### 3.315 $$\int \frac{2+x+3 x^2-5 x^3+4 x^4}{(d+e x) (3+2 x+5 x^2)^2} \, dx$$

Optimal. Leaf size=224 $-\frac{x (423 d-1367 e)+1367 d-293 e}{700 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )}-\frac{\left (-61 d^2 e+205 d^3+23 d e^2+14 e^3\right ) \log \left (5 x^2+2 x+3\right )}{50 \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{\left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)}{e \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{\left (-26423 d^2 e+6565 d^3+11089 d e^2-6623 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{700 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^2}$

[Out]

-(1367*d - 293*e + (423*d - 1367*e)*x)/(700*(5*d^2 - 2*d*e + 3*e^2)*(3 + 2*x + 5*x^2)) + ((6565*d^3 - 26423*d^
2*e + 11089*d*e^2 - 6623*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]])/(700*Sqrt[14]*(5*d^2 - 2*d*e + 3*e^2)^2) + ((4*d^4 +
5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*Log[d + e*x])/(e*(5*d^2 - 2*d*e + 3*e^2)^2) - ((205*d^3 - 61*d^2*e + 23*
d*e^2 + 14*e^3)*Log[3 + 2*x + 5*x^2])/(50*(5*d^2 - 2*d*e + 3*e^2)^2)

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Rubi [A]  time = 0.340275, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 38, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.158, Rules used = {1646, 1628, 634, 618, 204, 628} $-\frac{x (423 d-1367 e)+1367 d-293 e}{700 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )}-\frac{\left (-61 d^2 e+205 d^3+23 d e^2+14 e^3\right ) \log \left (5 x^2+2 x+3\right )}{50 \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{\left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)}{e \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{\left (-26423 d^2 e+6565 d^3+11089 d e^2-6623 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{700 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(1367*d - 293*e + (423*d - 1367*e)*x)/(700*(5*d^2 - 2*d*e + 3*e^2)*(3 + 2*x + 5*x^2)) + ((6565*d^3 - 26423*d^
2*e + 11089*d*e^2 - 6623*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]])/(700*Sqrt[14]*(5*d^2 - 2*d*e + 3*e^2)^2) + ((4*d^4 +
5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*Log[d + e*x])/(e*(5*d^2 - 2*d*e + 3*e^2)^2) - ((205*d^3 - 61*d^2*e + 23*
d*e^2 + 14*e^3)*Log[3 + 2*x + 5*x^2])/(50*(5*d^2 - 2*d*e + 3*e^2)^2)

Rule 1646

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

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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{2+x+3 x^2-5 x^3+4 x^4}{(d+e x) \left (3+2 x+5 x^2\right )^2} \, dx &=-\frac{1367 d-293 e+(423 d-1367 e) x}{700 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )}+\frac{1}{56} \int \frac{\frac{2 \left (369 d^2-421 d e+280 e^2\right )}{5 \left (5 d^2-2 d e+3 e^2\right )}-\frac{2 \left (924 d^2-285 d e+281 e^2\right ) x}{5 \left (5 d^2-2 d e+3 e^2\right )}+\frac{224 x^2}{5}}{(d+e x) \left (3+2 x+5 x^2\right )} \, dx\\ &=-\frac{1367 d-293 e+(423 d-1367 e) x}{700 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )}+\frac{1}{56} \int \left (\frac{56 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{\left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac{2 \left (165 d^3-4943 d^2 e+2089 d e^2-1403 e^3-28 \left (205 d^3-61 d^2 e+23 d e^2+14 e^3\right ) x\right )}{5 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=-\frac{1367 d-293 e+(423 d-1367 e) x}{700 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )}+\frac{\left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{\int \frac{165 d^3-4943 d^2 e+2089 d e^2-1403 e^3-28 \left (205 d^3-61 d^2 e+23 d e^2+14 e^3\right ) x}{3+2 x+5 x^2} \, dx}{140 \left (5 d^2-2 d e+3 e^2\right )^2}\\ &=-\frac{1367 d-293 e+(423 d-1367 e) x}{700 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )}+\frac{\left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{\left (6565 d^3-26423 d^2 e+11089 d e^2-6623 e^3\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{700 \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{\left (205 d^3-61 d^2 e+23 d e^2+14 e^3\right ) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{50 \left (5 d^2-2 d e+3 e^2\right )^2}\\ &=-\frac{1367 d-293 e+(423 d-1367 e) x}{700 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )}+\frac{\left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{\left (205 d^3-61 d^2 e+23 d e^2+14 e^3\right ) \log \left (3+2 x+5 x^2\right )}{50 \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{\left (6565 d^3-26423 d^2 e+11089 d e^2-6623 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{350 \left (5 d^2-2 d e+3 e^2\right )^2}\\ &=-\frac{1367 d-293 e+(423 d-1367 e) x}{700 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )}+\frac{\left (6565 d^3-26423 d^2 e+11089 d e^2-6623 e^3\right ) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{700 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^2}+\frac{\left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e \left (5 d^2-2 d e+3 e^2\right )^2}-\frac{\left (205 d^3-61 d^2 e+23 d e^2+14 e^3\right ) \log \left (3+2 x+5 x^2\right )}{50 \left (5 d^2-2 d e+3 e^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.159759, size = 186, normalized size = 0.83 $\frac{\frac{14 \left (5 d^2-2 d e+3 e^2\right ) (e (1367 x+293)-d (423 x+1367))}{5 x^2+2 x+3}-196 \left (-61 d^2 e+205 d^3+23 d e^2+14 e^3\right ) \log \left (5 x^2+2 x+3\right )+\frac{9800 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)}{e}+\sqrt{14} \left (-26423 d^2 e+6565 d^3+11089 d e^2-6623 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{9800 \left (5 d^2-2 d e+3 e^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((14*(5*d^2 - 2*d*e + 3*e^2)*(-(d*(1367 + 423*x)) + e*(293 + 1367*x)))/(3 + 2*x + 5*x^2) + Sqrt[14]*(6565*d^3
- 26423*d^2*e + 11089*d*e^2 - 6623*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]] + (9800*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^
3 + 2*e^4)*Log[d + e*x])/e - 196*(205*d^3 - 61*d^2*e + 23*d*e^2 + 14*e^3)*Log[3 + 2*x + 5*x^2])/(9800*(5*d^2 -
2*d*e + 3*e^2)^2)

