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

Optimal. Leaf size=143 \[ \frac{1}{21} x^{21} (d+10 e)+\frac{5}{19} x^{19} (2 d+9 e)+\frac{15}{17} x^{17} (3 d+8 e)+2 x^{15} (4 d+7 e)+\frac{42}{13} x^{13} (5 d+6 e)+\frac{42}{11} x^{11} (6 d+5 e)+\frac{10}{3} x^9 (7 d+4 e)+\frac{15}{7} x^7 (8 d+3 e)+x^5 (9 d+2 e)+\frac{1}{3} x^3 (10 d+e)+d x+\frac{e x^{23}}{23} \]

[Out]

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

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Rubi [A]  time = 0.0742943, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {28, 373} \[ \frac{1}{21} x^{21} (d+10 e)+\frac{5}{19} x^{19} (2 d+9 e)+\frac{15}{17} x^{17} (3 d+8 e)+2 x^{15} (4 d+7 e)+\frac{42}{13} x^{13} (5 d+6 e)+\frac{42}{11} x^{11} (6 d+5 e)+\frac{10}{3} x^9 (7 d+4 e)+\frac{15}{7} x^7 (8 d+3 e)+x^5 (9 d+2 e)+\frac{1}{3} x^3 (10 d+e)+d x+\frac{e x^{23}}{23} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 373

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

Rubi steps

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

Mathematica [A]  time = 0.0181159, size = 143, normalized size = 1. \[ \frac{1}{21} x^{21} (d+10 e)+\frac{5}{19} x^{19} (2 d+9 e)+\frac{15}{17} x^{17} (3 d+8 e)+2 x^{15} (4 d+7 e)+\frac{42}{13} x^{13} (5 d+6 e)+\frac{42}{11} x^{11} (6 d+5 e)+\frac{10}{3} x^9 (7 d+4 e)+\frac{15}{7} x^7 (8 d+3 e)+x^5 (9 d+2 e)+\frac{1}{3} x^3 (10 d+e)+d x+\frac{e x^{23}}{23} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 127, normalized size = 0.9 \begin{align*}{\frac{e{x}^{23}}{23}}+{\frac{ \left ( d+10\,e \right ){x}^{21}}{21}}+{\frac{ \left ( 10\,d+45\,e \right ){x}^{19}}{19}}+{\frac{ \left ( 45\,d+120\,e \right ){x}^{17}}{17}}+{\frac{ \left ( 120\,d+210\,e \right ){x}^{15}}{15}}+{\frac{ \left ( 210\,d+252\,e \right ){x}^{13}}{13}}+{\frac{ \left ( 252\,d+210\,e \right ){x}^{11}}{11}}+{\frac{ \left ( 210\,d+120\,e \right ){x}^{9}}{9}}+{\frac{ \left ( 120\,d+45\,e \right ){x}^{7}}{7}}+{\frac{ \left ( 45\,d+10\,e \right ){x}^{5}}{5}}+{\frac{ \left ( 10\,d+e \right ){x}^{3}}{3}}+dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.96456, size = 169, normalized size = 1.18 \begin{align*} \frac{1}{23} \, e x^{23} + \frac{1}{21} \,{\left (d + 10 \, e\right )} x^{21} + \frac{5}{19} \,{\left (2 \, d + 9 \, e\right )} x^{19} + \frac{15}{17} \,{\left (3 \, d + 8 \, e\right )} x^{17} + 2 \,{\left (4 \, d + 7 \, e\right )} x^{15} + \frac{42}{13} \,{\left (5 \, d + 6 \, e\right )} x^{13} + \frac{42}{11} \,{\left (6 \, d + 5 \, e\right )} x^{11} + \frac{10}{3} \,{\left (7 \, d + 4 \, e\right )} x^{9} + \frac{15}{7} \,{\left (8 \, d + 3 \, e\right )} x^{7} +{\left (9 \, d + 2 \, e\right )} x^{5} + \frac{1}{3} \,{\left (10 \, d + e\right )} x^{3} + d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.27074, size = 397, normalized size = 2.78 \begin{align*} \frac{1}{23} x^{23} e + \frac{10}{21} x^{21} e + \frac{1}{21} x^{21} d + \frac{45}{19} x^{19} e + \frac{10}{19} x^{19} d + \frac{120}{17} x^{17} e + \frac{45}{17} x^{17} d + 14 x^{15} e + 8 x^{15} d + \frac{252}{13} x^{13} e + \frac{210}{13} x^{13} d + \frac{210}{11} x^{11} e + \frac{252}{11} x^{11} d + \frac{40}{3} x^{9} e + \frac{70}{3} x^{9} d + \frac{45}{7} x^{7} e + \frac{120}{7} x^{7} d + 2 x^{5} e + 9 x^{5} d + \frac{1}{3} x^{3} e + \frac{10}{3} x^{3} d + x d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/23*x^23*e + 10/21*x^21*e + 1/21*x^21*d + 45/19*x^19*e + 10/19*x^19*d + 120/17*x^17*e + 45/17*x^17*d + 14*x^1
5*e + 8*x^15*d + 252/13*x^13*e + 210/13*x^13*d + 210/11*x^11*e + 252/11*x^11*d + 40/3*x^9*e + 70/3*x^9*d + 45/
7*x^7*e + 120/7*x^7*d + 2*x^5*e + 9*x^5*d + 1/3*x^3*e + 10/3*x^3*d + x*d

