3.6.79 \(\int \frac {(d+e x) (1+2 x+x^2)^5}{x^{13}} \, dx\) [579]

Optimal. Leaf size=31 \[ -\frac {d (1+x)^{11}}{12 x^{12}}+\frac {(d-12 e) (1+x)^{11}}{132 x^{11}} \]

[Out]

-1/12*d*(1+x)^11/x^12+1/132*(d-12*e)*(1+x)^11/x^11

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Rubi [A]
time = 0.00, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {27, 79, 37} \begin {gather*} \frac {(x+1)^{11} (d-12 e)}{132 x^{11}}-\frac {d (x+1)^{11}}{12 x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^13,x]

[Out]

-1/12*(d*(1 + x)^11)/x^12 + ((d - 12*e)*(1 + x)^11)/(132*x^11)

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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps

\begin {align*} \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{13}} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^{13}} \, dx\\ &=-\frac {d (1+x)^{11}}{12 x^{12}}-\frac {1}{12} (d-12 e) \int \frac {(1+x)^{10}}{x^{12}} \, dx\\ &=-\frac {d (1+x)^{11}}{12 x^{12}}+\frac {(d-12 e) (1+x)^{11}}{132 x^{11}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(114\) vs. \(2(31)=62\).
time = 0.02, size = 114, normalized size = 3.68 \begin {gather*} -\frac {12 e x \left (1+11 x+55 x^2+165 x^3+330 x^4+462 x^5+462 x^6+330 x^7+165 x^8+55 x^9+11 x^{10}\right )+d \left (11+120 x+594 x^2+1760 x^3+3465 x^4+4752 x^5+4620 x^6+3168 x^7+1485 x^8+440 x^9+66 x^{10}\right )}{132 x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^13,x]

[Out]

-1/132*(12*e*x*(1 + 11*x + 55*x^2 + 165*x^3 + 330*x^4 + 462*x^5 + 462*x^6 + 330*x^7 + 165*x^8 + 55*x^9 + 11*x^
10) + d*(11 + 120*x + 594*x^2 + 1760*x^3 + 3465*x^4 + 4752*x^5 + 4620*x^6 + 3168*x^7 + 1485*x^8 + 440*x^9 + 66
*x^10))/x^12

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(129\) vs. \(2(27)=54\).
time = 0.68, size = 130, normalized size = 4.19

method	result	size
norman	\(\frac {-e \,x^{11}+\left (-\frac {d}{2}-5 e \right ) x^{10}+\left (-\frac {10 d}{3}-15 e \right ) x^{9}+\left (-\frac {45 d}{4}-30 e \right ) x^{8}+\left (-24 d -42 e \right ) x^{7}+\left (-35 d -42 e \right ) x^{6}+\left (-36 d -30 e \right ) x^{5}+\left (-\frac {105 d}{4}-15 e \right ) x^{4}+\left (-\frac {40 d}{3}-5 e \right ) x^{3}+\left (-\frac {9 d}{2}-e \right ) x^{2}+\left (-\frac {10 d}{11}-\frac {e}{11}\right ) x -\frac {d}{12}}{x^{12}}\)	\(123\)
risch	\(\frac {-e \,x^{11}+\left (-\frac {d}{2}-5 e \right ) x^{10}+\left (-\frac {10 d}{3}-15 e \right ) x^{9}+\left (-\frac {45 d}{4}-30 e \right ) x^{8}+\left (-24 d -42 e \right ) x^{7}+\left (-35 d -42 e \right ) x^{6}+\left (-36 d -30 e \right ) x^{5}+\left (-\frac {105 d}{4}-15 e \right ) x^{4}+\left (-\frac {40 d}{3}-5 e \right ) x^{3}+\left (-\frac {9 d}{2}-e \right ) x^{2}+\left (-\frac {10 d}{11}-\frac {e}{11}\right ) x -\frac {d}{12}}{x^{12}}\)	\(123\)
default	\(-\frac {10 d +e}{11 x^{11}}-\frac {120 d +210 e}{5 x^{5}}-\frac {120 d +45 e}{9 x^{9}}-\frac {45 d +120 e}{4 x^{4}}-\frac {d}{12 x^{12}}-\frac {210 d +252 e}{6 x^{6}}-\frac {d +10 e}{2 x^{2}}-\frac {210 d +120 e}{8 x^{8}}-\frac {252 d +210 e}{7 x^{7}}-\frac {45 d +10 e}{10 x^{10}}-\frac {10 d +45 e}{3 x^{3}}-\frac {e}{x}\)	\(130\)
gosper	\(-\frac {132 e \,x^{11}+66 d \,x^{10}+660 e \,x^{10}+440 d \,x^{9}+1980 e \,x^{9}+1485 d \,x^{8}+3960 e \,x^{8}+3168 d \,x^{7}+5544 e \,x^{7}+4620 d \,x^{6}+5544 x^{6} e +4752 d \,x^{5}+3960 e \,x^{5}+3465 d \,x^{4}+1980 x^{4} e +1760 d \,x^{3}+660 e \,x^{3}+594 d \,x^{2}+132 e \,x^{2}+120 d x +12 e x +11 d}{132 x^{12}}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^13,x,method=_RETURNVERBOSE)

