Optimal. Leaf size=150 \[ a^2 A \log (x)+a^2 B x+\frac{1}{6} x^6 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{4} x^4 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac{1}{2} a x^2 (a C+2 A b)+\frac{1}{5} B x^5 \left (2 a c+b^2\right )+\frac{2}{3} a b B x^3+\frac{1}{8} c x^8 (A c+2 b C)+\frac{2}{7} b B c x^7+\frac{1}{9} B c^2 x^9+\frac{1}{10} c^2 C x^{10} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.106612, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {1628} \[ a^2 A \log (x)+a^2 B x+\frac{1}{6} x^6 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{4} x^4 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac{1}{2} a x^2 (a C+2 A b)+\frac{1}{5} B x^5 \left (2 a c+b^2\right )+\frac{2}{3} a b B x^3+\frac{1}{8} c x^8 (A c+2 b C)+\frac{2}{7} b B c x^7+\frac{1}{9} B c^2 x^9+\frac{1}{10} c^2 C x^{10} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1628
Rubi steps
\begin{align*} \int \frac{\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2}{x} \, dx &=\int \left (a^2 B+\frac{a^2 A}{x}+a (2 A b+a C) x+2 a b B x^2+\left (A \left (b^2+2 a c\right )+2 a b C\right ) x^3+B \left (b^2+2 a c\right ) x^4+\left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^5+2 b B c x^6+c (A c+2 b C) x^7+B c^2 x^8+c^2 C x^9\right ) \, dx\\ &=a^2 B x+\frac{1}{2} a (2 A b+a C) x^2+\frac{2}{3} a b B x^3+\frac{1}{4} \left (A \left (b^2+2 a c\right )+2 a b C\right ) x^4+\frac{1}{5} B \left (b^2+2 a c\right ) x^5+\frac{1}{6} \left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^6+\frac{2}{7} b B c x^7+\frac{1}{8} c (A c+2 b C) x^8+\frac{1}{9} B c^2 x^9+\frac{1}{10} c^2 C x^{10}+a^2 A \log (x)\\ \end{align*}
Mathematica [A] time = 0.039684, size = 150, normalized size = 1. \[ a^2 A \log (x)+a^2 B x+\frac{1}{6} x^6 \left (2 a c C+2 A b c+b^2 C\right )+\frac{1}{4} x^4 \left (2 a A c+2 a b C+A b^2\right )+\frac{1}{2} a x^2 (a C+2 A b)+\frac{1}{5} B x^5 \left (2 a c+b^2\right )+\frac{2}{3} a b B x^3+\frac{1}{8} c x^8 (A c+2 b C)+\frac{2}{7} b B c x^7+\frac{1}{9} B c^2 x^9+\frac{1}{10} c^2 C x^{10} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 149, normalized size = 1. \begin{align*}{\frac{{c}^{2}C{x}^{10}}{10}}+{\frac{B{c}^{2}{x}^{9}}{9}}+{\frac{A{x}^{8}{c}^{2}}{8}}+{\frac{C{x}^{8}bc}{4}}+{\frac{2\,bBc{x}^{7}}{7}}+{\frac{A{x}^{6}bc}{3}}+{\frac{C{x}^{6}ac}{3}}+{\frac{C{x}^{6}{b}^{2}}{6}}+{\frac{2\,B{x}^{5}ac}{5}}+{\frac{B{x}^{5}{b}^{2}}{5}}+{\frac{A{x}^{4}ac}{2}}+{\frac{A{x}^{4}{b}^{2}}{4}}+{\frac{C{x}^{4}ab}{2}}+{\frac{2\,abB{x}^{3}}{3}}+A{x}^{2}ab+{\frac{C{x}^{2}{a}^{2}}{2}}+{a}^{2}Bx+{a}^{2}A\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.955496, size = 186, normalized size = 1.24 \begin{align*} \frac{1}{10} \, C c^{2} x^{10} + \frac{1}{9} \, B c^{2} x^{9} + \frac{2}{7} \, B b c x^{7} + \frac{1}{8} \,{\left (2 \, C b c + A c^{2}\right )} x^{8} + \frac{1}{6} \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{6} + \frac{2}{3} \, B a b x^{3} + \frac{1}{5} \,{\left (B b^{2} + 2 \, B a c\right )} x^{5} + \frac{1}{4} \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} + B a^{2} x + A a^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (C a^{2} + 2 \, A a b\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.24675, size = 335, normalized size = 2.23 \begin{align*} \frac{1}{10} \, C c^{2} x^{10} + \frac{1}{9} \, B c^{2} x^{9} + \frac{2}{7} \, B b c x^{7} + \frac{1}{8} \,{\left (2 \, C b c + A c^{2}\right )} x^{8} + \frac{1}{6} \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{6} + \frac{2}{3} \, B a b x^{3} + \frac{1}{5} \,{\left (B b^{2} + 2 \, B a c\right )} x^{5} + \frac{1}{4} \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} + B a^{2} x + A a^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (C a^{2} + 2 \, A a b\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.499565, size = 156, normalized size = 1.04 \begin{align*} A a^{2} \log{\left (x \right )} + B a^{2} x + \frac{2 B a b x^{3}}{3} + \frac{2 B b c x^{7}}{7} + \frac{B c^{2} x^{9}}{9} + \frac{C c^{2} x^{10}}{10} + x^{8} \left (\frac{A c^{2}}{8} + \frac{C b c}{4}\right ) + x^{6} \left (\frac{A b c}{3} + \frac{C a c}{3} + \frac{C b^{2}}{6}\right ) + x^{5} \left (\frac{2 B a c}{5} + \frac{B b^{2}}{5}\right ) + x^{4} \left (\frac{A a c}{2} + \frac{A b^{2}}{4} + \frac{C a b}{2}\right ) + x^{2} \left (A a b + \frac{C a^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09492, size = 201, normalized size = 1.34 \begin{align*} \frac{1}{10} \, C c^{2} x^{10} + \frac{1}{9} \, B c^{2} x^{9} + \frac{1}{4} \, C b c x^{8} + \frac{1}{8} \, A c^{2} x^{8} + \frac{2}{7} \, B b c x^{7} + \frac{1}{6} \, C b^{2} x^{6} + \frac{1}{3} \, C a c x^{6} + \frac{1}{3} \, A b c x^{6} + \frac{1}{5} \, B b^{2} x^{5} + \frac{2}{5} \, B a c x^{5} + \frac{1}{2} \, C a b x^{4} + \frac{1}{4} \, A b^{2} x^{4} + \frac{1}{2} \, A a c x^{4} + \frac{2}{3} \, B a b x^{3} + \frac{1}{2} \, C a^{2} x^{2} + A a b x^{2} + B a^{2} x + A a^{2} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]