3.6.2 \(\int \frac {(d+e x) (1+2 x+x^2)^5}{x^9} \, dx\)

Optimal. Leaf size=138 \[ -\frac {10 d+e}{7 x^7}-\frac {5 (9 d+2 e)}{6 x^6}-\frac {3 (8 d+3 e)}{x^5}-\frac {15 (7 d+4 e)}{2 x^4}-\frac {14 (6 d+5 e)}{x^3}+\frac {1}{2} x^2 (d+10 e)-\frac {21 (5 d+6 e)}{x^2}+5 x (2 d+9 e)-\frac {30 (4 d+7 e)}{x}+15 (3 d+8 e) \log (x)-\frac {d}{8 x^8}+\frac {e x^3}{3} \]

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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}{2} x^2 (d+10 e)-\frac {21 (5 d+6 e)}{x^2}-\frac {14 (6 d+5 e)}{x^3}-\frac {15 (7 d+4 e)}{2 x^4}-\frac {3 (8 d+3 e)}{x^5}-\frac {5 (9 d+2 e)}{6 x^6}-\frac {10 d+e}{7 x^7}+5 x (2 d+9 e)-\frac {30 (4 d+7 e)}{x}+15 (3 d+8 e) \log (x)-\frac {d}{8 x^8}+\frac {e x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^9,x]

[Out]

-d/(8*x^8) - (10*d + e)/(7*x^7) - (5*(9*d + 2*e))/(6*x^6) - (3*(8*d + 3*e))/x^5 - (15*(7*d + 4*e))/(2*x^4) - (
14*(6*d + 5*e))/x^3 - (21*(5*d + 6*e))/x^2 - (30*(4*d + 7*e))/x + 5*(2*d + 9*e)*x + ((d + 10*e)*x^2)/2 + (e*x^
3)/3 + 15*(3*d + 8*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^9} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^9} \, dx\\ &=\int \left (5 (2 d+9 e)+\frac {d}{x^9}+\frac {10 d+e}{x^8}+\frac {5 (9 d+2 e)}{x^7}+\frac {15 (8 d+3 e)}{x^6}+\frac {30 (7 d+4 e)}{x^5}+\frac {42 (6 d+5 e)}{x^4}+\frac {42 (5 d+6 e)}{x^3}+\frac {30 (4 d+7 e)}{x^2}+\frac {15 (3 d+8 e)}{x}+(d+10 e) x+e x^2\right ) \, dx\\ &=-\frac {d}{8 x^8}-\frac {10 d+e}{7 x^7}-\frac {5 (9 d+2 e)}{6 x^6}-\frac {3 (8 d+3 e)}{x^5}-\frac {15 (7 d+4 e)}{2 x^4}-\frac {14 (6 d+5 e)}{x^3}-\frac {21 (5 d+6 e)}{x^2}-\frac {30 (4 d+7 e)}{x}+5 (2 d+9 e) x+\frac {1}{2} (d+10 e) x^2+\frac {e x^3}{3}+15 (3 d+8 e) \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 140, normalized size = 1.01 \begin {gather*} \frac {-10 d-e}{7 x^7}-\frac {5 (9 d+2 e)}{6 x^6}-\frac {3 (8 d+3 e)}{x^5}-\frac {15 (7 d+4 e)}{2 x^4}-\frac {14 (6 d+5 e)}{x^3}+\frac {1}{2} x^2 (d+10 e)-\frac {21 (5 d+6 e)}{x^2}+5 x (2 d+9 e)-\frac {30 (4 d+7 e)}{x}+15 (3 d+8 e) \log (x)-\frac {d}{8 x^8}+\frac {e x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^9,x]

[Out]

