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

Optimal. Leaf size=138 \[ \frac {1}{7} x^7 (d+10 e)+\frac {5}{6} x^6 (2 d+9 e)+3 x^5 (3 d+8 e)+\frac {15}{2} x^4 (4 d+7 e)+14 x^3 (5 d+6 e)+21 x^2 (6 d+5 e)-\frac {10 d+e}{2 x^2}+30 x (7 d+4 e)-\frac {5 (9 d+2 e)}{x}+15 (8 d+3 e) \log (x)-\frac {d}{3 x^3}+\frac {e x^8}{8} \]

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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} \begin {gather*} \frac {1}{7} x^7 (d+10 e)+\frac {5}{6} x^6 (2 d+9 e)+3 x^5 (3 d+8 e)+\frac {15}{2} x^4 (4 d+7 e)+14 x^3 (5 d+6 e)+21 x^2 (6 d+5 e)-\frac {10 d+e}{2 x^2}+30 x (7 d+4 e)-\frac {5 (9 d+2 e)}{x}+15 (8 d+3 e) \log (x)-\frac {d}{3 x^3}+\frac {e x^8}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.04, size = 140, normalized size = 1.01 \begin {gather*} \frac {1}{7} x^7 (d+10 e)+\frac {5}{6} x^6 (2 d+9 e)+3 x^5 (3 d+8 e)+\frac {15}{2} x^4 (4 d+7 e)+14 x^3 (5 d+6 e)+21 x^2 (6 d+5 e)+\frac {-10 d-e}{2 x^2}+30 x (7 d+4 e)-\frac {5 (9 d+2 e)}{x}+15 (8 d+3 e) \log (x)-\frac {d}{3 x^3}+\frac {e x^8}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*d/x^3 + (-10*d - e)/(2*x^2) - (5*(9*d + 2*e))/x + 30*(7*d + 4*e)*x + 21*(6*d + 5*e)*x^2 + 14*(5*d + 6*e)*
x^3 + (15*(4*d + 7*e)*x^4)/2 + 3*(3*d + 8*e)*x^5 + (5*(2*d + 9*e)*x^6)/6 + ((d + 10*e)*x^7)/7 + (e*x^8)/8 + 15
*(8*d + 3*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^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 131, normalized size = 0.95 \begin {gather*} \frac {21 \, e x^{11} + 24 \, {\left (d + 10 \, e\right )} x^{10} + 140 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 504 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 1260 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 2352 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 3528 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 5040 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 2520 \, {\left (8 \, d + 3 \, e\right )} x^{3} \log \relax (x) - 840 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 84 \, {\left (10 \, d + e\right )} x - 56 \, d}{168 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^4,x, algorithm="fricas")

[Out]

1/168*(21*e*x^11 + 24*(d + 10*e)*x^10 + 140*(2*d + 9*e)*x^9 + 504*(3*d + 8*e)*x^8 + 1260*(4*d + 7*e)*x^7 + 235
2*(5*d + 6*e)*x^6 + 3528*(6*d + 5*e)*x^5 + 5040*(7*d + 4*e)*x^4 + 2520*(8*d + 3*e)*x^3*log(x) - 840*(9*d + 2*e
)*x^2 - 84*(10*d + e)*x - 56*d)/x^3

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giac [A]  time = 0.16, size = 139, normalized size = 1.01 \begin {gather*} \frac {1}{8} \, x^{8} e + \frac {1}{7} \, d x^{7} + \frac {10}{7} \, x^{7} e + \frac {5}{3} \, d x^{6} + \frac {15}{2} \, x^{6} e + 9 \, d x^{5} + 24 \, x^{5} e + 30 \, d x^{4} + \frac {105}{2} \, x^{4} e + 70 \, d x^{3} + 84 \, x^{3} e + 126 \, d x^{2} + 105 \, x^{2} e + 210 \, d x + 120 \, x e + 15 \, {\left (8 \, d + 3 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {30 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 3 \, {\left (10 \, d + e\right )} x + 2 \, d}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^4,x, algorithm="giac")

[Out]

1/8*x^8*e + 1/7*d*x^7 + 10/7*x^7*e + 5/3*d*x^6 + 15/2*x^6*e + 9*d*x^5 + 24*x^5*e + 30*d*x^4 + 105/2*x^4*e + 70
*d*x^3 + 84*x^3*e + 126*d*x^2 + 105*x^2*e + 210*d*x + 120*x*e + 15*(8*d + 3*e)*log(abs(x)) - 1/6*(30*(9*d + 2*
e)*x^2 + 3*(10*d + e)*x + 2*d)/x^3

