Optimal. Leaf size=209 \[ -\frac{a^3 c}{4 x^4}-\frac{a^3 d}{3 x^3}-\frac{a^3 e}{2 x^2}-\frac{a^2 (a f+3 b c)}{x}+a^2 \log (x) (a g+3 b d)+a^2 x (a h+3 b e)+\frac{1}{5} b^2 x^5 (3 a f+b c)+\frac{1}{6} b^2 x^6 (3 a g+b d)+\frac{1}{7} b^2 x^7 (3 a h+b e)+\frac{3}{2} a b x^2 (a f+b c)+a b x^3 (a g+b d)+\frac{3}{4} a b x^4 (a h+b e)+\frac{1}{8} b^3 f x^8+\frac{1}{9} b^3 g x^9+\frac{1}{10} b^3 h x^{10} \]
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Rubi [A] time = 0.479195, 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 \[ -\frac{a^3 c}{4 x^4}-\frac{a^3 d}{3 x^3}-\frac{a^3 e}{2 x^2}-\frac{a^2 (a f+3 b c)}{x}+a^2 \log (x) (a g+3 b d)+a^2 x (a h+3 b e)+\frac{1}{5} b^2 x^5 (3 a f+b c)+\frac{1}{6} b^2 x^6 (3 a g+b d)+\frac{1}{7} b^2 x^7 (3 a h+b e)+\frac{3}{2} a b x^2 (a f+b c)+a b x^3 (a g+b d)+\frac{3}{4} a b x^4 (a h+b e)+\frac{1}{8} b^3 f x^8+\frac{1}{9} b^3 g x^9+\frac{1}{10} b^3 h x^{10} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^5,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{3} c}{4 x^{4}} - \frac{a^{3} d}{3 x^{3}} - \frac{a^{3} e}{2 x^{2}} + a^{2} \left (a g + 3 b d\right ) \log{\left (x \right )} - \frac{a^{2} \left (a f + 3 b c\right )}{x} + \frac{a^{2} \left (a h + 3 b e\right ) \int h\, dx}{h} + \frac{3 a b x^{4} \left (a h + b e\right )}{4} + a b x^{3} \left (a g + b d\right ) + 3 a b \left (a f + b c\right ) \int x\, dx + \frac{b^{3} f x^{8}}{8} + \frac{b^{3} g x^{9}}{9} + \frac{b^{3} h x^{10}}{10} + \frac{b^{2} x^{7} \left (3 a h + b e\right )}{7} + \frac{b^{2} x^{6} \left (3 a g + b d\right )}{6} + \frac{b^{2} x^{5} \left (3 a f + b c\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a)**3*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/x**5,x)
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Mathematica [A] time = 0.328199, size = 170, normalized size = 0.81 \[ a^2 \log (x) (a g+3 b d)+\frac{-210 a^3 \left (3 c+4 d x+6 x^2 \left (e+2 f x-2 h x^3\right )\right )+630 a^2 b x^3 \left (x^2 \left (12 e+6 f x+4 g x^2+3 h x^3\right )-12 c\right )+18 a b^2 x^6 \left (210 c+x \left (140 d+105 e x+84 f x^2+70 g x^3+60 h x^4\right )\right )+b^3 x^9 \left (504 c+x \left (420 d+360 e x+315 f x^2+280 g x^3+252 h x^4\right )\right )}{2520 x^4} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^5,x]
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Maple [A] time = 0.011, size = 220, normalized size = 1.1 \[{\frac{{b}^{3}h{x}^{10}}{10}}+{\frac{{b}^{3}g{x}^{9}}{9}}+{\frac{{b}^{3}f{x}^{8}}{8}}+{\frac{3\,{x}^{7}a{b}^{2}h}{7}}+{\frac{{x}^{7}{b}^{3}e}{7}}+{\frac{{x}^{6}a{b}^{2}g}{2}}+{\frac{{x}^{6}{b}^{3}d}{6}}+{\frac{3\,{x}^{5}a{b}^{2}f}{5}}+{\frac{{x}^{5}{b}^{3}c}{5}}+{\frac{3\,{x}^{4}{a}^{2}bh}{4}}+{\frac{3\,{x}^{4}a{b}^{2}e}{4}}+{x}^{3}{a}^{2}bg+a{b}^{2}d{x}^{3}+{\frac{3\,{x}^{2}{a}^{2}bf}{2}}+{\frac{3\,c{x}^{2}a{b}^{2}}{2}}+x{a}^{3}h+3\,{a}^{2}bex+\ln \left ( x \right ){a}^{3}g+3\,\ln \left ( x \right ){a}^{2}bd-{\frac{{a}^{3}c}{4\,{x}^{4}}}-{\frac{{a}^{3}d}{3\,{x}^{3}}}-{\frac{{a}^{3}e}{2\,{x}^{2}}}-{\frac{{a}^{3}f}{x}}-3\,{\frac{{a}^{2}bc}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^5,x)
[Out]