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Sympy [A]  time = 0.096666, size = 134, normalized size = 0.94 \begin{align*} d x + \frac{e x^{23}}{23} + x^{21} \left (\frac{d}{21} + \frac{10 e}{21}\right ) + x^{19} \left (\frac{10 d}{19} + \frac{45 e}{19}\right ) + x^{17} \left (\frac{45 d}{17} + \frac{120 e}{17}\right ) + x^{15} \left (8 d + 14 e\right ) + x^{13} \left (\frac{210 d}{13} + \frac{252 e}{13}\right ) + x^{11} \left (\frac{252 d}{11} + \frac{210 e}{11}\right ) + x^{9} \left (\frac{70 d}{3} + \frac{40 e}{3}\right ) + x^{7} \left (\frac{120 d}{7} + \frac{45 e}{7}\right ) + x^{5} \left (9 d + 2 e\right ) + x^{3} \left (\frac{10 d}{3} + \frac{e}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*x + e*x**23/23 + x**21*(d/21 + 10*e/21) + x**19*(10*d/19 + 45*e/19) + x**17*(45*d/17 + 120*e/17) + x**15*(8*
d + 14*e) + x**13*(210*d/13 + 252*e/13) + x**11*(252*d/11 + 210*e/11) + x**9*(70*d/3 + 40*e/3) + x**7*(120*d/7
+ 45*e/7) + x**5*(9*d + 2*e) + x**3*(10*d/3 + e/3)

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Giac [A]  time = 1.10528, size = 190, normalized size = 1.33 \begin{align*} \frac{1}{23} \, x^{23} e + \frac{1}{21} \, d x^{21} + \frac{10}{21} \, x^{21} e + \frac{10}{19} \, d x^{19} + \frac{45}{19} \, x^{19} e + \frac{45}{17} \, d x^{17} + \frac{120}{17} \, x^{17} e + 8 \, d x^{15} + 14 \, x^{15} e + \frac{210}{13} \, d x^{13} + \frac{252}{13} \, x^{13} e + \frac{252}{11} \, d x^{11} + \frac{210}{11} \, x^{11} e + \frac{70}{3} \, d x^{9} + \frac{40}{3} \, x^{9} e + \frac{120}{7} \, d x^{7} + \frac{45}{7} \, x^{7} e + 9 \, d x^{5} + 2 \, x^{5} e + \frac{10}{3} \, d x^{3} + \frac{1}{3} \, x^{3} e + d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/23*x^23*e + 1/21*d*x^21 + 10/21*x^21*e + 10/19*d*x^19 + 45/19*x^19*e + 45/17*d*x^17 + 120/17*x^17*e + 8*d*x^
15 + 14*x^15*e + 210/13*d*x^13 + 252/13*x^13*e + 252/11*d*x^11 + 210/11*x^11*e + 70/3*d*x^9 + 40/3*x^9*e + 120
/7*d*x^7 + 45/7*x^7*e + 9*d*x^5 + 2*x^5*e + 10/3*d*x^3 + 1/3*x^3*e + d*x