3.6.3 \(\int \frac {(d+e x) (1+2 x+x^2)^5}{x^{10}} \, dx\)

Optimal. Leaf size=137 \[ -\frac {10 d+e}{8 x^8}-\frac {5 (9 d+2 e)}{7 x^7}-\frac {5 (8 d+3 e)}{2 x^6}-\frac {6 (7 d+4 e)}{x^5}-\frac {21 (6 d+5 e)}{2 x^4}-\frac {14 (5 d+6 e)}{x^3}-\frac {15 (4 d+7 e)}{x^2}+x (d+10 e)-\frac {15 (3 d+8 e)}{x}+5 (2 d+9 e) \log (x)-\frac {d}{9 x^9}+\frac {e x^2}{2} \]

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Rubi [A]  time = 0.08, antiderivative size = 137, 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 {15 (4 d+7 e)}{x^2}-\frac {14 (5 d+6 e)}{x^3}-\frac {21 (6 d+5 e)}{2 x^4}-\frac {6 (7 d+4 e)}{x^5}-\frac {5 (8 d+3 e)}{2 x^6}-\frac {5 (9 d+2 e)}{7 x^7}-\frac {10 d+e}{8 x^8}+x (d+10 e)-\frac {15 (3 d+8 e)}{x}+5 (2 d+9 e) \log (x)-\frac {d}{9 x^9}+\frac {e x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^10,x]

[Out]

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

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Mathematica [A]  time = 0.04, size = 139, normalized size = 1.01 \begin {gather*} \frac {-10 d-e}{8 x^8}-\frac {5 (9 d+2 e)}{7 x^7}-\frac {5 (8 d+3 e)}{2 x^6}-\frac {6 (7 d+4 e)}{x^5}-\frac {21 (6 d+5 e)}{2 x^4}-\frac {14 (5 d+6 e)}{x^3}-\frac {15 (4 d+7 e)}{x^2}+x (d+10 e)-\frac {15 (3 d+8 e)}{x}+5 (2 d+9 e) \log (x)-\frac {d}{9 x^9}+\frac {e x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^10,x]

[Out]

-1/9*d/x^9 + (-10*d - e)/(8*x^8) - (5*(9*d + 2*e))/(7*x^7) - (5*(8*d + 3*e))/(2*x^6) - (6*(7*d + 4*e))/x^5 - (
21*(6*d + 5*e))/(2*x^4) - (14*(5*d + 6*e))/x^3 - (15*(4*d + 7*e))/x^2 - (15*(3*d + 8*e))/x + (d + 10*e)*x + (e
*x^2)/2 + 5*(2*d + 9*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^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^10,x]

[Out]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^10, x]

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fricas [A]  time = 0.42, size = 131, normalized size = 0.96 \begin {gather*} \frac {252 \, e x^{11} + 504 \, {\left (d + 10 \, e\right )} x^{10} + 2520 \, {\left (2 \, d + 9 \, e\right )} x^{9} \log \relax (x) - 7560 \, {\left (3 \, d + 8 \, e\right )} x^{8} - 7560 \, {\left (4 \, d + 7 \, e\right )} x^{7} - 7056 \, {\left (5 \, d + 6 \, e\right )} x^{6} - 5292 \, {\left (6 \, d + 5 \, e\right )} x^{5} - 3024 \, {\left (7 \, d + 4 \, e\right )} x^{4} - 1260 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 360 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 63 \, {\left (10 \, d + e\right )} x - 56 \, d}{504 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^10,x, algorithm="fricas")

[Out]

1/504*(252*e*x^11 + 504*(d + 10*e)*x^10 + 2520*(2*d + 9*e)*x^9*log(x) - 7560*(3*d + 8*e)*x^8 - 7560*(4*d + 7*e
)*x^7 - 7056*(5*d + 6*e)*x^6 - 5292*(6*d + 5*e)*x^5 - 3024*(7*d + 4*e)*x^4 - 1260*(8*d + 3*e)*x^3 - 360*(9*d +
2*e)*x^2 - 63*(10*d + e)*x - 56*d)/x^9

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giac [A]  time = 0.17, size = 138, normalized size = 1.01 \begin {gather*} \frac {1}{2} \, x^{2} e + d x + 10 \, x e + 5 \, {\left (2 \, d + 9 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {7560 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 7560 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 7056 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 5292 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 3024 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 1260 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 360 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 63 \, {\left (10 \, d + e\right )} x + 56 \, d}{504 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^10,x, algorithm="giac")

[Out]

1/2*x^2*e + d*x + 10*x*e + 5*(2*d + 9*e)*log(abs(x)) - 1/504*(7560*(3*d + 8*e)*x^8 + 7560*(4*d + 7*e)*x^7 + 70
56*(5*d + 6*e)*x^6 + 5292*(6*d + 5*e)*x^5 + 3024*(7*d + 4*e)*x^4 + 1260*(8*d + 3*e)*x^3 + 360*(9*d + 2*e)*x^2
+ 63*(10*d + e)*x + 56*d)/x^9

