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

Optimal. Leaf size=29 \[ \frac{1}{22} \left (x^2+1\right )^{11} (d-e)+\frac{1}{24} e \left (x^2+1\right )^{12} \]

[Out]

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

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Rubi [A]  time = 0.0493032, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {28, 444, 43} \[ \frac{1}{22} \left (x^2+1\right )^{11} (d-e)+\frac{1}{24} e \left (x^2+1\right )^{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

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

Mathematica [B]  time = 0.0135391, size = 149, normalized size = 5.14 \[ \frac{1}{22} x^{22} (d+10 e)+\frac{1}{4} x^{20} (2 d+9 e)+\frac{5}{6} x^{18} (3 d+8 e)+\frac{15}{8} x^{16} (4 d+7 e)+3 x^{14} (5 d+6 e)+\frac{7}{2} x^{12} (6 d+5 e)+3 x^{10} (7 d+4 e)+\frac{15}{8} x^8 (8 d+3 e)+\frac{5}{6} x^6 (9 d+2 e)+\frac{1}{4} x^4 (10 d+e)+\frac{d x^2}{2}+\frac{e x^{24}}{24} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x^2)/2 + ((10*d + e)*x^4)/4 + (5*(9*d + 2*e)*x^6)/6 + (15*(8*d + 3*e)*x^8)/8 + 3*(7*d + 4*e)*x^10 + (7*(6*d
+ 5*e)*x^12)/2 + 3*(5*d + 6*e)*x^14 + (15*(4*d + 7*e)*x^16)/8 + (5*(3*d + 8*e)*x^18)/6 + ((2*d + 9*e)*x^20)/4
+ ((d + 10*e)*x^22)/22 + (e*x^24)/24

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Maple [B]  time = 0.002, size = 130, normalized size = 4.5 \begin{align*}{\frac{e{x}^{24}}{24}}+{\frac{ \left ( d+10\,e \right ){x}^{22}}{22}}+{\frac{ \left ( 10\,d+45\,e \right ){x}^{20}}{20}}+{\frac{ \left ( 45\,d+120\,e \right ){x}^{18}}{18}}+{\frac{ \left ( 120\,d+210\,e \right ){x}^{16}}{16}}+{\frac{ \left ( 210\,d+252\,e \right ){x}^{14}}{14}}+{\frac{ \left ( 252\,d+210\,e \right ){x}^{12}}{12}}+{\frac{ \left ( 210\,d+120\,e \right ){x}^{10}}{10}}+{\frac{ \left ( 120\,d+45\,e \right ){x}^{8}}{8}}+{\frac{ \left ( 45\,d+10\,e \right ){x}^{6}}{6}}+{\frac{ \left ( 10\,d+e \right ){x}^{4}}{4}}+{\frac{d{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*e*x^24+1/22*(d+10*e)*x^22+1/20*(10*d+45*e)*x^20+1/18*(45*d+120*e)*x^18+1/16*(120*d+210*e)*x^16+1/14*(210*
d+252*e)*x^14+1/12*(252*d+210*e)*x^12+1/10*(210*d+120*e)*x^10+1/8*(120*d+45*e)*x^8+1/6*(45*d+10*e)*x^6+1/4*(10
*d+e)*x^4+1/2*d*x^2

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Maxima [B]  time = 0.973006, size = 174, normalized size = 6. \begin{align*} \frac{1}{24} \, e x^{24} + \frac{1}{22} \,{\left (d + 10 \, e\right )} x^{22} + \frac{1}{4} \,{\left (2 \, d + 9 \, e\right )} x^{20} + \frac{5}{6} \,{\left (3 \, d + 8 \, e\right )} x^{18} + \frac{15}{8} \,{\left (4 \, d + 7 \, e\right )} x^{16} + 3 \,{\left (5 \, d + 6 \, e\right )} x^{14} + \frac{7}{2} \,{\left (6 \, d + 5 \, e\right )} x^{12} + 3 \,{\left (7 \, d + 4 \, e\right )} x^{10} + \frac{15}{8} \,{\left (8 \, d + 3 \, e\right )} x^{8} + \frac{5}{6} \,{\left (9 \, d + 2 \, e\right )} x^{6} + \frac{1}{4} \,{\left (10 \, d + e\right )} x^{4} + \frac{1}{2} \, d x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*e*x^24 + 1/22*(d + 10*e)*x^22 + 1/4*(2*d + 9*e)*x^20 + 5/6*(3*d + 8*e)*x^18 + 15/8*(4*d + 7*e)*x^16 + 3*(
5*d + 6*e)*x^14 + 7/2*(6*d + 5*e)*x^12 + 3*(7*d + 4*e)*x^10 + 15/8*(8*d + 3*e)*x^8 + 5/6*(9*d + 2*e)*x^6 + 1/4
*(10*d + e)*x^4 + 1/2*d*x^2