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maple [A]  time = 0.05, size = 128, normalized size = 0.93 \begin {gather*} \frac {e \,x^{8}}{8}+\frac {d \,x^{7}}{7}+\frac {10 e \,x^{7}}{7}+\frac {5 d \,x^{6}}{3}+\frac {15 e \,x^{6}}{2}+9 d \,x^{5}+24 e \,x^{5}+30 d \,x^{4}+\frac {105 e \,x^{4}}{2}+70 d \,x^{3}+84 e \,x^{3}+126 d \,x^{2}+105 e \,x^{2}+210 d x +120 d \ln \relax (x )+120 e x +45 e \ln \relax (x )-\frac {45 d}{x}-\frac {10 e}{x}-\frac {5 d}{x^{2}}-\frac {e}{2 x^{2}}-\frac {d}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*e*x^8+1/7*d*x^7+10/7*e*x^7+5/3*d*x^6+15/2*e*x^6+9*d*x^5+24*e*x^5+30*d*x^4+105/2*e*x^4+70*d*x^3+84*e*x^3+12
6*d*x^2+105*e*x^2+210*d*x+120*e*x-1/3*d/x^3-5*d/x^2-1/2*e/x^2-45*d/x-10*e/x+120*d*ln(x)+45*e*ln(x)

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maxima [A]  time = 0.51, size = 127, normalized size = 0.92 \begin {gather*} \frac {1}{8} \, e x^{8} + \frac {1}{7} \, {\left (d + 10 \, e\right )} x^{7} + \frac {5}{6} \, {\left (2 \, d + 9 \, e\right )} x^{6} + 3 \, {\left (3 \, d + 8 \, e\right )} x^{5} + \frac {15}{2} \, {\left (4 \, d + 7 \, e\right )} x^{4} + 14 \, {\left (5 \, d + 6 \, e\right )} x^{3} + 21 \, {\left (6 \, d + 5 \, e\right )} x^{2} + 30 \, {\left (7 \, d + 4 \, e\right )} x + 15 \, {\left (8 \, d + 3 \, e\right )} \log \relax (x) - \frac {30 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 3 \, {\left (10 \, d + e\right )} x + 2 \, d}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^4,x, algorithm="maxima")

[Out]

1/8*e*x^8 + 1/7*(d + 10*e)*x^7 + 5/6*(2*d + 9*e)*x^6 + 3*(3*d + 8*e)*x^5 + 15/2*(4*d + 7*e)*x^4 + 14*(5*d + 6*
e)*x^3 + 21*(6*d + 5*e)*x^2 + 30*(7*d + 4*e)*x + 15*(8*d + 3*e)*log(x) - 1/6*(30*(9*d + 2*e)*x^2 + 3*(10*d + e
)*x + 2*d)/x^3

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mupad [B]  time = 0.07, size = 121, normalized size = 0.88 \begin {gather*} x^6\,\left (\frac {5\,d}{3}+\frac {15\,e}{2}\right )+x^7\,\left (\frac {d}{7}+\frac {10\,e}{7}\right )+x^5\,\left (9\,d+24\,e\right )+x^4\,\left (30\,d+\frac {105\,e}{2}\right )+x^3\,\left (70\,d+84\,e\right )+x^2\,\left (126\,d+105\,e\right )+\ln \relax (x)\,\left (120\,d+45\,e\right )-\frac {\left (45\,d+10\,e\right )\,x^2+\left (5\,d+\frac {e}{2}\right )\,x+\frac {d}{3}}{x^3}+\frac {e\,x^8}{8}+x\,\left (210\,d+120\,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^4,x)

[Out]

x^6*((5*d)/3 + (15*e)/2) + x^7*(d/7 + (10*e)/7) + x^5*(9*d + 24*e) + x^4*(30*d + (105*e)/2) + x^3*(70*d + 84*e
) + x^2*(126*d + 105*e) + log(x)*(120*d + 45*e) - (d/3 + x^2*(45*d + 10*e) + x*(5*d + e/2))/x^3 + (e*x^8)/8 +
x*(210*d + 120*e)

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sympy [A]  time = 0.65, size = 124, normalized size = 0.90 \begin {gather*} \frac {e x^{8}}{8} + x^{7} \left (\frac {d}{7} + \frac {10 e}{7}\right ) + x^{6} \left (\frac {5 d}{3} + \frac {15 e}{2}\right ) + x^{5} \left (9 d + 24 e\right ) + x^{4} \left (30 d + \frac {105 e}{2}\right ) + x^{3} \left (70 d + 84 e\right ) + x^{2} \left (126 d + 105 e\right ) + x \left (210 d + 120 e\right ) + 15 \left (8 d + 3 e\right ) \log {\relax (x )} + \frac {- 2 d + x^{2} \left (- 270 d - 60 e\right ) + x \left (- 30 d - 3 e\right )}{6 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**4,x)

[Out]

e*x**8/8 + x**7*(d/7 + 10*e/7) + x**6*(5*d/3 + 15*e/2) + x**5*(9*d + 24*e) + x**4*(30*d + 105*e/2) + x**3*(70*
d + 84*e) + x**2*(126*d + 105*e) + x*(210*d + 120*e) + 15*(8*d + 3*e)*log(x) + (-2*d + x**2*(-270*d - 60*e) +
x*(-30*d - 3*e))/(6*x**3)

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