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

Optimal. Leaf size=138 \[ -\frac {10 d+e}{6 x^6}-\frac {9 d+2 e}{x^5}-\frac {15 (8 d+3 e)}{4 x^4}+\frac {1}{3} x^3 (d+10 e)-\frac {10 (7 d+4 e)}{x^3}+\frac {5}{2} x^2 (2 d+9 e)-\frac {21 (6 d+5 e)}{x^2}+15 x (3 d+8 e)-\frac {42 (5 d+6 e)}{x}+30 (4 d+7 e) \log (x)-\frac {d}{7 x^7}+\frac {e x^4}{4} \]

________________________________________________________________________________________

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}{3} x^3 (d+10 e)+\frac {5}{2} x^2 (2 d+9 e)-\frac {21 (6 d+5 e)}{x^2}-\frac {10 (7 d+4 e)}{x^3}-\frac {15 (8 d+3 e)}{4 x^4}-\frac {9 d+2 e}{x^5}-\frac {10 d+e}{6 x^6}+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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-d/(7*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 76

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*} \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^8} \, dx &=\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]  time = 0.04, size = 139, normalized size = 1.01 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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]

________________________________________________________________________________________

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^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 131, normalized size = 0.95 \begin {gather*} \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 \relax (x) - 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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

giac [A]  time = 0.15, size = 139, normalized size = 1.01 \begin {gather*} \frac {1}{4} \, x^{4} e + \frac {1}{3} \, d x^{3} + \frac {10}{3} \, x^{3} e + 5 \, d x^{2} + \frac {45}{2} \, x^{2} e + 45 \, d x + 120 \, x e + 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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*e + 1/3*d*x^3 + 10/3*x^3*e + 5*d*x^2 + 45/2*x^2*e + 45*d*x + 120*x*e + 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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 128, normalized size = 0.93 \begin {gather*} \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 d \ln \relax (x )+120 e x +210 e \ln \relax (x )-\frac {210 d}{x}-\frac {252 e}{x}-\frac {126 d}{x^{2}}-\frac {105 e}{x^{2}}-\frac {70 d}{x^{3}}-\frac {40 e}{x^{3}}-\frac {30 d}{x^{4}}-\frac {45 e}{4 x^{4}}-\frac {9 d}{x^{5}}-\frac {2 e}{x^{5}}-\frac {5 d}{3 x^{6}}-\frac {e}{6 x^{6}}-\frac {d}{7 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

________________________________________________________________________________________

maxima [A]  time = 0.59, size = 127, normalized size = 0.92 \begin {gather*} \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 \relax (x) - \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

mupad [B]  time = 1.07, size = 121, normalized size = 0.88 \begin {gather*} x^3\,\left (\frac {d}{3}+\frac {10\,e}{3}\right )+x^2\,\left (5\,d+\frac {45\,e}{2}\right )+\ln \relax (x)\,\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 2.99, size = 128, normalized size = 0.93 \begin {gather*} \frac {e x^{4}}{4} + x^{3} \left (\frac {d}{3} + \frac {10 e}{3}\right ) + x^{2} \left (5 d + \frac {45 e}{2}\right ) + x \left (45 d + 120 e\right ) + 30 \left (4 d + 7 e\right ) \log {\relax (x )} + \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________