3.5.89 \(\int x^4 (d+e x) (1+2 x+x^2)^5 \, dx\)

Optimal. Leaf size=87 \[ \frac {1}{15} (x+1)^{15} (d-5 e)-\frac {1}{7} (x+1)^{14} (2 d-5 e)+\frac {2}{13} (x+1)^{13} (3 d-5 e)-\frac {1}{12} (x+1)^{12} (4 d-5 e)+\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{16} e (x+1)^{16} \]

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Rubi [A]  time = 0.06, antiderivative size = 87, 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}{15} (x+1)^{15} (d-5 e)-\frac {1}{7} (x+1)^{14} (2 d-5 e)+\frac {2}{13} (x+1)^{13} (3 d-5 e)-\frac {1}{12} (x+1)^{12} (4 d-5 e)+\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{16} e (x+1)^{16} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

((d - e)*(1 + x)^11)/11 - ((4*d - 5*e)*(1 + x)^12)/12 + (2*(3*d - 5*e)*(1 + x)^13)/13 - ((2*d - 5*e)*(1 + x)^1
4)/7 + ((d - 5*e)*(1 + x)^15)/15 + (e*(1 + x)^16)/16

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 x^4 (d+e x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^4 (1+x)^{10} (d+e x) \, dx\\ &=\int \left ((d-e) (1+x)^{10}+(-4 d+5 e) (1+x)^{11}+2 (3 d-5 e) (1+x)^{12}-2 (2 d-5 e) (1+x)^{13}+(d-5 e) (1+x)^{14}+e (1+x)^{15}\right ) \, dx\\ &=\frac {1}{11} (d-e) (1+x)^{11}-\frac {1}{12} (4 d-5 e) (1+x)^{12}+\frac {2}{13} (3 d-5 e) (1+x)^{13}-\frac {1}{7} (2 d-5 e) (1+x)^{14}+\frac {1}{15} (d-5 e) (1+x)^{15}+\frac {1}{16} e (1+x)^{16}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 153, normalized size = 1.76 \begin {gather*} \frac {1}{15} x^{15} (d+10 e)+\frac {5}{14} x^{14} (2 d+9 e)+\frac {15}{13} x^{13} (3 d+8 e)+\frac {5}{2} x^{12} (4 d+7 e)+\frac {42}{11} x^{11} (5 d+6 e)+\frac {21}{5} x^{10} (6 d+5 e)+\frac {10}{3} x^9 (7 d+4 e)+\frac {15}{8} x^8 (8 d+3 e)+\frac {5}{7} x^7 (9 d+2 e)+\frac {1}{6} x^6 (10 d+e)+\frac {d x^5}{5}+\frac {e x^{16}}{16} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

(d*x^5)/5 + ((10*d + e)*x^6)/6 + (5*(9*d + 2*e)*x^7)/7 + (15*(8*d + 3*e)*x^8)/8 + (10*(7*d + 4*e)*x^9)/3 + (21
*(6*d + 5*e)*x^10)/5 + (42*(5*d + 6*e)*x^11)/11 + (5*(4*d + 7*e)*x^12)/2 + (15*(3*d + 8*e)*x^13)/13 + (5*(2*d
+ 9*e)*x^14)/14 + ((d + 10*e)*x^15)/15 + (e*x^16)/16

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^4 (d+e x) \left (1+2 x+x^2\right )^5 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^4*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

IntegrateAlgebraic[x^4*(d + e*x)*(1 + 2*x + x^2)^5, x]

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fricas [A]  time = 0.38, size = 133, normalized size = 1.53 \begin {gather*} \frac {1}{16} x^{16} e + \frac {2}{3} x^{15} e + \frac {1}{15} x^{15} d + \frac {45}{14} x^{14} e + \frac {5}{7} x^{14} d + \frac {120}{13} x^{13} e + \frac {45}{13} x^{13} d + \frac {35}{2} x^{12} e + 10 x^{12} d + \frac {252}{11} x^{11} e + \frac {210}{11} x^{11} d + 21 x^{10} e + \frac {126}{5} x^{10} d + \frac {40}{3} x^{9} e + \frac {70}{3} x^{9} d + \frac {45}{8} x^{8} e + 15 x^{8} d + \frac {10}{7} x^{7} e + \frac {45}{7} x^{7} d + \frac {1}{6} x^{6} e + \frac {5}{3} x^{6} d + \frac {1}{5} x^{5} d \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")

