3.320 \(\int \frac{(d+e x) (2+x+3 x^2-5 x^3+4 x^4)}{(3+2 x+5 x^2)^3} \, dx\)

Optimal. Leaf size=103 \[ -\frac{(423 x+1367) (d+e x)}{7000 \left (5 x^2+2 x+3\right )^2}+\frac{x (11015 d+36353 e)+34347 d-6511 e}{196000 \left (5 x^2+2 x+3\right )}+\frac{(42375 d-34207 e) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{196000 \sqrt{14}}+\frac{2}{125} e \log \left (5 x^2+2 x+3\right ) \]

[Out]

-((1367 + 423*x)*(d + e*x))/(7000*(3 + 2*x + 5*x^2)^2) + (34347*d - 6511*e + (11015*d + 36353*e)*x)/(196000*(3
+ 2*x + 5*x^2)) + ((42375*d - 34207*e)*ArcTan[(1 + 5*x)/Sqrt[14]])/(196000*Sqrt[14]) + (2*e*Log[3 + 2*x + 5*x
^2])/125

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Rubi [A]  time = 0.145351, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1644, 1660, 634, 618, 204, 628} \[ -\frac{(423 x+1367) (d+e x)}{7000 \left (5 x^2+2 x+3\right )^2}+\frac{x (11015 d+36353 e)+34347 d-6511 e}{196000 \left (5 x^2+2 x+3\right )}+\frac{(42375 d-34207 e) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{196000 \sqrt{14}}+\frac{2}{125} e \log \left (5 x^2+2 x+3\right ) \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2)^3,x]

[Out]

-((1367 + 423*x)*(d + e*x))/(7000*(3 + 2*x + 5*x^2)^2) + (34347*d - 6511*e + (11015*d + 36353*e)*x)/(196000*(3
+ 2*x + 5*x^2)) + ((42375*d - 34207*e)*ArcTan[(1 + 5*x)/Sqrt[14]])/(196000*Sqrt[14]) + (2*e*Log[3 + 2*x + 5*x
^2])/125

Rule 1644

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[Po
lynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g +
(2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x
+ c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(d + e*x)*Q + g*(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c
*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && N
eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[p] ||  !IntegerQ[m
] ||  !RationalQ[a, b, c, d, e]) &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2,
0]))

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*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[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 634

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

Rule 618

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{(d+e x) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{\left (3+2 x+5 x^2\right )^3} \, dx &=-\frac{(1367+423 x) (d+e x)}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{1}{112} \int \frac{\frac{2}{125} (3267 d+1367 e)-\frac{12}{25} (308 d-123 e) x+\frac{112}{25} (20 d-33 e) x^2+\frac{448 e x^3}{5}}{\left (3+2 x+5 x^2\right )^2} \, dx\\ &=-\frac{(1367+423 x) (d+e x)}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{34347 d-6511 e+(11015 d+36353 e) x}{196000 \left (3+2 x+5 x^2\right )}+\frac{\int \frac{\frac{4}{25} (8475 d-5587 e)+\frac{25088 e x}{25}}{3+2 x+5 x^2} \, dx}{6272}\\ &=-\frac{(1367+423 x) (d+e x)}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{34347 d-6511 e+(11015 d+36353 e) x}{196000 \left (3+2 x+5 x^2\right )}+\frac{(42375 d-34207 e) \int \frac{1}{3+2 x+5 x^2} \, dx}{196000}+\frac{1}{125} (2 e) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx\\ &=-\frac{(1367+423 x) (d+e x)}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{34347 d-6511 e+(11015 d+36353 e) x}{196000 \left (3+2 x+5 x^2\right )}+\frac{2}{125} e \log \left (3+2 x+5 x^2\right )+\frac{(-42375 d+34207 e) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{98000}\\ &=-\frac{(1367+423 x) (d+e x)}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{34347 d-6511 e+(11015 d+36353 e) x}{196000 \left (3+2 x+5 x^2\right )}+\frac{(42375 d-34207 e) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{196000 \sqrt{14}}+\frac{2}{125} e \log \left (3+2 x+5 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0804348, size = 107, normalized size = 1.04 \[ \frac{-2115 d x-6835 d-5989 e x+1269 e}{35000 \left (5 x^2+2 x+3\right )^2}+\frac{55075 d x+171735 d+181765 e x-44399 e}{980000 \left (5 x^2+2 x+3\right )}+\frac{(42375 d-34207 e) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{196000 \sqrt{14}}+\frac{2}{125} e \log \left (5 x^2+2 x+3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2)^3,x]

