3.58 \(\int x^3 (d+e x^2) (1+2 x^2+x^4)^5 \, dx\)

Optimal. Leaf size=45 \[ \frac{1}{24} \left (x^2+1\right )^{12} (d-2 e)-\frac{1}{22} \left (x^2+1\right )^{11} (d-e)+\frac{1}{26} e \left (x^2+1\right )^{13} \]

[Out]

-((d - e)*(1 + x^2)^11)/22 + ((d - 2*e)*(1 + x^2)^12)/24 + (e*(1 + x^2)^13)/26

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Rubi [A]  time = 0.123396, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {28, 446, 76} \[ \frac{1}{24} \left (x^2+1\right )^{12} (d-2 e)-\frac{1}{22} \left (x^2+1\right )^{11} (d-e)+\frac{1}{26} e \left (x^2+1\right )^{13} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d + e*x^2)*(1 + 2*x^2 + x^4)^5,x]

[Out]

-((d - e)*(1 + x^2)^11)/22 + ((d - 2*e)*(1 + x^2)^12)/24 + (e*(1 + x^2)^13)/26

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

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 \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int x^3 \left (1+x^2\right )^{10} \left (d+e x^2\right ) \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int x (1+x)^{10} (d+e x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left ((-d+e) (1+x)^{10}+(d-2 e) (1+x)^{11}+e (1+x)^{12}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{22} (d-e) \left (1+x^2\right )^{11}+\frac{1}{24} (d-2 e) \left (1+x^2\right )^{12}+\frac{1}{26} e \left (1+x^2\right )^{13}\\ \end{align*}

Mathematica [B]  time = 0.0211536, size = 151, normalized size = 3.36 \[ \frac{1}{24} x^{24} (d+10 e)+\frac{5}{22} x^{22} (2 d+9 e)+\frac{3}{4} x^{20} (3 d+8 e)+\frac{5}{3} x^{18} (4 d+7 e)+\frac{21}{8} x^{16} (5 d+6 e)+3 x^{14} (6 d+5 e)+\frac{5}{2} x^{12} (7 d+4 e)+\frac{3}{2} x^{10} (8 d+3 e)+\frac{5}{8} x^8 (9 d+2 e)+\frac{1}{6} x^6 (10 d+e)+\frac{d x^4}{4}+\frac{e x^{26}}{26} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d + e*x^2)*(1 + 2*x^2 + x^4)^5,x]

[Out]

(d*x^4)/4 + ((10*d + e)*x^6)/6 + (5*(9*d + 2*e)*x^8)/8 + (3*(8*d + 3*e)*x^10)/2 + (5*(7*d + 4*e)*x^12)/2 + 3*(
6*d + 5*e)*x^14 + (21*(5*d + 6*e)*x^16)/8 + (5*(4*d + 7*e)*x^18)/3 + (3*(3*d + 8*e)*x^20)/4 + (5*(2*d + 9*e)*x
^22)/22 + ((d + 10*e)*x^24)/24 + (e*x^26)/26

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Maple [B]  time = 0., size = 130, normalized size = 2.9 \begin{align*}{\frac{e{x}^{26}}{26}}+{\frac{ \left ( d+10\,e \right ){x}^{24}}{24}}+{\frac{ \left ( 10\,d+45\,e \right ){x}^{22}}{22}}+{\frac{ \left ( 45\,d+120\,e \right ){x}^{20}}{20}}+{\frac{ \left ( 120\,d+210\,e \right ){x}^{18}}{18}}+{\frac{ \left ( 210\,d+252\,e \right ){x}^{16}}{16}}+{\frac{ \left ( 252\,d+210\,e \right ){x}^{14}}{14}}+{\frac{ \left ( 210\,d+120\,e \right ){x}^{12}}{12}}+{\frac{ \left ( 120\,d+45\,e \right ){x}^{10}}{10}}+{\frac{ \left ( 45\,d+10\,e \right ){x}^{8}}{8}}+{\frac{ \left ( 10\,d+e \right ){x}^{6}}{6}}+{\frac{d{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(e*x^2+d)*(x^4+2*x^2+1)^5,x)

[Out]

1/26*e*x^26+1/24*(d+10*e)*x^24+1/22*(10*d+45*e)*x^22+1/20*(45*d+120*e)*x^20+1/18*(120*d+210*e)*x^18+1/16*(210*
d+252*e)*x^16+1/14*(252*d+210*e)*x^14+1/12*(210*d+120*e)*x^12+1/10*(120*d+45*e)*x^10+1/8*(45*d+10*e)*x^8+1/6*(
10*d+e)*x^6+1/4*d*x^4

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Maxima [B]  time = 0.984403, size = 174, normalized size = 3.87 \begin{align*} \frac{1}{26} \, e x^{26} + \frac{1}{24} \,{\left (d + 10 \, e\right )} x^{24} + \frac{5}{22} \,{\left (2 \, d + 9 \, e\right )} x^{22} + \frac{3}{4} \,{\left (3 \, d + 8 \, e\right )} x^{20} + \frac{5}{3} \,{\left (4 \, d + 7 \, e\right )} x^{18} + \frac{21}{8} \,{\left (5 \, d + 6 \, e\right )} x^{16} + 3 \,{\left (6 \, d + 5 \, e\right )} x^{14} + \frac{5}{2} \,{\left (7 \, d + 4 \, e\right )} x^{12} + \frac{3}{2} \,{\left (8 \, d + 3 \, e\right )} x^{10} + \frac{5}{8} \,{\left (9 \, d + 2 \, e\right )} x^{8} + \frac{1}{6} \,{\left (10 \, d + e\right )} x^{6} + \frac{1}{4} \, d x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)*(x^4+2*x^2+1)^5,x, algorithm="maxima")

