Optimal. Leaf size=152 \[ -\frac{a^2 c}{3 x^3}-\frac{a^2 d}{2 x^2}-\frac{a^2 e}{x}+\frac{1}{3} b x^3 (2 a f+b c)+a \log (x) (a f+2 b c)+\frac{1}{4} b x^4 (2 a g+b d)+a x (a g+2 b d)+\frac{1}{5} b x^5 (2 a h+b e)+\frac{1}{2} a x^2 (a h+2 b e)+\frac{1}{6} b^2 f x^6+\frac{1}{7} b^2 g x^7+\frac{1}{8} b^2 h x^8 \]
[Out]
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Rubi [A] time = 0.307914, antiderivative size = 152, 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 \[ -\frac{a^2 c}{3 x^3}-\frac{a^2 d}{2 x^2}-\frac{a^2 e}{x}+\frac{1}{3} b x^3 (2 a f+b c)+a \log (x) (a f+2 b c)+\frac{1}{4} b x^4 (2 a g+b d)+a x (a g+2 b d)+\frac{1}{5} b x^5 (2 a h+b e)+\frac{1}{2} a x^2 (a h+2 b e)+\frac{1}{6} b^2 f x^6+\frac{1}{7} b^2 g x^7+\frac{1}{8} b^2 h x^8 \]
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^4,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} c}{3 x^{3}} - \frac{a^{2} d}{2 x^{2}} - \frac{a^{2} e}{x} + a \left (a f + 2 b c\right ) \log{\left (x \right )} + a \left (a h + 2 b e\right ) \int x\, dx + \frac{a \left (a g + 2 b d\right ) \int g\, dx}{g} + \frac{b^{2} f x^{6}}{6} + \frac{b^{2} g x^{7}}{7} + \frac{b^{2} h x^{8}}{8} + \frac{b x^{5} \left (2 a h + b e\right )}{5} + \frac{b x^{4} \left (2 a g + b d\right )}{4} + \frac{b x^{3} \left (2 a f + b c\right )}{3} \]
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**4,x)
[Out]
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Mathematica [A] time = 0.183313, size = 123, normalized size = 0.81 \[ -\frac{a^2 \left (2 c+3 x \left (d+2 e x+x^3 (-(2 g+h x))\right )\right )}{6 x^3}+a \log (x) (a f+2 b c)+\frac{1}{30} a b x \left (60 d+x \left (30 e+x \left (20 f+15 g x+12 h x^2\right )\right )\right )+\frac{1}{840} b^2 x^3 \left (280 c+x \left (210 d+x \left (168 e+140 f x+120 g x^2+105 h x^3\right )\right )\right ) \]
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^4,x]
[Out]
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Maple [A] time = 0.011, size = 149, normalized size = 1. \[{\frac{{b}^{2}h{x}^{8}}{8}}+{\frac{{b}^{2}g{x}^{7}}{7}}+{\frac{{b}^{2}f{x}^{6}}{6}}+{\frac{2\,{x}^{5}abh}{5}}+{\frac{{x}^{5}{b}^{2}e}{5}}+{\frac{{x}^{4}abg}{2}}+{\frac{{b}^{2}d{x}^{4}}{4}}+{\frac{2\,{x}^{3}abf}{3}}+{\frac{{x}^{3}{b}^{2}c}{3}}+{\frac{{x}^{2}{a}^{2}h}{2}}+{x}^{2}abe+x{a}^{2}g+2\,xabd+\ln \left ( x \right ){a}^{2}f+2\,\ln \left ( x \right ) abc-{\frac{{a}^{2}c}{3\,{x}^{3}}}-{\frac{{a}^{2}d}{2\,{x}^{2}}}-{\frac{e{a}^{2}}{x}} \]
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^4,x)
[Out]
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Maxima [A] time = 7.45838, size = 198, normalized size = 1.3 \[ \frac{1}{8} \, b^{2} h x^{8} + \frac{1}{7} \, b^{2} g x^{7} + \frac{1}{6} \, b^{2} f x^{6} + \frac{1}{5} \,{\left (b^{2} e + 2 \, a b h\right )} x^{5} + \frac{1}{4} \,{\left (b^{2} d + 2 \, a b g\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} c + 2 \, a b f\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b e + a^{2} h\right )} x^{2} +{\left (2 \, a b d + a^{2} g\right )} x +{\left (2 \, a b c + a^{2} f\right )} \log \left (x\right ) - \frac{6 \, a^{2} e x^{2} + 3 \, a^{2} d x + 2 \, a^{2} c}{6 \, x^{3}} \]
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^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.249359, size = 207, normalized size = 1.36 \[ \frac{105 \, b^{2} h x^{11} + 120 \, b^{2} g x^{10} + 140 \, b^{2} f x^{9} + 168 \,{\left (b^{2} e + 2 \, a b h\right )} x^{8} + 210 \,{\left (b^{2} d + 2 \, a b g\right )} x^{7} + 280 \,{\left (b^{2} c + 2 \, a b f\right )} x^{6} + 420 \,{\left (2 \, a b e + a^{2} h\right )} x^{5} - 840 \, a^{2} e x^{2} + 840 \,{\left (2 \, a b d + a^{2} g\right )} x^{4} + 840 \,{\left (2 \, a b c + a^{2} f\right )} x^{3} \log \left (x\right ) - 420 \, a^{2} d x - 280 \, a^{2} c}{840 \, x^{3}} \]
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^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.86847, size = 156, normalized size = 1.03 \[ a \left (a f + 2 b c\right ) \log{\left (x \right )} + \frac{b^{2} f x^{6}}{6} + \frac{b^{2} g x^{7}}{7} + \frac{b^{2} h x^{8}}{8} + x^{5} \left (\frac{2 a b h}{5} + \frac{b^{2} e}{5}\right ) + x^{4} \left (\frac{a b g}{2} + \frac{b^{2} d}{4}\right ) + x^{3} \left (\frac{2 a b f}{3} + \frac{b^{2} c}{3}\right ) + x^{2} \left (\frac{a^{2} h}{2} + a b e\right ) + x \left (a^{2} g + 2 a b d\right ) - \frac{2 a^{2} c + 3 a^{2} d x + 6 a^{2} e x^{2}}{6 x^{3}} \]
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**4,x)
[Out]
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GIAC/XCAS [A] time = 0.219669, size = 207, normalized size = 1.36 \[ \frac{1}{8} \, b^{2} h x^{8} + \frac{1}{7} \, b^{2} g x^{7} + \frac{1}{6} \, b^{2} f x^{6} + \frac{2}{5} \, a b h x^{5} + \frac{1}{5} \, b^{2} x^{5} e + \frac{1}{4} \, b^{2} d x^{4} + \frac{1}{2} \, a b g x^{4} + \frac{1}{3} \, b^{2} c x^{3} + \frac{2}{3} \, a b f x^{3} + \frac{1}{2} \, a^{2} h x^{2} + a b x^{2} e + 2 \, a b d x + a^{2} g x +{\left (2 \, a b c + a^{2} f\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{6 \, a^{2} x^{2} e + 3 \, a^{2} d x + 2 \, a^{2} c}{6 \, x^{3}} \]
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^4,x, algorithm="giac")
[Out]