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

Optimal. Leaf size=102 \[ \frac {620 (4 x+5)}{2 x^2+5 x+3}-\frac {155 (4 x+5)}{3 \left (2 x^2+5 x+3\right )^2}+\frac {62 x+73}{3 \left (2 x^2+5 x+3\right )^3}+\frac {(2 x+1) (6 x+7)}{4 \left (2 x^2+5 x+3\right )^4}+2480 \log (x+1)-2480 \log (2 x+3) \]

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Rubi [A]  time = 0.04, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {738, 638, 614, 616, 31} \[ \frac {620 (4 x+5)}{2 x^2+5 x+3}-\frac {155 (4 x+5)}{3 \left (2 x^2+5 x+3\right )^2}+\frac {62 x+73}{3 \left (2 x^2+5 x+3\right )^3}+\frac {(2 x+1) (6 x+7)}{4 \left (2 x^2+5 x+3\right )^4}+2480 \log (x+1)-2480 \log (2 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + 2*x)*(7 + 6*x))/(4*(3 + 5*x + 2*x^2)^4) + (73 + 62*x)/(3*(3 + 5*x + 2*x^2)^3) - (155*(5 + 4*x))/(3*(3 +
5*x + 2*x^2)^2) + (620*(5 + 4*x))/(3 + 5*x + 2*x^2) + 2480*Log[1 + x] - 2480*Log[3 + 2*x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rubi steps

\begin {align*} \int \frac {(1+2 x)^2}{\left (3+5 x+2 x^2\right )^5} \, dx &=\frac {(1+2 x) (7+6 x)}{4 \left (3+5 x+2 x^2\right )^4}-\frac {1}{4} \int \frac {-28-72 x}{\left (3+5 x+2 x^2\right )^4} \, dx\\ &=\frac {(1+2 x) (7+6 x)}{4 \left (3+5 x+2 x^2\right )^4}+\frac {73+62 x}{3 \left (3+5 x+2 x^2\right )^3}+\frac {310}{3} \int \frac {1}{\left (3+5 x+2 x^2\right )^3} \, dx\\ &=\frac {(1+2 x) (7+6 x)}{4 \left (3+5 x+2 x^2\right )^4}+\frac {73+62 x}{3 \left (3+5 x+2 x^2\right )^3}-\frac {155 (5+4 x)}{3 \left (3+5 x+2 x^2\right )^2}-620 \int \frac {1}{\left (3+5 x+2 x^2\right )^2} \, dx\\ &=\frac {(1+2 x) (7+6 x)}{4 \left (3+5 x+2 x^2\right )^4}+\frac {73+62 x}{3 \left (3+5 x+2 x^2\right )^3}-\frac {155 (5+4 x)}{3 \left (3+5 x+2 x^2\right )^2}+\frac {620 (5+4 x)}{3+5 x+2 x^2}+2480 \int \frac {1}{3+5 x+2 x^2} \, dx\\ &=\frac {(1+2 x) (7+6 x)}{4 \left (3+5 x+2 x^2\right )^4}+\frac {73+62 x}{3 \left (3+5 x+2 x^2\right )^3}-\frac {155 (5+4 x)}{3 \left (3+5 x+2 x^2\right )^2}+\frac {620 (5+4 x)}{3+5 x+2 x^2}+4960 \int \frac {1}{2+2 x} \, dx-4960 \int \frac {1}{3+2 x} \, dx\\ &=\frac {(1+2 x) (7+6 x)}{4 \left (3+5 x+2 x^2\right )^4}+\frac {73+62 x}{3 \left (3+5 x+2 x^2\right )^3}-\frac {155 (5+4 x)}{3 \left (3+5 x+2 x^2\right )^2}+\frac {620 (5+4 x)}{3+5 x+2 x^2}+2480 \log (1+x)-2480 \log (3+2 x)\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 99, normalized size = 0.97 \[ \frac {620 (4 x+5)}{2 x^2+5 x+3}-\frac {155 (4 x+5)}{3 \left (2 x^2+5 x+3\right )^2}+\frac {31 (4 x+5)}{6 \left (2 x^2+5 x+3\right )^3}-\frac {10 x+11}{4 \left (2 x^2+5 x+3\right )^4}+2480 \log (2 (x+1))-2480 \log (2 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(11 + 10*x)/(3 + 5*x + 2*x^2)^4 + (31*(5 + 4*x))/(6*(3 + 5*x + 2*x^2)^3) - (155*(5 + 4*x))/(3*(3 + 5*x +
2*x^2)^2) + (620*(5 + 4*x))/(3 + 5*x + 2*x^2) + 2480*Log[2*(1 + x)] - 2480*Log[3 + 2*x]

