\(\int \frac {(d+e x) (1+2 x+x^2)^5}{x^8} \, dx\) [574]

Optimal result

Integrand size = 19, antiderivative size = 138 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^8} \, dx=-\frac {d}{7 x^7}-\frac {10 d+e}{6 x^6}-\frac {9 d+2 e}{x^5}-\frac {15 (8 d+3 e)}{4 x^4}-\frac {10 (7 d+4 e)}{x^3}-\frac {21 (6 d+5 e)}{x^2}-\frac {42 (5 d+6 e)}{x}+15 (3 d+8 e) x+\frac {5}{2} (2 d+9 e) x^2+\frac {1}{3} (d+10 e) x^3+\frac {e x^4}{4}+30 (4 d+7 e) \log (x) \]

[Out]

-1/7*d/x^7+1/6*(-10*d-e)/x^6+(-9*d-2*e)/x^5-15/4*(8*d+3*e)/x^4-10*(7*d+4*e)/x^3-21*(6*d+5*e)/x^2-42*(5*d+6*e)/
x+15*(3*d+8*e)*x+5/2*(2*d+9*e)*x^2+1/3*(d+10*e)*x^3+1/4*e*x^4+30*(4*d+7*e)*ln(x)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {27, 77} \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^8} \, dx=-\frac {10 d+e}{6 x^6}-\frac {9 d+2 e}{x^5}-\frac {15 (8 d+3 e)}{4 x^4}+\frac {1}{3} x^3 (d+10 e)-\frac {10 (7 d+4 e)}{x^3}+\frac {5}{2} x^2 (2 d+9 e)-\frac {21 (6 d+5 e)}{x^2}+15 x (3 d+8 e)-\frac {42 (5 d+6 e)}{x}+30 (4 d+7 e) \log (x)-\frac {d}{7 x^7}+\frac {e x^4}{4} \]

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^8,x]

[Out]

-1/7*d/x^7 - (10*d + e)/(6*x^6) - (9*d + 2*e)/x^5 - (15*(8*d + 3*e))/(4*x^4) - (10*(7*d + 4*e))/x^3 - (21*(6*d
+ 5*e))/x^2 - (42*(5*d + 6*e))/x + 15*(3*d + 8*e)*x + (5*(2*d + 9*e)*x^2)/2 + ((d + 10*e)*x^3)/3 + (e*x^4)/4
+ 30*(4*d + 7*e)*Log[x]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps \begin{align*} \text {integral}& = \int \frac {(1+x)^{10} (d+e x)}{x^8} \, dx \\ & = \int \left (15 (3 d+8 e)+\frac {d}{x^8}+\frac {10 d+e}{x^7}+\frac {5 (9 d+2 e)}{x^6}+\frac {15 (8 d+3 e)}{x^5}+\frac {30 (7 d+4 e)}{x^4}+\frac {42 (6 d+5 e)}{x^3}+\frac {42 (5 d+6 e)}{x^2}+\frac {30 (4 d+7 e)}{x}+5 (2 d+9 e) x+(d+10 e) x^2+e x^3\right ) \, dx \\ & = -\frac {d}{7 x^7}-\frac {10 d+e}{6 x^6}-\frac {9 d+2 e}{x^5}-\frac {15 (8 d+3 e)}{4 x^4}-\frac {10 (7 d+4 e)}{x^3}-\frac {21 (6 d+5 e)}{x^2}-\frac {42 (5 d+6 e)}{x}+15 (3 d+8 e) x+\frac {5}{2} (2 d+9 e) x^2+\frac {1}{3} (d+10 e) x^3+\frac {e x^4}{4}+30 (4 d+7 e) \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^8} \, dx=-\frac {d}{7 x^7}+\frac {-10 d-e}{6 x^6}+\frac {-9 d-2 e}{x^5}-\frac {15 (8 d+3 e)}{4 x^4}-\frac {10 (7 d+4 e)}{x^3}-\frac {21 (6 d+5 e)}{x^2}-\frac {42 (5 d+6 e)}{x}+15 (3 d+8 e) x+\frac {5}{2} (2 d+9 e) x^2+\frac {1}{3} (d+10 e) x^3+\frac {e x^4}{4}+30 (4 d+7 e) \log (x) \]

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^8,x]

[Out]

-1/7*d/x^7 + (-10*d - e)/(6*x^6) + (-9*d - 2*e)/x^5 - (15*(8*d + 3*e))/(4*x^4) - (10*(7*d + 4*e))/x^3 - (21*(6
*d + 5*e))/x^2 - (42*(5*d + 6*e))/x + 15*(3*d + 8*e)*x + (5*(2*d + 9*e)*x^2)/2 + ((d + 10*e)*x^3)/3 + (e*x^4)/
4 + 30*(4*d + 7*e)*Log[x]

