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

Optimal. Leaf size=93 \[ \frac{d x^{20}}{20}+\frac{5 d x^{18}}{9}+\frac{45 d x^{16}}{16}+\frac{60 d x^{14}}{7}+\frac{35 d x^{12}}{2}+\frac{126 d x^{10}}{5}+\frac{105 d x^8}{4}+20 d x^6+\frac{45 d x^4}{4}+5 d x^2+d \log (x)+\frac{1}{22} e \left (x^2+1\right )^{11} \]

[Out]

5*d*x^2 + (45*d*x^4)/4 + 20*d*x^6 + (105*d*x^8)/4 + (126*d*x^10)/5 + (35*d*x^12)/2 + (60*d*x^14)/7 + (45*d*x^1
6)/16 + (5*d*x^18)/9 + (d*x^20)/20 + (e*(1 + x^2)^11)/22 + d*Log[x]

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Rubi [A]  time = 0.0547076, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {28, 446, 80, 43} \[ \frac{d x^{20}}{20}+\frac{5 d x^{18}}{9}+\frac{45 d x^{16}}{16}+\frac{60 d x^{14}}{7}+\frac{35 d x^{12}}{2}+\frac{126 d x^{10}}{5}+\frac{105 d x^8}{4}+20 d x^6+\frac{45 d x^4}{4}+5 d x^2+d \log (x)+\frac{1}{22} e \left (x^2+1\right )^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

5*d*x^2 + (45*d*x^4)/4 + 20*d*x^6 + (105*d*x^8)/4 + (126*d*x^10)/5 + (35*d*x^12)/2 + (60*d*x^14)/7 + (45*d*x^1
6)/16 + (5*d*x^18)/9 + (d*x^20)/20 + (e*(1 + x^2)^11)/22 + d*Log[x]

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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
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 \frac{\left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5}{x} \, dx &=\int \frac{\left (1+x^2\right )^{10} \left (d+e x^2\right )}{x} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(1+x)^{10} (d+e x)}{x} \, dx,x,x^2\right )\\ &=\frac{1}{22} e \left (1+x^2\right )^{11}+\frac{1}{2} d \operatorname{Subst}\left (\int \frac{(1+x)^{10}}{x} \, dx,x,x^2\right )\\ &=\frac{1}{22} e \left (1+x^2\right )^{11}+\frac{1}{2} d \operatorname{Subst}\left (\int \left (10+\frac{1}{x}+45 x+120 x^2+210 x^3+252 x^4+210 x^5+120 x^6+45 x^7+10 x^8+x^9\right ) \, dx,x,x^2\right )\\ &=5 d x^2+\frac{45 d x^4}{4}+20 d x^6+\frac{105 d x^8}{4}+\frac{126 d x^{10}}{5}+\frac{35 d x^{12}}{2}+\frac{60 d x^{14}}{7}+\frac{45 d x^{16}}{16}+\frac{5 d x^{18}}{9}+\frac{d x^{20}}{20}+\frac{1}{22} e \left (1+x^2\right )^{11}+d \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0276129, size = 149, normalized size = 1.6 \[ \frac{1}{20} x^{20} (d+10 e)+\frac{5}{18} x^{18} (2 d+9 e)+\frac{15}{16} x^{16} (3 d+8 e)+\frac{15}{7} x^{14} (4 d+7 e)+\frac{7}{2} x^{12} (5 d+6 e)+\frac{21}{5} x^{10} (6 d+5 e)+\frac{15}{4} x^8 (7 d+4 e)+\frac{5}{2} x^6 (8 d+3 e)+\frac{5}{4} x^4 (9 d+2 e)+\frac{1}{2} x^2 (10 d+e)+d \log (x)+\frac{e x^{22}}{22} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((10*d + e)*x^2)/2 + (5*(9*d + 2*e)*x^4)/4 + (5*(8*d + 3*e)*x^6)/2 + (15*(7*d + 4*e)*x^8)/4 + (21*(6*d + 5*e)*
x^10)/5 + (7*(5*d + 6*e)*x^12)/2 + (15*(4*d + 7*e)*x^14)/7 + (15*(3*d + 8*e)*x^16)/16 + (5*(2*d + 9*e)*x^18)/1
8 + ((d + 10*e)*x^20)/20 + (e*x^22)/22 + d*Log[x]