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IntegrateAlgebraic [A]  time = 0.07, size = 66, normalized size = 0.65 \[ \frac {238080 x^7+2083200 x^6+7757440 x^5+15934000 x^4+19495776 x^3+14209160 x^2+5712464 x+977397}{12 \left (2 x^2+5 x+3\right )^4}+2480 \log (x+1)-2480 \log (2 x+3) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + 2*x)^2/(3 + 5*x + 2*x^2)^5,x]

[Out]

(977397 + 5712464*x + 14209160*x^2 + 19495776*x^3 + 15934000*x^4 + 7757440*x^5 + 2083200*x^6 + 238080*x^7)/(12
*(3 + 5*x + 2*x^2)^4) + 2480*Log[1 + x] - 2480*Log[3 + 2*x]

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fricas [A]  time = 0.89, size = 173, normalized size = 1.70 \[ \frac {238080 \, x^{7} + 2083200 \, x^{6} + 7757440 \, x^{5} + 15934000 \, x^{4} + 19495776 \, x^{3} + 14209160 \, x^{2} - 29760 \, {\left (16 \, x^{8} + 160 \, x^{7} + 696 \, x^{6} + 1720 \, x^{5} + 2641 \, x^{4} + 2580 \, x^{3} + 1566 \, x^{2} + 540 \, x + 81\right )} \log \left (2 \, x + 3\right ) + 29760 \, {\left (16 \, x^{8} + 160 \, x^{7} + 696 \, x^{6} + 1720 \, x^{5} + 2641 \, x^{4} + 2580 \, x^{3} + 1566 \, x^{2} + 540 \, x + 81\right )} \log \left (x + 1\right ) + 5712464 \, x + 977397}{12 \, {\left (16 \, x^{8} + 160 \, x^{7} + 696 \, x^{6} + 1720 \, x^{5} + 2641 \, x^{4} + 2580 \, x^{3} + 1566 \, x^{2} + 540 \, x + 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(238080*x^7 + 2083200*x^6 + 7757440*x^5 + 15934000*x^4 + 19495776*x^3 + 14209160*x^2 - 29760*(16*x^8 + 16
0*x^7 + 696*x^6 + 1720*x^5 + 2641*x^4 + 2580*x^3 + 1566*x^2 + 540*x + 81)*log(2*x + 3) + 29760*(16*x^8 + 160*x
^7 + 696*x^6 + 1720*x^5 + 2641*x^4 + 2580*x^3 + 1566*x^2 + 540*x + 81)*log(x + 1) + 5712464*x + 977397)/(16*x^
8 + 160*x^7 + 696*x^6 + 1720*x^5 + 2641*x^4 + 2580*x^3 + 1566*x^2 + 540*x + 81)

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giac [A]  time = 1.07, size = 66, normalized size = 0.65 \[ \frac {238080 \, x^{7} + 2083200 \, x^{6} + 7757440 \, x^{5} + 15934000 \, x^{4} + 19495776 \, x^{3} + 14209160 \, x^{2} + 5712464 \, x + 977397}{12 \, {\left (2 \, x^{2} + 5 \, x + 3\right )}^{4}} - 2480 \, \log \left ({\left | 2 \, x + 3 \right |}\right ) + 2480 \, \log \left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(238080*x^7 + 2083200*x^6 + 7757440*x^5 + 15934000*x^4 + 19495776*x^3 + 14209160*x^2 + 5712464*x + 977397
)/(2*x^2 + 5*x + 3)^4 - 2480*log(abs(2*x + 3)) + 2480*log(abs(x + 1))

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maple [A]  time = 0.30, size = 64, normalized size = 0.63

