3.5.93 \(\int (d+e x) (1+2 x+x^2)^5 \, dx\)

Optimal. Leaf size=25 \[ \frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{12} e (x+1)^{12} \]

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {27, 43} \begin {gather*} \frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{12} e (x+1)^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

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

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [B]  time = 0.02, size = 113, normalized size = 4.52 \begin {gather*} d \left (\frac {x^{11}}{11}+x^{10}+5 x^9+15 x^8+30 x^7+42 x^6+42 x^5+30 x^4+15 x^3+5 x^2+x\right )+\frac {1}{132} e \left (11 x^{10}+120 x^9+594 x^8+1760 x^7+3465 x^6+4752 x^5+4620 x^4+3168 x^3+1485 x^2+440 x+66\right ) x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x^2*(66 + 440*x + 1485*x^2 + 3168*x^3 + 4620*x^4 + 4752*x^5 + 3465*x^6 + 1760*x^7 + 594*x^8 + 120*x^9 + 11*
x^10))/132 + d*(x + 5*x^2 + 15*x^3 + 30*x^4 + 42*x^5 + 42*x^6 + 30*x^7 + 15*x^8 + 5*x^9 + x^10 + x^11/11)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x) \left (1+2 x+x^2\right )^5 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.35, size = 129, normalized size = 5.16 \begin {gather*} \frac {1}{12} x^{12} e + \frac {10}{11} x^{11} e + \frac {1}{11} x^{11} d + \frac {9}{2} x^{10} e + x^{10} d + \frac {40}{3} x^{9} e + 5 x^{9} d + \frac {105}{4} x^{8} e + 15 x^{8} d + 36 x^{7} e + 30 x^{7} d + 35 x^{6} e + 42 x^{6} d + 24 x^{5} e + 42 x^{5} d + \frac {45}{4} x^{4} e + 30 x^{4} d + \frac {10}{3} x^{3} e + 15 x^{3} d + \frac {1}{2} x^{2} e + 5 x^{2} d + x d \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x^12*e + 10/11*x^11*e + 1/11*x^11*d + 9/2*x^10*e + x^10*d + 40/3*x^9*e + 5*x^9*d + 105/4*x^8*e + 15*x^8*d
+ 36*x^7*e + 30*x^7*d + 35*x^6*e + 42*x^6*d + 24*x^5*e + 42*x^5*d + 45/4*x^4*e + 30*x^4*d + 10/3*x^3*e + 15*x
^3*d + 1/2*x^2*e + 5*x^2*d + x*d

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giac [B]  time = 0.19, size = 140, normalized size = 5.60 \begin {gather*} \frac {1}{12} \, x^{12} e + \frac {1}{11} \, d x^{11} + \frac {10}{11} \, x^{11} e + d x^{10} + \frac {9}{2} \, x^{10} e + 5 \, d x^{9} + \frac {40}{3} \, x^{9} e + 15 \, d x^{8} + \frac {105}{4} \, x^{8} e + 30 \, d x^{7} + 36 \, x^{7} e + 42 \, d x^{6} + 35 \, x^{6} e + 42 \, d x^{5} + 24 \, x^{5} e + 30 \, d x^{4} + \frac {45}{4} \, x^{4} e + 15 \, d x^{3} + \frac {10}{3} \, x^{3} e + 5 \, d x^{2} + \frac {1}{2} \, x^{2} e + d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x^12*e + 1/11*d*x^11 + 10/11*x^11*e + d*x^10 + 9/2*x^10*e + 5*d*x^9 + 40/3*x^9*e + 15*d*x^8 + 105/4*x^8*e
+ 30*d*x^7 + 36*x^7*e + 42*d*x^6 + 35*x^6*e + 42*d*x^5 + 24*x^5*e + 30*d*x^4 + 45/4*x^4*e + 15*d*x^3 + 10/3*x
^3*e + 5*d*x^2 + 1/2*x^2*e + d*x

