3.64 \(\int \frac{(d+e x^2) (1+2 x^2+x^4)^5}{x^3} \, dx\)

Optimal. Leaf size=147 \[ \frac{1}{18} x^{18} (d+10 e)+\frac{5}{16} x^{16} (2 d+9 e)+\frac{15}{14} x^{14} (3 d+8 e)+\frac{5}{2} x^{12} (4 d+7 e)+\frac{21}{5} x^{10} (5 d+6 e)+\frac{21}{4} x^8 (6 d+5 e)+5 x^6 (7 d+4 e)+\frac{15}{4} x^4 (8 d+3 e)+\frac{5}{2} x^2 (9 d+2 e)+(10 d+e) \log (x)-\frac{d}{2 x^2}+\frac{e x^{20}}{20} \]

[Out]

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

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Rubi [A]  time = 0.135537, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {28, 446, 76} \[ \frac{1}{18} x^{18} (d+10 e)+\frac{5}{16} x^{16} (2 d+9 e)+\frac{15}{14} x^{14} (3 d+8 e)+\frac{5}{2} x^{12} (4 d+7 e)+\frac{21}{5} x^{10} (5 d+6 e)+\frac{21}{4} x^8 (6 d+5 e)+5 x^6 (7 d+4 e)+\frac{15}{4} x^4 (8 d+3 e)+\frac{5}{2} x^2 (9 d+2 e)+(10 d+e) \log (x)-\frac{d}{2 x^2}+\frac{e x^{20}}{20} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)*(1 + 2*x^2 + x^4)^5)/x^3,x]

[Out]

-d/(2*x^2) + (5*(9*d + 2*e)*x^2)/2 + (15*(8*d + 3*e)*x^4)/4 + 5*(7*d + 4*e)*x^6 + (21*(6*d + 5*e)*x^8)/4 + (21
*(5*d + 6*e)*x^10)/5 + (5*(4*d + 7*e)*x^12)/2 + (15*(3*d + 8*e)*x^14)/14 + (5*(2*d + 9*e)*x^16)/16 + ((d + 10*
e)*x^18)/18 + (e*x^20)/20 + (10*d + e)*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 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

Mathematica [A]  time = 0.0399448, size = 147, normalized size = 1. \[ \frac{1}{18} x^{18} (d+10 e)+\frac{5}{16} x^{16} (2 d+9 e)+\frac{15}{14} x^{14} (3 d+8 e)+\frac{5}{2} x^{12} (4 d+7 e)+\frac{21}{5} x^{10} (5 d+6 e)+\frac{21}{4} x^8 (6 d+5 e)+5 x^6 (7 d+4 e)+\frac{15}{4} x^4 (8 d+3 e)+\frac{5}{2} x^2 (9 d+2 e)+(10 d+e) \log (x)-\frac{d}{2 x^2}+\frac{e x^{20}}{20} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)*(1 + 2*x^2 + x^4)^5)/x^3,x]

[Out]

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

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Maple [A]  time = 0.006, size = 131, normalized size = 0.9 \begin{align*}{\frac{e{x}^{20}}{20}}+{\frac{d{x}^{18}}{18}}+{\frac{5\,{x}^{18}e}{9}}+{\frac{5\,d{x}^{16}}{8}}+{\frac{45\,{x}^{16}e}{16}}+{\frac{45\,d{x}^{14}}{14}}+{\frac{60\,{x}^{14}e}{7}}+10\,d{x}^{12}+{\frac{35\,{x}^{12}e}{2}}+21\,d{x}^{10}+{\frac{126\,{x}^{10}e}{5}}+{\frac{63\,d{x}^{8}}{2}}+{\frac{105\,{x}^{8}e}{4}}+35\,d{x}^{6}+20\,{x}^{6}e+30\,d{x}^{4}+{\frac{45\,{x}^{4}e}{4}}+{\frac{45\,d{x}^{2}}{2}}+5\,e{x}^{2}+10\,d\ln \left ( x \right ) +\ln \left ( x \right ) e-{\frac{d}{2\,{x}^{2}}} \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^3,x)

[Out]

1/20*e*x^20+1/18*d*x^18+5/9*x^18*e+5/8*d*x^16+45/16*x^16*e+45/14*d*x^14+60/7*x^14*e+10*d*x^12+35/2*x^12*e+21*d
*x^10+126/5*x^10*e+63/2*d*x^8+105/4*x^8*e+35*d*x^6+20*x^6*e+30*d*x^4+45/4*x^4*e+45/2*d*x^2+5*e*x^2+10*d*ln(x)+
ln(x)*e-1/2*d/x^2