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maple [A]  time = 0.06, size = 127, normalized size = 0.93 \begin {gather*} \frac {e \,x^{2}}{2}+d x +10 d \ln \relax (x )+10 e x +45 e \ln \relax (x )-\frac {45 d}{x}-\frac {120 e}{x}-\frac {60 d}{x^{2}}-\frac {105 e}{x^{2}}-\frac {70 d}{x^{3}}-\frac {84 e}{x^{3}}-\frac {63 d}{x^{4}}-\frac {105 e}{2 x^{4}}-\frac {42 d}{x^{5}}-\frac {24 e}{x^{5}}-\frac {20 d}{x^{6}}-\frac {15 e}{2 x^{6}}-\frac {45 d}{7 x^{7}}-\frac {10 e}{7 x^{7}}-\frac {5 d}{4 x^{8}}-\frac {e}{8 x^{8}}-\frac {d}{9 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^10,x)

[Out]

1/2*e*x^2+d*x+10*e*x-42*d/x^5-24*e/x^5-63*d/x^4-105/2*e/x^4-70*d/x^3-84*e/x^3-5/4*d/x^8-1/8*e/x^8-60*d/x^2-105
*e/x^2-1/9*d/x^9-45/7*d/x^7-10/7*e/x^7-20*d/x^6-15/2*e/x^6-45*d/x-120*e/x+10*d*ln(x)+45*e*ln(x)

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maxima [A]  time = 0.51, size = 126, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, e x^{2} + {\left (d + 10 \, e\right )} x + 5 \, {\left (2 \, d + 9 \, e\right )} \log \relax (x) - \frac {7560 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 7560 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 7056 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 5292 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 3024 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 1260 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 360 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 63 \, {\left (10 \, d + e\right )} x + 56 \, d}{504 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^10,x, algorithm="maxima")

[Out]

1/2*e*x^2 + (d + 10*e)*x + 5*(2*d + 9*e)*log(x) - 1/504*(7560*(3*d + 8*e)*x^8 + 7560*(4*d + 7*e)*x^7 + 7056*(5
*d + 6*e)*x^6 + 5292*(6*d + 5*e)*x^5 + 3024*(7*d + 4*e)*x^4 + 1260*(8*d + 3*e)*x^3 + 360*(9*d + 2*e)*x^2 + 63*
(10*d + e)*x + 56*d)/x^9

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mupad [B]  time = 0.07, size = 119, normalized size = 0.87 \begin {gather*} \ln \relax (x)\,\left (10\,d+45\,e\right )+x\,\left (d+10\,e\right )+\frac {e\,x^2}{2}-\frac {\left (45\,d+120\,e\right )\,x^8+\left (60\,d+105\,e\right )\,x^7+\left (70\,d+84\,e\right )\,x^6+\left (63\,d+\frac {105\,e}{2}\right )\,x^5+\left (42\,d+24\,e\right )\,x^4+\left (20\,d+\frac {15\,e}{2}\right )\,x^3+\left (\frac {45\,d}{7}+\frac {10\,e}{7}\right )\,x^2+\left (\frac {5\,d}{4}+\frac {e}{8}\right )\,x+\frac {d}{9}}{x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x)*(2*x + x^2 + 1)^5)/x^10,x)

[Out]

log(x)*(10*d + 45*e) + x*(d + 10*e) + (e*x^2)/2 - (d/9 + x^3*(20*d + (15*e)/2) + x^4*(42*d + 24*e) + x^2*((45*
d)/7 + (10*e)/7) + x^6*(70*d + 84*e) + x^7*(60*d + 105*e) + x^8*(45*d + 120*e) + x^5*(63*d + (105*e)/2) + x*((
5*d)/4 + e/8))/x^9

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sympy [A]  time = 5.59, size = 126, normalized size = 0.92 \begin {gather*} \frac {e x^{2}}{2} + x \left (d + 10 e\right ) + 5 \left (2 d + 9 e\right ) \log {\relax (x )} + \frac {- 56 d + x^{8} \left (- 22680 d - 60480 e\right ) + x^{7} \left (- 30240 d - 52920 e\right ) + x^{6} \left (- 35280 d - 42336 e\right ) + x^{5} \left (- 31752 d - 26460 e\right ) + x^{4} \left (- 21168 d - 12096 e\right ) + x^{3} \left (- 10080 d - 3780 e\right ) + x^{2} \left (- 3240 d - 720 e\right ) + x \left (- 630 d - 63 e\right )}{504 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**10,x)

[Out]

e*x**2/2 + x*(d + 10*e) + 5*(2*d + 9*e)*log(x) + (-56*d + x**8*(-22680*d - 60480*e) + x**7*(-30240*d - 52920*e
) + x**6*(-35280*d - 42336*e) + x**5*(-31752*d - 26460*e) + x**4*(-21168*d - 12096*e) + x**3*(-10080*d - 3780*
e) + x**2*(-3240*d - 720*e) + x*(-630*d - 63*e))/(504*x**9)

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