Optimal. Leaf size=198 \[ a^2 x (a f+3 b c)+\frac{1}{2} a^2 x^2 (a g+3 b d)+a^2 b e x^3-\frac{a^3 c}{2 x^2}-\frac{a^3 d}{x}+a^3 e \log (x)+\frac{1}{7} b^2 x^7 (3 a f+b c)+\frac{1}{8} b^2 x^8 (3 a g+b d)+\frac{1}{2} a b^2 e x^6+\frac{3}{4} a b x^4 (a f+b c)+\frac{3}{5} a b x^5 (a g+b d)+\frac{h \left (a+b x^3\right )^4}{12 b}+\frac{1}{9} b^3 e x^9+\frac{1}{10} b^3 f x^{10}+\frac{1}{11} b^3 g x^{11} \]
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Rubi [A] time = 0.196774, antiderivative size = 198, 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} \[ a^2 x (a f+3 b c)+\frac{1}{2} a^2 x^2 (a g+3 b d)+a^2 b e x^3-\frac{a^3 c}{2 x^2}-\frac{a^3 d}{x}+a^3 e \log (x)+\frac{1}{7} b^2 x^7 (3 a f+b c)+\frac{1}{8} b^2 x^8 (3 a g+b d)+\frac{1}{2} a b^2 e x^6+\frac{3}{4} a b x^4 (a f+b c)+\frac{3}{5} a b x^5 (a g+b d)+\frac{h \left (a+b x^3\right )^4}{12 b}+\frac{1}{9} b^3 e x^9+\frac{1}{10} b^3 f x^{10}+\frac{1}{11} b^3 g x^{11} \]
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^3} \, dx &=\frac{h \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+f x^3+g x^4\right )}{x^3} \, dx\\ &=\frac{h \left (a+b x^3\right )^4}{12 b}+\int \left (a^2 (3 b c+a f)+\frac{a^3 c}{x^3}+\frac{a^3 d}{x^2}+\frac{a^3 e}{x}+a^2 (3 b d+a g) x+3 a^2 b e x^2+3 a b (b c+a f) x^3+3 a b (b d+a g) x^4+3 a b^2 e x^5+b^2 (b c+3 a f) x^6+b^2 (b d+3 a g) x^7+b^3 e x^8+b^3 f x^9+b^3 g x^{10}\right ) \, dx\\ &=-\frac{a^3 c}{2 x^2}-\frac{a^3 d}{x}+a^2 (3 b c+a f) x+\frac{1}{2} a^2 (3 b d+a g) x^2+a^2 b e x^3+\frac{3}{4} a b (b c+a f) x^4+\frac{3}{5} a b (b d+a g) x^5+\frac{1}{2} a b^2 e x^6+\frac{1}{7} b^2 (b c+3 a f) x^7+\frac{1}{8} b^2 (b d+3 a g) x^8+\frac{1}{9} b^3 e x^9+\frac{1}{10} b^3 f x^{10}+\frac{1}{11} b^3 g x^{11}+\frac{h \left (a+b x^3\right )^4}{12 b}+a^3 e \log (x)\\ \end{align*}
Mathematica [A] time = 0.117362, size = 174, normalized size = 0.88 \[ \frac{1}{20} a^2 b x \left (60 c+x \left (30 d+x \left (20 e+15 f x+12 g x^2+10 h x^3\right )\right )\right )+\frac{a^3 \left (-3 c-6 d x+x^3 \left (6 f+3 g x+2 h x^2\right )\right )}{6 x^2}+a^3 e \log (x)+\frac{1}{840} a b^2 x^4 (630 c+x (504 d+5 x (84 e+x (72 f+7 x (9 g+8 h x)))))+\frac{b^3 x^7 \left (3960 c+7 x \left (495 d+440 e x+6 x^2 \left (66 f+60 g x+55 h x^2\right )\right )\right )}{27720} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 222, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}h{x}^{12}}{12}}+{\frac{{b}^{3}g{x}^{11}}{11}}+{\frac{{b}^{3}f{x}^{10}}{10}}+{\frac{{x}^{9}a{b}^{2}h}{3}}+{\frac{{b}^{3}e{x}^{9}}{9}}+{\frac{3\,{x}^{8}a{b}^{2}g}{8}}+{\frac{{b}^{3}d{x}^{8}}{8}}+{\frac{3\,{x}^{7}a{b}^{2}f}{7}}+{\frac{{b}^{3}c{x}^{7}}{7}}+{\frac{{x}^{6}{a}^{2}bh}{2}}+{\frac{a{b}^{2}e{x}^{6}}{2}}+{\frac{3\,{x}^{5}{a}^{2}bg}{5}}+{\frac{3\,a{b}^{2}d{x}^{5}}{5}}+{\frac{3\,{x}^{4}{a}^{2}bf}{4}}+{\frac{3\,a{b}^{2}c{x}^{4}}{4}}+{\frac{{x}^{3}{a}^{3}h}{3}}+{a}^{2}be{x}^{3}+{\frac{{x}^{2}{a}^{3}g}{2}}+{\frac{3\,{a}^{2}bd{x}^{2}}{2}}+{a}^{3}fx+3\,{a}^{2}bcx+{a}^{3}e\ln \left ( x \right ) -{\frac{{a}^{3}c}{2\,{x}^{2}}}-{\frac{{a}^{3}d}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.951091, size = 286, normalized size = 1.44 \begin{align*} \frac{1}{12} \, b^{3} h