[Out]

-1/11*(10*d+e)/x^11-1/5*(120*d+210*e)/x^5-1/9*(120*d+45*e)/x^9-1/4*(45*d+120*e)/x^4-1/12*d/x^12-1/6*(210*d+252
*e)/x^6-1/2*(d+10*e)/x^2-1/8*(210*d+120*e)/x^8-1/7*(252*d+210*e)/x^7-1/10*(45*d+10*e)/x^10-1/3*(10*d+45*e)/x^3
-e/x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (28) = 56\).
time = 0.27, size = 140, normalized size = 4.52 \begin {gather*} -\frac {132 \, x^{11} e + 66 \, {\left (d + 10 \, e\right )} x^{10} + 220 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 495 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 792 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 924 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 792 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 495 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 220 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 66 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 12 \, {\left (10 \, d + e\right )} x + 11 \, d}{132 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^13,x, algorithm="maxima")

[Out]

-1/132*(132*x^11*e + 66*(d + 10*e)*x^10 + 220*(2*d + 9*e)*x^9 + 495*(3*d + 8*e)*x^8 + 792*(4*d + 7*e)*x^7 + 92
4*(5*d + 6*e)*x^6 + 792*(6*d + 5*e)*x^5 + 495*(7*d + 4*e)*x^4 + 220*(8*d + 3*e)*x^3 + 66*(9*d + 2*e)*x^2 + 12*
(10*d + e)*x + 11*d)/x^12

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (28) = 56\).
time = 1.92, size = 123, normalized size = 3.97 \begin {gather*} -\frac {66 \, d x^{10} + 440 \, d x^{9} + 1485 \, d x^{8} + 3168 \, d x^{7} + 4620 \, d x^{6} + 4752 \, d x^{5} + 3465 \, d x^{4} + 1760 \, d x^{3} + 594 \, d x^{2} + 120 \, d x + 12 \, {\left (11 \, x^{11} + 55 \, x^{10} + 165 \, x^{9} + 330 \, x^{8} + 462 \, x^{7} + 462 \, x^{6} + 330 \, x^{5} + 165 \, x^{4} + 55 \, x^{3} + 11 \, x^{2} + x\right )} e + 11 \, d}{132 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^13,x, algorithm="fricas")

[Out]

-1/132*(66*d*x^10 + 440*d*x^9 + 1485*d*x^8 + 3168*d*x^7 + 4620*d*x^6 + 4752*d*x^5 + 3465*d*x^4 + 1760*d*x^3 +
594*d*x^2 + 120*d*x + 12*(11*x^11 + 55*x^10 + 165*x^9 + 330*x^8 + 462*x^7 + 462*x^6 + 330*x^5 + 165*x^4 + 55*x
^3 + 11*x^2 + x)*e + 11*d)/x^12