[Out]

1/26*e*x^26 + 1/24*(d + 10*e)*x^24 + 5/22*(2*d + 9*e)*x^22 + 3/4*(3*d + 8*e)*x^20 + 5/3*(4*d + 7*e)*x^18 + 21/
8*(5*d + 6*e)*x^16 + 3*(6*d + 5*e)*x^14 + 5/2*(7*d + 4*e)*x^12 + 3/2*(8*d + 3*e)*x^10 + 5/8*(9*d + 2*e)*x^8 +
1/6*(10*d + e)*x^6 + 1/4*d*x^4

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Fricas [B]  time = 1.24393, size = 387, normalized size = 8.6 \begin{align*} \frac{1}{26} x^{26} e + \frac{5}{12} x^{24} e + \frac{1}{24} x^{24} d + \frac{45}{22} x^{22} e + \frac{5}{11} x^{22} d + 6 x^{20} e + \frac{9}{4} x^{20} d + \frac{35}{3} x^{18} e + \frac{20}{3} x^{18} d + \frac{63}{4} x^{16} e + \frac{105}{8} x^{16} d + 15 x^{14} e + 18 x^{14} d + 10 x^{12} e + \frac{35}{2} x^{12} d + \frac{9}{2} x^{10} e + 12 x^{10} d + \frac{5}{4} x^{8} e + \frac{45}{8} x^{8} d + \frac{1}{6} x^{6} e + \frac{5}{3} x^{6} d + \frac{1}{4} x^{4} d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)*(x^4+2*x^2+1)^5,x, algorithm="fricas")

[Out]

1/26*x^26*e + 5/12*x^24*e + 1/24*x^24*d + 45/22*x^22*e + 5/11*x^22*d + 6*x^20*e + 9/4*x^20*d + 35/3*x^18*e + 2
0/3*x^18*d + 63/4*x^16*e + 105/8*x^16*d + 15*x^14*e + 18*x^14*d + 10*x^12*e + 35/2*x^12*d + 9/2*x^10*e + 12*x^
10*d + 5/4*x^8*e + 45/8*x^8*d + 1/6*x^6*e + 5/3*x^6*d + 1/4*x^4*d

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Sympy [B]  time = 0.095724, size = 136, normalized size = 3.02 \begin{align*} \frac{d x^{4}}{4} + \frac{e x^{26}}{26} + x^{24} \left (\frac{d}{24} + \frac{5 e}{12}\right ) + x^{22} \left (\frac{5 d}{11} + \frac{45 e}{22}\right ) + x^{20} \left (\frac{9 d}{4} + 6 e\right ) + x^{18} \left (\frac{20 d}{3} + \frac{35 e}{3}\right ) + x^{16} \left (\frac{105 d}{8} + \frac{63 e}{4}\right ) + x^{14} \left (18 d + 15 e\right ) + x^{12} \left (\frac{35 d}{2} + 10 e\right ) + x^{10} \left (12 d + \frac{9 e}{2}\right ) + x^{8} \left (\frac{45 d}{8} + \frac{5 e}{4}\right ) + x^{6} \left (\frac{5 d}{3} + \frac{e}{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x**2+d)*(x**4+2*x**2+1)**5,x)

[Out]

d*x**4/4 + e*x**26/26 + x**24*(d/24 + 5*e/12) + x**22*(5*d/11 + 45*e/22) + x**20*(9*d/4 + 6*e) + x**18*(20*d/3
+ 35*e/3) + x**16*(105*d/8 + 63*e/4) + x**14*(18*d + 15*e) + x**12*(35*d/2 + 10*e) + x**10*(12*d + 9*e/2) + x
**8*(45*d/8 + 5*e/4) + x**6*(5*d/3 + e/6)

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Giac [B]  time = 1.1107, size = 194, normalized size = 4.31 \begin{align*} \frac{1}{26} \, x^{26} e + \frac{1}{24} \, d x^{24} + \frac{5}{12} \, x^{24} e + \frac{5}{11} \, d x^{22} + \frac{45}{22} \, x^{22} e + \frac{9}{4} \, d x^{20} + 6 \, x^{20} e + \frac{20}{3} \, d x^{18} + \frac{35}{3} \, x^{18} e + \frac{105}{8} \, d x^{16} + \frac{63}{4} \, x^{16} e + 18 \, d x^{14} + 15 \, x^{14} e + \frac{35}{2} \, d x^{12} + 10 \, x^{12} e + 12 \, d x^{10} + \frac{9}{2} \, x^{10} e + \frac{45}{8} \, d x^{8} + \frac{5}{4} \, x^{8} e + \frac{5}{3} \, d x^{6} + \frac{1}{6} \, x^{6} e + \frac{1}{4} \, d x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)*(x^4+2*x^2+1)^5,x, algorithm="giac")

[Out]

1/26*x^26*e + 1/24*d*x^24 + 5/12*x^24*e + 5/11*d*x^22 + 45/22*x^22*e + 9/4*d*x^20 + 6*x^20*e + 20/3*d*x^18 + 3
5/3*x^18*e + 105/8*d*x^16 + 63/4*x^16*e + 18*d*x^14 + 15*x^14*e + 35/2*d*x^12 + 10*x^12*e + 12*d*x^10 + 9/2*x^
10*e + 45/8*d*x^8 + 5/4*x^8*e + 5/3*d*x^6 + 1/6*x^6*e + 1/4*d*x^4