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

Optimal. Leaf size=153 \[ \frac{1}{23} x^{23} (d+10 e)+\frac{5}{21} x^{21} (2 d+9 e)+\frac{15}{19} x^{19} (3 d+8 e)+\frac{30}{17} x^{17} (4 d+7 e)+\frac{14}{5} x^{15} (5 d+6 e)+\frac{42}{13} x^{13} (6 d+5 e)+\frac{30}{11} x^{11} (7 d+4 e)+\frac{5}{3} x^9 (8 d+3 e)+\frac{5}{7} x^7 (9 d+2 e)+\frac{1}{5} x^5 (10 d+e)+\frac{d x^3}{3}+\frac{e x^{25}}{25} \]

[Out]

(d*x^3)/3 + ((10*d + e)*x^5)/5 + (5*(9*d + 2*e)*x^7)/7 + (5*(8*d + 3*e)*x^9)/3 + (30*(7*d + 4*e)*x^11)/11 + (4
2*(6*d + 5*e)*x^13)/13 + (14*(5*d + 6*e)*x^15)/5 + (30*(4*d + 7*e)*x^17)/17 + (15*(3*d + 8*e)*x^19)/19 + (5*(2
*d + 9*e)*x^21)/21 + ((d + 10*e)*x^23)/23 + (e*x^25)/25

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Rubi [A]  time = 0.0845725, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {28, 448} \[ \frac{1}{23} x^{23} (d+10 e)+\frac{5}{21} x^{21} (2 d+9 e)+\frac{15}{19} x^{19} (3 d+8 e)+\frac{30}{17} x^{17} (4 d+7 e)+\frac{14}{5} x^{15} (5 d+6 e)+\frac{42}{13} x^{13} (6 d+5 e)+\frac{30}{11} x^{11} (7 d+4 e)+\frac{5}{3} x^9 (8 d+3 e)+\frac{5}{7} x^7 (9 d+2 e)+\frac{1}{5} x^5 (10 d+e)+\frac{d x^3}{3}+\frac{e x^{25}}{25} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x^3)/3 + ((10*d + e)*x^5)/5 + (5*(9*d + 2*e)*x^7)/7 + (5*(8*d + 3*e)*x^9)/3 + (30*(7*d + 4*e)*x^11)/11 + (4
2*(6*d + 5*e)*x^13)/13 + (14*(5*d + 6*e)*x^15)/5 + (30*(4*d + 7*e)*x^17)/17 + (15*(3*d + 8*e)*x^19)/19 + (5*(2
*d + 9*e)*x^21)/21 + ((d + 10*e)*x^23)/23 + (e*x^25)/25

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 448

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

Rubi steps

\begin{align*} \int x^2 \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int x^2 \left (1+x^2\right )^{10} \left (d+e x^2\right ) \, dx\\ &=\int \left (d x^2+(10 d+e) x^4+5 (9 d+2 e) x^6+15 (8 d+3 e) x^8+30 (7 d+4 e) x^{10}+42 (6 d+5 e) x^{12}+42 (5 d+6 e) x^{14}+30 (4 d+7 e) x^{16}+15 (3 d+8 e) x^{18}+5 (2 d+9 e) x^{20}+(d+10 e) x^{22}+e x^{24}\right ) \, dx\\ &=\frac{d x^3}{3}+\frac{1}{5} (10 d+e) x^5+\frac{5}{7} (9 d+2 e) x^7+\frac{5}{3} (8 d+3 e) x^9+\frac{30}{11} (7 d+4 e) x^{11}+\frac{42}{13} (6 d+5 e) x^{13}+\frac{14}{5} (5 d+6 e) x^{15}+\frac{30}{17} (4 d+7 e) x^{17}+\frac{15}{19} (3 d+8 e) x^{19}+\frac{5}{21} (2 d+9 e) x^{21}+\frac{1}{23} (d+10 e) x^{23}+\frac{e x^{25}}{25}\\ \end{align*}

