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

Optimal. Leaf size=138 \[ \frac {1}{8} x^8 (d+10 e)+\frac {5}{7} x^7 (2 d+9 e)+\frac {5}{2} x^6 (3 d+8 e)+6 x^5 (4 d+7 e)+\frac {21}{2} x^4 (5 d+6 e)+14 x^3 (6 d+5 e)+15 x^2 (7 d+4 e)+15 x (8 d+3 e)-\frac {10 d+e}{x}+5 (9 d+2 e) \log (x)-\frac {d}{2 x^2}+\frac {e x^9}{9} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 138, 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} \[ \frac {1}{8} x^8 (d+10 e)+\frac {5}{7} x^7 (2 d+9 e)+\frac {5}{2} x^6 (3 d+8 e)+6 x^5 (4 d+7 e)+\frac {21}{2} x^4 (5 d+6 e)+14 x^3 (6 d+5 e)+15 x^2 (7 d+4 e)+15 x (8 d+3 e)-\frac {10 d+e}{x}+5 (9 d+2 e) \log (x)-\frac {d}{2 x^2}+\frac {e x^9}{9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.04, size = 139, normalized size = 1.01 \[ \frac {1}{8} x^8 (d+10 e)+\frac {5}{7} x^7 (2 d+9 e)+\frac {5}{2} x^6 (3 d+8 e)+6 x^5 (4 d+7 e)+\frac {21}{2} x^4 (5 d+6 e)+14 x^3 (6 d+5 e)+15 x^2 (7 d+4 e)+15 x (8 d+3 e)+\frac {-10 d-e}{x}+5 (9 d+2 e) \log (x)-\frac {d}{2 x^2}+\frac {e x^9}{9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.77, size = 131, normalized size = 0.95 \[ \frac {56 \, e x^{11} + 63 \, {\left (d + 10 \, e\right )} x^{10} + 360 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 1260 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 3024 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 5292 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 7056 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 7560 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 7560 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 2520 \, {\left (9 \, d + 2 \, e\right )} x^{2} \log \relax (x) - 504 \, {\left (10 \, d + e\right )} x - 252 \, d}{504 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^3,x, algorithm="fricas")

[Out]

1/504*(56*e*x^11 + 63*(d + 10*e)*x^10 + 360*(2*d + 9*e)*x^9 + 1260*(3*d + 8*e)*x^8 + 3024*(4*d + 7*e)*x^7 + 52
92*(5*d + 6*e)*x^6 + 7056*(6*d + 5*e)*x^5 + 7560*(7*d + 4*e)*x^4 + 7560*(8*d + 3*e)*x^3 + 2520*(9*d + 2*e)*x^2
*log(x) - 504*(10*d + e)*x - 252*d)/x^2

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giac [A]  time = 0.16, size = 137, normalized size = 0.99 \[ \frac {1}{9} \, x^{9} e + \frac {1}{8} \, d x^{8} + \frac {5}{4} \, x^{8} e + \frac {10}{7} \, d x^{7} + \frac {45}{7} \, x^{7} e + \frac {15}{2} \, d x^{6} + 20 \, x^{6} e + 24 \, d x^{5} + 42 \, x^{5} e + \frac {105}{2} \, d x^{4} + 63 \, x^{4} e + 84 \, d x^{3} + 70 \, x^{3} e + 105 \, d x^{2} + 60 \, x^{2} e + 120 \, d x + 45 \, x e + 5 \, {\left (9 \, d + 2 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {2 \, {\left (10 \, d + e\right )} x + d}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^3,x, algorithm="giac")

[Out]

1/9*x^9*e + 1/8*d*x^8 + 5/4*x^8*e + 10/7*d*x^7 + 45/7*x^7*e + 15/2*d*x^6 + 20*x^6*e + 24*d*x^5 + 42*x^5*e + 10
5/2*d*x^4 + 63*x^4*e + 84*d*x^3 + 70*x^3*e + 105*d*x^2 + 60*x^2*e + 120*d*x + 45*x*e + 5*(9*d + 2*e)*log(abs(x
)) - 1/2*(2*(10*d + e)*x + d)/x^2