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.88

method	result	size
risch	\(\frac {e \,x^{4}}{4}+\frac {d \,x^{3}}{3}+\frac {10 e \,x^{3}}{3}+5 d \,x^{2}+\frac {45 e \,x^{2}}{2}+45 d x +120 e x +\frac {\left (-210 d -252 e \right ) x^{6}+\left (-126 d -105 e \right ) x^{5}+\left (-70 d -40 e \right ) x^{4}+\left (-30 d -\frac {45 e}{4}\right ) x^{3}+\left (-9 d -2 e \right ) x^{2}+\left (-\frac {5 d}{3}-\frac {e}{6}\right ) x -\frac {d}{7}}{x^{7}}+120 d \ln \left (x \right )+210 e \ln \left (x \right )\)	\(122\)
norman	\(\frac {\left (-210 d -252 e \right ) x^{6}+\left (-126 d -105 e \right ) x^{5}+\left (-70 d -40 e \right ) x^{4}+\left (-30 d -\frac {45 e}{4}\right ) x^{3}+\left (-9 d -2 e \right ) x^{2}+\left (5 d +\frac {45 e}{2}\right ) x^{9}+\left (45 d +120 e \right ) x^{8}+\left (-\frac {5 d}{3}-\frac {e}{6}\right ) x +\left (\frac {d}{3}+\frac {10 e}{3}\right ) x^{10}-\frac {d}{7}+\frac {e \,x^{11}}{4}}{x^{7}}+\left (120 d +210 e \right ) \ln \left (x \right )\)	\(123\)
default	\(\frac {e \,x^{4}}{4}+\frac {d \,x^{3}}{3}+\frac {10 e \,x^{3}}{3}+5 d \,x^{2}+\frac {45 e \,x^{2}}{2}+45 d x +120 e x -\frac {10 d +e}{6 x^{6}}-\frac {120 d +45 e}{4 x^{4}}-\frac {45 d +10 e}{5 x^{5}}+\left (120 d +210 e \right ) \ln \left (x \right )-\frac {252 d +210 e}{2 x^{2}}-\frac {d}{7 x^{7}}-\frac {210 d +252 e}{x}-\frac {210 d +120 e}{3 x^{3}}\)	\(126\)
parallelrisch	\(\frac {21 e \,x^{11}+28 d \,x^{10}+280 e \,x^{10}+420 d \,x^{9}+1890 e \,x^{9}+10080 \ln \left (x \right ) x^{7} d +17640 \ln \left (x \right ) x^{7} e +3780 d \,x^{8}+10080 e \,x^{8}-17640 d \,x^{6}-21168 e \,x^{6}-10584 d \,x^{5}-8820 e \,x^{5}-5880 d \,x^{4}-3360 e \,x^{4}-2520 d \,x^{3}-945 e \,x^{3}-756 d \,x^{2}-168 e \,x^{2}-140 d x -14 e x -12 d}{84 x^{7}}\)	\(136\)

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^8,x,method=_RETURNVERBOSE)

[Out]

1/4*e*x^4+1/3*d*x^3+10/3*e*x^3+5*d*x^2+45/2*e*x^2+45*d*x+120*e*x+((-210*d-252*e)*x^6+(-126*d-105*e)*x^5+(-70*d
-40*e)*x^4+(-30*d-45/4*e)*x^3+(-9*d-2*e)*x^2+(-5/3*d-1/6*e)*x-1/7*d)/x^7+120*d*ln(x)+210*e*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^8} \, dx=\frac {21 \, e x^{11} + 28 \, {\left (d + 10 \, e\right )} x^{10} + 210 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 1260 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 2520 \, {\left (4 \, d + 7 \, e\right )} x^{7} \log \left (x\right ) - 3528 \, {\left (5 \, d + 6 \, e\right )} x^{6} - 1764 \, {\left (6 \, d + 5 \, e\right )} x^{5} - 840 \, {\left (7 \, d + 4 \, e\right )} x^{4} - 315 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 84 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 14 \, {\left (10 \, d + e\right )} x - 12 \, d}{84 \, x^{7}} \]

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^8,x, algorithm="fricas")

[Out]

1/84*(21*e*x^11 + 28*(d + 10*e)*x^10 + 210*(2*d + 9*e)*x^9 + 1260*(3*d + 8*e)*x^8 + 2520*(4*d + 7*e)*x^7*log(x
) - 3528*(5*d + 6*e)*x^6 - 1764*(6*d + 5*e)*x^5 - 840*(7*d + 4*e)*x^4 - 315*(8*d + 3*e)*x^3 - 84*(9*d + 2*e)*x
^2 - 14*(10*d + e)*x - 12*d)/x^7

Sympy [A] (verification not implemented)

