Optimal. Leaf size=86 \[ \log (x) (a f+b c)+x (a g+b d)+\frac{1}{2} x^2 (a h+b e)-\frac{a c}{3 x^3}-\frac{a d}{2 x^2}-\frac{a e}{x}+\frac{1}{3} b f x^3+\frac{1}{4} b g x^4+\frac{1}{5} b h x^5 \]
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Rubi [A] time = 0.155345, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.028 \[ \log (x) (a f+b c)+x (a g+b d)+\frac{1}{2} x^2 (a h+b e)-\frac{a c}{3 x^3}-\frac{a d}{2 x^2}-\frac{a e}{x}+\frac{1}{3} b f x^3+\frac{1}{4} b g x^4+\frac{1}{5} b h x^5 \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^3)*(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. \[ - \frac{a c}{3 x^{3}} - \frac{a d}{2 x^{2}} - \frac{a e}{x} + \frac{b f x^{3}}{3} + \frac{b g x^{4}}{4} + \frac{b h x^{5}}{5} + \left (a f + b c\right ) \log{\left (x \right )} + \left (a h + b e\right ) \int x\, dx + \frac{\left (a g + b d\right ) \int a\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a)*(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 = 0.148415, size = 76, normalized size = 0.88 \[ \log (x) (a f+b c)-\frac{a \left (2 c+3 x \left (d+2 e x+x^3 (-(2 g+h x))\right )\right )}{6 x^3}+\frac{1}{60} b x \left (60 d+x \left (30 e+x \left (20 f+15 g x+12 h x^2\right )\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^3)*(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 = 0.01, size = 76, normalized size = 0.9 \[{\frac{bh{x}^{5}}{5}}+{\frac{bg{x}^{4}}{4}}+{\frac{bf{x}^{3}}{3}}+{\frac{{x}^{2}ah}{2}}+{\frac{be{x}^{2}}{2}}+xag+xbd+\ln \left ( x \right ) af+\ln \left ( x \right ) bc-{\frac{ac}{3\,{x}^{3}}}-{\frac{ad}{2\,{x}^{2}}}-{\frac{ae}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a)*(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 = 1.3717, size = 101, normalized size = 1.17 \[ \frac{1}{5} \, b h x^{5} + \frac{1}{4} \, b g x^{4} + \frac{1}{3} \, b f x^{3} + \frac{1}{2} \,{\left (b e + a h\right )} x^{2} +{\left (b d + a g\right )} x +{\left (b c + a f\right )} \log \left (x\right ) - \frac{6 \, a e x^{2} + 3 \, a d x + 2 \, a c}{6 \, x^{3}} \]
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)/x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240668, size = 109, normalized size = 1.27 \[ \frac{12 \, b h x^{8} + 15 \, b g x^{7} + 20 \, b f x^{6} + 30 \,{\left (b e + a h\right )} x^{5} + 60 \,{\left (b d + a g\right )} x^{4} + 60 \,{\left (b c + a f\right )} x^{3} \log \left (x\right ) - 60 \, a e x^{2} - 30 \, a d x - 20 \, a c}{60 \, x^{3}} \]
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)/x^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.51655, size = 82, normalized size = 0.95 \[ \frac{b f x^{3}}{3} + \frac{b g x^{4}}{4} + \frac{b h x^{5}}{5} + x^{2} \left (\frac{a h}{2} + \frac{b e}{2}\right ) + x \left (a g + b d\right ) + \left (a f + b c\right ) \log{\left (x \right )} - \frac{2 a c + 3 a d x + 6 a e x^{2}}{6 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**3+a)*(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 = 0.21093, size = 107, normalized size = 1.24 \[ \frac{1}{5} \, b h x^{5} + \frac{1}{4} \, b g x^{4} + \frac{1}{3} \, b f x^{3} + \frac{1}{2} \, a h x^{2} + \frac{1}{2} \, b x^{2} e + b d x + a g x +{\left (b c + a f\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{6 \, a x^{2} e + 3 \, a d x + 2 \, a c}{6 \, x^{3}} \]
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)/x^4,x, algorithm="giac")
[Out]