[Out]

(-6835*d + 1269*e - 2115*d*x - 5989*e*x)/(35000*(3 + 2*x + 5*x^2)^2) + (171735*d - 44399*e + 55075*d*x + 18176
5*e*x)/(980000*(3 + 2*x + 5*x^2)) + ((42375*d - 34207*e)*ArcTan[(1 + 5*x)/Sqrt[14]])/(196000*Sqrt[14]) + (2*e*
Log[3 + 2*x + 5*x^2])/125

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Maple [A]  time = 0.053, size = 102, normalized size = 1. \begin{align*} 25\,{\frac{1}{ \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{2}} \left ( \left ({\frac{36353\,e}{980000}}+{\frac{2203\,d}{196000}} \right ){x}^{3}+ \left ({\frac{28307\,e}{4900000}}+{\frac{38753\,d}{980000}} \right ){x}^{2}+ \left ({\frac{57761\,e}{4900000}}+{\frac{17979\,d}{980000}} \right ) x+{\frac{12953\,d}{980000}}-{\frac{19533\,e}{4900000}} \right ) }+{\frac{2\,e\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) }{125}}+{\frac{339\,\sqrt{14}d}{21952}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }-{\frac{34207\,\sqrt{14}e}{2744000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^3,x)

[Out]

25*((36353/980000*e+2203/196000*d)*x^3+(28307/4900000*e+38753/980000*d)*x^2+(57761/4900000*e+17979/980000*d)*x
+12953/980000*d-19533/4900000*e)/(5*x^2+2*x+3)^2+2/125*e*ln(5*x^2+2*x+3)+339/21952*14^(1/2)*arctan(1/28*(10*x+
2)*14^(1/2))*d-34207/2744000*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*e

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Maxima [A]  time = 1.48458, size = 136, normalized size = 1.32 \begin{align*} \frac{1}{2744000} \, \sqrt{14}{\left (42375 \, d - 34207 \, e\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{2}{125} \, e \log \left (5 \, x^{2} + 2 \, x + 3\right ) + \frac{5 \,{\left (11015 \, d + 36353 \, e\right )} x^{3} +{\left (193765 \, d + 28307 \, e\right )} x^{2} +{\left (89895 \, d + 57761 \, e\right )} x + 64765 \, d - 19533 \, e}{196000 \,{\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^3,x, algorithm="maxima")

[Out]

1/2744000*sqrt(14)*(42375*d - 34207*e)*arctan(1/14*sqrt(14)*(5*x + 1)) + 2/125*e*log(5*x^2 + 2*x + 3) + 1/1960
00*(5*(11015*d + 36353*e)*x^3 + (193765*d + 28307*e)*x^2 + (89895*d + 57761*e)*x + 64765*d - 19533*e)/(25*x^4
+ 20*x^3 + 34*x^2 + 12*x + 9)

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Fricas [A]  time = 1.26263, size = 558, normalized size = 5.42 \begin{align*} \frac{70 \,{\left (11015 \, d + 36353 \, e\right )} x^{3} + 14 \,{\left (193765 \, d + 28307 \, e\right )} x^{2} + \sqrt{14}{\left (25 \,{\left (42375 \, d - 34207 \, e\right )} x^{4} + 20 \,{\left (42375 \, d - 34207 \, e\right )} x^{3} + 34 \,{\left (42375 \, d - 34207 \, e\right )} x^{2} + 12 \,{\left (42375 \, d - 34207 \, e\right )} x + 381375 \, d - 307863 \, e\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + 14 \,{\left (89895 \, d + 57761 \, e\right )} x + 43904 \,{\left (25 \, e x^{4} + 20 \, e x^{3} + 34 \, e x^{2} + 12 \, e x + 9 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + 906710 \, d - 273462 \, e}{2744000 \,{\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^3,x, algorithm="fricas")

