Optimal. Leaf size=209 \[ -\frac{a^2 (a f+3 b c)}{x}+a^2 \log (x) (a g+3 b d)+a^2 x (a h+3 b e)-\frac{a^3 c}{4 x^4}-\frac{a^3 d}{3 x^3}-\frac{a^3 e}{2 x^2}+\frac{1}{5} b^2 x^5 (3 a f+b c)+\frac{1}{6} b^2 x^6 (3 a g+b d)+\frac{1}{7} b^2 x^7 (3 a h+b e)+\frac{3}{2} a b x^2 (a f+b c)+a b x^3 (a g+b d)+\frac{3}{4} a b x^4 (a h+b e)+\frac{1}{8} b^3 f x^8+\frac{1}{9} b^3 g x^9+\frac{1}{10} b^3 h x^{10} \]
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Rubi [A] time = 0.176742, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {1820} \[ -\frac{a^2 (a f+3 b c)}{x}+a^2 \log (x) (a g+3 b d)+a^2 x (a h+3 b e)-\frac{a^3 c}{4 x^4}-\frac{a^3 d}{3 x^3}-\frac{a^3 e}{2 x^2}+\frac{1}{5} b^2 x^5 (3 a f+b c)+\frac{1}{6} b^2 x^6 (3 a g+b d)+\frac{1}{7} b^2 x^7 (3 a h+b e)+\frac{3}{2} a b x^2 (a f+b c)+a b x^3 (a g+b d)+\frac{3}{4} a b x^4 (a h+b e)+\frac{1}{8} b^3 f x^8+\frac{1}{9} b^3 g x^9+\frac{1}{10} b^3 h x^{10} \]
Antiderivative was successfully verified.
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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^5} \, dx &=\int \left (a^2 (3 b e+a h)+\frac{a^3 c}{x^5}+\frac{a^3 d}{x^4}+\frac{a^3 e}{x^3}+\frac{a^2 (3 b c+a f)}{x^2}+\frac{a^2 (3 b d+a g)}{x}+3 a b (b c+a f) x+3 a b (b d+a g) x^2+3 a b (b e+a h) x^3+b^2 (b c+3 a f) x^4+b^2 (b d+3 a g) x^5+b^2 (b e+3 a h) x^6+b^3 f x^7+b^3 g x^8+b^3 h x^9\right ) \, dx\\ &=-\frac{a^3 c}{4 x^4}-\frac{a^3 d}{3 x^3}-\frac{a^3 e}{2 x^2}-\frac{a^2 (3 b c+a f)}{x}+a^2 (3 b e+a h) x+\frac{3}{2} a b (b c+a f) x^2+a b (b d+a g) x^3+\frac{3}{4} a b (b e+a h) x^4+\frac{1}{5} b^2 (b c+3 a f) x^5+\frac{1}{6} b^2 (b d+3 a g) x^6+\frac{1}{7} b^2 (b e+3 a h) x^7+\frac{1}{8} b^3 f x^8+\frac{1}{9} b^3 g x^9+\frac{1}{10} b^3 h x^{10}+a^2 (3 b d+a g) \log (x)\\ \end{align*}
Mathematica [A] time = 0.120628, size = 170, normalized size = 0.81 \[ \frac{630 a^2 b x^3 \left (x^2 \left (12 e+6 f x+4 g x^2+3 h x^3\right )-12 c\right )-210 a^3 \left (3 c+4 d x+6 x^2 \left (e+2 f x-2 h x^3\right )\right )+18 a b^2 x^6 \left (210 c+x \left (140 d+105 e x+84 f x^2+70 g x^3+60 h x^4\right )\right )+b^3 x^9 \left (504 c+x \left (420 d+360 e x+315 f x^2+280 g x^3+252 h x^4\right )\right )}{2520 x^4}+a^2 \log (x) (a g+3 b d) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 220, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}h{x}^{10}}{10}}+{\frac{{b}^{3}g{x}^{9}}{9}}+{\frac{{b}^{3}f{x}^{8}}{8}}+{\frac{3\,{x}^{7}a{b}^{2}h}{7}}+{\frac{{x}^{7}{b}^{3}e}{7}}+{\frac{{x}^{6}a{b}^{2}g}{2}}+{\frac{{x}^{6}{b}^{3}d}{6}}+{\frac{3\,{x}^{5}a{b}^{2}f}{5}}+{\frac{{x}^{5}{b}^{3}c}{5}}+{\frac{3\,{x}^{4}{a}^{2}bh}{4}}+{\frac{3\,{x}^{4}a{b}^{2}e}{4}}+{x}^{3}{a}^{2}bg+a{b}^{2}d{x}^{3}+{\frac{3\,{x}^{2}{a}^{2}bf}{2}}+{\frac{3\,{x}^{2}a{b}^{2}c}{2}}+{a}^{3}hx+3\,{a}^{2}bex+\ln \left ( x \right ){a}^{3}g+3\,\ln \left ( x \right ){a}^{2}bd-{\frac{{a}^{3}d}{3\,{x}^{3}}}-{\frac{{a}^{3}c}{4\,{x}^{4}}}-{\frac{{a}^{3}e}{2\,{x}^{2}}}-{\frac{{a}^{3}f}{x}}-3\,{\frac{b{a}^{2}c}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.947768, size = 286, normalized size = 1.37 \begin{align*} \frac{1}{10} \, b^{3} h x^{10} + \frac{1}{9} \, b^{3} g