Optimal. Leaf size=200 \[ a^2 b c x^3+\frac{1}{4} a^2 x^4 (a g+3 b d)+\frac{1}{5} a^2 x^5 (a h+3 b e)+a^3 c \log (x)+a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{2} a b^2 c x^6+\frac{1}{10} b^2 x^{10} (3 a g+b d)+\frac{1}{11} b^2 x^{11} (3 a h+b e)+\frac{3}{7} a b x^7 (a g+b d)+\frac{3}{8} a b x^8 (a h+b e)+\frac{f \left (a+b x^3\right )^4}{12 b}+\frac{1}{9} b^3 c x^9+\frac{1}{13} b^3 g x^{13}+\frac{1}{14} b^3 h x^{14} \]
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Rubi [A] time = 0.145609, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {1583, 1820} \[ a^2 b c x^3+\frac{1}{4} a^2 x^4 (a g+3 b d)+\frac{1}{5} a^2 x^5 (a h+3 b e)+a^3 c \log (x)+a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{2} a b^2 c x^6+\frac{1}{10} b^2 x^{10} (3 a g+b d)+\frac{1}{11} b^2 x^{11} (3 a h+b e)+\frac{3}{7} a b x^7 (a g+b d)+\frac{3}{8} a b x^8 (a h+b e)+\frac{f \left (a+b x^3\right )^4}{12 b}+\frac{1}{9} b^3 c x^9+\frac{1}{13} b^3 g x^{13}+\frac{1}{14} b^3 h x^{14} \]
Antiderivative was successfully verified.
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Rule 1583
Rule 1820
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{x} \, dx &=\frac{f \left (a+b x^3\right )^4}{12 b}+\int \frac{\left (a+b x^3\right )^3 \left (c+d x+e x^2+g x^4+h x^5\right )}{x} \, dx\\ &=\frac{f \left (a+b x^3\right )^4}{12 b}+\int \left (a^3 d+\frac{a^3 c}{x}+a^3 e x+3 a^2 b c x^2+a^2 (3 b d+a g) x^3+a^2 (3 b e+a h) x^4+3 a b^2 c x^5+3 a b (b d+a g) x^6+3 a b (b e+a h) x^7+b^3 c x^8+b^2 (b d+3 a g) x^9+b^2 (b e+3 a h) x^{10}+b^3 g x^{12}+b^3 h x^{13}\right ) \, dx\\ &=a^3 d x+\frac{1}{2} a^3 e x^2+a^2 b c x^3+\frac{1}{4} a^2 (3 b d+a g) x^4+\frac{1}{5} a^2 (3 b e+a h) x^5+\frac{1}{2} a b^2 c x^6+\frac{3}{7} a b (b d+a g) x^7+\frac{3}{8} a b (b e+a h) x^8+\frac{1}{9} b^3 c x^9+\frac{1}{10} b^2 (b d+3 a g) x^{10}+\frac{1}{11} b^2 (b e+3 a h) x^{11}+\frac{1}{13} b^3 g x^{13}+\frac{1}{14} b^3 h x^{14}+\frac{f \left (a+b x^3\right )^4}{12 b}+a^3 c \log (x)\\ \end{align*}
Mathematica [A] time = 0.0777477, size = 214, normalized size = 1.07 \[ \frac{1}{3} a^2 x^3 (a f+3 b c)+\frac{1}{4} a^2 x^4 (a g+3 b d)+\frac{1}{5} a^2 x^5 (a h+3 b e)+a^3 c \log (x)+a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{9} b^2 x^9 (3 a f+b c)+\frac{1}{10} b^2 x^{10} (3 a g+b d)+\frac{1}{11} b^2 x^{11} (3 a h+b e)+\frac{1}{2} a b x^6 (a f+b c)+\frac{3}{7} a b x^7 (a g+b d)+\frac{3}{8} a b x^8 (a h+b e)+\frac{1}{12} b^3 f x^{12}+\frac{1}{13} b^3 g x^{13}+\frac{1}{14} b^3 h x^{14} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 224, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}h{x}^{14}}{14}}+{\frac{{b}^{3}g{x}^{13}}{13}}+{\frac{f{x}^{12}{b}^{3}}{12}}+{\frac{3\,{x}^{11}a{b}^{2}h}{11}}+{\frac{{b}^{3}e{x}^{11}}{11}}+{\frac{3\,{x}^{10}a{b}^{2}g}{10}}+{\frac{{b}^{3}d{x}^{10}}{10}}+{\frac{{x}^{9}a{b}^{2}f}{3}}+{\frac{{b}^{3}c{x}^{9}}{9}}+{\frac{3\,{x}^{8}{a}^{2}bh}{8}}+{\frac{3\,a{b}^{2}e{x}^{8}}{8}}+{\frac{3\,{x}^{7}{a}^{2}bg}{7}}+{\frac{3\,a{b}^{2}d{x}^{7}}{7}}+{\frac{{x}^{6}{a}^{2}bf}{2}}+{\frac{a{b}^{2}c{x}^{6}}{2}}+{\frac{{x}^{5}{a}^{3}h}{5}}+{\frac{3\,{a}^{2}be{x}^{5}}{5}}+{\frac{{x}^{4}{a}^{3}g}{4}}+{\frac{3\,{a}^{2}bd{x}^{4}}{4}}+{\frac{{a}^{3}f{x}^{3}}{3}}+{a}^{2}bc{x}^{3}+{\frac{{a}^{3}e{x}^{2}}{2}}+{a}^{3}dx+{a}^{3}c\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960838, size = 286, normalized size = 1.43 \begin{align*} \frac{1}{14} \, b^{3} h x^{14} + \frac{1}{13} \, b^{3} g x^{13} + \frac{1}{12} \, b^{3} f x^{12} + \frac{1}{11} \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + \frac{1}{10} \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + \frac{1}{9} \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + \frac{3}{8} \,{\left (a b^{2} e + a^{2} b h\right )} x^{8} + \frac{3}{7} \,{\left (a b^{2} d + a^{2} b g\right )} x^{7} + \frac{1}{2} \,{\left (a b^{2} c + a^{2} b f\right )} x^{6} + \frac{1}{2} \, a^{3} e x^{2} + \frac{1}{5} \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} + a^{3} d x + \frac{1}{4} \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} + a^{3} c \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.997019, size = 497, normalized size = 2.48 \begin{align*} \frac{1}{14} \, b^{3} h x^{14} + \frac{1}{13} \, b^{3} g x^{13} + \frac{1}{12} \, b^{3} f x^{12} + \frac{1}{11} \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + \frac{1}{10} \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + \frac{1}{9} \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + \frac{3}{8} \,{\left (a b^{2} e + a^{2} b h\right )} x^{8} + \frac{3}{7} \,{\left (a b^{2} d + a^{2} b g\right )} x^{7} + \frac{1}{2} \,{\left (a b^{2} c + a^{2} b f\right )} x^{6} + \frac{1}{2} \, a^{3} e x^{2} + \frac{1}{5} \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} + a^{3} d x + \frac{1}{4} \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} + a^{3} c \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.604787, size = 240, normalized size = 1.2 \begin{align*} a^{3} c \log{\left (x \right )} + a^{3} d x + \frac{a^{3} e x^{2}}{2} + \frac{b^{3} f x^{12}}{12} + \frac{b^{3} g x^{13}}{13} + \frac{b^{3} h x^{14}}{14} + x^{11} \left (\frac{3 a b^{2} h}{11} + \frac{b^{3} e}{11}\right ) + x^{10} \left (\frac{3 a b^{2} g}{10} + \frac{b^{3} d}{10}\right ) + x^{9} \left (\frac{a b^{2} f}{3} + \frac{b^{3} c}{9}\right ) + x^{8} \left (\frac{3 a^{2} b h}{8} + \frac{3 a b^{2} e}{8}\right ) + x^{7} \left (\frac{3 a^{2} b g}{7} + \frac{3 a b^{2} d}{7}\right ) + x^{6} \left (\frac{a^{2} b f}{2} + \frac{a b^{2} c}{2}\right ) + x^{5} \left (\frac{a^{3} h}{5} + \frac{3 a^{2} b e}{5}\right ) + x^{4} \left (\frac{a^{3} g}{4} + \frac{3 a^{2} b d}{4}\right ) + x^{3} \left (\frac{a^{3} f}{3} + a^{2} b c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06867, size = 308, normalized size = 1.54 \begin{align*} \frac{1}{14} \, b^{3} h x^{14} + \frac{1}{13} \, b^{3} g x^{13} + \frac{1}{12} \, b^{3} f x^{12} + \frac{3}{11} \, a b^{2} h x^{11} + \frac{1}{11} \, b^{3} x^{11} e + \frac{1}{10} \, b^{3} d x^{10} + \frac{3}{10} \, a b^{2} g x^{10} + \frac{1}{9} \, b^{3} c x^{9} + \frac{1}{3} \, a b^{2} f x^{9} + \frac{3}{8} \, a^{2} b h x^{8} + \frac{3}{8} \, a b^{2} x^{8} e + \frac{3}{7} \, a b^{2} d x^{7} + \frac{3}{7} \, a^{2} b g x^{7} + \frac{1}{2} \, a b^{2} c x^{6} + \frac{1}{2} \, a^{2} b f x^{6} + \frac{1}{5} \, a^{3} h x^{5} + \frac{3}{5} \, a^{2} b x^{5} e + \frac{3}{4} \, a^{2} b d x^{4} + \frac{1}{4} \, a^{3} g x^{4} + a^{2} b c x^{3} + \frac{1}{3} \, a^{3} f x^{3} + \frac{1}{2} \, a^{3} x^{2} e + a^{3} d x + a^{3} c \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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