3.563 \(\int x^3 (d+e x) (1+2 x+x^2)^5 \, dx\)

Optimal. Leaf size=69 \[ \frac {1}{14} (x+1)^{14} (d-4 e)-\frac {3}{13} (x+1)^{13} (d-2 e)+\frac {1}{12} (x+1)^{12} (3 d-4 e)-\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{15} e (x+1)^{15} \]

[Out]

-1/11*(d-e)*(1+x)^11+1/12*(3*d-4*e)*(1+x)^12-3/13*(d-2*e)*(1+x)^13+1/14*(d-4*e)*(1+x)^14+1/15*e*(1+x)^15

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Rubi [A]  time = 0.05, antiderivative size = 69, 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} \[ \frac {1}{14} (x+1)^{14} (d-4 e)-\frac {3}{13} (x+1)^{13} (d-2 e)+\frac {1}{12} (x+1)^{12} (3 d-4 e)-\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{15} e (x+1)^{15} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

-((d - e)*(1 + x)^11)/11 + ((3*d - 4*e)*(1 + x)^12)/12 - (3*(d - 2*e)*(1 + x)^13)/13 + ((d - 4*e)*(1 + x)^14)/
14 + (e*(1 + x)^15)/15

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

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Mathematica [B]  time = 0.02, size = 153, normalized size = 2.22 \[ \frac {1}{14} x^{14} (d+10 e)+\frac {5}{13} x^{13} (2 d+9 e)+\frac {5}{4} x^{12} (3 d+8 e)+\frac {30}{11} x^{11} (4 d+7 e)+\frac {21}{5} x^{10} (5 d+6 e)+\frac {14}{3} x^9 (6 d+5 e)+\frac {15}{4} x^8 (7 d+4 e)+\frac {15}{7} x^7 (8 d+3 e)+\frac {5}{6} x^6 (9 d+2 e)+\frac {1}{5} x^5 (10 d+e)+\frac {d x^4}{4}+\frac {e x^{15}}{15} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

(d*x^4)/4 + ((10*d + e)*x^5)/5 + (5*(9*d + 2*e)*x^6)/6 + (15*(8*d + 3*e)*x^7)/7 + (15*(7*d + 4*e)*x^8)/4 + (14
*(6*d + 5*e)*x^9)/3 + (21*(5*d + 6*e)*x^10)/5 + (30*(4*d + 7*e)*x^11)/11 + (5*(3*d + 8*e)*x^12)/4 + (5*(2*d +
9*e)*x^13)/13 + ((d + 10*e)*x^14)/14 + (e*x^15)/15

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fricas [B]  time = 0.75, size = 133, normalized size = 1.93 \[ \frac {1}{15} x^{15} e + \frac {5}{7} x^{14} e + \frac {1}{14} x^{14} d + \frac {45}{13} x^{13} e + \frac {10}{13} x^{13} d + 10 x^{12} e + \frac {15}{4} x^{12} d + \frac {210}{11} x^{11} e + \frac {120}{11} x^{11} d + \frac {126}{5} x^{10} e + 21 x^{10} d + \frac {70}{3} x^{9} e + 28 x^{9} d + 15 x^{8} e + \frac {105}{4} x^{8} d + \frac {45}{7} x^{7} e + \frac {120}{7} x^{7} d + \frac {5}{3} x^{6} e + \frac {15}{2} x^{6} d + \frac {1}{5} x^{5} e + 2 x^{5} d + \frac {1}{4} x^{4} d \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")

[Out]

1/15*x^15*e + 5/7*x^14*e + 1/14*x^14*d + 45/13*x^13*e + 10/13*x^13*d + 10*x^12*e + 15/4*x^12*d + 210/11*x^11*e
+ 120/11*x^11*d + 126/5*x^10*e + 21*x^10*d + 70/3*x^9*e + 28*x^9*d + 15*x^8*e + 105/4*x^8*d + 45/7*x^7*e + 12
0/7*x^7*d + 5/3*x^6*e + 15/2*x^6*d + 1/5*x^5*e + 2*x^5*d + 1/4*x^4*d

