Optimal. Leaf size=166 \[ a^2 b^2 c x^6+\frac{6}{7} a^2 b^2 d x^7+\frac{3}{4} a^2 b^2 e x^8+\frac{4}{3} a^3 b c x^3+a^3 b d x^4+\frac{4}{5} a^3 b e x^5+a^4 c \log (x)+a^4 d x+\frac{1}{2} a^4 e x^2+\frac{4}{9} a b^3 c x^9+\frac{2}{5} a b^3 d x^{10}+\frac{4}{11} a b^3 e x^{11}+\frac{1}{12} b^4 c x^{12}+\frac{1}{13} b^4 d x^{13}+\frac{1}{14} b^4 e x^{14} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.108534, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1628} \[ a^2 b^2 c x^6+\frac{6}{7} a^2 b^2 d x^7+\frac{3}{4} a^2 b^2 e x^8+\frac{4}{3} a^3 b c x^3+a^3 b d x^4+\frac{4}{5} a^3 b e x^5+a^4 c \log (x)+a^4 d x+\frac{1}{2} a^4 e x^2+\frac{4}{9} a b^3 c x^9+\frac{2}{5} a b^3 d x^{10}+\frac{4}{11} a b^3 e x^{11}+\frac{1}{12} b^4 c x^{12}+\frac{1}{13} b^4 d x^{13}+\frac{1}{14} b^4 e x^{14} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1628
Rubi steps
\begin{align*} \int \frac{\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x} \, dx &=\int \left (a^4 d+\frac{a^4 c}{x}+a^4 e x+4 a^3 b c x^2+4 a^3 b d x^3+4 a^3 b e x^4+6 a^2 b^2 c x^5+6 a^2 b^2 d x^6+6 a^2 b^2 e x^7+4 a b^3 c x^8+4 a b^3 d x^9+4 a b^3 e x^{10}+b^4 c x^{11}+b^4 d x^{12}+b^4 e x^{13}\right ) \, dx\\ &=a^4 d x+\frac{1}{2} a^4 e x^2+\frac{4}{3} a^3 b c x^3+a^3 b d x^4+\frac{4}{5} a^3 b e x^5+a^2 b^2 c x^6+\frac{6}{7} a^2 b^2 d x^7+\frac{3}{4} a^2 b^2 e x^8+\frac{4}{9} a b^3 c x^9+\frac{2}{5} a b^3 d x^{10}+\frac{4}{11} a b^3 e x^{11}+\frac{1}{12} b^4 c x^{12}+\frac{1}{13} b^4 d x^{13}+\frac{1}{14} b^4 e x^{14}+a^4 c \log (x)\\ \end{align*}
Mathematica [A] time = 0.0088974, size = 166, normalized size = 1. \[ a^2 b^2 c x^6+\frac{6}{7} a^2 b^2 d x^7+\frac{3}{4} a^2 b^2 e x^8+\frac{4}{3} a^3 b c x^3+a^3 b d x^4+\frac{4}{5} a^3 b e x^5+a^4 c \log (x)+a^4 d x+\frac{1}{2} a^4 e x^2+\frac{4}{9} a b^3 c x^9+\frac{2}{5} a b^3 d x^{10}+\frac{4}{11} a b^3 e x^{11}+\frac{1}{12} b^4 c x^{12}+\frac{1}{13} b^4 d x^{13}+\frac{1}{14} b^4 e x^{14} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 145, normalized size = 0.9 \begin{align*}{a}^{4}dx+{\frac{{a}^{4}e{x}^{2}}{2}}+{\frac{4\,{a}^{3}bc{x}^{3}}{3}}+{a}^{3}bd{x}^{4}+{\frac{4\,{a}^{3}be{x}^{5}}{5}}+{a}^{2}{b}^{2}c{x}^{6}+{\frac{6\,{a}^{2}{b}^{2}d{x}^{7}}{7}}+{\frac{3\,{a}^{2}{b}^{2}e{x}^{8}}{4}}+{\frac{4\,a{b}^{3}c{x}^{9}}{9}}+{\frac{2\,a{b}^{3}d{x}^{10}}{5}}+{\frac{4\,a{b}^{3}e{x}^{11}}{11}}+{\frac{{b}^{4}c{x}^{12}}{12}}+{\frac{{b}^{4}d{x}^{13}}{13}}+{\frac{{b}^{4}e{x}^{14}}{14}}+{a}^{4}c\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.936197, size = 194, normalized size = 1.17 \begin{align*} \frac{1}{14} \, b^{4} e x^{14} + \frac{1}{13} \, b^{4} d x^{13} + \frac{1}{12} \, b^{4} c x^{12} + \frac{4}{11} \, a b^{3} e x^{11} + \frac{2}{5} \, a b^{3} d x^{10} + \frac{4}{9} \, a b^{3} c x^{9} + \frac{3}{4} \, a^{2} b^{2} e x^{8} + \frac{6}{7} \, a^{2} b^{2} d x^{7} + a^{2} b^{2} c x^{6} + \frac{4}{5} \, a^{3} b e x^{5} + a^{3} b d x^{4} + \frac{4}{3} \, a^{3} b c x^{3} + \frac{1}{2} \, a^{4} e x^{2} + a^{4} d x + a^{4} c \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.46817, size = 347, normalized size = 2.09 \begin{align*} \frac{1}{14} \, b^{4} e x^{14} + \frac{1}{13} \, b^{4} d x^{13} + \frac{1}{12} \, b^{4} c x^{12} + \frac{4}{11} \, a b^{3} e x^{11} + \frac{2}{5} \, a b^{3} d x^{10} + \frac{4}{9} \, a b^{3} c x^{9} + \frac{3}{4} \, a^{2} b^{2} e x^{8} + \frac{6}{7} \, a^{2} b^{2} d x^{7} + a^{2} b^{2} c x^{6} + \frac{4}{5} \, a^{3} b e x^{5} + a^{3} b d x^{4} + \frac{4}{3} \, a^{3} b c x^{3} + \frac{1}{2} \, a^{4} e x^{2} + a^{4} d x + a^{4} c \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.456358, size = 175, normalized size = 1.05 \begin{align*} a^{4} c \log{\left (x \right )} + a^{4} d x + \frac{a^{4} e x^{2}}{2} + \frac{4 a^{3} b c x^{3}}{3} + a^{3} b d x^{4} + \frac{4 a^{3} b e x^{5}}{5} + a^{2} b^{2} c x^{6} + \frac{6 a^{2} b^{2} d x^{7}}{7} + \frac{3 a^{2} b^{2} e x^{8}}{4} + \frac{4 a b^{3} c x^{9}}{9} + \frac{2 a b^{3} d x^{10}}{5} + \frac{4 a b^{3} e x^{11}}{11} + \frac{b^{4} c x^{12}}{12} + \frac{b^{4} d x^{13}}{13} + \frac{b^{4} e x^{14}}{14} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.06076, size = 203, normalized size = 1.22 \begin{align*} \frac{1}{14} \, b^{4} x^{14} e + \frac{1}{13} \, b^{4} d x^{13} + \frac{1}{12} \, b^{4} c x^{12} + \frac{4}{11} \, a b^{3} x^{11} e + \frac{2}{5} \, a b^{3} d x^{10} + \frac{4}{9} \, a b^{3} c x^{9} + \frac{3}{4} \, a^{2} b^{2} x^{8} e + \frac{6}{7} \, a^{2} b^{2} d x^{7} + a^{2} b^{2} c x^{6} + \frac{4}{5} \, a^{3} b x^{5} e + a^{3} b d x^{4} + \frac{4}{3} \, a^{3} b c x^{3} + \frac{1}{2} \, a^{4} x^{2} e + a^{4} d x + a^{4} c \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]