[Out]

1/16*x^16*e + 2/3*x^15*e + 1/15*x^15*d + 45/14*x^14*e + 5/7*x^14*d + 120/13*x^13*e + 45/13*x^13*d + 35/2*x^12*
e + 10*x^12*d + 252/11*x^11*e + 210/11*x^11*d + 21*x^10*e + 126/5*x^10*d + 40/3*x^9*e + 70/3*x^9*d + 45/8*x^8*
e + 15*x^8*d + 10/7*x^7*e + 45/7*x^7*d + 1/6*x^6*e + 5/3*x^6*d + 1/5*x^5*d

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giac [A]  time = 0.16, size = 144, normalized size = 1.66 \begin {gather*} \frac {1}{16} \, x^{16} e + \frac {1}{15} \, d x^{15} + \frac {2}{3} \, x^{15} e + \frac {5}{7} \, d x^{14} + \frac {45}{14} \, x^{14} e + \frac {45}{13} \, d x^{13} + \frac {120}{13} \, x^{13} e + 10 \, d x^{12} + \frac {35}{2} \, x^{12} e + \frac {210}{11} \, d x^{11} + \frac {252}{11} \, x^{11} e + \frac {126}{5} \, d x^{10} + 21 \, x^{10} e + \frac {70}{3} \, d x^{9} + \frac {40}{3} \, x^{9} e + 15 \, d x^{8} + \frac {45}{8} \, x^{8} e + \frac {45}{7} \, d x^{7} + \frac {10}{7} \, x^{7} e + \frac {5}{3} \, d x^{6} + \frac {1}{6} \, x^{6} e + \frac {1}{5} \, d x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")

[Out]

1/16*x^16*e + 1/15*d*x^15 + 2/3*x^15*e + 5/7*d*x^14 + 45/14*x^14*e + 45/13*d*x^13 + 120/13*x^13*e + 10*d*x^12
+ 35/2*x^12*e + 210/11*d*x^11 + 252/11*x^11*e + 126/5*d*x^10 + 21*x^10*e + 70/3*d*x^9 + 40/3*x^9*e + 15*d*x^8
+ 45/8*x^8*e + 45/7*d*x^7 + 10/7*x^7*e + 5/3*d*x^6 + 1/6*x^6*e + 1/5*d*x^5

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maple [A]  time = 0.05, size = 130, normalized size = 1.49 \begin {gather*} \frac {e \,x^{16}}{16}+\frac {\left (d +10 e \right ) x^{15}}{15}+\frac {\left (10 d +45 e \right ) x^{14}}{14}+\frac {\left (45 d +120 e \right ) x^{13}}{13}+\frac {\left (120 d +210 e \right ) x^{12}}{12}+\frac {\left (210 d +252 e \right ) x^{11}}{11}+\frac {\left (252 d +210 e \right ) x^{10}}{10}+\frac {\left (210 d +120 e \right ) x^{9}}{9}+\frac {\left (120 d +45 e \right ) x^{8}}{8}+\frac {\left (45 d +10 e \right ) x^{7}}{7}+\frac {d \,x^{5}}{5}+\frac {\left (10 d +e \right ) x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x+d)*(x^2+2*x+1)^5,x)

[Out]

1/16*e*x^16+1/15*(d+10*e)*x^15+1/14*(10*d+45*e)*x^14+1/13*(45*d+120*e)*x^13+1/12*(120*d+210*e)*x^12+1/11*(210*
d+252*e)*x^11+1/10*(252*d+210*e)*x^10+1/9*(210*d+120*e)*x^9+1/8*(120*d+45*e)*x^8+1/7*(45*d+10*e)*x^7+1/6*(10*d
+e)*x^6+1/5*d*x^5

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maxima [A]  time = 0.57, size = 129, normalized size = 1.48 \begin {gather*} \frac {1}{16} \, e x^{16} + \frac {1}{15} \, {\left (d + 10 \, e\right )} x^{15} + \frac {5}{14} \, {\left (2 \, d + 9 \, e\right )} x^{14} + \frac {15}{13} \, {\left (3 \, d + 8 \, e\right )} x^{13} + \frac {5}{2} \, {\left (4 \, d + 7 \, e\right )} x^{12} + \frac {42}{11} \, {\left (5 \, d + 6 \, e\right )} x^{11} + \frac {21}{5} \, {\left (6 \, d + 5 \, e\right )} x^{10} + \frac {10}{3} \, {\left (7 \, d + 4 \, e\right )} x^{9} + \frac {15}{8} \, {\left (8 \, d + 3 \, e\right )} x^{8} + \frac {5}{7} \, {\left (9 \, d + 2 \, e\right )} x^{7} + \frac {1}{6} \, {\left (10 \, d + e\right )} x^{6} + \frac {1}{5} \, d x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")