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giac [B]  time = 0.15, size = 144, normalized size = 2.09 \[ \frac {1}{15} \, x^{15} e + \frac {1}{14} \, d x^{14} + \frac {5}{7} \, x^{14} e + \frac {10}{13} \, d x^{13} + \frac {45}{13} \, x^{13} e + \frac {15}{4} \, d x^{12} + 10 \, x^{12} e + \frac {120}{11} \, d x^{11} + \frac {210}{11} \, x^{11} e + 21 \, d x^{10} + \frac {126}{5} \, x^{10} e + 28 \, d x^{9} + \frac {70}{3} \, x^{9} e + \frac {105}{4} \, d x^{8} + 15 \, x^{8} e + \frac {120}{7} \, d x^{7} + \frac {45}{7} \, x^{7} e + \frac {15}{2} \, d x^{6} + \frac {5}{3} \, x^{6} e + 2 \, d x^{5} + \frac {1}{5} \, x^{5} e + \frac {1}{4} \, d x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")

[Out]

1/15*x^15*e + 1/14*d*x^14 + 5/7*x^14*e + 10/13*d*x^13 + 45/13*x^13*e + 15/4*d*x^12 + 10*x^12*e + 120/11*d*x^11
+ 210/11*x^11*e + 21*d*x^10 + 126/5*x^10*e + 28*d*x^9 + 70/3*x^9*e + 105/4*d*x^8 + 15*x^8*e + 120/7*d*x^7 + 4
5/7*x^7*e + 15/2*d*x^6 + 5/3*x^6*e + 2*d*x^5 + 1/5*x^5*e + 1/4*d*x^4

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maple [B]  time = 0.04, size = 130, normalized size = 1.88 \[ \frac {e \,x^{15}}{15}+\frac {\left (d +10 e \right ) x^{14}}{14}+\frac {\left (10 d +45 e \right ) x^{13}}{13}+\frac {\left (45 d +120 e \right ) x^{12}}{12}+\frac {\left (120 d +210 e \right ) x^{11}}{11}+\frac {\left (210 d +252 e \right ) x^{10}}{10}+\frac {\left (252 d +210 e \right ) x^{9}}{9}+\frac {\left (210 d +120 e \right ) x^{8}}{8}+\frac {\left (120 d +45 e \right ) x^{7}}{7}+\frac {\left (45 d +10 e \right ) x^{6}}{6}+\frac {d \,x^{4}}{4}+\frac {\left (10 d +e \right ) x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(e*x+d)*(x^2+2*x+1)^5,x)

[Out]

1/15*e*x^15+1/14*(d+10*e)*x^14+1/13*(10*d+45*e)*x^13+1/12*(45*d+120*e)*x^12+1/11*(120*d+210*e)*x^11+1/10*(210*
d+252*e)*x^10+1/9*(252*d+210*e)*x^9+1/8*(210*d+120*e)*x^8+1/7*(120*d+45*e)*x^7+1/6*(45*d+10*e)*x^6+1/5*(10*d+e
)*x^5+1/4*d*x^4

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maxima [B]  time = 0.54, size = 129, normalized size = 1.87 \[ \frac {1}{15} \, e x^{15} + \frac {1}{14} \, {\left (d + 10 \, e\right )} x^{14} + \frac {5}{13} \, {\left (2 \, d + 9 \, e\right )} x^{13} + \frac {5}{4} \, {\left (3 \, d + 8 \, e\right )} x^{12} + \frac {30}{11} \, {\left (4 \, d + 7 \, e\right )} x^{11} + \frac {21}{5} \, {\left (5 \, d + 6 \, e\right )} x^{10} + \frac {14}{3} \, {\left (6 \, d + 5 \, e\right )} x^{9} + \frac {15}{4} \, {\left (7 \, d + 4 \, e\right )} x^{8} + \frac {15}{7} \, {\left (8 \, d + 3 \, e\right )} x^{7} + \frac {5}{6} \, {\left (9 \, d + 2 \, e\right )} x^{6} + \frac {1}{5} \, {\left (10 \, d + e\right )} x^{5} + \frac {1}{4} \, d x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")