-1/8*d/x^8 + (-10*d - e)/(7*x^7) - (5*(9*d + 2*e))/(6*x^6) - (3*(8*d + 3*e))/x^5 - (15*(7*d + 4*e))/(2*x^4) -
(14*(6*d + 5*e))/x^3 - (21*(5*d + 6*e))/x^2 - (30*(4*d + 7*e))/x + 5*(2*d + 9*e)*x + ((d + 10*e)*x^2)/2 + (e*x
^3)/3 + 15*(3*d + 8*e)*Log[x]

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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^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^9,x]

[Out]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^9, x]

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fricas [A]  time = 0.42, size = 131, normalized size = 0.95 \begin {gather*} \frac {56 \, e x^{11} + 84 \, {\left (d + 10 \, e\right )} x^{10} + 840 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 2520 \, {\left (3 \, d + 8 \, e\right )} x^{8} \log \relax (x) - 5040 \, {\left (4 \, d + 7 \, e\right )} x^{7} - 3528 \, {\left (5 \, d + 6 \, e\right )} x^{6} - 2352 \, {\left (6 \, d + 5 \, e\right )} x^{5} - 1260 \, {\left (7 \, d + 4 \, e\right )} x^{4} - 504 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 140 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 24 \, {\left (10 \, d + e\right )} x - 21 \, d}{168 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^9,x, algorithm="fricas")

[Out]

1/168*(56*e*x^11 + 84*(d + 10*e)*x^10 + 840*(2*d + 9*e)*x^9 + 2520*(3*d + 8*e)*x^8*log(x) - 5040*(4*d + 7*e)*x
^7 - 3528*(5*d + 6*e)*x^6 - 2352*(6*d + 5*e)*x^5 - 1260*(7*d + 4*e)*x^4 - 504*(8*d + 3*e)*x^3 - 140*(9*d + 2*e
)*x^2 - 24*(10*d + e)*x - 21*d)/x^8

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giac [A]  time = 0.16, size = 139, normalized size = 1.01 \begin {gather*} \frac {1}{3} \, x^{3} e + \frac {1}{2} \, d x^{2} + 5 \, x^{2} e + 10 \, d x + 45 \, x e + 15 \, {\left (3 \, d + 8 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {5040 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 3528 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 2352 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 1260 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 504 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 140 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 24 \, {\left (10 \, d + e\right )} x + 21 \, d}{168 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^9,x, algorithm="giac")

[Out]

1/3*x^3*e + 1/2*d*x^2 + 5*x^2*e + 10*d*x + 45*x*e + 15*(3*d + 8*e)*log(abs(x)) - 1/168*(5040*(4*d + 7*e)*x^7 +
3528*(5*d + 6*e)*x^6 + 2352*(6*d + 5*e)*x^5 + 1260*(7*d + 4*e)*x^4 + 504*(8*d + 3*e)*x^3 + 140*(9*d + 2*e)*x^
2 + 24*(10*d + e)*x + 21*d)/x^8

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maple [A]  time = 0.06, size = 128, normalized size = 0.93 \begin {gather*} \frac {e \,x^{3}}{3}+\frac {d \,x^{2}}{2}+5 e \,x^{2}+10 d x +45 d \ln \relax (x )+45 e x +120 e \ln \relax (x )-\frac {120 d}{x}-\frac {210 e}{x}-\frac {105 d}{x^{2}}-\frac {126 e}{x^{2}}-\frac {84 d}{x^{3}}-\frac {70 e}{x^{3}}-\frac {105 d}{2 x^{4}}-\frac {30 e}{x^{4}}-\frac {24 d}{x^{5}}-\frac {9 e}{x^{5}}-\frac {15 d}{2 x^{6}}-\frac {5 e}{3 x^{6}}-\frac {10 d}{7 x^{7}}-\frac {e}{7 x^{7}}-\frac {d}{8 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^9,x)

[Out]

1/3*e*x^3+1/2*d*x^2+5*e*x^2+10*d*x+45*e*x-24*d/x^5-9*e/x^5-105/2*d/x^4-30*e/x^4-84*d/x^3-70*e/x^3-1/8*d/x^8-10
5*d/x^2-126*e/x^2-10/7*d/x^7-1/7*e/x^7-15/2*d/x^6-5/3*e/x^6-120*d/x-210*e/x+45*d*ln(x)+120*e*ln(x)