[Out]

1/16*e*x^16 + 1/15*(d + 10*e)*x^15 + 5/14*(2*d + 9*e)*x^14 + 15/13*(3*d + 8*e)*x^13 + 5/2*(4*d + 7*e)*x^12 + 4
2/11*(5*d + 6*e)*x^11 + 21/5*(6*d + 5*e)*x^10 + 10/3*(7*d + 4*e)*x^9 + 15/8*(8*d + 3*e)*x^8 + 5/7*(9*d + 2*e)*
x^7 + 1/6*(10*d + e)*x^6 + 1/5*d*x^5

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mupad [B]  time = 0.08, size = 123, normalized size = 1.41 \begin {gather*} \frac {e\,x^{16}}{16}+\left (\frac {d}{15}+\frac {2\,e}{3}\right )\,x^{15}+\left (\frac {5\,d}{7}+\frac {45\,e}{14}\right )\,x^{14}+\left (\frac {45\,d}{13}+\frac {120\,e}{13}\right )\,x^{13}+\left (10\,d+\frac {35\,e}{2}\right )\,x^{12}+\left (\frac {210\,d}{11}+\frac {252\,e}{11}\right )\,x^{11}+\left (\frac {126\,d}{5}+21\,e\right )\,x^{10}+\left (\frac {70\,d}{3}+\frac {40\,e}{3}\right )\,x^9+\left (15\,d+\frac {45\,e}{8}\right )\,x^8+\left (\frac {45\,d}{7}+\frac {10\,e}{7}\right )\,x^7+\left (\frac {5\,d}{3}+\frac {e}{6}\right )\,x^6+\frac {d\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d + e*x)*(2*x + x^2 + 1)^5,x)

[Out]

x^6*((5*d)/3 + e/6) + x^15*(d/15 + (2*e)/3) + x^12*(10*d + (35*e)/2) + x^7*((45*d)/7 + (10*e)/7) + x^8*(15*d +
(45*e)/8) + x^14*((5*d)/7 + (45*e)/14) + x^9*((70*d)/3 + (40*e)/3) + x^10*((126*d)/5 + 21*e) + x^13*((45*d)/1
3 + (120*e)/13) + x^11*((210*d)/11 + (252*e)/11) + (d*x^5)/5 + (e*x^16)/16

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sympy [A]  time = 0.10, size = 139, normalized size = 1.60 \begin {gather*} \frac {d x^{5}}{5} + \frac {e x^{16}}{16} + x^{15} \left (\frac {d}{15} + \frac {2 e}{3}\right ) + x^{14} \left (\frac {5 d}{7} + \frac {45 e}{14}\right ) + x^{13} \left (\frac {45 d}{13} + \frac {120 e}{13}\right ) + x^{12} \left (10 d + \frac {35 e}{2}\right ) + x^{11} \left (\frac {210 d}{11} + \frac {252 e}{11}\right ) + x^{10} \left (\frac {126 d}{5} + 21 e\right ) + x^{9} \left (\frac {70 d}{3} + \frac {40 e}{3}\right ) + x^{8} \left (15 d + \frac {45 e}{8}\right ) + x^{7} \left (\frac {45 d}{7} + \frac {10 e}{7}\right ) + x^{6} \left (\frac {5 d}{3} + \frac {e}{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x+d)*(x**2+2*x+1)**5,x)

[Out]

d*x**5/5 + e*x**16/16 + x**15*(d/15 + 2*e/3) + x**14*(5*d/7 + 45*e/14) + x**13*(45*d/13 + 120*e/13) + x**12*(1
0*d + 35*e/2) + x**11*(210*d/11 + 252*e/11) + x**10*(126*d/5 + 21*e) + x**9*(70*d/3 + 40*e/3) + x**8*(15*d + 4
5*e/8) + x**7*(45*d/7 + 10*e/7) + x**6*(5*d/3 + e/6)

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