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maple [A]  time = 0.08, size = 128, normalized size = 0.93 \[ \frac {e \,x^{9}}{9}+\frac {d \,x^{8}}{8}+\frac {5 e \,x^{8}}{4}+\frac {10 d \,x^{7}}{7}+\frac {45 e \,x^{7}}{7}+\frac {15 d \,x^{6}}{2}+20 e \,x^{6}+24 d \,x^{5}+42 e \,x^{5}+\frac {105 d \,x^{4}}{2}+63 e \,x^{4}+84 d \,x^{3}+70 e \,x^{3}+105 d \,x^{2}+60 e \,x^{2}+120 d x +45 d \ln \relax (x )+45 e x +10 e \ln \relax (x )-\frac {10 d}{x}-\frac {e}{x}-\frac {d}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*e*x^9+1/8*d*x^8+5/4*e*x^8+10/7*d*x^7+45/7*e*x^7+15/2*d*x^6+20*e*x^6+24*d*x^5+42*e*x^5+105/2*d*x^4+63*e*x^4
+84*d*x^3+70*e*x^3+105*d*x^2+60*e*x^2+120*d*x+45*e*x-1/2*d/x^2-10*d/x-e/x+45*d*ln(x)+10*e*ln(x)

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maxima [A]  time = 0.51, size = 125, normalized size = 0.91 \[ \frac {1}{9} \, e x^{9} + \frac {1}{8} \, {\left (d + 10 \, e\right )} x^{8} + \frac {5}{7} \, {\left (2 \, d + 9 \, e\right )} x^{7} + \frac {5}{2} \, {\left (3 \, d + 8 \, e\right )} x^{6} + 6 \, {\left (4 \, d + 7 \, e\right )} x^{5} + \frac {21}{2} \, {\left (5 \, d + 6 \, e\right )} x^{4} + 14 \, {\left (6 \, d + 5 \, e\right )} x^{3} + 15 \, {\left (7 \, d + 4 \, e\right )} x^{2} + 15 \, {\left (8 \, d + 3 \, e\right )} x + 5 \, {\left (9 \, d + 2 \, e\right )} \log \relax (x) - \frac {2 \, {\left (10 \, d + e\right )} x + d}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.08, size = 119, normalized size = 0.86 \[ x^8\,\left (\frac {d}{8}+\frac {5\,e}{4}\right )+x^6\,\left (\frac {15\,d}{2}+20\,e\right )+x^5\,\left (24\,d+42\,e\right )+x^7\,\left (\frac {10\,d}{7}+\frac {45\,e}{7}\right )+x^3\,\left (84\,d+70\,e\right )+x^2\,\left (105\,d+60\,e\right )+x^4\,\left (\frac {105\,d}{2}+63\,e\right )+\ln \relax (x)\,\left (45\,d+10\,e\right )+\frac {e\,x^9}{9}-\frac {\frac {d}{2}+x\,\left (10\,d+e\right )}{x^2}+x\,\left (120\,d+45\,e\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^8*(d/8 + (5*e)/4) + x^6*((15*d)/2 + 20*e) + x^5*(24*d + 42*e) + x^7*((10*d)/7 + (45*e)/7) + x^3*(84*d + 70*e
) + x^2*(105*d + 60*e) + x^4*((105*d)/2 + 63*e) + log(x)*(45*d + 10*e) + (e*x^9)/9 - (d/2 + x*(10*d + e))/x^2
+ x*(120*d + 45*e)

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sympy [A]  time = 0.45, size = 122, normalized size = 0.88 \[ \frac {e x^{9}}{9} + x^{8} \left (\frac {d}{8} + \frac {5 e}{4}\right ) + x^{7} \left (\frac {10 d}{7} + \frac {45 e}{7}\right ) + x^{6} \left (\frac {15 d}{2} + 20 e\right ) + x^{5} \left (24 d + 42 e\right ) + x^{4} \left (\frac {105 d}{2} + 63 e\right ) + x^{3} \left (84 d + 70 e\right ) + x^{2} \left (105 d + 60 e\right ) + x \left (120 d + 45 e\right ) + 5 \left (9 d + 2 e\right ) \log {\relax (x )} + \frac {- d + x \left (- 20 d - 2 e\right )}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**3,x)

[Out]

e*x**9/9 + x**8*(d/8 + 5*e/4) + x**7*(10*d/7 + 45*e/7) + x**6*(15*d/2 + 20*e) + x**5*(24*d + 42*e) + x**4*(105
*d/2 + 63*e) + x**3*(84*d + 70*e) + x**2*(105*d + 60*e) + x*(120*d + 45*e) + 5*(9*d + 2*e)*log(x) + (-d + x*(-
20*d - 2*e))/(2*x**2)

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