x^{9} + \frac{1}{8} \, b^{3} f x^{8} + \frac{1}{7} \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{7} + \frac{1}{6} \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{6} + \frac{1}{5} \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{5} + \frac{3}{4} \,{\left (a b^{2} e + a^{2} b h\right )} x^{4} +{\left (a b^{2} d + a^{2} b g\right )} x^{3} + \frac{3}{2} \,{\left (a b^{2} c + a^{2} b f\right )} x^{2} +{\left (3 \, a^{2} b e + a^{3} h\right )} x +{\left (3 \, a^{2} b d + a^{3} g\right )} \log \left (x\right ) - \frac{6 \, a^{3} e x^{2} + 4 \, a^{3} d x + 3 \, a^{3} c + 12 \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{3}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.97709, size = 528, normalized size = 2.53 \begin{align*} \frac{252 \, b^{3} h x^{14} + 280 \, b^{3} g x^{13} + 315 \, b^{3} f x^{12} + 360 \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + 420 \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + 504 \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + 1890 \,{\left (a b^{2} e + a^{2} b h\right )} x^{8} + 2520 \,{\left (a b^{2} d + a^{2} b g\right )} x^{7} + 3780 \,{\left (a b^{2} c + a^{2} b f\right )} x^{6} - 1260 \, a^{3} e x^{2} + 2520 \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} + 2520 \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} \log \left (x\right ) - 840 \, a^{3} d x - 630 \, a^{3} c - 2520 \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{3}}{2520 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.97899, size = 233, normalized size = 1.11 \begin{align*} a^{2} \left (a g + 3 b d\right ) \log{\left (x \right )} + \frac{b^{3} f x^{8}}{8} + \frac{b^{3} g x^{9}}{9} + \frac{b^{3} h x^{10}}{10} + x^{7} \left (\frac{3 a b^{2} h}{7} + \frac{b^{3} e}{7}\right ) + x^{6} \left (\frac{a b^{2} g}{2} + \frac{b^{3} d}{6}\right ) + x^{5} \left (\frac{3 a b^{2} f}{5} + \frac{b^{3} c}{5}\right ) + x^{4} \left (\frac{3 a^{2} b h}{4} + \frac{3 a b^{2} e}{4}\right ) + x^{3} \left (a^{2} b g + a b^{2} d\right ) + x^{2} \left (\frac{3 a^{2} b f}{2} + \frac{3 a b^{2} c}{2}\right ) + x \left (a^{3} h + 3 a^{2} b e\right ) - \frac{3 a^{3} c + 4 a^{3} d x + 6 a^{3} e x^{2} + x^{3} \left (12 a^{3} f + 36 a^{2} b c\right )}{12 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07367, size = 302, normalized size = 1.44 \begin{align*} \frac{1}{10} \, b^{3} h x^{10} + \frac{1}{9} \, b^{3} g x^{9} + \frac{1}{8} \, b^{3} f x^{8} + \frac{3}{7} \, a b^{2} h x^{7} + \frac{1}{7} \, b^{3} x^{7} e + \frac{1}{6} \, b^{3} d x^{6} + \frac{1}{2} \, a b^{2} g x^{6} + \frac{1}{5} \, b^{3} c x^{5} + \frac{3}{5} \, a b^{2} f x^{5} + \frac{3}{4} \, a^{2} b h x^{4} + \frac{3}{4} \, a b^{2} x^{4} e + a b^{2} d x^{3} + a^{2} b g x^{3} + \frac{3}{2} \, a b^{2} c x^{2} + \frac{3}{2} \, a^{2} b f x^{2} + a^{3} h x + 3 \, a^{2} b x e +{\left (3 \, a^{2} b d + a^{3} g\right )} \log \left ({\left | x \right |}\right ) - \frac{6 \, a^{3} x^{2} e + 4 \, a^{3} d x + 3 \, a^{3} c + 12 \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{3}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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