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Maple [A]  time = 0.003, size = 132, normalized size = 1.4 \begin{align*}{\frac{e{x}^{22}}{22}}+{\frac{d{x}^{20}}{20}}+{\frac{e{x}^{20}}{2}}+{\frac{5\,d{x}^{18}}{9}}+{\frac{5\,{x}^{18}e}{2}}+{\frac{45\,d{x}^{16}}{16}}+{\frac{15\,{x}^{16}e}{2}}+{\frac{60\,d{x}^{14}}{7}}+15\,{x}^{14}e+{\frac{35\,d{x}^{12}}{2}}+21\,{x}^{12}e+{\frac{126\,d{x}^{10}}{5}}+21\,{x}^{10}e+{\frac{105\,d{x}^{8}}{4}}+15\,{x}^{8}e+20\,d{x}^{6}+{\frac{15\,{x}^{6}e}{2}}+{\frac{45\,d{x}^{4}}{4}}+{\frac{5\,{x}^{4}e}{2}}+5\,d{x}^{2}+{\frac{e{x}^{2}}{2}}+d\ln \left ( x \right ) \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,x)

[Out]

1/22*e*x^22+1/20*d*x^20+1/2*e*x^20+5/9*d*x^18+5/2*x^18*e+45/16*d*x^16+15/2*x^16*e+60/7*d*x^14+15*x^14*e+35/2*d
*x^12+21*x^12*e+126/5*d*x^10+21*x^10*e+105/4*d*x^8+15*x^8*e+20*d*x^6+15/2*x^6*e+45/4*d*x^4+5/2*x^4*e+5*d*x^2+1
/2*e*x^2+d*ln(x)

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Maxima [A]  time = 0.955849, size = 176, normalized size = 1.89 \begin{align*} \frac{1}{22} \, e x^{22} + \frac{1}{20} \,{\left (d + 10 \, e\right )} x^{20} + \frac{5}{18} \,{\left (2 \, d + 9 \, e\right )} x^{18} + \frac{15}{16} \,{\left (3 \, d + 8 \, e\right )} x^{16} + \frac{15}{7} \,{\left (4 \, d + 7 \, e\right )} x^{14} + \frac{7}{2} \,{\left (5 \, d + 6 \, e\right )} x^{12} + \frac{21}{5} \,{\left (6 \, d + 5 \, e\right )} x^{10} + \frac{15}{4} \,{\left (7 \, d + 4 \, e\right )} x^{8} + \frac{5}{2} \,{\left (8 \, d + 3 \, e\right )} x^{6} + \frac{5}{4} \,{\left (9 \, d + 2 \, e\right )} x^{4} + \frac{1}{2} \,{\left (10 \, d + e\right )} x^{2} + \frac{1}{2} \, d \log \left (x^{2}\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,x, algorithm="maxima")

[Out]

1/22*e*x^22 + 1/20*(d + 10*e)*x^20 + 5/18*(2*d + 9*e)*x^18 + 15/16*(3*d + 8*e)*x^16 + 15/7*(4*d + 7*e)*x^14 +
7/2*(5*d + 6*e)*x^12 + 21/5*(6*d + 5*e)*x^10 + 15/4*(7*d + 4*e)*x^8 + 5/2*(8*d + 3*e)*x^6 + 5/4*(9*d + 2*e)*x^
4 + 1/2*(10*d + e)*x^2 + 1/2*d*log(x^2)

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Fricas [A]  time = 1.42698, size = 344, normalized size = 3.7 \begin{align*} \frac{1}{22} \, e x^{22} + \frac{1}{20} \,{\left (d + 10 \, e\right )} x^{20} + \frac{5}{18} \,{\left (2 \, d + 9 \, e\right )} x^{18} + \frac{15}{16} \,{\left (3 \, d + 8 \, e\right )} x^{16} + \frac{15}{7} \,{\left (4 \, d + 7 \, e\right )} x^{14} + \frac{7}{2} \,{\left (5 \, d + 6 \, e\right )} x^{12} + \frac{21}{5} \,{\left (6 \, d + 5 \, e\right )} x^{10} + \frac{15}{4} \,{\left (7 \, d + 4 \, e\right )} x^{8} + \frac{5}{2} \,{\left (8 \, d + 3 \, e\right )} x^{6} + \frac{5}{4} \,{\left (9 \, d + 2 \, e\right )} x^{4} + \frac{1}{2} \,{\left (10 \, d + e\right )} x^{2} + d \log \left (x\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,x, algorithm="fricas")

