\(\int \frac {(d+e x) (1+2 x+x^2)^5}{x^7} \, dx\) [573]

Optimal result

Integrand size = 19, antiderivative size = 140 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=-\frac {d}{6 x^6}-\frac {10 d+e}{5 x^5}-\frac {5 (9 d+2 e)}{4 x^4}-\frac {5 (8 d+3 e)}{x^3}-\frac {15 (7 d+4 e)}{x^2}-\frac {42 (6 d+5 e)}{x}+30 (4 d+7 e) x+\frac {15}{2} (3 d+8 e) x^2+\frac {5}{3} (2 d+9 e) x^3+\frac {1}{4} (d+10 e) x^4+\frac {e x^5}{5}+42 (5 d+6 e) \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 140, 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} \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=-\frac {10 d+e}{5 x^5}+\frac {1}{4} x^4 (d+10 e)-\frac {5 (9 d+2 e)}{4 x^4}+\frac {5}{3} x^3 (2 d+9 e)-\frac {5 (8 d+3 e)}{x^3}+\frac {15}{2} x^2 (3 d+8 e)-\frac {15 (7 d+4 e)}{x^2}+30 x (4 d+7 e)-\frac {42 (6 d+5 e)}{x}+42 (5 d+6 e) \log (x)-\frac {d}{6 x^6}+\frac {e x^5}{5} \]

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^7,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=-\frac {d}{6 x^6}+\frac {-10 d-e}{5 x^5}-\frac {5 (9 d+2 e)}{4 x^4}-\frac {5 (8 d+3 e)}{x^3}-\frac {15 (7 d+4 e)}{x^2}-\frac {42 (6 d+5 e)}{x}+30 (4 d+7 e) x+\frac {15}{2} (3 d+8 e) x^2+\frac {5}{3} (2 d+9 e) x^3+\frac {1}{4} (d+10 e) x^4+\frac {e x^5}{5}+42 (5 d+6 e) \log (x) \]

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^7,x]

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.88

method	result	size
norman	\(\frac {\left (-252 d -210 e \right ) x^{5}+\left (-105 d -60 e \right ) x^{4}+\left (-40 d -15 e \right ) x^{3}+\left (-2 d -\frac {e}{5}\right ) x +\left (120 d +210 e \right ) x^{7}+\left (-\frac {45 d}{4}-\frac {5 e}{2}\right ) x^{2}+\left (\frac {d}{4}+\frac {5 e}{2}\right ) x^{10}+\left (\frac {10 d}{3}+15 e \right ) x^{9}+\left (\frac {45 d}{2}+60 e \right ) x^{8}-\frac {d}{6}+\frac {e \,x^{11}}{5}}{x^{6}}+\left (210 d +252 e \right ) \ln \left (x \right )\)	\(123\)
risch	\(\frac {e \,x^{5}}{5}+\frac {d \,x^{4}}{4}+\frac {5 e \,x^{4}}{2}+\frac {10 d \,x^{3}}{3}+15 e \,x^{3}+\frac {45 d \,x^{2}}{2}+60 e \,x^{2}+120 d x +210 e x +\frac {\left (-252 d -210 e \right ) x^{5}+\left (-105 d -60 e \right ) x^{4}+\left (-40 d -15 e \right ) x^{3}+\left (-\frac {45 d}{4}-\frac {5 e}{2}\right ) x^{2}+\left (-2 d -\frac {e}{5}\right ) x -\frac {d}{6}}{x^{6}}+210 d \ln \left (x \right )+252 e \ln \left (x \right )\)	\(123\)
default	\(\frac {e \,x^{5}}{5}+\frac {d \,x^{4}}{4}+\frac {5 e \,x^{4}}{2}+\frac {10 d \,x^{3}}{3}+15 e \,x^{3}+\frac {45 d \,x^{2}}{2}+60 e \,x^{2}+120 d x +210 e x -\frac {d}{6 x^{6}}-\frac {45 d +10 e}{4 x^{4}}-\frac {10 d +e}{5 x^{5}}+\left (210 d +252 e \right ) \ln \left (x \right )-\frac {210 d +120 e}{2 x^{2}}-\frac {252 d +210 e}{x}-\frac {120 d +45 e}{3 x^{3}}\)	\(126\)
parallelrisch	\(\frac {12 e \,x^{11}+15 d \,x^{10}+150 e \,x^{10}+200 d \,x^{9}+900 e \,x^{9}+1350 d \,x^{8}+3600 e \,x^{8}+12600 \ln \left (x \right ) x^{6} d +15120 \ln \left (x \right ) x^{6} e +7200 d \,x^{7}+12600 e \,x^{7}-15120 d \,x^{5}-12600 e \,x^{5}-6300 d \,x^{4}-3600 e \,x^{4}-2400 d \,x^{3}-900 e \,x^{3}-675 d \,x^{2}-150 e \,x^{2}-120 d x -12 e x -10 d}{60 x^{6}}\)	\(136\)