Mathematica [A]  time = 0.0205113, size = 153, normalized size = 1. \[ \frac{1}{23} x^{23} (d+10 e)+\frac{5}{21} x^{21} (2 d+9 e)+\frac{15}{19} x^{19} (3 d+8 e)+\frac{30}{17} x^{17} (4 d+7 e)+\frac{14}{5} x^{15} (5 d+6 e)+\frac{42}{13} x^{13} (6 d+5 e)+\frac{30}{11} x^{11} (7 d+4 e)+\frac{5}{3} x^9 (8 d+3 e)+\frac{5}{7} x^7 (9 d+2 e)+\frac{1}{5} x^5 (10 d+e)+\frac{d x^3}{3}+\frac{e x^{25}}{25} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x^3)/3 + ((10*d + e)*x^5)/5 + (5*(9*d + 2*e)*x^7)/7 + (5*(8*d + 3*e)*x^9)/3 + (30*(7*d + 4*e)*x^11)/11 + (4
2*(6*d + 5*e)*x^13)/13 + (14*(5*d + 6*e)*x^15)/5 + (30*(4*d + 7*e)*x^17)/17 + (15*(3*d + 8*e)*x^19)/19 + (5*(2
*d + 9*e)*x^21)/21 + ((d + 10*e)*x^23)/23 + (e*x^25)/25

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Maple [A]  time = 0., size = 130, normalized size = 0.9 \begin{align*}{\frac{e{x}^{25}}{25}}+{\frac{ \left ( d+10\,e \right ){x}^{23}}{23}}+{\frac{ \left ( 10\,d+45\,e \right ){x}^{21}}{21}}+{\frac{ \left ( 45\,d+120\,e \right ){x}^{19}}{19}}+{\frac{ \left ( 120\,d+210\,e \right ){x}^{17}}{17}}+{\frac{ \left ( 210\,d+252\,e \right ){x}^{15}}{15}}+{\frac{ \left ( 252\,d+210\,e \right ){x}^{13}}{13}}+{\frac{ \left ( 210\,d+120\,e \right ){x}^{11}}{11}}+{\frac{ \left ( 120\,d+45\,e \right ){x}^{9}}{9}}+{\frac{ \left ( 45\,d+10\,e \right ){x}^{7}}{7}}+{\frac{ \left ( 10\,d+e \right ){x}^{5}}{5}}+{\frac{d{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*e*x^25+1/23*(d+10*e)*x^23+1/21*(10*d+45*e)*x^21+1/19*(45*d+120*e)*x^19+1/17*(120*d+210*e)*x^17+1/15*(210*
d+252*e)*x^15+1/13*(252*d+210*e)*x^13+1/11*(210*d+120*e)*x^11+1/9*(120*d+45*e)*x^9+1/7*(45*d+10*e)*x^7+1/5*(10
*d+e)*x^5+1/3*d*x^3

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Maxima [A]  time = 0.965705, size = 174, normalized size = 1.14 \begin{align*} \frac{1}{25} \, e x^{25} + \frac{1}{23} \,{\left (d + 10 \, e\right )} x^{23} + \frac{5}{21} \,{\left (2 \, d + 9 \, e\right )} x^{21} + \frac{15}{19} \,{\left (3 \, d + 8 \, e\right )} x^{19} + \frac{30}{17} \,{\left (4 \, d + 7 \, e\right )} x^{17} + \frac{14}{5} \,{\left (5 \, d + 6 \, e\right )} x^{15} + \frac{42}{13} \,{\left (6 \, d + 5 \, e\right )} x^{13} + \frac{30}{11} \,{\left (7 \, d + 4 \, e\right )} x^{11} + \frac{5}{3} \,{\left (8 \, d + 3 \, e\right )} x^{9} + \frac{5}{7} \,{\left (9 \, d + 2 \, e\right )} x^{7} + \frac{1}{5} \,{\left (10 \, d + e\right )} x^{5} + \frac{1}{3} \, d x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*e*x^25 + 1/23*(d + 10*e)*x^23 + 5/21*(2*d + 9*e)*x^21 + 15/19*(3*d + 8*e)*x^19 + 30/17*(4*d + 7*e)*x^17 +
14/5*(5*d + 6*e)*x^15 + 42/13*(6*d + 5*e)*x^13 + 30/11*(7*d + 4*e)*x^11 + 5/3*(8*d + 3*e)*x^9 + 5/7*(9*d + 2*
e)*x^7 + 1/5*(10*d + e)*x^5 + 1/3*d*x^3