[Out]

1/22*e*x^22 + 1/20*(d + 10*e)*x^20 + 5/18*(2*d + 9*e)*x^18 + 15/16*(3*d + 8*e)*x^16 + 15/7*(4*d + 7*e)*x^14 +
7/2*(5*d + 6*e)*x^12 + 21/5*(6*d + 5*e)*x^10 + 15/4*(7*d + 4*e)*x^8 + 5/2*(8*d + 3*e)*x^6 + 5/4*(9*d + 2*e)*x^
4 + 1/2*(10*d + e)*x^2 + d*log(x)

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Sympy [A]  time = 0.321105, size = 131, normalized size = 1.41 \begin{align*} d \log{\left (x \right )} + \frac{e x^{22}}{22} + x^{20} \left (\frac{d}{20} + \frac{e}{2}\right ) + x^{18} \left (\frac{5 d}{9} + \frac{5 e}{2}\right ) + x^{16} \left (\frac{45 d}{16} + \frac{15 e}{2}\right ) + x^{14} \left (\frac{60 d}{7} + 15 e\right ) + x^{12} \left (\frac{35 d}{2} + 21 e\right ) + x^{10} \left (\frac{126 d}{5} + 21 e\right ) + x^{8} \left (\frac{105 d}{4} + 15 e\right ) + x^{6} \left (20 d + \frac{15 e}{2}\right ) + x^{4} \left (\frac{45 d}{4} + \frac{5 e}{2}\right ) + x^{2} \left (5 d + \frac{e}{2}\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,x)

[Out]

d*log(x) + e*x**22/22 + x**20*(d/20 + e/2) + x**18*(5*d/9 + 5*e/2) + x**16*(45*d/16 + 15*e/2) + x**14*(60*d/7
+ 15*e) + x**12*(35*d/2 + 21*e) + x**10*(126*d/5 + 21*e) + x**8*(105*d/4 + 15*e) + x**6*(20*d + 15*e/2) + x**4
*(45*d/4 + 5*e/2) + x**2*(5*d + e/2)

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Giac [A]  time = 1.11416, size = 196, normalized size = 2.11 \begin{align*} \frac{1}{22} \, x^{22} e + \frac{1}{20} \, d x^{20} + \frac{1}{2} \, x^{20} e + \frac{5}{9} \, d x^{18} + \frac{5}{2} \, x^{18} e + \frac{45}{16} \, d x^{16} + \frac{15}{2} \, x^{16} e + \frac{60}{7} \, d x^{14} + 15 \, x^{14} e + \frac{35}{2} \, d x^{12} + 21 \, x^{12} e + \frac{126}{5} \, d x^{10} + 21 \, x^{10} e + \frac{105}{4} \, d x^{8} + 15 \, x^{8} e + 20 \, d x^{6} + \frac{15}{2} \, x^{6} e + \frac{45}{4} \, d x^{4} + \frac{5}{2} \, x^{4} e + 5 \, d x^{2} + \frac{1}{2} \, x^{2} e + \frac{1}{2} \, d \log \left (x^{2}\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,x, algorithm="giac")

[Out]

1/22*x^22*e + 1/20*d*x^20 + 1/2*x^20*e + 5/9*d*x^18 + 5/2*x^18*e + 45/16*d*x^16 + 15/2*x^16*e + 60/7*d*x^14 +
15*x^14*e + 35/2*d*x^12 + 21*x^12*e + 126/5*d*x^10 + 21*x^10*e + 105/4*d*x^8 + 15*x^8*e + 20*d*x^6 + 15/2*x^6*
e + 45/4*d*x^4 + 5/2*x^4*e + 5*d*x^2 + 1/2*x^2*e + 1/2*d*log(x^2)