Optimal. Leaf size=90 \[ \log (x+1) (d-e+f-g+h-i)-\log (x+2) (d-2 e+4 f-8 g+16 h-32 i)+x (f-3 g+7 h-15 i)+\frac {1}{2} x^2 (g-3 h+7 i)+\frac {1}{3} x^3 (h-3 i)+\frac {i x^4}{4} \]
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Rubi [A] time = 0.11, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1586, 1657, 632, 31} \[ \log (x+1) (d-e+f-g+h-i)-\log (x+2) (d-2 e+4 f-8 g+16 h-32 i)+x (f-3 g+7 h-15 i)+\frac {1}{2} x^2 (g-3 h+7 i)+\frac {1}{3} x^3 (h-3 i)+\frac {i x^4}{4} \]
Antiderivative was successfully verified.
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Rule 31
Rule 632
Rule 1586
Rule 1657
Rubi steps
\begin {align*} \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+78 x^5\right )}{4-5 x^2+x^4} \, dx &=\int \frac {d+e x+f x^2+g x^3+h x^4+78 x^5}{2+3 x+x^2} \, dx\\ &=\int \left (-1170+f-3 g+7 h+(546+g-3 h) x-(234-h) x^2+78 x^3+\frac {2340+d-2 f+6 g-14 h+(2418+e-3 f+7 g-15 h) x}{2+3 x+x^2}\right ) \, dx\\ &=-(1170-f+3 g-7 h) x+\frac {1}{2} (546+g-3 h) x^2-\frac {1}{3} (234-h) x^3+\frac {39 x^4}{2}+\int \frac {2340+d-2 f+6 g-14 h+(2418+e-3 f+7 g-15 h) x}{2+3 x+x^2} \, dx\\ &=-(1170-f+3 g-7 h) x+\frac {1}{2} (546+g-3 h) x^2-\frac {1}{3} (234-h) x^3+\frac {39 x^4}{2}+(-78+d-e+f-g+h) \int \frac {1}{1+x} \, dx-(-2496+d-2 e+4 f-8 g+16 h) \int \frac {1}{2+x} \, dx\\ &=-(1170-f+3 g-7 h) x+\frac {1}{2} (546+g-3 h) x^2-\frac {1}{3} (234-h) x^3+\frac {39 x^4}{2}-(78-d+e-f+g-h) \log (1+x)+(2496-d+2 e-4 f+8 g-16 h) \log (2+x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 91, normalized size = 1.01 \[ \log (x+1) (d-e+f-g+h-i)+\log (x+2) (-d+2 e-4 f+8 g-16 h+32 i)+x (f-3 g+7 h-15 i)+\frac {1}{2} x^2 (g-3 h+7 i)+\frac {1}{3} x^3 (h-3 i)+\frac {i x^4}{4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 84, normalized size = 0.93 \[ \frac {1}{4} \, i x^{4} + \frac {1}{3} \, {\left (h - 3 \, i\right )} x^{3} + \frac {1}{2} \, {\left (g - 3 \, h + 7 \, i\right )} x^{2} + {\left (f - 3 \, g + 7 \, h - 15 \, i\right )} x - {\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i\right )} \log \left (x + 2\right ) + {\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 97, normalized size = 1.08 \[ \frac {1}{4} \, i x^{4} + \frac {1}{3} \, h x^{3} - i x^{3} + \frac {1}{2} \, g x^{2} - \frac {3}{2} \, h x^{2} + \frac {7}{2} \, i x^{2} + f x - 3 \, g x + 7 \, h x - 15 \, i x - {\left (d + 4 \, f - 8 \, g + 16 \, h - 32 \, i - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + {\left (d + f - g + h - i - e\right )} \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 134, normalized size = 1.49 \[ \frac {i \,x^{4}}{4}+\frac {h \,x^{3}}{3}-i \,x^{3}+\frac {g \,x^{2}}{2}-\frac {3 h \,x^{2}}{2}+\frac {7 i \,x^{2}}{2}-d \ln \left (x +2\right )+d \ln \left (x +1\right )+2 e \ln \left (x +2\right )-e \ln \left (x +1\right )+f x -4 f \ln \left (x +2\right )+f \ln \left (x +1\right )-3 g x +8 g \ln \left (x +2\right )-g \ln \left (x +1\right )+7 h x -16 h \ln \left (x +2\right )+h \ln \left (x +1\right )-15 i x +32 i \ln \left (x +2\right )-i \ln \left (x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 84, normalized size = 0.93 \[ \frac {1}{4} \, i x^{4} + \frac {1}{3} \, {\left (h - 3 \, i\right )} x^{3} + \frac {1}{2} \, {\left (g - 3 \, h + 7 \, i\right )} x^{2} + {\left (f - 3 \, g + 7 \, h - 15 \, i\right )} x - {\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i\right )} \log \left (x + 2\right ) + {\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 86, normalized size = 0.96 \[ x^3\,\left (\frac {h}{3}-i\right )-\ln \left (x+2\right )\,\left (d-2\,e+4\,f-8\,g+16\,h-32\,i\right )+\ln \left (x+1\right )\,\left (d-e+f-g+h-i\right )+\frac {i\,x^4}{4}+x^2\,\left (\frac {g}{2}-\frac {3\,h}{2}+\frac {7\,i}{2}\right )+x\,\left (f-3\,g+7\,h-15\,i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.59, size = 122, normalized size = 1.36 \[ \frac {i x^{4}}{4} + x^{3} \left (\frac {h}{3} - i\right ) + x^{2} \left (\frac {g}{2} - \frac {3 h}{2} + \frac {7 i}{2}\right ) + x \left (f - 3 g + 7 h - 15 i\right ) + \left (- d + 2 e - 4 f + 8 g - 16 h + 32 i\right ) \log {\left (x + \frac {4 d - 6 e + 10 f - 18 g + 34 h - 66 i}{2 d - 3 e + 5 f - 9 g + 17 h - 33 i} \right )} + \left (d - e + f - g + h - i\right ) \log {\left (x + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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