method	result	size
norman	\(\frac {173600 x^{6}+\frac {1428116}{3} x +19840 x^{7}+\frac {1939360}{3} x^{5}+1624648 x^{3}+\frac {3552290}{3} x^{2}+\frac {3983500}{3} x^{4}+\frac {325799}{4}}{\left (2 x^{2}+5 x +3\right )^{4}}+2480 \ln \left (1+x \right )-2480 \ln \left (3+2 x \right )\)	\(64\)
risch	\(\frac {173600 x^{6}+\frac {1428116}{3} x +19840 x^{7}+\frac {1939360}{3} x^{5}+1624648 x^{3}+\frac {3552290}{3} x^{2}+\frac {3983500}{3} x^{4}+\frac {325799}{4}}{\left (2 x^{2}+5 x +3\right )^{4}}+2480 \ln \left (1+x \right )-2480 \ln \left (3+2 x \right )\)	\(65\)
default	\(\frac {16}{\left (3+2 x \right )^{4}}+\frac {256}{3 \left (3+2 x \right )^{3}}+\frac {328}{\left (3+2 x \right )^{2}}+\frac {1360}{3+2 x}-2480 \ln \left (3+2 x \right )-\frac {1}{4 \left (1+x \right )^{4}}+\frac {14}{3 \left (1+x \right )^{3}}-\frac {52}{\left (1+x \right )^{2}}+\frac {560}{1+x}+2480 \ln \left (1+x \right )\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+2*x)^2/(2*x^2+5*x+3)^5,x,method=_RETURNVERBOSE)

[Out]

(173600*x^6+1428116/3*x+19840*x^7+1939360/3*x^5+1624648*x^3+3552290/3*x^2+3983500/3*x^4+325799/4)/(2*x^2+5*x+3
)^4+2480*ln(1+x)-2480*ln(3+2*x)

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maxima [A]  time = 0.62, size = 94, normalized size = 0.92 \[ \frac {238080 \, x^{7} + 2083200 \, x^{6} + 7757440 \, x^{5} + 15934000 \, x^{4} + 19495776 \, x^{3} + 14209160 \, x^{2} + 5712464 \, x + 977397}{12 \, {\left (16 \, x^{8} + 160 \, x^{7} + 696 \, x^{6} + 1720 \, x^{5} + 2641 \, x^{4} + 2580 \, x^{3} + 1566 \, x^{2} + 540 \, x + 81\right )}} - 2480 \, \log \left (2 \, x + 3\right ) + 2480 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(238080*x^7 + 2083200*x^6 + 7757440*x^5 + 15934000*x^4 + 19495776*x^3 + 14209160*x^2 + 5712464*x + 977397
)/(16*x^8 + 160*x^7 + 696*x^6 + 1720*x^5 + 2641*x^4 + 2580*x^3 + 1566*x^2 + 540*x + 81) - 2480*log(2*x + 3) +
2480*log(x + 1)

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mupad [B]  time = 0.21, size = 85, normalized size = 0.83 \[ \frac {1240\,x^7+10850\,x^6+\frac {121210\,x^5}{3}+\frac {995875\,x^4}{12}+\frac {203081\,x^3}{2}+\frac {1776145\,x^2}{24}+\frac {357029\,x}{12}+\frac {325799}{64}}{x^8+10\,x^7+\frac {87\,x^6}{2}+\frac {215\,x^5}{2}+\frac {2641\,x^4}{16}+\frac {645\,x^3}{4}+\frac {783\,x^2}{8}+\frac {135\,x}{4}+\frac {81}{16}}-4960\,\mathrm {atanh}\left (4\,x+5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((357029*x)/12 + (1776145*x^2)/24 + (203081*x^3)/2 + (995875*x^4)/12 + (121210*x^5)/3 + 10850*x^6 + 1240*x^7 +
325799/64)/((135*x)/4 + (783*x^2)/8 + (645*x^3)/4 + (2641*x^4)/16 + (215*x^5)/2 + (87*x^6)/2 + 10*x^7 + x^8 +
81/16) - 4960*atanh(4*x + 5)

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sympy [A]  time = 0.23, size = 90, normalized size = 0.88 \[ \frac {238080 x^{7} + 2083200 x^{6} + 7757440 x^{5} + 15934000 x^{4} + 19495776 x^{3} + 14209160 x^{2} + 5712464 x + 977397}{192 x^{8} + 1920 x^{7} + 8352 x^{6} + 20640 x^{5} + 31692 x^{4} + 30960 x^{3} + 18792 x^{2} + 6480 x + 972} + 2480 \log {\left (x + 1 \right )} - 2480 \log {\left (x + \frac {3}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(238080*x**7 + 2083200*x**6 + 7757440*x**5 + 15934000*x**4 + 19495776*x**3 + 14209160*x**2 + 5712464*x + 97739
7)/(192*x**8 + 1920*x**7 + 8352*x**6 + 20640*x**5 + 31692*x**4 + 30960*x**3 + 18792*x**2 + 6480*x + 972) + 248
0*log(x + 1) - 2480*log(x + 3/2)

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