x^{12} + \frac{1}{11} \, b^{3} g x^{11} + \frac{1}{10} \, b^{3} f x^{10} + \frac{1}{9} \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{9} + \frac{1}{8} \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{8} + \frac{1}{7} \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{7} + \frac{1}{2} \,{\left (a b^{2} e + a^{2} b h\right )} x^{6} + \frac{3}{5} \,{\left (a b^{2} d + a^{2} b g\right )} x^{5} + \frac{3}{4} \,{\left (a b^{2} c + a^{2} b f\right )} x^{4} + a^{3} e \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{2} +{\left (3 \, a^{2} b c + a^{3} f\right )} x - \frac{2 \, a^{3} d x + a^{3} c}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.01098, size = 551, normalized size = 2.78 \begin{align*} \frac{2310 \, b^{3} h x^{14} + 2520 \, b^{3} g x^{13} + 2772 \, b^{3} f x^{12} + 3080 \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + 3465 \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + 3960 \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + 13860 \,{\left (a b^{2} e + a^{2} b h\right )} x^{8} + 16632 \,{\left (a b^{2} d + a^{2} b g\right )} x^{7} + 20790 \,{\left (a b^{2} c + a^{2} b f\right )} x^{6} + 27720 \, a^{3} e x^{2} \log \left (x\right ) + 9240 \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} - 27720 \, a^{3} d x + 13860 \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} - 13860 \, a^{3} c + 27720 \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{3}}{27720 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.716463, size = 236, normalized size = 1.19 \begin{align*} a^{3} e \log{\left (x \right )} + \frac{b^{3} f x^{10}}{10} + \frac{b^{3} g x^{11}}{11} + \frac{b^{3} h x^{12}}{12} + x^{9} \left (\frac{a b^{2} h}{3} + \frac{b^{3} e}{9}\right ) + x^{8} \left (\frac{3 a b^{2} g}{8} + \frac{b^{3} d}{8}\right ) + x^{7} \left (\frac{3 a b^{2} f}{7} + \frac{b^{3} c}{7}\right ) + x^{6} \left (\frac{a^{2} b h}{2} + \frac{a b^{2} e}{2}\right ) + x^{5} \left (\frac{3 a^{2} b g}{5} + \frac{3 a b^{2} d}{5}\right ) + x^{4} \left (\frac{3 a^{2} b f}{4} + \frac{3 a b^{2} c}{4}\right ) + x^{3} \left (\frac{a^{3} h}{3} + a^{2} b e\right ) + x^{2} \left (\frac{a^{3} g}{2} + \frac{3 a^{2} b d}{2}\right ) + x \left (a^{3} f + 3 a^{2} b c\right ) - \frac{a^{3} c + 2 a^{3} d x}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06294, size = 305, normalized size = 1.54 \begin{align*} \frac{1}{12} \, b^{3} h x^{12} + \frac{1}{11} \, b^{3} g x^{11} + \frac{1}{10} \, b^{3} f x^{10} + \frac{1}{3} \, a b^{2} h x^{9} + \frac{1}{9} \, b^{3} x^{9} e + \frac{1}{8} \, b^{3} d x^{8} + \frac{3}{8} \, a b^{2} g x^{8} + \frac{1}{7} \, b^{3} c x^{7} + \frac{3}{7} \, a b^{2} f x^{7} + \frac{1}{2} \, a^{2} b h x^{6} + \frac{1}{2} \, a b^{2} x^{6} e + \frac{3}{5} \, a b^{2} d x^{5} + \frac{3}{5} \, a^{2} b g x^{5} + \frac{3}{4} \, a b^{2} c x^{4} + \frac{3}{4} \, a^{2} b f x^{4} + \frac{1}{3} \, a^{3} h x^{3} + a^{2} b x^{3} e + \frac{3}{2} \, a^{2} b d x^{2} + \frac{1}{2} \, a^{3} g x^{2} + 3 \, a^{2} b c x + a^{3} f x + a^{3} e \log \left ({\left | x \right |}\right ) - \frac{2 \, a^{3} d x + a^{3} c}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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