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Fricas [B]  time = 1.23295, size = 381, normalized size = 13.14 \begin{align*} \frac{1}{24} x^{24} e + \frac{5}{11} x^{22} e + \frac{1}{22} x^{22} d + \frac{9}{4} x^{20} e + \frac{1}{2} x^{20} d + \frac{20}{3} x^{18} e + \frac{5}{2} x^{18} d + \frac{105}{8} x^{16} e + \frac{15}{2} x^{16} d + 18 x^{14} e + 15 x^{14} d + \frac{35}{2} x^{12} e + 21 x^{12} d + 12 x^{10} e + 21 x^{10} d + \frac{45}{8} x^{8} e + 15 x^{8} d + \frac{5}{3} x^{6} e + \frac{15}{2} x^{6} d + \frac{1}{4} x^{4} e + \frac{5}{2} x^{4} d + \frac{1}{2} x^{2} d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*x^24*e + 5/11*x^22*e + 1/22*x^22*d + 9/4*x^20*e + 1/2*x^20*d + 20/3*x^18*e + 5/2*x^18*d + 105/8*x^16*e +
15/2*x^16*d + 18*x^14*e + 15*x^14*d + 35/2*x^12*e + 21*x^12*d + 12*x^10*e + 21*x^10*d + 45/8*x^8*e + 15*x^8*d
+ 5/3*x^6*e + 15/2*x^6*d + 1/4*x^4*e + 5/2*x^4*d + 1/2*x^2*d

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Sympy [B]  time = 0.094275, size = 133, normalized size = 4.59 \begin{align*} \frac{d x^{2}}{2} + \frac{e x^{24}}{24} + x^{22} \left (\frac{d}{22} + \frac{5 e}{11}\right ) + x^{20} \left (\frac{d}{2} + \frac{9 e}{4}\right ) + x^{18} \left (\frac{5 d}{2} + \frac{20 e}{3}\right ) + x^{16} \left (\frac{15 d}{2} + \frac{105 e}{8}\right ) + x^{14} \left (15 d + 18 e\right ) + x^{12} \left (21 d + \frac{35 e}{2}\right ) + x^{10} \left (21 d + 12 e\right ) + x^{8} \left (15 d + \frac{45 e}{8}\right ) + x^{6} \left (\frac{15 d}{2} + \frac{5 e}{3}\right ) + x^{4} \left (\frac{5 d}{2} + \frac{e}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*x**2/2 + e*x**24/24 + x**22*(d/22 + 5*e/11) + x**20*(d/2 + 9*e/4) + x**18*(5*d/2 + 20*e/3) + x**16*(15*d/2 +
105*e/8) + x**14*(15*d + 18*e) + x**12*(21*d + 35*e/2) + x**10*(21*d + 12*e) + x**8*(15*d + 45*e/8) + x**6*(1
5*d/2 + 5*e/3) + x**4*(5*d/2 + e/4)

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Giac [B]  time = 1.13029, size = 194, normalized size = 6.69 \begin{align*} \frac{1}{24} \, x^{24} e + \frac{1}{22} \, d x^{22} + \frac{5}{11} \, x^{22} e + \frac{1}{2} \, d x^{20} + \frac{9}{4} \, x^{20} e + \frac{5}{2} \, d x^{18} + \frac{20}{3} \, x^{18} e + \frac{15}{2} \, d x^{16} + \frac{105}{8} \, x^{16} e + 15 \, d x^{14} + 18 \, x^{14} e + 21 \, d x^{12} + \frac{35}{2} \, x^{12} e + 21 \, d x^{10} + 12 \, x^{10} e + 15 \, d x^{8} + \frac{45}{8} \, x^{8} e + \frac{15}{2} \, d x^{6} + \frac{5}{3} \, x^{6} e + \frac{5}{2} \, d x^{4} + \frac{1}{4} \, x^{4} e + \frac{1}{2} \, d x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*x^24*e + 1/22*d*x^22 + 5/11*x^22*e + 1/2*d*x^20 + 9/4*x^20*e + 5/2*d*x^18 + 20/3*x^18*e + 15/2*d*x^16 + 1
05/8*x^16*e + 15*d*x^14 + 18*x^14*e + 21*d*x^12 + 35/2*x^12*e + 21*d*x^10 + 12*x^10*e + 15*d*x^8 + 45/8*x^8*e
+ 15/2*d*x^6 + 5/3*x^6*e + 5/2*d*x^4 + 1/4*x^4*e + 1/2*d*x^2