3.5.100 \(\int \frac {(d+e x) (1+2 x+x^2)^5}{x^7} \, dx\)

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

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

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^7,x]

[Out]

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

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

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^7,x]

[Out]

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^7,x]

[Out]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^7, x]

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fricas [A]  time = 0.40, size = 131, normalized size = 0.94 \begin {gather*} \frac {12 \, e x^{11} + 15 \, {\left (d + 10 \, e\right )} x^{10} + 100 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 450 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 1800 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 2520 \, {\left (5 \, d + 6 \, e\right )} x^{6} \log \relax (x) - 2520 \, {\left (6 \, d + 5 \, e\right )} x^{5} - 900 \, {\left (7 \, d + 4 \, e\right )} x^{4} - 300 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 75 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 12 \, {\left (10 \, d + e\right )} x - 10 \, d}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^7,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 139, normalized size = 0.99 \begin {gather*} \frac {1}{5} \, x^{5} e + \frac {1}{4} \, d x^{4} + \frac {5}{2} \, x^{4} e + \frac {10}{3} \, d x^{3} + 15 \, x^{3} e + \frac {45}{2} \, d x^{2} + 60 \, x^{2} e + 120 \, d x + 210 \, x e + 42 \, {\left (5 \, d + 6 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {2520 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 900 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 300 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 75 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 12 \, {\left (10 \, d + e\right )} x + 10 \, d}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^7,x, algorithm="giac")

[Out]

1/5*x^5*e + 1/4*d*x^4 + 5/2*x^4*e + 10/3*d*x^3 + 15*x^3*e + 45/2*d*x^2 + 60*x^2*e + 120*d*x + 210*x*e + 42*(5*
d + 6*e)*log(abs(x)) - 1/60*(2520*(6*d + 5*e)*x^5 + 900*(7*d + 4*e)*x^4 + 300*(8*d + 3*e)*x^3 + 75*(9*d + 2*e)
*x^2 + 12*(10*d + e)*x + 10*d)/x^6

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maple [A]  time = 0.06, size = 128, normalized size = 0.91 \begin {gather*} \frac {e \,x^{5}}{5}+\frac {d \,x^{4}}{4}+\frac {5 e \,x^{4}}{2}+\frac {10 d \,x^{3}}{3}+15 e \,x^{3}+\frac {45 d \,x^{2}}{2}+60 e \,x^{2}+120 d x +210 d \ln \relax (x )+210 e x +252 e \ln \relax (x )-\frac {252 d}{x}-\frac {210 e}{x}-\frac {105 d}{x^{2}}-\frac {60 e}{x^{2}}-\frac {40 d}{x^{3}}-\frac {15 e}{x^{3}}-\frac {45 d}{4 x^{4}}-\frac {5 e}{2 x^{4}}-\frac {2 d}{x^{5}}-\frac {e}{5 x^{5}}-\frac {d}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^7,x)

[Out]

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

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maxima [A]  time = 0.54, size = 127, normalized size = 0.91 \begin {gather*} \frac {1}{5} \, e x^{5} + \frac {1}{4} \, {\left (d + 10 \, e\right )} x^{4} + \frac {5}{3} \, {\left (2 \, d + 9 \, e\right )} x^{3} + \frac {15}{2} \, {\left (3 \, d + 8 \, e\right )} x^{2} + 30 \, {\left (4 \, d + 7 \, e\right )} x + 42 \, {\left (5 \, d + 6 \, e\right )} \log \relax (x) - \frac {2520 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 900 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 300 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 75 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 12 \, {\left (10 \, d + e\right )} x + 10 \, d}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^7,x, algorithm="maxima")

[Out]

1/5*e*x^5 + 1/4*(d + 10*e)*x^4 + 5/3*(2*d + 9*e)*x^3 + 15/2*(3*d + 8*e)*x^2 + 30*(4*d + 7*e)*x + 42*(5*d + 6*e
)*log(x) - 1/60*(2520*(6*d + 5*e)*x^5 + 900*(7*d + 4*e)*x^4 + 300*(8*d + 3*e)*x^3 + 75*(9*d + 2*e)*x^2 + 12*(1
0*d + e)*x + 10*d)/x^6

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mupad [B]  time = 1.07, size = 121, normalized size = 0.86 \begin {gather*} x^4\,\left (\frac {d}{4}+\frac {5\,e}{2}\right )+x^3\,\left (\frac {10\,d}{3}+15\,e\right )+x^2\,\left (\frac {45\,d}{2}+60\,e\right )+\ln \relax (x)\,\left (210\,d+252\,e\right )-\frac {\left (252\,d+210\,e\right )\,x^5+\left (105\,d+60\,e\right )\,x^4+\left (40\,d+15\,e\right )\,x^3+\left (\frac {45\,d}{4}+\frac {5\,e}{2}\right )\,x^2+\left (2\,d+\frac {e}{5}\right )\,x+\frac {d}{6}}{x^6}+\frac {e\,x^5}{5}+x\,\left (120\,d+210\,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^7,x)

[Out]

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

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sympy [A]  time = 2.17, size = 128, normalized size = 0.91 \begin {gather*} \frac {e x^{5}}{5} + x^{4} \left (\frac {d}{4} + \frac {5 e}{2}\right ) + x^{3} \left (\frac {10 d}{3} + 15 e\right ) + x^{2} \left (\frac {45 d}{2} + 60 e\right ) + x \left (120 d + 210 e\right ) + 42 \left (5 d + 6 e\right ) \log {\relax (x )} + \frac {- 10 d + x^{5} \left (- 15120 d - 12600 e\right ) + x^{4} \left (- 6300 d - 3600 e\right ) + x^{3} \left (- 2400 d - 900 e\right ) + x^{2} \left (- 675 d - 150 e\right ) + x \left (- 120 d - 12 e\right )}{60 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**7,x)

[Out]

e*x**5/5 + x**4*(d/4 + 5*e/2) + x**3*(10*d/3 + 15*e) + x**2*(45*d/2 + 60*e) + x*(120*d + 210*e) + 42*(5*d + 6*
e)*log(x) + (-10*d + x**5*(-15120*d - 12600*e) + x**4*(-6300*d - 3600*e) + x**3*(-2400*d - 900*e) + x**2*(-675
*d - 150*e) + x*(-120*d - 12*e))/(60*x**6)

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