Optimal. Leaf size=86 \[ -\frac {a f+b c}{x}+\log (x) (a g+b d)+x (a h+b e)-\frac {a c}{4 x^4}-\frac {a d}{3 x^3}-\frac {a e}{2 x^2}+\frac {1}{2} b f x^2+\frac {1}{3} b g x^3+\frac {1}{4} b h x^4 \]
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Rubi [A] time = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {1820} \[ -\frac {a f+b c}{x}+\log (x) (a g+b d)+x (a h+b e)-\frac {a c}{4 x^4}-\frac {a d}{3 x^3}-\frac {a e}{2 x^2}+\frac {1}{2} b f x^2+\frac {1}{3} b g x^3+\frac {1}{4} b h x^4 \]
Antiderivative was successfully verified.
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Rule 1820
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{x^5} \, dx &=\int \left (b e \left (1+\frac {a h}{b e}\right )+\frac {a c}{x^5}+\frac {a d}{x^4}+\frac {a e}{x^3}+\frac {b c+a f}{x^2}+\frac {b d+a g}{x}+b f x+b g x^2+b h x^3\right ) \, dx\\ &=-\frac {a c}{4 x^4}-\frac {a d}{3 x^3}-\frac {a e}{2 x^2}-\frac {b c+a f}{x}+(b e+a h) x+\frac {1}{2} b f x^2+\frac {1}{3} b g x^3+\frac {1}{4} b h x^4+(b d+a g) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.08, size = 77, normalized size = 0.90 \[ \log (x) (a g+b d)-\frac {a \left (3 c+4 d x+6 x^2 \left (e+2 f x-2 h x^3\right )\right )}{12 x^4}+b \left (-\frac {c}{x}+e x+\frac {1}{12} x^2 \left (6 f+4 g x+3 h x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 81, normalized size = 0.94 \[ \frac {3 \, b h x^{8} + 4 \, b g x^{7} + 6 \, b f x^{6} + 12 \, {\left (b e + a h\right )} x^{5} + 12 \, {\left (b d + a g\right )} x^{4} \log \relax (x) - 6 \, a e x^{2} - 12 \, {\left (b c + a f\right )} x^{3} - 4 \, a d x - 3 \, a c}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 77, normalized size = 0.90 \[ \frac {1}{4} \, b h x^{4} + \frac {1}{3} \, b g x^{3} + \frac {1}{2} \, b f x^{2} + a h x + b x e + {\left (b d + a g\right )} \log \left ({\left | x \right |}\right ) - \frac {12 \, {\left (b c + a f\right )} x^{3} + 6 \, a x^{2} e + 4 \, a d x + 3 \, a c}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 76, normalized size = 0.88 \[ \frac {b h \,x^{4}}{4}+\frac {b g \,x^{3}}{3}+\frac {b f \,x^{2}}{2}+a g \ln \relax (x )+a h x +b d \ln \relax (x )+b e x -\frac {a f}{x}-\frac {b c}{x}-\frac {a e}{2 x^{2}}-\frac {a d}{3 x^{3}}-\frac {a c}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 75, normalized size = 0.87 \[ \frac {1}{4} \, b h x^{4} + \frac {1}{3} \, b g x^{3} + \frac {1}{2} \, b f x^{2} + {\left (b e + a h\right )} x + {\left (b d + a g\right )} \log \relax (x) - \frac {6 \, a e x^{2} + 12 \, {\left (b c + a f\right )} x^{3} + 4 \, a d x + 3 \, a c}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.98, size = 74, normalized size = 0.86 \[ x\,\left (b\,e+a\,h\right )-\frac {\left (b\,c+a\,f\right )\,x^3+\frac {a\,e\,x^2}{2}+\frac {a\,d\,x}{3}+\frac {a\,c}{4}}{x^4}+\ln \relax (x)\,\left (b\,d+a\,g\right )+\frac {b\,h\,x^4}{4}+\frac {b\,f\,x^2}{2}+\frac {b\,g\,x^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.57, size = 83, normalized size = 0.97 \[ \frac {b f x^{2}}{2} + \frac {b g x^{3}}{3} + \frac {b h x^{4}}{4} + x \left (a h + b e\right ) + \left (a g + b d\right ) \log {\relax (x )} + \frac {- 3 a c - 4 a d x - 6 a e x^{2} + x^{3} \left (- 12 a f - 12 b c\right )}{12 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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