2\,{a}^{3}b{x}^{2}+{\frac{3\,{a}^{2}{b}^{2}{x}^{4}}{2}}+{\frac{2\,a{b}^{3}{x}^{6}}{3}}+{\frac{{b}^{4}{x}^{8}}{8}}+{a}^{4}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.976765, size = 63, normalized size = 1.26 \begin{align*} \frac{1}{8} \, b^{4} x^{8} + \frac{2}{3} \, a b^{3} x^{6} + \frac{3}{2} \, a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + \frac{1}{2} \, a^{4} \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.67549, size = 100, normalized size = 2. \begin{align*} \frac{1}{8} \, b^{4} x^{8} + \frac{2}{3} \, a b^{3} x^{6} + \frac{3}{2} \, a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.274536, size = 49, normalized size = 0.98 \begin{align*} a^{4} \log{\left (x \right )} + 2 a^{3} b x^{2} + \frac{3 a^{2} b^{2} x^{4}}{2} + \frac{2 a b^{3} x^{6}}{3} + \frac{b^{4} x^{8}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11139, size = 63, normalized size = 1.26 \begin{align*} \frac{1}{8} \, b^{4} x^{8} + \frac{2}{3} \, a b^{3} x^{6} + \frac{3}{2} \, a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + \frac{1}{2} \, a^{4} \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]