Optimal. Leaf size=110 \[ \frac{3}{2} a^2 A c^2 x^4+2 a^3 A c x^2+a^4 A \log (x)+\frac{6}{5} a^2 B c^2 x^5+\frac{4}{3} a^3 B c x^3+a^4 B x+\frac{2}{3} a A c^3 x^6+\frac{4}{7} a B c^3 x^7+\frac{1}{8} A c^4 x^8+\frac{1}{9} B c^4 x^9 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.052258, antiderivative size = 110, 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 = {766} \[ \frac{3}{2} a^2 A c^2 x^4+2 a^3 A c x^2+a^4 A \log (x)+\frac{6}{5} a^2 B c^2 x^5+\frac{4}{3} a^3 B c x^3+a^4 B x+\frac{2}{3} a A c^3 x^6+\frac{4}{7} a B c^3 x^7+\frac{1}{8} A c^4 x^8+\frac{1}{9} B c^4 x^9 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 766
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^4}{x} \, dx &=\int \left (a^4 B+\frac{a^4 A}{x}+4 a^3 A c x+4 a^3 B c x^2+6 a^2 A c^2 x^3+6 a^2 B c^2 x^4+4 a A c^3 x^5+4 a B c^3 x^6+A c^4 x^7+B c^4 x^8\right ) \, dx\\ &=a^4 B x+2 a^3 A c x^2+\frac{4}{3} a^3 B c x^3+\frac{3}{2} a^2 A c^2 x^4+\frac{6}{5} a^2 B c^2 x^5+\frac{2}{3} a A c^3 x^6+\frac{4}{7} a B c^3 x^7+\frac{1}{8} A c^4 x^8+\frac{1}{9} B c^4 x^9+a^4 A \log (x)\\ \end{align*}
Mathematica [A] time = 0.0071966, size = 110, normalized size = 1. \[ \frac{3}{2} a^2 A c^2 x^4+2 a^3 A c x^2+a^4 A \log (x)+\frac{6}{5} a^2 B c^2 x^5+\frac{4}{3} a^3 B c x^3+a^4 B x+\frac{2}{3} a A c^3 x^6+\frac{4}{7} a B c^3 x^7+\frac{1}{8} A c^4 x^8+\frac{1}{9} B c^4 x^9 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 97, normalized size = 0.9 \begin{align*}{a}^{4}Bx+2\,{a}^{3}Ac{x}^{2}+{\frac{4\,{a}^{3}Bc{x}^{3}}{3}}+{\frac{3\,{a}^{2}A{c}^{2}{x}^{4}}{2}}+{\frac{6\,{a}^{2}B{c}^{2}{x}^{5}}{5}}+{\frac{2\,aA{c}^{3}{x}^{6}}{3}}+{\frac{4\,aB{c}^{3}{x}^{7}}{7}}+{\frac{A{c}^{4}{x}^{8}}{8}}+{\frac{B{c}^{4}{x}^{9}}{9}}+{a}^{4}A\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.08801, size = 130, normalized size = 1.18 \begin{align*} \frac{1}{9} \, B c^{4} x^{9} + \frac{1}{8} \, A c^{4} x^{8} + \frac{4}{7} \, B a c^{3} x^{7} + \frac{2}{3} \, A a c^{3} x^{6} + \frac{6}{5} \, B a^{2} c^{2} x^{5} + \frac{3}{2} \, A a^{2} c^{2} x^{4} + \frac{4}{3} \, B a^{3} c x^{3} + 2 \, A a^{3} c x^{2} + B a^{4} x + A a^{4} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.55654, size = 224, normalized size = 2.04 \begin{align*} \frac{1}{9} \, B c^{4} x^{9} + \frac{1}{8} \, A c^{4} x^{8} + \frac{4}{7} \, B a c^{3} x^{7} + \frac{2}{3} \, A a c^{3} x^{6} + \frac{6}{5} \, B a^{2} c^{2} x^{5} + \frac{3}{2} \, A a^{2} c^{2} x^{4} + \frac{4}{3} \, B a^{3} c x^{3} + 2 \, A a^{3} c x^{2} + B a^{4} x + A a^{4} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.423524, size = 117, normalized size = 1.06 \begin{align*} A a^{4} \log{\left (x \right )} + 2 A a^{3} c x^{2} + \frac{3 A a^{2} c^{2} x^{4}}{2} + \frac{2 A a c^{3} x^{6}}{3} + \frac{A c^{4} x^{8}}{8} + B a^{4} x + \frac{4 B a^{3} c x^{3}}{3} + \frac{6 B a^{2} c^{2} x^{5}}{5} + \frac{4 B a c^{3} x^{7}}{7} + \frac{B c^{4} x^{9}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18686, size = 131, normalized size = 1.19 \begin{align*} \frac{1}{9} \, B c^{4} x^{9} + \frac{1}{8} \, A c^{4} x^{8} + \frac{4}{7} \, B a c^{3} x^{7} + \frac{2}{3} \, A a c^{3} x^{6} + \frac{6}{5} \, B a^{2} c^{2} x^{5} + \frac{3}{2} \, A a^{2} c^{2} x^{4} + \frac{4}{3} \, B a^{3} c x^{3} + 2 \, A a^{3} c x^{2} + B a^{4} x + A a^{4} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]