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maple [B]  time = 0.05, size = 127, normalized size = 5.08 \begin {gather*} \frac {e \,x^{12}}{12}+\frac {\left (d +10 e \right ) x^{11}}{11}+\frac {\left (10 d +45 e \right ) x^{10}}{10}+\frac {\left (45 d +120 e \right ) x^{9}}{9}+\frac {\left (120 d +210 e \right ) x^{8}}{8}+\frac {\left (210 d +252 e \right ) x^{7}}{7}+\frac {\left (252 d +210 e \right ) x^{6}}{6}+\frac {\left (210 d +120 e \right ) x^{5}}{5}+\frac {\left (120 d +45 e \right ) x^{4}}{4}+\frac {\left (45 d +10 e \right ) x^{3}}{3}+d x +\frac {\left (10 d +e \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.51, size = 126, normalized size = 5.04 \begin {gather*} \frac {1}{12} \, e x^{12} + \frac {1}{11} \, {\left (d + 10 \, e\right )} x^{11} + \frac {1}{2} \, {\left (2 \, d + 9 \, e\right )} x^{10} + \frac {5}{3} \, {\left (3 \, d + 8 \, e\right )} x^{9} + \frac {15}{4} \, {\left (4 \, d + 7 \, e\right )} x^{8} + 6 \, {\left (5 \, d + 6 \, e\right )} x^{7} + 7 \, {\left (6 \, d + 5 \, e\right )} x^{6} + 6 \, {\left (7 \, d + 4 \, e\right )} x^{5} + \frac {15}{4} \, {\left (8 \, d + 3 \, e\right )} x^{4} + \frac {5}{3} \, {\left (9 \, d + 2 \, e\right )} x^{3} + \frac {1}{2} \, {\left (10 \, d + e\right )} x^{2} + d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*e*x^12 + 1/11*(d + 10*e)*x^11 + 1/2*(2*d + 9*e)*x^10 + 5/3*(3*d + 8*e)*x^9 + 15/4*(4*d + 7*e)*x^8 + 6*(5*
d + 6*e)*x^7 + 7*(6*d + 5*e)*x^6 + 6*(7*d + 4*e)*x^5 + 15/4*(8*d + 3*e)*x^4 + 5/3*(9*d + 2*e)*x^3 + 1/2*(10*d
+ e)*x^2 + d*x

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mupad [B]  time = 0.08, size = 118, normalized size = 4.72 \begin {gather*} \frac {e\,x^{12}}{12}+\left (\frac {d}{11}+\frac {10\,e}{11}\right )\,x^{11}+\left (d+\frac {9\,e}{2}\right )\,x^{10}+\left (5\,d+\frac {40\,e}{3}\right )\,x^9+\left (15\,d+\frac {105\,e}{4}\right )\,x^8+\left (30\,d+36\,e\right )\,x^7+\left (42\,d+35\,e\right )\,x^6+\left (42\,d+24\,e\right )\,x^5+\left (30\,d+\frac {45\,e}{4}\right )\,x^4+\left (15\,d+\frac {10\,e}{3}\right )\,x^3+\left (5\,d+\frac {e}{2}\right )\,x^2+d\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)*(2*x + x^2 + 1)^5,x)

[Out]

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

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sympy [B]  time = 0.10, size = 119, normalized size = 4.76 \begin {gather*} d x + \frac {e x^{12}}{12} + x^{11} \left (\frac {d}{11} + \frac {10 e}{11}\right ) + x^{10} \left (d + \frac {9 e}{2}\right ) + x^{9} \left (5 d + \frac {40 e}{3}\right ) + x^{8} \left (15 d + \frac {105 e}{4}\right ) + x^{7} \left (30 d + 36 e\right ) + x^{6} \left (42 d + 35 e\right ) + x^{5} \left (42 d + 24 e\right ) + x^{4} \left (30 d + \frac {45 e}{4}\right ) + x^{3} \left (15 d + \frac {10 e}{3}\right ) + x^{2} \left (5 d + \frac {e}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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