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^7,x,method=_RETURNVERBOSE)

[Out]

((-252*d-210*e)*x^5+(-105*d-60*e)*x^4+(-40*d-15*e)*x^3+(-2*d-1/5*e)*x+(120*d+210*e)*x^7+(-45/4*d-5/2*e)*x^2+(1
/4*d+5/2*e)*x^10+(10/3*d+15*e)*x^9+(45/2*d+60*e)*x^8-1/6*d+1/5*e*x^11)/x^6+(210*d+252*e)*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=\frac {12 \, e x^{11} + 15 \, {\left (d + 10 \, e\right )} x^{10} + 100 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 450 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 1800 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 2520 \, {\left (5 \, d + 6 \, e\right )} x^{6} \log \left (x\right ) - 2520 \, {\left (6 \, d + 5 \, e\right )} x^{5} - 900 \, {\left (7 \, d + 4 \, e\right )} x^{4} - 300 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 75 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 12 \, {\left (10 \, d + e\right )} x - 10 \, d}{60 \, x^{6}} \]

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^7,x, algorithm="fricas")

[Out]

1/60*(12*e*x^11 + 15*(d + 10*e)*x^10 + 100*(2*d + 9*e)*x^9 + 450*(3*d + 8*e)*x^8 + 1800*(4*d + 7*e)*x^7 + 2520
*(5*d + 6*e)*x^6*log(x) - 2520*(6*d + 5*e)*x^5 - 900*(7*d + 4*e)*x^4 - 300*(8*d + 3*e)*x^3 - 75*(9*d + 2*e)*x^
2 - 12*(10*d + e)*x - 10*d)/x^6

Sympy [A] (verification not implemented)

Time = 1.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=\frac {e x^{5}}{5} + x^{4} \left (\frac {d}{4} + \frac {5 e}{2}\right ) + x^{3} \cdot \left (\frac {10 d}{3} + 15 e\right ) + x^{2} \cdot \left (\frac {45 d}{2} + 60 e\right ) + x \left (120 d + 210 e\right ) + 42 \cdot \left (5 d + 6 e\right ) \log {\left (x \right )} + \frac {- 10 d + x^{5} \left (- 15120 d - 12600 e\right ) + x^{4} \left (- 6300 d - 3600 e\right ) + x^{3} \left (- 2400 d - 900 e\right ) + x^{2} \left (- 675 d - 150 e\right ) + x \left (- 120 d - 12 e\right )}{60 x^{6}} \]

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**7,x)

