3.63 \(\int \frac{(d+e x^2) (1+2 x^2+x^4)^5}{x^2} \, dx\)

Optimal. Leaf size=141 \[ \frac{1}{19} x^{19} (d+10 e)+\frac{5}{17} x^{17} (2 d+9 e)+x^{15} (3 d+8 e)+\frac{30}{13} x^{13} (4 d+7 e)+\frac{42}{11} x^{11} (5 d+6 e)+\frac{14}{3} x^9 (6 d+5 e)+\frac{30}{7} x^7 (7 d+4 e)+3 x^5 (8 d+3 e)+\frac{5}{3} x^3 (9 d+2 e)+x (10 d+e)-\frac{d}{x}+\frac{e x^{21}}{21} \]

[Out]

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

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Rubi [A]  time = 0.0821492, antiderivative size = 141, 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} \[ \frac{1}{19} x^{19} (d+10 e)+\frac{5}{17} x^{17} (2 d+9 e)+x^{15} (3 d+8 e)+\frac{30}{13} x^{13} (4 d+7 e)+\frac{42}{11} x^{11} (5 d+6 e)+\frac{14}{3} x^9 (6 d+5 e)+\frac{30}{7} x^7 (7 d+4 e)+3 x^5 (8 d+3 e)+\frac{5}{3} x^3 (9 d+2 e)+x (10 d+e)-\frac{d}{x}+\frac{e x^{21}}{21} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0281304, size = 141, normalized size = 1. \[ \frac{1}{19} x^{19} (d+10 e)+\frac{5}{17} x^{17} (2 d+9 e)+x^{15} (3 d+8 e)+\frac{30}{13} x^{13} (4 d+7 e)+\frac{42}{11} x^{11} (5 d+6 e)+\frac{14}{3} x^9 (6 d+5 e)+\frac{30}{7} x^7 (7 d+4 e)+3 x^5 (8 d+3 e)+\frac{5}{3} x^3 (9 d+2 e)+x (10 d+e)-\frac{d}{x}+\frac{e x^{21}}{21} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 129, normalized size = 0.9 \begin{align*}{\frac{e{x}^{21}}{21}}+{\frac{{x}^{19}d}{19}}+{\frac{10\,{x}^{19}e}{19}}+{\frac{10\,{x}^{17}d}{17}}+{\frac{45\,{x}^{17}e}{17}}+3\,{x}^{15}d+8\,{x}^{15}e+{\frac{120\,{x}^{13}d}{13}}+{\frac{210\,{x}^{13}e}{13}}+{\frac{210\,{x}^{11}d}{11}}+{\frac{252\,{x}^{11}e}{11}}+28\,{x}^{9}d+{\frac{70\,{x}^{9}e}{3}}+30\,{x}^{7}d+{\frac{120\,{x}^{7}e}{7}}+24\,d{x}^{5}+9\,{x}^{5}e+15\,d{x}^{3}+{\frac{10\,{x}^{3}e}{3}}+10\,dx+ex-{\frac{d}{x}} \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^2,x)

[Out]

1/21*e*x^21+1/19*x^19*d+10/19*x^19*e+10/17*x^17*d+45/17*x^17*e+3*x^15*d+8*x^15*e+120/13*x^13*d+210/13*x^13*e+2
10/11*x^11*d+252/11*x^11*e+28*x^9*d+70/3*x^9*e+30*x^7*d+120/7*x^7*e+24*d*x^5+9*x^5*e+15*d*x^3+10/3*x^3*e+10*d*
x+e*x-d/x