[Out]

1/2744000*(70*(11015*d + 36353*e)*x^3 + 14*(193765*d + 28307*e)*x^2 + sqrt(14)*(25*(42375*d - 34207*e)*x^4 + 2
0*(42375*d - 34207*e)*x^3 + 34*(42375*d - 34207*e)*x^2 + 12*(42375*d - 34207*e)*x + 381375*d - 307863*e)*arcta
n(1/14*sqrt(14)*(5*x + 1)) + 14*(89895*d + 57761*e)*x + 43904*(25*e*x^4 + 20*e*x^3 + 34*e*x^2 + 12*e*x + 9*e)*
log(5*x^2 + 2*x + 3) + 906710*d - 273462*e)/(25*x^4 + 20*x^3 + 34*x^2 + 12*x + 9)

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Sympy [C]  time = 1.65198, size = 163, normalized size = 1.58 \begin{align*} \left (\frac{2 e}{125} - \frac{\sqrt{14} i \left (42375 d - 34207 e\right )}{5488000}\right ) \log{\left (x + \frac{8475 d - \frac{34207 e}{5} - \frac{\sqrt{14} i \left (42375 d - 34207 e\right )}{5}}{42375 d - 34207 e} \right )} + \left (\frac{2 e}{125} + \frac{\sqrt{14} i \left (42375 d - 34207 e\right )}{5488000}\right ) \log{\left (x + \frac{8475 d - \frac{34207 e}{5} + \frac{\sqrt{14} i \left (42375 d - 34207 e\right )}{5}}{42375 d - 34207 e} \right )} + \frac{64765 d - 19533 e + x^{3} \left (55075 d + 181765 e\right ) + x^{2} \left (193765 d + 28307 e\right ) + x \left (89895 d + 57761 e\right )}{4900000 x^{4} + 3920000 x^{3} + 6664000 x^{2} + 2352000 x + 1764000} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x**4-5*x**3+3*x**2+x+2)/(5*x**2+2*x+3)**3,x)

[Out]

(2*e/125 - sqrt(14)*I*(42375*d - 34207*e)/5488000)*log(x + (8475*d - 34207*e/5 - sqrt(14)*I*(42375*d - 34207*e
)/5)/(42375*d - 34207*e)) + (2*e/125 + sqrt(14)*I*(42375*d - 34207*e)/5488000)*log(x + (8475*d - 34207*e/5 + s
qrt(14)*I*(42375*d - 34207*e)/5)/(42375*d - 34207*e)) + (64765*d - 19533*e + x**3*(55075*d + 181765*e) + x**2*
(193765*d + 28307*e) + x*(89895*d + 57761*e))/(4900000*x**4 + 3920000*x**3 + 6664000*x**2 + 2352000*x + 176400
0)

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Giac [A]  time = 1.16485, size = 131, normalized size = 1.27 \begin{align*} \frac{1}{2744000} \, \sqrt{14}{\left (42375 \, d - 34207 \, e\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{2}{125} \, e \log \left (5 \, x^{2} + 2 \, x + 3\right ) + \frac{5 \,{\left (11015 \, d + 36353 \, e\right )} x^{3} +{\left (193765 \, d + 28307 \, e\right )} x^{2} +{\left (89895 \, d + 57761 \, e\right )} x + 64765 \, d - 19533 \, e}{196000 \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^3,x, algorithm="giac")

[Out]

1/2744000*sqrt(14)*(42375*d - 34207*e)*arctan(1/14*sqrt(14)*(5*x + 1)) + 2/125*e*log(5*x^2 + 2*x + 3) + 1/1960
00*(5*(11015*d + 36353*e)*x^3 + (193765*d + 28307*e)*x^2 + (89895*d + 57761*e)*x + 64765*d - 19533*e)/(5*x^2 +
2*x + 3)^2