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maxima [A]  time = 0.47, size = 127, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, e x^{3} + \frac {1}{2} \, {\left (d + 10 \, e\right )} x^{2} + 5 \, {\left (2 \, d + 9 \, e\right )} x + 15 \, {\left (3 \, d + 8 \, e\right )} \log \relax (x) - \frac {5040 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 3528 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 2352 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 1260 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 504 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 140 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 24 \, {\left (10 \, d + e\right )} x + 21 \, d}{168 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^9,x, algorithm="maxima")

[Out]

1/3*e*x^3 + 1/2*(d + 10*e)*x^2 + 5*(2*d + 9*e)*x + 15*(3*d + 8*e)*log(x) - 1/168*(5040*(4*d + 7*e)*x^7 + 3528*
(5*d + 6*e)*x^6 + 2352*(6*d + 5*e)*x^5 + 1260*(7*d + 4*e)*x^4 + 504*(8*d + 3*e)*x^3 + 140*(9*d + 2*e)*x^2 + 24
*(10*d + e)*x + 21*d)/x^8

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mupad [B]  time = 1.08, size = 121, normalized size = 0.88 \begin {gather*} x^2\,\left (\frac {d}{2}+5\,e\right )+\ln \relax (x)\,\left (45\,d+120\,e\right )+\frac {e\,x^3}{3}+x\,\left (10\,d+45\,e\right )-\frac {\left (120\,d+210\,e\right )\,x^7+\left (105\,d+126\,e\right )\,x^6+\left (84\,d+70\,e\right )\,x^5+\left (\frac {105\,d}{2}+30\,e\right )\,x^4+\left (24\,d+9\,e\right )\,x^3+\left (\frac {15\,d}{2}+\frac {5\,e}{3}\right )\,x^2+\left (\frac {10\,d}{7}+\frac {e}{7}\right )\,x+\frac {d}{8}}{x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x)*(2*x + x^2 + 1)^5)/x^9,x)

[Out]

x^2*(d/2 + 5*e) + log(x)*(45*d + 120*e) + (e*x^3)/3 + x*(10*d + 45*e) - (d/8 + x^2*((15*d)/2 + (5*e)/3) + x^3*
(24*d + 9*e) + x^4*((105*d)/2 + 30*e) + x^5*(84*d + 70*e) + x^6*(105*d + 126*e) + x^7*(120*d + 210*e) + x*((10
*d)/7 + e/7))/x^8

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sympy [A]  time = 4.27, size = 126, normalized size = 0.91 \begin {gather*} \frac {e x^{3}}{3} + x^{2} \left (\frac {d}{2} + 5 e\right ) + x \left (10 d + 45 e\right ) + 15 \left (3 d + 8 e\right ) \log {\relax (x )} + \frac {- 21 d + x^{7} \left (- 20160 d - 35280 e\right ) + x^{6} \left (- 17640 d - 21168 e\right ) + x^{5} \left (- 14112 d - 11760 e\right ) + x^{4} \left (- 8820 d - 5040 e\right ) + x^{3} \left (- 4032 d - 1512 e\right ) + x^{2} \left (- 1260 d - 280 e\right ) + x \left (- 240 d - 24 e\right )}{168 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**9,x)

[Out]

e*x**3/3 + x**2*(d/2 + 5*e) + x*(10*d + 45*e) + 15*(3*d + 8*e)*log(x) + (-21*d + x**7*(-20160*d - 35280*e) + x
**6*(-17640*d - 21168*e) + x**5*(-14112*d - 11760*e) + x**4*(-8820*d - 5040*e) + x**3*(-4032*d - 1512*e) + x**
2*(-1260*d - 280*e) + x*(-240*d - 24*e))/(168*x**8)

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