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Maxima [A]  time = 0.946439, size = 176, normalized size = 1.2 \begin{align*} \frac{1}{20} \, e x^{20} + \frac{1}{18} \,{\left (d + 10 \, e\right )} x^{18} + \frac{5}{16} \,{\left (2 \, d + 9 \, e\right )} x^{16} + \frac{15}{14} \,{\left (3 \, d + 8 \, e\right )} x^{14} + \frac{5}{2} \,{\left (4 \, d + 7 \, e\right )} x^{12} + \frac{21}{5} \,{\left (5 \, d + 6 \, e\right )} x^{10} + \frac{21}{4} \,{\left (6 \, d + 5 \, e\right )} x^{8} + 5 \,{\left (7 \, d + 4 \, e\right )} x^{6} + \frac{15}{4} \,{\left (8 \, d + 3 \, e\right )} x^{4} + \frac{5}{2} \,{\left (9 \, d + 2 \, e\right )} x^{2} + \frac{1}{2} \,{\left (10 \, d + e\right )} \log \left (x^{2}\right ) - \frac{d}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.43393, size = 378, normalized size = 2.57 \begin{align*} \frac{252 \, e x^{22} + 280 \,{\left (d + 10 \, e\right )} x^{20} + 1575 \,{\left (2 \, d + 9 \, e\right )} x^{18} + 5400 \,{\left (3 \, d + 8 \, e\right )} x^{16} + 12600 \,{\left (4 \, d + 7 \, e\right )} x^{14} + 21168 \,{\left (5 \, d + 6 \, e\right )} x^{12} + 26460 \,{\left (6 \, d + 5 \, e\right )} x^{10} + 25200 \,{\left (7 \, d + 4 \, e\right )} x^{8} + 18900 \,{\left (8 \, d + 3 \, e\right )} x^{6} + 12600 \,{\left (9 \, d + 2 \, e\right )} x^{4} + 5040 \,{\left (10 \, d + e\right )} x^{2} \log \left (x\right ) - 2520 \, d}{5040 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

1/5040*(252*e*x^22 + 280*(d + 10*e)*x^20 + 1575*(2*d + 9*e)*x^18 + 5400*(3*d + 8*e)*x^16 + 12600*(4*d + 7*e)*x
^14 + 21168*(5*d + 6*e)*x^12 + 26460*(6*d + 5*e)*x^10 + 25200*(7*d + 4*e)*x^8 + 18900*(8*d + 3*e)*x^6 + 12600*
(9*d + 2*e)*x^4 + 5040*(10*d + e)*x^2*log(x) - 2520*d)/x^2

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Sympy [A]  time = 0.533572, size = 131, normalized size = 0.89 \begin{align*} - \frac{d}{2 x^{2}} + \frac{e x^{20}}{20} + x^{18} \left (\frac{d}{18} + \frac{5 e}{9}\right ) + x^{16} \left (\frac{5 d}{8} + \frac{45 e}{16}\right ) + x^{14} \left (\frac{45 d}{14} + \frac{60 e}{7}\right ) + x^{12} \left (10 d + \frac{35 e}{2}\right ) + x^{10} \left (21 d + \frac{126 e}{5}\right ) + x^{8} \left (\frac{63 d}{2} + \frac{105 e}{4}\right ) + x^{6} \left (35 d + 20 e\right ) + x^{4} \left (30 d + \frac{45 e}{4}\right ) + x^{2} \left (\frac{45 d}{2} + 5 e\right ) + \left (10 d + e\right ) \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**3,x)

[Out]

-d/(2*x**2) + e*x**20/20 + x**18*(d/18 + 5*e/9) + x**16*(5*d/8 + 45*e/16) + x**14*(45*d/14 + 60*e/7) + x**12*(
10*d + 35*e/2) + x**10*(21*d + 126*e/5) + x**8*(63*d/2 + 105*e/4) + x**6*(35*d + 20*e) + x**4*(30*d + 45*e/4)
+ x**2*(45*d/2 + 5*e) + (10*d + e)*log(x)

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Giac [A]  time = 1.11002, size = 211, normalized size = 1.44 \begin{align*} \frac{1}{20} \, x^{20} e + \frac{1}{18} \, d x^{18} + \frac{5}{9} \, x^{18} e + \frac{5}{8} \, d x^{16} + \frac{45}{16} \, x^{16} e + \frac{45}{14} \, d x^{14} + \frac{60}{7} \, x^{14} e + 10 \, d x^{12} + \frac{35}{2} \, x^{12} e + 21 \, d x^{10} + \frac{126}{5} \, x^{10} e + \frac{63}{2} \, d x^{8} + \frac{105}{4} \, x^{8} e + 35 \, d x^{6} + 20 \, x^{6} e + 30 \, d x^{4} + \frac{45}{4} \, x^{4} e + \frac{45}{2} \, d x^{2} + 5 \, x^{2} e + \frac{1}{2} \,{\left (10 \, d + e\right )} \log \left (x^{2}\right ) - \frac{10 \, d x^{2} + x^{2} e + d}{2 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

1/20*x^20*e + 1/18*d*x^18 + 5/9*x^18*e + 5/8*d*x^16 + 45/16*x^16*e + 45/14*d*x^14 + 60/7*x^14*e + 10*d*x^12 +
35/2*x^12*e + 21*d*x^10 + 126/5*x^10*e + 63/2*d*x^8 + 105/4*x^8*e + 35*d*x^6 + 20*x^6*e + 30*d*x^4 + 45/4*x^4*
e + 45/2*d*x^2 + 5*x^2*e + 1/2*(10*d + e)*log(x^2) - 1/2*(10*d*x^2 + x^2*e + d)/x^2