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Maxima [A] time = 1.38383, size = 286, normalized size = 1.37 \[ \frac{1}{10} \, b^{3} h x^{10} + \frac{1}{9} \, b^{3} g x^{9} + \frac{1}{8} \, b^{3} f x^{8} + \frac{1}{7} \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{7} + \frac{1}{6} \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{6} + \frac{1}{5} \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{5} + \frac{3}{4} \,{\left (a b^{2} e + a^{2} b h\right )} x^{4} +{\left (a b^{2} d + a^{2} b g\right )} x^{3} + \frac{3}{2} \,{\left (a b^{2} c + a^{2} b f\right )} x^{2} +{\left (3 \, a^{2} b e + a^{3} h\right )} x +{\left (3 \, a^{2} b d + a^{3} g\right )} \log \left (x\right ) - \frac{6 \, a^{3} e x^{2} + 4 \, a^{3} d x + 3 \, a^{3} c + 12 \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{3}}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^3/x^5,x, algorithm="maxima")
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Fricas [A] time = 0.246749, size = 296, normalized size = 1.42 \[ \frac{252 \, b^{3} h x^{14} + 280 \, b^{3} g x^{13} + 315 \, b^{3} f x^{12} + 360 \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + 420 \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + 504 \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + 1890 \,{\left (a b^{2} e + a^{2} b h\right )} x^{8} + 2520 \,{\left (a b^{2} d + a^{2} b g\right )} x^{7} + 3780 \,{\left (a b^{2} c + a^{2} b f\right )} x^{6} - 1260 \, a^{3} e x^{2} + 2520 \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} + 2520 \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} \log \left (x\right ) - 840 \, a^{3} d x - 630 \, a^{3} c - 2520 \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{3}}{2520 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^3/x^5,x, algorithm="fricas")
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Sympy [A] time = 6.36861, size = 233, normalized size = 1.11 \[ a^{2} \left (a g + 3 b d\right ) \log{\left (x \right )} + \frac{b^{3} f x^{8}}{8} + \frac{b^{3} g x^{9}}{9} + \frac{b^{3} h x^{10}}{10} + x^{7} \left (\frac{3 a b^{2} h}{7} + \frac{b^{3} e}{7}\right ) + x^{6} \left (\frac{a b^{2} g}{2} + \frac{b^{3} d}{6}\right ) + x^{5} \left (\frac{3 a b^{2} f}{5} + \frac{b^{3} c}{5}\right ) + x^{4} \left (\frac{3 a^{2} b h}{4} + \frac{3 a b^{2} e}{4}\right ) + x^{3} \left (a^{2} b g + a b^{2} d\right ) + x^{2} \left (\frac{3 a^{2} b f}{2} + \frac{3 a b^{2} c}{2}\right ) + x \left (a^{3} h + 3 a^{2} b e\right ) - \frac{3 a^{3} c + 4 a^{3} d x + 6 a^{3} e x^{2} + x^{3} \left (12 a^{3} f + 36 a^{2} b c\right )}{12 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**3+a)**3*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/x**5,x)
[Out]
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GIAC/XCAS [A] time = 0.220889, size = 302, normalized size = 1.44 \[ \frac{1}{10} \, b^{3} h x^{10} + \frac{1}{9} \, b^{3} g x^{9} + \frac{1}{8} \, b^{3} f x^{8} + \frac{3}{7} \, a b^{2} h x^{7} + \frac{1}{7} \, b^{3} x^{7} e + \frac{1}{6} \, b^{3} d x^{6} + \frac{1}{2} \, a b^{2} g x^{6} + \frac{1}{5} \, b^{3} c x^{5} + \frac{3}{5} \, a b^{2} f x^{5} + \frac{3}{4} \, a^{2} b h x^{4} + \frac{3}{4} \, a b^{2} x^{4} e + a b^{2} d x^{3} + a^{2} b g x^{3} + \frac{3}{2} \, a b^{2} c x^{2} + \frac{3}{2} \, a^{2} b f x^{2} + a^{3} h x + 3 \, a^{2} b x e +{\left (3 \, a^{2} b d + a^{3} g\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{6 \, a^{3} x^{2} e + 4 \, a^{3} d x + 3 \, a^{3} c + 12 \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{3}}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^3/x^5,x, algorithm="giac")
[Out]