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (26) = 52\).
time = 8.44, size = 131, normalized size = 4.23 \begin {gather*} \frac {- 11 d - 132 e x^{11} + x^{10} \left (- 66 d - 660 e\right ) + x^{9} \left (- 440 d - 1980 e\right ) + x^{8} \left (- 1485 d - 3960 e\right ) + x^{7} \left (- 3168 d - 5544 e\right ) + x^{6} \left (- 4620 d - 5544 e\right ) + x^{5} \left (- 4752 d - 3960 e\right ) + x^{4} \left (- 3465 d - 1980 e\right ) + x^{3} \left (- 1760 d - 660 e\right ) + x^{2} \left (- 594 d - 132 e\right ) + x \left (- 120 d - 12 e\right )}{132 x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**13,x)

[Out]

(-11*d - 132*e*x**11 + x**10*(-66*d - 660*e) + x**9*(-440*d - 1980*e) + x**8*(-1485*d - 3960*e) + x**7*(-3168*
d - 5544*e) + x**6*(-4620*d - 5544*e) + x**5*(-4752*d - 3960*e) + x**4*(-3465*d - 1980*e) + x**3*(-1760*d - 66
0*e) + x**2*(-594*d - 132*e) + x*(-120*d - 12*e))/(132*x**12)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (28) = 56\).
time = 1.58, size = 142, normalized size = 4.58 \begin {gather*} -\frac {132 \, x^{11} e + 66 \, d x^{10} + 660 \, x^{10} e + 440 \, d x^{9} + 1980 \, x^{9} e + 1485 \, d x^{8} + 3960 \, x^{8} e + 3168 \, d x^{7} + 5544 \, x^{7} e + 4620 \, d x^{6} + 5544 \, x^{6} e + 4752 \, d x^{5} + 3960 \, x^{5} e + 3465 \, d x^{4} + 1980 \, x^{4} e + 1760 \, d x^{3} + 660 \, x^{3} e + 594 \, d x^{2} + 132 \, x^{2} e + 120 \, d x + 12 \, x e + 11 \, d}{132 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^13,x, algorithm="giac")

[Out]

-1/132*(132*x^11*e + 66*d*x^10 + 660*x^10*e + 440*d*x^9 + 1980*x^9*e + 1485*d*x^8 + 3960*x^8*e + 3168*d*x^7 +
5544*x^7*e + 4620*d*x^6 + 5544*x^6*e + 4752*d*x^5 + 3960*x^5*e + 3465*d*x^4 + 1980*x^4*e + 1760*d*x^3 + 660*x^
3*e + 594*d*x^2 + 132*x^2*e + 120*d*x + 12*x*e + 11*d)/x^12

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Mupad [B]
time = 0.12, size = 120, normalized size = 3.87 \begin {gather*} -\frac {e\,x^{11}+\left (\frac {d}{2}+5\,e\right )\,x^{10}+\left (\frac {10\,d}{3}+15\,e\right )\,x^9+\left (\frac {45\,d}{4}+30\,e\right )\,x^8+\left (24\,d+42\,e\right )\,x^7+\left (35\,d+42\,e\right )\,x^6+\left (36\,d+30\,e\right )\,x^5+\left (\frac {105\,d}{4}+15\,e\right )\,x^4+\left (\frac {40\,d}{3}+5\,e\right )\,x^3+\left (\frac {9\,d}{2}+e\right )\,x^2+\left (\frac {10\,d}{11}+\frac {e}{11}\right )\,x+\frac {d}{12}}{x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x)*(2*x + x^2 + 1)^5)/x^13,x)

[Out]

-(d/12 + x^10*(d/2 + 5*e) + x^9*((10*d)/3 + 15*e) + x^3*((40*d)/3 + 5*e) + x^5*(36*d + 30*e) + x^7*(24*d + 42*
e) + x^6*(35*d + 42*e) + x^8*((45*d)/4 + 30*e) + x^4*((105*d)/4 + 15*e) + e*x^11 + x*((10*d)/11 + e/11) + x^2*
((9*d)/2 + e))/x^12

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