3.5.99 \(\int \frac {(d+e x) (1+2 x+x^2)^5}{x^6} \, dx\)

Optimal. Leaf size=140 \[ \frac {1}{5} x^5 (d+10 e)+\frac {5}{4} x^4 (2 d+9 e)-\frac {10 d+e}{4 x^4}+5 x^3 (3 d+8 e)-\frac {5 (9 d+2 e)}{3 x^3}+15 x^2 (4 d+7 e)-\frac {15 (8 d+3 e)}{2 x^2}+42 x (5 d+6 e)-\frac {30 (7 d+4 e)}{x}+42 (6 d+5 e) \log (x)-\frac {d}{5 x^5}+\frac {e x^6}{6} \]

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Rubi [A]  time = 0.07, antiderivative size = 140, 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}{5} x^5 (d+10 e)+\frac {5}{4} x^4 (2 d+9 e)+5 x^3 (3 d+8 e)+15 x^2 (4 d+7 e)-\frac {15 (8 d+3 e)}{2 x^2}-\frac {5 (9 d+2 e)}{3 x^3}-\frac {10 d+e}{4 x^4}+42 x (5 d+6 e)-\frac {30 (7 d+4 e)}{x}+42 (6 d+5 e) \log (x)-\frac {d}{5 x^5}+\frac {e x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^6,x]

[Out]

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

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Mathematica [A]  time = 0.04, size = 142, normalized size = 1.01 \begin {gather*} \frac {1}{5} x^5 (d+10 e)+\frac {5}{4} x^4 (2 d+9 e)+\frac {-10 d-e}{4 x^4}+5 x^3 (3 d+8 e)-\frac {5 (9 d+2 e)}{3 x^3}+15 x^2 (4 d+7 e)-\frac {15 (8 d+3 e)}{2 x^2}+42 x (5 d+6 e)-\frac {30 (7 d+4 e)}{x}+42 (6 d+5 e) \log (x)-\frac {d}{5 x^5}+\frac {e x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^6,x]

[Out]

-1/5*d/x^5 + (-10*d - e)/(4*x^4) - (5*(9*d + 2*e))/(3*x^3) - (15*(8*d + 3*e))/(2*x^2) - (30*(7*d + 4*e))/x + 4
2*(5*d + 6*e)*x + 15*(4*d + 7*e)*x^2 + 5*(3*d + 8*e)*x^3 + (5*(2*d + 9*e)*x^4)/4 + ((d + 10*e)*x^5)/5 + (e*x^6
)/6 + 42*(6*d + 5*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^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^6,x]

[Out]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^6, x]

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fricas [A]  time = 0.41, size = 131, normalized size = 0.94 \begin {gather*} \frac {10 \, e x^{11} + 12 \, {\left (d + 10 \, e\right )} x^{10} + 75 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 300 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 900 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 2520 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 2520 \, {\left (6 \, d + 5 \, e\right )} x^{5} \log \relax (x) - 1800 \, {\left (7 \, d + 4 \, e\right )} x^{4} - 450 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 100 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 15 \, {\left (10 \, d + e\right )} x - 12 \, d}{60 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^6,x, algorithm="fricas")

[Out]

1/60*(10*e*x^11 + 12*(d + 10*e)*x^10 + 75*(2*d + 9*e)*x^9 + 300*(3*d + 8*e)*x^8 + 900*(4*d + 7*e)*x^7 + 2520*(
5*d + 6*e)*x^6 + 2520*(6*d + 5*e)*x^5*log(x) - 1800*(7*d + 4*e)*x^4 - 450*(8*d + 3*e)*x^3 - 100*(9*d + 2*e)*x^
2 - 15*(10*d + e)*x - 12*d)/x^5

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giac [A]  time = 0.15, size = 139, normalized size = 0.99 \begin {gather*} \frac {1}{6} \, x^{6} e + \frac {1}{5} \, d x^{5} + 2 \, x^{5} e + \frac {5}{2} \, d x^{4} + \frac {45}{4} \, x^{4} e + 15 \, d x^{3} + 40 \, x^{3} e + 60 \, d x^{2} + 105 \, x^{2} e + 210 \, d x + 252 \, x e + 42 \, {\left (6 \, d + 5 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {1800 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 450 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 100 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 15 \, {\left (10 \, d + e\right )} x + 12 \, d}{60 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^6,x, algorithm="giac")

[Out]

1/6*x^6*e + 1/5*d*x^5 + 2*x^5*e + 5/2*d*x^4 + 45/4*x^4*e + 15*d*x^3 + 40*x^3*e + 60*d*x^2 + 105*x^2*e + 210*d*
x + 252*x*e + 42*(6*d + 5*e)*log(abs(x)) - 1/60*(1800*(7*d + 4*e)*x^4 + 450*(8*d + 3*e)*x^3 + 100*(9*d + 2*e)*
x^2 + 15*(10*d + e)*x + 12*d)/x^5

