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

Optimal. Leaf size=139 \[ \frac {1}{9} x^9 (d+10 e)+\frac {5}{8} x^8 (2 d+9 e)+\frac {15}{7} x^7 (3 d+8 e)+5 x^6 (4 d+7 e)+\frac {42}{5} x^5 (5 d+6 e)+\frac {21}{2} x^4 (6 d+5 e)+10 x^3 (7 d+4 e)+\frac {15}{2} x^2 (8 d+3 e)+5 x (9 d+2 e)+(10 d+e) \log (x)-\frac {d}{x}+\frac {e x^{10}}{10} \]

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Rubi [A]  time = 0.07, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {27, 76} \begin {gather*} \frac {1}{9} x^9 (d+10 e)+\frac {5}{8} x^8 (2 d+9 e)+\frac {15}{7} x^7 (3 d+8 e)+5 x^6 (4 d+7 e)+\frac {42}{5} x^5 (5 d+6 e)+\frac {21}{2} x^4 (6 d+5 e)+10 x^3 (7 d+4 e)+\frac {15}{2} x^2 (8 d+3 e)+5 x (9 d+2 e)+(10 d+e) \log (x)-\frac {d}{x}+\frac {e x^{10}}{10} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(d/x) + 5*(9*d + 2*e)*x + (15*(8*d + 3*e)*x^2)/2 + 10*(7*d + 4*e)*x^3 + (21*(6*d + 5*e)*x^4)/2 + (42*(5*d + 6
*e)*x^5)/5 + 5*(4*d + 7*e)*x^6 + (15*(3*d + 8*e)*x^7)/7 + (5*(2*d + 9*e)*x^8)/8 + ((d + 10*e)*x^9)/9 + (e*x^10
)/10 + (10*d + e)*Log[x]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && 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])

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 139, normalized size = 1.00 \begin {gather*} \frac {1}{9} x^9 (d+10 e)+\frac {5}{8} x^8 (2 d+9 e)+\frac {15}{7} x^7 (3 d+8 e)+5 x^6 (4 d+7 e)+\frac {42}{5} x^5 (5 d+6 e)+\frac {21}{2} x^4 (6 d+5 e)+10 x^3 (7 d+4 e)+\frac {15}{2} x^2 (8 d+3 e)+5 x (9 d+2 e)+(10 d+e) \log (x)-\frac {d}{x}+\frac {e x^{10}}{10} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(d/x) + 5*(9*d + 2*e)*x + (15*(8*d + 3*e)*x^2)/2 + 10*(7*d + 4*e)*x^3 + (21*(6*d + 5*e)*x^4)/2 + (42*(5*d + 6
*e)*x^5)/5 + 5*(4*d + 7*e)*x^6 + (15*(3*d + 8*e)*x^7)/7 + (5*(2*d + 9*e)*x^8)/8 + ((d + 10*e)*x^9)/9 + (e*x^10
)/10 + (10*d + e)*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 131, normalized size = 0.94 \begin {gather*} \frac {252 \, e x^{11} + 280 \, {\left (d + 10 \, e\right )} x^{10} + 1575 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 5400 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 12600 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 21168 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 26460 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 25200 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 18900 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 12600 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 2520 \, {\left (10 \, d + e\right )} x \log \relax (x) - 2520 \, d}{2520 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*(252*e*x^11 + 280*(d + 10*e)*x^10 + 1575*(2*d + 9*e)*x^9 + 5400*(3*d + 8*e)*x^8 + 12600*(4*d + 7*e)*x^7
+ 21168*(5*d + 6*e)*x^6 + 26460*(6*d + 5*e)*x^5 + 25200*(7*d + 4*e)*x^4 + 18900*(8*d + 3*e)*x^3 + 12600*(9*d
+ 2*e)*x^2 + 2520*(10*d + e)*x*log(x) - 2520*d)/x

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giac [A]  time = 0.16, size = 137, normalized size = 0.99 \begin {gather*} \frac {1}{10} \, x^{10} e + \frac {1}{9} \, d x^{9} + \frac {10}{9} \, x^{9} e + \frac {5}{4} \, d x^{8} + \frac {45}{8} \, x^{8} e + \frac {45}{7} \, d x^{7} + \frac {120}{7} \, x^{7} e + 20 \, d x^{6} + 35 \, x^{6} e + 42 \, d x^{5} + \frac {252}{5} \, x^{5} e + 63 \, d x^{4} + \frac {105}{2} \, x^{4} e + 70 \, d x^{3} + 40 \, x^{3} e + 60 \, d x^{2} + \frac {45}{2} \, x^{2} e + 45 \, d x + 10 \, x e + {\left (10 \, d + e\right )} \log \left ({\left | x \right |}\right ) - \frac {d}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*e + 1/9*d*x^9 + 10/9*x^9*e + 5/4*d*x^8 + 45/8*x^8*e + 45/7*d*x^7 + 120/7*x^7*e + 20*d*x^6 + 35*x^6*e
+ 42*d*x^5 + 252/5*x^5*e + 63*d*x^4 + 105/2*x^4*e + 70*d*x^3 + 40*x^3*e + 60*d*x^2 + 45/2*x^2*e + 45*d*x + 10
*x*e + (10*d + e)*log(abs(x)) - d/x