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Fricas [A]  time = 1.27225, size = 414, normalized size = 2.71 \begin{align*} \frac{1}{25} x^{25} e + \frac{10}{23} x^{23} e + \frac{1}{23} x^{23} d + \frac{15}{7} x^{21} e + \frac{10}{21} x^{21} d + \frac{120}{19} x^{19} e + \frac{45}{19} x^{19} d + \frac{210}{17} x^{17} e + \frac{120}{17} x^{17} d + \frac{84}{5} x^{15} e + 14 x^{15} d + \frac{210}{13} x^{13} e + \frac{252}{13} x^{13} d + \frac{120}{11} x^{11} e + \frac{210}{11} x^{11} d + 5 x^{9} e + \frac{40}{3} x^{9} d + \frac{10}{7} x^{7} e + \frac{45}{7} x^{7} d + \frac{1}{5} x^{5} e + 2 x^{5} d + \frac{1}{3} x^{3} d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*x^25*e + 10/23*x^23*e + 1/23*x^23*d + 15/7*x^21*e + 10/21*x^21*d + 120/19*x^19*e + 45/19*x^19*d + 210/17*
x^17*e + 120/17*x^17*d + 84/5*x^15*e + 14*x^15*d + 210/13*x^13*e + 252/13*x^13*d + 120/11*x^11*e + 210/11*x^11
*d + 5*x^9*e + 40/3*x^9*d + 10/7*x^7*e + 45/7*x^7*d + 1/5*x^5*e + 2*x^5*d + 1/3*x^3*d

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Sympy [A]  time = 0.097982, size = 139, normalized size = 0.91 \begin{align*} \frac{d x^{3}}{3} + \frac{e x^{25}}{25} + x^{23} \left (\frac{d}{23} + \frac{10 e}{23}\right ) + x^{21} \left (\frac{10 d}{21} + \frac{15 e}{7}\right ) + x^{19} \left (\frac{45 d}{19} + \frac{120 e}{19}\right ) + x^{17} \left (\frac{120 d}{17} + \frac{210 e}{17}\right ) + x^{15} \left (14 d + \frac{84 e}{5}\right ) + x^{13} \left (\frac{252 d}{13} + \frac{210 e}{13}\right ) + x^{11} \left (\frac{210 d}{11} + \frac{120 e}{11}\right ) + x^{9} \left (\frac{40 d}{3} + 5 e\right ) + x^{7} \left (\frac{45 d}{7} + \frac{10 e}{7}\right ) + x^{5} \left (2 d + \frac{e}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*x**3/3 + e*x**25/25 + x**23*(d/23 + 10*e/23) + x**21*(10*d/21 + 15*e/7) + x**19*(45*d/19 + 120*e/19) + x**17
*(120*d/17 + 210*e/17) + x**15*(14*d + 84*e/5) + x**13*(252*d/13 + 210*e/13) + x**11*(210*d/11 + 120*e/11) + x
**9*(40*d/3 + 5*e) + x**7*(45*d/7 + 10*e/7) + x**5*(2*d + e/5)

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Giac [A]  time = 1.12289, size = 194, normalized size = 1.27 \begin{align*} \frac{1}{25} \, x^{25} e + \frac{1}{23} \, d x^{23} + \frac{10}{23} \, x^{23} e + \frac{10}{21} \, d x^{21} + \frac{15}{7} \, x^{21} e + \frac{45}{19} \, d x^{19} + \frac{120}{19} \, x^{19} e + \frac{120}{17} \, d x^{17} + \frac{210}{17} \, x^{17} e + 14 \, d x^{15} + \frac{84}{5} \, x^{15} e + \frac{252}{13} \, d x^{13} + \frac{210}{13} \, x^{13} e + \frac{210}{11} \, d x^{11} + \frac{120}{11} \, x^{11} e + \frac{40}{3} \, d x^{9} + 5 \, x^{9} e + \frac{45}{7} \, d x^{7} + \frac{10}{7} \, x^{7} e + 2 \, d x^{5} + \frac{1}{5} \, x^{5} e + \frac{1}{3} \, d x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*x^25*e + 1/23*d*x^23 + 10/23*x^23*e + 10/21*d*x^21 + 15/7*x^21*e + 45/19*d*x^19 + 120/19*x^19*e + 120/17*
d*x^17 + 210/17*x^17*e + 14*d*x^15 + 84/5*x^15*e + 252/13*d*x^13 + 210/13*x^13*e + 210/11*d*x^11 + 120/11*x^11
*e + 40/3*d*x^9 + 5*x^9*e + 45/7*d*x^7 + 10/7*x^7*e + 2*d*x^5 + 1/5*x^5*e + 1/3*d*x^3