[Out]

1/15*e*x^15 + 1/14*(d + 10*e)*x^14 + 5/13*(2*d + 9*e)*x^13 + 5/4*(3*d + 8*e)*x^12 + 30/11*(4*d + 7*e)*x^11 + 2
1/5*(5*d + 6*e)*x^10 + 14/3*(6*d + 5*e)*x^9 + 15/4*(7*d + 4*e)*x^8 + 15/7*(8*d + 3*e)*x^7 + 5/6*(9*d + 2*e)*x^
6 + 1/5*(10*d + e)*x^5 + 1/4*d*x^4

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mupad [B]  time = 0.08, size = 123, normalized size = 1.78 \[ \frac {e\,x^{15}}{15}+\left (\frac {d}{14}+\frac {5\,e}{7}\right )\,x^{14}+\left (\frac {10\,d}{13}+\frac {45\,e}{13}\right )\,x^{13}+\left (\frac {15\,d}{4}+10\,e\right )\,x^{12}+\left (\frac {120\,d}{11}+\frac {210\,e}{11}\right )\,x^{11}+\left (21\,d+\frac {126\,e}{5}\right )\,x^{10}+\left (28\,d+\frac {70\,e}{3}\right )\,x^9+\left (\frac {105\,d}{4}+15\,e\right )\,x^8+\left (\frac {120\,d}{7}+\frac {45\,e}{7}\right )\,x^7+\left (\frac {15\,d}{2}+\frac {5\,e}{3}\right )\,x^6+\left (2\,d+\frac {e}{5}\right )\,x^5+\frac {d\,x^4}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(d + e*x)*(2*x + x^2 + 1)^5,x)

[Out]

x^5*(2*d + e/5) + x^6*((15*d)/2 + (5*e)/3) + x^12*((15*d)/4 + 10*e) + x^14*(d/14 + (5*e)/7) + x^13*((10*d)/13
+ (45*e)/13) + x^9*(28*d + (70*e)/3) + x^8*((105*d)/4 + 15*e) + x^10*(21*d + (126*e)/5) + x^7*((120*d)/7 + (45
*e)/7) + x^11*((120*d)/11 + (210*e)/11) + (d*x^4)/4 + (e*x^15)/15

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sympy [B]  time = 0.10, size = 136, normalized size = 1.97 \[ \frac {d x^{4}}{4} + \frac {e x^{15}}{15} + x^{14} \left (\frac {d}{14} + \frac {5 e}{7}\right ) + x^{13} \left (\frac {10 d}{13} + \frac {45 e}{13}\right ) + x^{12} \left (\frac {15 d}{4} + 10 e\right ) + x^{11} \left (\frac {120 d}{11} + \frac {210 e}{11}\right ) + x^{10} \left (21 d + \frac {126 e}{5}\right ) + x^{9} \left (28 d + \frac {70 e}{3}\right ) + x^{8} \left (\frac {105 d}{4} + 15 e\right ) + x^{7} \left (\frac {120 d}{7} + \frac {45 e}{7}\right ) + x^{6} \left (\frac {15 d}{2} + \frac {5 e}{3}\right ) + x^{5} \left (2 d + \frac {e}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x+d)*(x**2+2*x+1)**5,x)

[Out]

d*x**4/4 + e*x**15/15 + x**14*(d/14 + 5*e/7) + x**13*(10*d/13 + 45*e/13) + x**12*(15*d/4 + 10*e) + x**11*(120*
d/11 + 210*e/11) + x**10*(21*d + 126*e/5) + x**9*(28*d + 70*e/3) + x**8*(105*d/4 + 15*e) + x**7*(120*d/7 + 45*
e/7) + x**6*(15*d/2 + 5*e/3) + x**5*(2*d + e/5)

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