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Maxima [A]  time = 0.938122, size = 169, normalized size = 1.2 \begin{align*} \frac{1}{21} \, e x^{21} + \frac{1}{19} \,{\left (d + 10 \, e\right )} x^{19} + \frac{5}{17} \,{\left (2 \, d + 9 \, e\right )} x^{17} +{\left (3 \, d + 8 \, e\right )} x^{15} + \frac{30}{13} \,{\left (4 \, d + 7 \, e\right )} x^{13} + \frac{42}{11} \,{\left (5 \, d + 6 \, e\right )} x^{11} + \frac{14}{3} \,{\left (6 \, d + 5 \, e\right )} x^{9} + \frac{30}{7} \,{\left (7 \, d + 4 \, e\right )} x^{7} + 3 \,{\left (8 \, d + 3 \, e\right )} x^{5} + \frac{5}{3} \,{\left (9 \, d + 2 \, e\right )} x^{3} +{\left (10 \, d + e\right )} x - \frac{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^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.4223, size = 401, normalized size = 2.84 \begin{align*} \frac{46189 \, e x^{22} + 51051 \,{\left (d + 10 \, e\right )} x^{20} + 285285 \,{\left (2 \, d + 9 \, e\right )} x^{18} + 969969 \,{\left (3 \, d + 8 \, e\right )} x^{16} + 2238390 \,{\left (4 \, d + 7 \, e\right )} x^{14} + 3703518 \,{\left (5 \, d + 6 \, e\right )} x^{12} + 4526522 \,{\left (6 \, d + 5 \, e\right )} x^{10} + 4157010 \,{\left (7 \, d + 4 \, e\right )} x^{8} + 2909907 \,{\left (8 \, d + 3 \, e\right )} x^{6} + 1616615 \,{\left (9 \, d + 2 \, e\right )} x^{4} + 969969 \,{\left (10 \, d + e\right )} x^{2} - 969969 \, d}{969969 \, 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^2,x, algorithm="fricas")

[Out]

1/969969*(46189*e*x^22 + 51051*(d + 10*e)*x^20 + 285285*(2*d + 9*e)*x^18 + 969969*(3*d + 8*e)*x^16 + 2238390*(
4*d + 7*e)*x^14 + 3703518*(5*d + 6*e)*x^12 + 4526522*(6*d + 5*e)*x^10 + 4157010*(7*d + 4*e)*x^8 + 2909907*(8*d
+ 3*e)*x^6 + 1616615*(9*d + 2*e)*x^4 + 969969*(10*d + e)*x^2 - 969969*d)/x

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Sympy [A]  time = 0.473909, size = 124, normalized size = 0.88 \begin{align*} - \frac{d}{x} + \frac{e x^{21}}{21} + x^{19} \left (\frac{d}{19} + \frac{10 e}{19}\right ) + x^{17} \left (\frac{10 d}{17} + \frac{45 e}{17}\right ) + x^{15} \left (3 d + 8 e\right ) + x^{13} \left (\frac{120 d}{13} + \frac{210 e}{13}\right ) + x^{11} \left (\frac{210 d}{11} + \frac{252 e}{11}\right ) + x^{9} \left (28 d + \frac{70 e}{3}\right ) + x^{7} \left (30 d + \frac{120 e}{7}\right ) + x^{5} \left (24 d + 9 e\right ) + x^{3} \left (15 d + \frac{10 e}{3}\right ) + x \left (10 d + e\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**2,x)

[Out]

-d/x + e*x**21/21 + x**19*(d/19 + 10*e/19) + x**17*(10*d/17 + 45*e/17) + x**15*(3*d + 8*e) + x**13*(120*d/13 +
210*e/13) + x**11*(210*d/11 + 252*e/11) + x**9*(28*d + 70*e/3) + x**7*(30*d + 120*e/7) + x**5*(24*d + 9*e) +
x**3*(15*d + 10*e/3) + x*(10*d + e)

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Giac [A]  time = 1.12163, size = 188, normalized size = 1.33 \begin{align*} \frac{1}{21} \, x^{21} e + \frac{1}{19} \, d x^{19} + \frac{10}{19} \, x^{19} e + \frac{10}{17} \, d x^{17} + \frac{45}{17} \, x^{17} e + 3 \, d x^{15} + 8 \, x^{15} e + \frac{120}{13} \, d x^{13} + \frac{210}{13} \, x^{13} e + \frac{210}{11} \, d x^{11} + \frac{252}{11} \, x^{11} e + 28 \, d x^{9} + \frac{70}{3} \, x^{9} e + 30 \, d x^{7} + \frac{120}{7} \, x^{7} e + 24 \, d x^{5} + 9 \, x^{5} e + 15 \, d x^{3} + \frac{10}{3} \, x^{3} e + 10 \, d x + x e - \frac{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^2,x, algorithm="giac")

[Out]

1/21*x^21*e + 1/19*d*x^19 + 10/19*x^19*e + 10/17*d*x^17 + 45/17*x^17*e + 3*d*x^15 + 8*x^15*e + 120/13*d*x^13 +
210/13*x^13*e + 210/11*d*x^11 + 252/11*x^11*e + 28*d*x^9 + 70/3*x^9*e + 30*d*x^7 + 120/7*x^7*e + 24*d*x^5 + 9
*x^5*e + 15*d*x^3 + 10/3*x^3*e + 10*d*x + x*e - d/x