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]

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

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Rubi [A]  time = 0.14, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.03, size = 147, normalized size = 1.00 \[ \frac {1}{18} x^{18} (d+10 e)+\frac {5}{16} x^{16} (2 d+9 e)+\frac {15}{14} x^{14} (3 d+8 e)+\frac {5}{2} x^{12} (4 d+7 e)+\frac {21}{5} x^{10} (5 d+6 e)+\frac {21}{4} x^8 (6 d+5 e)+5 x^6 (7 d+4 e)+\frac {15}{4} x^4 (8 d+3 e)+\frac {5}{2} x^2 (9 d+2 e)+(10 d+e) \log (x)-\frac {d}{2 x^2}+\frac {e x^{20}}{20} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*d/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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fricas [A]  time = 0.76, size = 133, normalized size = 0.90 \[ \frac {252 \, e x^{22} + 280 \, {\left (d + 10 \, e\right )} x^{20} + 1575 \, {\left (2 \, d + 9 \, e\right )} x^{18} + 5400 \, {\left (3 \, d + 8 \, e\right )} x^{16} + 12600 \, {\left (4 \, d + 7 \, e\right )} x^{14} + 21168 \, {\left (5 \, d + 6 \, e\right )} x^{12} + 26460 \, {\left (6 \, d + 5 \, e\right )} x^{10} + 25200 \, {\left (7 \, d + 4 \, e\right )} x^{8} + 18900 \, {\left (8 \, d + 3 \, e\right )} x^{6} + 12600 \, {\left (9 \, d + 2 \, e\right )} x^{4} + 5040 \, {\left (10 \, d + e\right )} x^{2} \log \relax (x) - 2520 \, d}{5040 \, x^{2}} \]

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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giac [A]  time = 0.26, size = 156, normalized size = 1.06 \[ \frac {1}{20} \, x^{20} e + \frac {1}{18} \, d x^{18} + \frac {5}{9} \, x^{18} e + \frac {5}{8} \, d x^{16} + \frac {45}{16} \, x^{16} e + \frac {45}{14} \, d x^{14} + \frac {60}{7} \, x^{14} e + 10 \, d x^{12} + \frac {35}{2} \, x^{12} e + 21 \, d x^{10} + \frac {126}{5} \, x^{10} e + \frac {63}{2} \, d x^{8} + \frac {105}{4} \, x^{8} e + 35 \, d x^{6} + 20 \, x^{6} e + 30 \, d x^{4} + \frac {45}{4} \, x^{4} e + \frac {45}{2} \, d x^{2} + 5 \, x^{2} e + \frac {1}{2} \, {\left (10 \, d + e\right )} \log \left (x^{2}\right ) - \frac {10 \, d x^{2} + x^{2} e + d}{2 \, x^{2}} \]

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

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maple [A]  time = 0.01, size = 131, normalized size = 0.89 \[ \frac {e \,x^{20}}{20}+\frac {d \,x^{18}}{18}+\frac {5 e \,x^{18}}{9}+\frac {5 d \,x^{16}}{8}+\frac {45 e \,x^{16}}{16}+\frac {45 d \,x^{14}}{14}+\frac {60 e \,x^{14}}{7}+10 d \,x^{12}+\frac {35 e \,x^{12}}{2}+21 d \,x^{10}+\frac {126 e \,x^{10}}{5}+\frac {63 d \,x^{8}}{2}+\frac {105 e \,x^{8}}{4}+35 d \,x^{6}+20 e \,x^{6}+30 d \,x^{4}+\frac {45 e \,x^{4}}{4}+\frac {45 d \,x^{2}}{2}+5 e \,x^{2}+10 d \ln \relax (x )+e \ln \relax (x )-\frac {d}{2 x^{2}} \]

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*e*x^18+5/8*d*x^16+45/16*e*x^16+45/14*d*x^14+60/7*e*x^14+10*d*x^12+35/2*e*x^12+21*d
*x^10+126/5*e*x^10+63/2*d*x^8+105/4*e*x^8+35*d*x^6+20*e*x^6+30*d*x^4+45/4*e*x^4+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 = 0.70, size = 130, normalized size = 0.88 \[ \frac {1}{20} \, e x^{20} + \frac {1}{18} \, {\left (d + 10 \, e\right )} x^{18} + \frac {5}{16} \, {\left (2 \, d + 9 \, e\right )} x^{16} + \frac {15}{14} \, {\left (3 \, d + 8 \, e\right )} x^{14} + \frac {5}{2} \, {\left (4 \, d + 7 \, e\right )} x^{12} + \frac {21}{5} \, {\left (5 \, d + 6 \, e\right )} x^{10} + \frac {21}{4} \, {\left (6 \, d + 5 \, e\right )} x^{8} + 5 \, {\left (7 \, d + 4 \, e\right )} x^{6} + \frac {15}{4} \, {\left (8 \, d + 3 \, e\right )} x^{4} + \frac {5}{2} \, {\left (9 \, d + 2 \, e\right )} x^{2} + \frac {1}{2} \, {\left (10 \, d + e\right )} \log \left (x^{2}\right ) - \frac {d}{2 \, x^{2}} \]

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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mupad [B]  time = 0.08, size = 120, normalized size = 0.82 \[ x^{18}\,\left (\frac {d}{18}+\frac {5\,e}{9}\right )+x^2\,\left (\frac {45\,d}{2}+5\,e\right )+x^{12}\,\left (10\,d+\frac {35\,e}{2}\right )+x^6\,\left (35\,d+20\,e\right )+x^4\,\left (30\,d+\frac {45\,e}{4}\right )+x^{16}\,\left (\frac {5\,d}{8}+\frac {45\,e}{16}\right )+x^{14}\,\left (\frac {45\,d}{14}+\frac {60\,e}{7}\right )+x^{10}\,\left (21\,d+\frac {126\,e}{5}\right )+x^8\,\left (\frac {63\,d}{2}+\frac {105\,e}{4}\right )-\frac {d}{2\,x^2}+\frac {e\,x^{20}}{20}+\ln \relax (x)\,\left (10\,d+e\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)*(2*x^2 + x^4 + 1)^5)/x^3,x)

[Out]

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

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sympy [A]  time = 0.39, size = 131, normalized size = 0.89 \[ - \frac {d}{2 x^{2}} + \frac {e x^{20}}{20} + x^{18} \left (\frac {d}{18} + \frac {5 e}{9}\right ) + x^{16} \left (\frac {5 d}{8} + \frac {45 e}{16}\right ) + x^{14} \left (\frac {45 d}{14} + \frac {60 e}{7}\right ) + x^{12} \left (10 d + \frac {35 e}{2}\right ) + x^{10} \left (21 d + \frac {126 e}{5}\right ) + x^{8} \left (\frac {63 d}{2} + \frac {105 e}{4}\right ) + x^{6} \left (35 d + 20 e\right ) + x^{4} \left (30 d + \frac {45 e}{4}\right ) + x^{2} \left (\frac {45 d}{2} + 5 e\right ) + \left (10 d + e\right ) \log {\relax (x )} \]

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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