[Out]

e*x**5/5 + x**4*(d/4 + 5*e/2) + x**3*(10*d/3 + 15*e) + x**2*(45*d/2 + 60*e) + x*(120*d + 210*e) + 42*(5*d + 6*
e)*log(x) + (-10*d + x**5*(-15120*d - 12600*e) + x**4*(-6300*d - 3600*e) + x**3*(-2400*d - 900*e) + x**2*(-675
*d - 150*e) + x*(-120*d - 12*e))/(60*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=\frac {1}{5} \, e x^{5} + \frac {1}{4} \, {\left (d + 10 \, e\right )} x^{4} + \frac {5}{3} \, {\left (2 \, d + 9 \, e\right )} x^{3} + \frac {15}{2} \, {\left (3 \, d + 8 \, e\right )} x^{2} + 30 \, {\left (4 \, d + 7 \, e\right )} x + 42 \, {\left (5 \, d + 6 \, e\right )} \log \left (x\right ) - \frac {2520 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 900 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 300 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 75 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 12 \, {\left (10 \, d + e\right )} x + 10 \, d}{60 \, x^{6}} \]

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^7,x, algorithm="maxima")

[Out]

1/5*e*x^5 + 1/4*(d + 10*e)*x^4 + 5/3*(2*d + 9*e)*x^3 + 15/2*(3*d + 8*e)*x^2 + 30*(4*d + 7*e)*x + 42*(5*d + 6*e
)*log(x) - 1/60*(2520*(6*d + 5*e)*x^5 + 900*(7*d + 4*e)*x^4 + 300*(8*d + 3*e)*x^3 + 75*(9*d + 2*e)*x^2 + 12*(1
0*d + e)*x + 10*d)/x^6

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=\frac {1}{5} \, e x^{5} + \frac {1}{4} \, d x^{4} + \frac {5}{2} \, e x^{4} + \frac {10}{3} \, d x^{3} + 15 \, e x^{3} + \frac {45}{2} \, d x^{2} + 60 \, e x^{2} + 120 \, d x + 210 \, e x + 42 \, {\left (5 \, d + 6 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {2520 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 900 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 300 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 75 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 12 \, {\left (10 \, d + e\right )} x + 10 \, d}{60 \, x^{6}} \]

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^7,x, algorithm="giac")

[Out]

1/5*e*x^5 + 1/4*d*x^4 + 5/2*e*x^4 + 10/3*d*x^3 + 15*e*x^3 + 45/2*d*x^2 + 60*e*x^2 + 120*d*x + 210*e*x + 42*(5*
d + 6*e)*log(abs(x)) - 1/60*(2520*(6*d + 5*e)*x^5 + 900*(7*d + 4*e)*x^4 + 300*(8*d + 3*e)*x^3 + 75*(9*d + 2*e)
*x^2 + 12*(10*d + e)*x + 10*d)/x^6

Mupad [B] (verification not implemented)

Time = 9.96 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=x^4\,\left (\frac {d}{4}+\frac {5\,e}{2}\right )+x^3\,\left (\frac {10\,d}{3}+15\,e\right )+x^2\,\left (\frac {45\,d}{2}+60\,e\right )+\ln \left (x\right )\,\left (210\,d+252\,e\right )-\frac {\left (252\,d+210\,e\right )\,x^5+\left (105\,d+60\,e\right )\,x^4+\left (40\,d+15\,e\right )\,x^3+\left (\frac {45\,d}{4}+\frac {5\,e}{2}\right )\,x^2+\left (2\,d+\frac {e}{5}\right )\,x+\frac {d}{6}}{x^6}+\frac {e\,x^5}{5}+x\,\left (120\,d+210\,e\right ) \]

[In]

int(((d + e*x)*(2*x + x^2 + 1)^5)/x^7,x)

[Out]

x^4*(d/4 + (5*e)/2) + x^3*((10*d)/3 + 15*e) + x^2*((45*d)/2 + 60*e) + log(x)*(210*d + 252*e) - (d/6 + x^2*((45
*d)/4 + (5*e)/2) + x^3*(40*d + 15*e) + x^4*(105*d + 60*e) + x^5*(252*d + 210*e) + x*(2*d + e/5))/x^6 + (e*x^5)
/5 + x*(120*d + 210*e)