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maple [A]  time = 0.05, size = 128, normalized size = 0.91 \begin {gather*} \frac {e \,x^{6}}{6}+\frac {d \,x^{5}}{5}+2 e \,x^{5}+\frac {5 d \,x^{4}}{2}+\frac {45 e \,x^{4}}{4}+15 d \,x^{3}+40 e \,x^{3}+60 d \,x^{2}+105 e \,x^{2}+210 d x +252 d \ln \relax (x )+252 e x +210 e \ln \relax (x )-\frac {210 d}{x}-\frac {120 e}{x}-\frac {60 d}{x^{2}}-\frac {45 e}{2 x^{2}}-\frac {15 d}{x^{3}}-\frac {10 e}{3 x^{3}}-\frac {5 d}{2 x^{4}}-\frac {e}{4 x^{4}}-\frac {d}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^6,x)

[Out]

1/6*e*x^6+1/5*d*x^5+2*e*x^5+5/2*d*x^4+45/4*e*x^4+15*d*x^3+40*e*x^3+60*d*x^2+105*e*x^2+210*d*x+252*e*x-1/5*d/x^
5-5/2*d/x^4-1/4*e/x^4-15*d/x^3-10/3*e/x^3-60*d/x^2-45/2*e/x^2-210*d/x-120*e/x+252*d*ln(x)+210*e*ln(x)

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maxima [A]  time = 0.54, size = 127, normalized size = 0.91 \begin {gather*} \frac {1}{6} \, e x^{6} + \frac {1}{5} \, {\left (d + 10 \, e\right )} x^{5} + \frac {5}{4} \, {\left (2 \, d + 9 \, e\right )} x^{4} + 5 \, {\left (3 \, d + 8 \, e\right )} x^{3} + 15 \, {\left (4 \, d + 7 \, e\right )} x^{2} + 42 \, {\left (5 \, d + 6 \, e\right )} x + 42 \, {\left (6 \, d + 5 \, e\right )} \log \relax (x) - \frac {1800 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 450 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 100 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 15 \, {\left (10 \, d + e\right )} x + 12 \, d}{60 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^6,x, algorithm="maxima")

[Out]

1/6*e*x^6 + 1/5*(d + 10*e)*x^5 + 5/4*(2*d + 9*e)*x^4 + 5*(3*d + 8*e)*x^3 + 15*(4*d + 7*e)*x^2 + 42*(5*d + 6*e)
*x + 42*(6*d + 5*e)*log(x) - 1/60*(1800*(7*d + 4*e)*x^4 + 450*(8*d + 3*e)*x^3 + 100*(9*d + 2*e)*x^2 + 15*(10*d
+ e)*x + 12*d)/x^5

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mupad [B]  time = 1.08, size = 121, normalized size = 0.86 \begin {gather*} x^5\,\left (\frac {d}{5}+2\,e\right )+x^3\,\left (15\,d+40\,e\right )+x^4\,\left (\frac {5\,d}{2}+\frac {45\,e}{4}\right )+x^2\,\left (60\,d+105\,e\right )+\ln \relax (x)\,\left (252\,d+210\,e\right )-\frac {\left (210\,d+120\,e\right )\,x^4+\left (60\,d+\frac {45\,e}{2}\right )\,x^3+\left (15\,d+\frac {10\,e}{3}\right )\,x^2+\left (\frac {5\,d}{2}+\frac {e}{4}\right )\,x+\frac {d}{5}}{x^5}+\frac {e\,x^6}{6}+x\,\left (210\,d+252\,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^6,x)

[Out]

x^5*(d/5 + 2*e) + x^3*(15*d + 40*e) + x^4*((5*d)/2 + (45*e)/4) + x^2*(60*d + 105*e) + log(x)*(252*d + 210*e) -
(d/5 + x^2*(15*d + (10*e)/3) + x^3*(60*d + (45*e)/2) + x^4*(210*d + 120*e) + x*((5*d)/2 + e/4))/x^5 + (e*x^6)
/6 + x*(210*d + 252*e)

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sympy [A]  time = 1.42, size = 124, normalized size = 0.89 \begin {gather*} \frac {e x^{6}}{6} + x^{5} \left (\frac {d}{5} + 2 e\right ) + x^{4} \left (\frac {5 d}{2} + \frac {45 e}{4}\right ) + x^{3} \left (15 d + 40 e\right ) + x^{2} \left (60 d + 105 e\right ) + x \left (210 d + 252 e\right ) + 42 \left (6 d + 5 e\right ) \log {\relax (x )} + \frac {- 12 d + x^{4} \left (- 12600 d - 7200 e\right ) + x^{3} \left (- 3600 d - 1350 e\right ) + x^{2} \left (- 900 d - 200 e\right ) + x \left (- 150 d - 15 e\right )}{60 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**6,x)

[Out]

e*x**6/6 + x**5*(d/5 + 2*e) + x**4*(5*d/2 + 45*e/4) + x**3*(15*d + 40*e) + x**2*(60*d + 105*e) + x*(210*d + 25
2*e) + 42*(6*d + 5*e)*log(x) + (-12*d + x**4*(-12600*d - 7200*e) + x**3*(-3600*d - 1350*e) + x**2*(-900*d - 20
0*e) + x*(-150*d - 15*e))/(60*x**5)

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