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maple [A]  time = 0.06, size = 127, normalized size = 0.91 \begin {gather*} \frac {e \,x^{10}}{10}+\frac {d \,x^{9}}{9}+\frac {10 e \,x^{9}}{9}+\frac {5 d \,x^{8}}{4}+\frac {45 e \,x^{8}}{8}+\frac {45 d \,x^{7}}{7}+\frac {120 e \,x^{7}}{7}+20 d \,x^{6}+35 e \,x^{6}+42 d \,x^{5}+\frac {252 e \,x^{5}}{5}+63 d \,x^{4}+\frac {105 e \,x^{4}}{2}+70 d \,x^{3}+40 e \,x^{3}+60 d \,x^{2}+\frac {45 e \,x^{2}}{2}+45 d x +10 d \ln \relax (x )+10 e x +e \ln \relax (x )-\frac {d}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*e*x^10+1/9*d*x^9+10/9*e*x^9+5/4*d*x^8+45/8*e*x^8+45/7*d*x^7+120/7*e*x^7+20*d*x^6+35*e*x^6+42*d*x^5+252/5*
e*x^5+63*d*x^4+105/2*e*x^4+70*d*x^3+40*e*x^3+60*d*x^2+45/2*e*x^2+45*d*x+10*e*x-d/x+10*d*ln(x)+e*ln(x)

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maxima [A]  time = 0.66, size = 125, normalized size = 0.90 \begin {gather*} \frac {1}{10} \, e x^{10} + \frac {1}{9} \, {\left (d + 10 \, e\right )} x^{9} + \frac {5}{8} \, {\left (2 \, d + 9 \, e\right )} x^{8} + \frac {15}{7} \, {\left (3 \, d + 8 \, e\right )} x^{7} + 5 \, {\left (4 \, d + 7 \, e\right )} x^{6} + \frac {42}{5} \, {\left (5 \, d + 6 \, e\right )} x^{5} + \frac {21}{2} \, {\left (6 \, d + 5 \, e\right )} x^{4} + 10 \, {\left (7 \, d + 4 \, e\right )} x^{3} + \frac {15}{2} \, {\left (8 \, d + 3 \, e\right )} x^{2} + 5 \, {\left (9 \, d + 2 \, e\right )} x + {\left (10 \, d + e\right )} \log \relax (x) - \frac {d}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*e*x^10 + 1/9*(d + 10*e)*x^9 + 5/8*(2*d + 9*e)*x^8 + 15/7*(3*d + 8*e)*x^7 + 5*(4*d + 7*e)*x^6 + 42/5*(5*d
+ 6*e)*x^5 + 21/2*(6*d + 5*e)*x^4 + 10*(7*d + 4*e)*x^3 + 15/2*(8*d + 3*e)*x^2 + 5*(9*d + 2*e)*x + (10*d + e)*l
og(x) - d/x

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mupad [B]  time = 0.09, size = 118, normalized size = 0.85 \begin {gather*} x^9\,\left (\frac {d}{9}+\frac {10\,e}{9}\right )+x^6\,\left (20\,d+35\,e\right )+x^8\,\left (\frac {5\,d}{4}+\frac {45\,e}{8}\right )+x^2\,\left (60\,d+\frac {45\,e}{2}\right )+x^3\,\left (70\,d+40\,e\right )+x^4\,\left (63\,d+\frac {105\,e}{2}\right )+x^7\,\left (\frac {45\,d}{7}+\frac {120\,e}{7}\right )+x^5\,\left (42\,d+\frac {252\,e}{5}\right )-\frac {d}{x}+\frac {e\,x^{10}}{10}+x\,\left (45\,d+10\,e\right )+\ln \relax (x)\,\left (10\,d+e\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^9*(d/9 + (10*e)/9) + x^6*(20*d + 35*e) + x^8*((5*d)/4 + (45*e)/8) + x^2*(60*d + (45*e)/2) + x^3*(70*d + 40*e
) + x^4*(63*d + (105*e)/2) + x^7*((45*d)/7 + (120*e)/7) + x^5*(42*d + (252*e)/5) - d/x + (e*x^10)/10 + x*(45*d
+ 10*e) + log(x)*(10*d + e)

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sympy [A]  time = 0.36, size = 121, normalized size = 0.87 \begin {gather*} - \frac {d}{x} + \frac {e x^{10}}{10} + x^{9} \left (\frac {d}{9} + \frac {10 e}{9}\right ) + x^{8} \left (\frac {5 d}{4} + \frac {45 e}{8}\right ) + x^{7} \left (\frac {45 d}{7} + \frac {120 e}{7}\right ) + x^{6} \left (20 d + 35 e\right ) + x^{5} \left (42 d + \frac {252 e}{5}\right ) + x^{4} \left (63 d + \frac {105 e}{2}\right ) + x^{3} \left (70 d + 40 e\right ) + x^{2} \left (60 d + \frac {45 e}{2}\right ) + x \left (45 d + 10 e\right ) + \left (10 d + e\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d/x + e*x**10/10 + x**9*(d/9 + 10*e/9) + x**8*(5*d/4 + 45*e/8) + x**7*(45*d/7 + 120*e/7) + x**6*(20*d + 35*e)
+ x**5*(42*d + 252*e/5) + x**4*(63*d + 105*e/2) + x**3*(70*d + 40*e) + x**2*(60*d + 45*e/2) + x*(45*d + 10*e)
+ (10*d + e)*log(x)

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