Optimal. Leaf size=138 \[ \frac {1}{7} x^7 (d+10 e)+\frac {5}{6} x^6 (2 d+9 e)+3 x^5 (3 d+8 e)+\frac {15}{2} x^4 (4 d+7 e)+14 x^3 (5 d+6 e)+21 x^2 (6 d+5 e)-\frac {10 d+e}{2 x^2}+30 x (7 d+4 e)-\frac {5 (9 d+2 e)}{x}+15 (8 d+3 e) \log (x)-\frac {d}{3 x^3}+\frac {e x^8}{8} \]
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Rubi [A] time = 0.07, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {27, 76} \begin {gather*} \frac {1}{7} x^7 (d+10 e)+\frac {5}{6} x^6 (2 d+9 e)+3 x^5 (3 d+8 e)+\frac {15}{2} x^4 (4 d+7 e)+14 x^3 (5 d+6 e)+21 x^2 (6 d+5 e)-\frac {10 d+e}{2 x^2}+30 x (7 d+4 e)-\frac {5 (9 d+2 e)}{x}+15 (8 d+3 e) \log (x)-\frac {d}{3 x^3}+\frac {e x^8}{8} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^4} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^4} \, dx\\ &=\int \left (30 (7 d+4 e)+\frac {d}{x^4}+\frac {10 d+e}{x^3}+\frac {5 (9 d+2 e)}{x^2}+\frac {15 (8 d+3 e)}{x}+42 (6 d+5 e) x+42 (5 d+6 e) x^2+30 (4 d+7 e) x^3+15 (3 d+8 e) x^4+5 (2 d+9 e) x^5+(d+10 e) x^6+e x^7\right ) \, dx\\ &=-\frac {d}{3 x^3}-\frac {10 d+e}{2 x^2}-\frac {5 (9 d+2 e)}{x}+30 (7 d+4 e) x+21 (6 d+5 e) x^2+14 (5 d+6 e) x^3+\frac {15}{2} (4 d+7 e) x^4+3 (3 d+8 e) x^5+\frac {5}{6} (2 d+9 e) x^6+\frac {1}{7} (d+10 e) x^7+\frac {e x^8}{8}+15 (8 d+3 e) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 140, normalized size = 1.01 \begin {gather*} \frac {1}{7} x^7 (d+10 e)+\frac {5}{6} x^6 (2 d+9 e)+3 x^5 (3 d+8 e)+\frac {15}{2} x^4 (4 d+7 e)+14 x^3 (5 d+6 e)+21 x^2 (6 d+5 e)+\frac {-10 d-e}{2 x^2}+30 x (7 d+4 e)-\frac {5 (9 d+2 e)}{x}+15 (8 d+3 e) \log (x)-\frac {d}{3 x^3}+\frac {e x^8}{8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 131, normalized size = 0.95 \begin {gather*} \frac {21 \, e x^{11} + 24 \, {\left (d + 10 \, e\right )} x^{10} + 140 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 504 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 1260 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 2352 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 3528 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 5040 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 2520 \, {\left (8 \, d + 3 \, e\right )} x^{3} \log \relax (x) - 840 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 84 \, {\left (10 \, d + e\right )} x - 56 \, d}{168 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 139, normalized size = 1.01 \begin {gather*} \frac {1}{8} \, x^{8} e + \frac {1}{7} \, d x^{7} + \frac {10}{7} \, x^{7} e + \frac {5}{3} \, d x^{6} + \frac {15}{2} \, x^{6} e + 9 \, d x^{5} + 24 \, x^{5} e + 30 \, d x^{4} + \frac {105}{2} \, x^{4} e + 70 \, d x^{3} + 84 \, x^{3} e + 126 \, d x^{2} + 105 \, x^{2} e + 210 \, d x + 120 \, x e + 15 \, {\left (8 \, d + 3 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {30 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 3 \, {\left (10 \, d + e\right )} x + 2 \, d}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 128, normalized size = 0.93 \begin {gather*} \frac {e \,x^{8}}{8}+\frac {d \,x^{7}}{7}+\frac {10 e \,x^{7}}{7}+\frac {5 d \,x^{6}}{3}+\frac {15 e \,x^{6}}{2}+9 d \,x^{5}+24 e \,x^{5}+30 d \,x^{4}+\frac {105 e \,x^{4}}{2}+70 d \,x^{3}+84 e \,x^{3}+126 d \,x^{2}+105 e \,x^{2}+210 d x +120 d \ln \relax (x )+120 e x +45 e \ln \relax (x )-\frac {45 d}{x}-\frac {10 e}{x}-\frac {5 d}{x^{2}}-\frac {e}{2 x^{2}}-\frac {d}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 127, normalized size = 0.92 \begin {gather*} \frac {1}{8} \, e x^{8} + \frac {1}{7} \, {\left (d + 10 \, e\right )} x^{7} + \frac {5}{6} \, {\left (2 \, d + 9 \, e\right )} x^{6} + 3 \, {\left (3 \, d + 8 \, e\right )} x^{5} + \frac {15}{2} \, {\left (4 \, d + 7 \, e\right )} x^{4} + 14 \, {\left (5 \, d + 6 \, e\right )} x^{3} + 21 \, {\left (6 \, d + 5 \, e\right )} x^{2} + 30 \, {\left (7 \, d + 4 \, e\right )} x + 15 \, {\left (8 \, d + 3 \, e\right )} \log \relax (x) - \frac {30 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 3 \, {\left (10 \, d + e\right )} x + 2 \, d}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 121, normalized size = 0.88 \begin {gather*} x^6\,\left (\frac {5\,d}{3}+\frac {15\,e}{2}\right )+x^7\,\left (\frac {d}{7}+\frac {10\,e}{7}\right )+x^5\,\left (9\,d+24\,e\right )+x^4\,\left (30\,d+\frac {105\,e}{2}\right )+x^3\,\left (70\,d+84\,e\right )+x^2\,\left (126\,d+105\,e\right )+\ln \relax (x)\,\left (120\,d+45\,e\right )-\frac {\left (45\,d+10\,e\right )\,x^2+\left (5\,d+\frac {e}{2}\right )\,x+\frac {d}{3}}{x^3}+\frac {e\,x^8}{8}+x\,\left (210\,d+120\,e\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 124, normalized size = 0.90 \begin {gather*} \frac {e x^{8}}{8} + x^{7} \left (\frac {d}{7} + \frac {10 e}{7}\right ) + x^{6} \left (\frac {5 d}{3} + \frac {15 e}{2}\right ) + x^{5} \left (9 d + 24 e\right ) + x^{4} \left (30 d + \frac {105 e}{2}\right ) + x^{3} \left (70 d + 84 e\right ) + x^{2} \left (126 d + 105 e\right ) + x \left (210 d + 120 e\right ) + 15 \left (8 d + 3 e\right ) \log {\relax (x )} + \frac {- 2 d + x^{2} \left (- 270 d - 60 e\right ) + x \left (- 30 d - 3 e\right )}{6 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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