Optimal. Leaf size=147 \[ -\frac{a^2 c}{2 x^2}-\frac{a^2 d}{x}+a^2 e \log (x)+\frac{1}{4} b x^4 (2 a f+b c)+a x (a f+2 b c)+\frac{1}{5} b x^5 (2 a g+b d)+\frac{1}{2} a x^2 (a g+2 b d)+\frac{2}{3} a b e x^3+\frac{h \left (a+b x^3\right )^3}{9 b}+\frac{1}{6} b^2 e x^6+\frac{1}{7} b^2 f x^7+\frac{1}{8} b^2 g x^8 \]
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Rubi [A] time = 0.338188, antiderivative size = 147, 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 \[ -\frac{a^2 c}{2 x^2}-\frac{a^2 d}{x}+a^2 e \log (x)+\frac{1}{4} b x^4 (2 a f+b c)+a x (a f+2 b c)+\frac{1}{5} b x^5 (2 a g+b d)+\frac{1}{2} a x^2 (a g+2 b d)+\frac{2}{3} a b e x^3+\frac{h \left (a+b x^3\right )^3}{9 b}+\frac{1}{6} b^2 e x^6+\frac{1}{7} b^2 f x^7+\frac{1}{8} b^2 g x^8 \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^3)^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^3,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} c}{2 x^{2}} - \frac{a^{2} d}{x} + a^{2} e \log{\left (x \right )} + \frac{a x^{3} \left (a h + 2 b e\right )}{3} + a \left (a g + 2 b d\right ) \int x\, dx + \frac{a \left (a f + 2 b c\right ) \int f\, dx}{f} + \frac{b^{2} f x^{7}}{7} + \frac{b^{2} g x^{8}}{8} + \frac{b^{2} h x^{9}}{9} + \frac{b x^{6} \left (2 a h + b e\right )}{6} + \frac{b x^{5} \left (2 a g + b d\right )}{5} + \frac{b x^{4} \left (2 a f + b c\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a)**2*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/x**3,x)
[Out]
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Mathematica [A] time = 0.185175, size = 127, normalized size = 0.86 \[ \frac{a^2 \left (-3 c-6 d x+x^3 \left (6 f+3 g x+2 h x^2\right )\right )}{6 x^2}+a^2 e \log (x)+\frac{1}{30} a b x \left (60 c+x \left (30 d+x \left (20 e+15 f x+12 g x^2+10 h x^3\right )\right )\right )+\frac{b^2 x^4 (630 c+x (504 d+5 x (84 e+x (72 f+7 x (9 g+8 h x)))))}{2520} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^3)^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^3,x]
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Maple [A] time = 0.01, size = 150, normalized size = 1. \[{\frac{{b}^{2}h{x}^{9}}{9}}+{\frac{{b}^{2}g{x}^{8}}{8}}+{\frac{{b}^{2}f{x}^{7}}{7}}+{\frac{{x}^{6}abh}{3}}+{\frac{{b}^{2}e{x}^{6}}{6}}+{\frac{2\,{x}^{5}abg}{5}}+{\frac{{b}^{2}d{x}^{5}}{5}}+{\frac{{x}^{4}abf}{2}}+{\frac{{x}^{4}{b}^{2}c}{4}}+{\frac{{x}^{3}{a}^{2}h}{3}}+{\frac{2\,abe{x}^{3}}{3}}+{\frac{{x}^{2}{a}^{2}g}{2}}+{x}^{2}abd+{a}^{2}fx+2\,xabc+{a}^{2}e\ln \left ( x \right ) -{\frac{{a}^{2}c}{2\,{x}^{2}}}-{\frac{{a}^{2}d}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a)^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^3,x)
[Out]
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Maxima [A] time = 1.38427, size = 197, normalized size = 1.34 \[ \frac{1}{9} \, b^{2} h x^{9} + \frac{1}{8} \, b^{2} g x^{8} + \frac{1}{7} \, b^{2} f x^{7} + \frac{1}{6} \,{\left (b^{2} e + 2 \, a b h\right )} x^{6} + \frac{1}{5} \,{\left (b^{2} d + 2 \, a b g\right )} x^{5} + \frac{1}{4} \,{\left (b^{2} c + 2 \, a b f\right )} x^{4} + \frac{1}{3} \,{\left (2 \, a b e + a^{2} h\right )} x^{3} + a^{2} e \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, a b d + a^{2} g\right )} x^{2} +{\left (2 \, a b c + a^{2} f\right )} x - \frac{2 \, a^{2} d x + a^{2} c}{2 \, x^{2}} \]
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)^2/x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25107, size = 207, normalized size = 1.41 \[ \frac{280 \, b^{2} h x^{11} + 315 \, b^{2} g x^{10} + 360 \, b^{2} f x^{9} + 420 \,{\left (b^{2} e + 2 \, a b h\right )} x^{8} + 504 \,{\left (b^{2} d + 2 \, a b g\right )} x^{7} + 630 \,{\left (b^{2} c + 2 \, a b f\right )} x^{6} + 840 \,{\left (2 \, a b e + a^{2} h\right )} x^{5} + 2520 \, a^{2} e x^{2} \log \left (x\right ) + 1260 \,{\left (2 \, a b d + a^{2} g\right )} x^{4} - 2520 \, a^{2} d x + 2520 \,{\left (2 \, a b c + a^{2} f\right )} x^{3} - 1260 \, a^{2} c}{2520 \, x^{2}} \]
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)^2/x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.11201, size = 156, normalized size = 1.06 \[ a^{2} e \log{\left (x \right )} + \frac{b^{2} f x^{7}}{7} + \frac{b^{2} g x^{8}}{8} + \frac{b^{2} h x^{9}}{9} + x^{6} \left (\frac{a b h}{3} + \frac{b^{2} e}{6}\right ) + x^{5} \left (\frac{2 a b g}{5} + \frac{b^{2} d}{5}\right ) + x^{4} \left (\frac{a b f}{2} + \frac{b^{2} c}{4}\right ) + x^{3} \left (\frac{a^{2} h}{3} + \frac{2 a b e}{3}\right ) + x^{2} \left (\frac{a^{2} g}{2} + a b d\right ) + x \left (a^{2} f + 2 a b c\right ) - \frac{a^{2} c + 2 a^{2} d x}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**3+a)**2*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.211995, size = 207, normalized size = 1.41 \[ \frac{1}{9} \, b^{2} h x^{9} + \frac{1}{8} \, b^{2} g x^{8} + \frac{1}{7} \, b^{2} f x^{7} + \frac{1}{3} \, a b h x^{6} + \frac{1}{6} \, b^{2} x^{6} e + \frac{1}{5} \, b^{2} d x^{5} + \frac{2}{5} \, a b g x^{5} + \frac{1}{4} \, b^{2} c x^{4} + \frac{1}{2} \, a b f x^{4} + \frac{1}{3} \, a^{2} h x^{3} + \frac{2}{3} \, a b x^{3} e + a b d x^{2} + \frac{1}{2} \, a^{2} g x^{2} + 2 \, a b c x + a^{2} f x + a^{2} e{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, a^{2} d x + a^{2} c}{2 \, x^{2}} \]
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)^2/x^3,x, algorithm="giac")
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