Time = 1.83 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^8} \, dx=\frac {e x^{4}}{4} + x^{3} \left (\frac {d}{3} + \frac {10 e}{3}\right ) + x^{2} \cdot \left (5 d + \frac {45 e}{2}\right ) + x \left (45 d + 120 e\right ) + 30 \cdot \left (4 d + 7 e\right ) \log {\left (x \right )} + \frac {- 12 d + x^{6} \left (- 17640 d - 21168 e\right ) + x^{5} \left (- 10584 d - 8820 e\right ) + x^{4} \left (- 5880 d - 3360 e\right ) + x^{3} \left (- 2520 d - 945 e\right ) + x^{2} \left (- 756 d - 168 e\right ) + x \left (- 140 d - 14 e\right )}{84 x^{7}} \]

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**8,x)

[Out]

e*x**4/4 + x**3*(d/3 + 10*e/3) + x**2*(5*d + 45*e/2) + x*(45*d + 120*e) + 30*(4*d + 7*e)*log(x) + (-12*d + x**
6*(-17640*d - 21168*e) + x**5*(-10584*d - 8820*e) + x**4*(-5880*d - 3360*e) + x**3*(-2520*d - 945*e) + x**2*(-
756*d - 168*e) + x*(-140*d - 14*e))/(84*x**7)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^8} \, dx=\frac {1}{4} \, e x^{4} + \frac {1}{3} \, {\left (d + 10 \, e\right )} x^{3} + \frac {5}{2} \, {\left (2 \, d + 9 \, e\right )} x^{2} + 15 \, {\left (3 \, d + 8 \, e\right )} x + 30 \, {\left (4 \, d + 7 \, e\right )} \log \left (x\right ) - \frac {3528 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 1764 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 840 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 315 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 84 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 14 \, {\left (10 \, d + e\right )} x + 12 \, d}{84 \, x^{7}} \]

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^8,x, algorithm="maxima")

[Out]

1/4*e*x^4 + 1/3*(d + 10*e)*x^3 + 5/2*(2*d + 9*e)*x^2 + 15*(3*d + 8*e)*x + 30*(4*d + 7*e)*log(x) - 1/84*(3528*(
5*d + 6*e)*x^6 + 1764*(6*d + 5*e)*x^5 + 840*(7*d + 4*e)*x^4 + 315*(8*d + 3*e)*x^3 + 84*(9*d + 2*e)*x^2 + 14*(1
0*d + e)*x + 12*d)/x^7

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^8} \, dx=\frac {1}{4} \, e x^{4} + \frac {1}{3} \, d x^{3} + \frac {10}{3} \, e x^{3} + 5 \, d x^{2} + \frac {45}{2} \, e x^{2} + 45 \, d x + 120 \, e x + 30 \, {\left (4 \, d + 7 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {3528 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 1764 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 840 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 315 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 84 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 14 \, {\left (10 \, d + e\right )} x + 12 \, d}{84 \, x^{7}} \]

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^8,x, algorithm="giac")

[Out]

1/4*e*x^4 + 1/3*d*x^3 + 10/3*e*x^3 + 5*d*x^2 + 45/2*e*x^2 + 45*d*x + 120*e*x + 30*(4*d + 7*e)*log(abs(x)) - 1/
84*(3528*(5*d + 6*e)*x^6 + 1764*(6*d + 5*e)*x^5 + 840*(7*d + 4*e)*x^4 + 315*(8*d + 3*e)*x^3 + 84*(9*d + 2*e)*x
^2 + 14*(10*d + e)*x + 12*d)/x^7

Mupad [B] (verification not implemented)

Time = 9.97 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^8} \, dx=x^3\,\left (\frac {d}{3}+\frac {10\,e}{3}\right )+x^2\,\left (5\,d+\frac {45\,e}{2}\right )+\ln \left (x\right )\,\left (120\,d+210\,e\right )+\frac {e\,x^4}{4}-\frac {\left (210\,d+252\,e\right )\,x^6+\left (126\,d+105\,e\right )\,x^5+\left (70\,d+40\,e\right )\,x^4+\left (30\,d+\frac {45\,e}{4}\right )\,x^3+\left (9\,d+2\,e\right )\,x^2+\left (\frac {5\,d}{3}+\frac {e}{6}\right )\,x+\frac {d}{7}}{x^7}+x\,\left (45\,d+120\,e\right ) \]

[In]

int(((d + e*x)*(2*x + x^2 + 1)^5)/x^8,x)

[Out]

x^3*(d/3 + (10*e)/3) + x^2*(5*d + (45*e)/2) + log(x)*(120*d + 210*e) + (e*x^4)/4 - (d/7 + x^2*(9*d + 2*e) + x^
3*(30*d + (45*e)/4) + x^4*(70*d + 40*e) + x^5*(126*d + 105*e) + x^6*(210*d + 252*e) + x*((5*d)/3 + e/6))/x^7 +
x*(45*d + 120*e)