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Maple [B]  time = 0.1, size = 691, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-423/700/(5*d^2-2*d*e+3*e^2)^2/(x^2+2/5*x+3/5)*d^3*x+7681/3500/(5*d^2-2*d*e+3*e^2)^2/(x^2+2/5*x+3/5)*x*d^2*e-4
003/3500/(5*d^2-2*d*e+3*e^2)^2/(x^2+2/5*x+3/5)*x*d*e^2+4101/3500/(5*d^2-2*d*e+3*e^2)^2/(x^2+2/5*x+3/5)*x*e^3-1
367/700/(5*d^2-2*d*e+3*e^2)^2/(x^2+2/5*x+3/5)*d^3+4199/3500/(5*d^2-2*d*e+3*e^2)^2/(x^2+2/5*x+3/5)*d^2*e-4687/3
500/(5*d^2-2*d*e+3*e^2)^2/(x^2+2/5*x+3/5)*d*e^2+879/3500/(5*d^2-2*d*e+3*e^2)^2/(x^2+2/5*x+3/5)*e^3-41/10/(5*d^
2-2*d*e+3*e^2)^2*ln(5*x^2+2*x+3)*d^3+61/50/(5*d^2-2*d*e+3*e^2)^2*ln(5*x^2+2*x+3)*d^2*e-23/50/(5*d^2-2*d*e+3*e^
2)^2*ln(5*x^2+2*x+3)*d*e^2-7/25/(5*d^2-2*d*e+3*e^2)^2*ln(5*x^2+2*x+3)*e^3+1313/1960/(5*d^2-2*d*e+3*e^2)^2*14^(
1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d^3-26423/9800/(5*d^2-2*d*e+3*e^2)^2*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2
))*d^2*e+11089/9800/(5*d^2-2*d*e+3*e^2)^2*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*d*e^2-6623/9800/(5*d^2-2*d*e
+3*e^2)^2*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))*e^3+4/(5*d^2-2*d*e+3*e^2)^2/e*ln(e*x+d)*d^4+5/(5*d^2-2*d*e+3
*e^2)^2*ln(e*x+d)*d^3+3/(5*d^2-2*d*e+3*e^2)^2*e*ln(e*x+d)*d^2-1/(5*d^2-2*d*e+3*e^2)^2*e^2*ln(e*x+d)*d+2/(5*d^2
-2*d*e+3*e^2)^2*e^3*ln(e*x+d)

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Maxima [A]  time = 1.60398, size = 390, normalized size = 1.74 \begin{align*} \frac{\sqrt{14}{\left (6565 \, d^{3} - 26423 \, d^{2} e + 11089 \, d e^{2} - 6623 \, e^{3}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right )}{9800 \,{\left (25 \, d^{4} - 20 \, d^{3} e + 34 \, d^{2} e^{2} - 12 \, d e^{3} + 9 \, e^{4}\right )}} + \frac{{\left (4 \, d^{4} + 5 \, d^{3} e + 3 \, d^{2} e^{2} - d e^{3} + 2 \, e^{4}\right )} \log \left (e x + d\right )}{25 \, d^{4} e - 20 \, d^{3} e^{2} + 34 \, d^{2} e^{3} - 12 \, d e^{4} + 9 \, e^{5}} - \frac{{\left (205 \, d^{3} - 61 \, d^{2} e + 23 \, d e^{2} + 14 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{50 \,{\left (25 \, d^{4} - 20 \, d^{3} e + 34 \, d^{2} e^{2} - 12 \, d e^{3} + 9 \, e^{4}\right )}} - \frac{{\left (423 \, d - 1367 \, e\right )} x + 1367 \, d - 293 \, e}{700 \,{\left (5 \,{\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )} x^{2} + 15 \, d^{2} - 6 \, d e + 9 \, e^{2} + 2 \,{\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9800*sqrt(14)*(6565*d^3 - 26423*d^2*e + 11089*d*e^2 - 6623*e^3)*arctan(1/14*sqrt(14)*(5*x + 1))/(25*d^4 - 20
*d^3*e + 34*d^2*e^2 - 12*d*e^3 + 9*e^4) + (4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*log(e*x + d)/(25*d^4*e
- 20*d^3*e^2 + 34*d^2*e^3 - 12*d*e^4 + 9*e^5) - 1/50*(205*d^3 - 61*d^2*e + 23*d*e^2 + 14*e^3)*log(5*x^2 + 2*x
+ 3)/(25*d^4 - 20*d^3*e + 34*d^2*e^2 - 12*d*e^3 + 9*e^4) - 1/700*((423*d - 1367*e)*x + 1367*d - 293*e)/(5*(5*
d^2 - 2*d*e + 3*e^2)*x^2 + 15*d^2 - 6*d*e + 9*e^2 + 2*(5*d^2 - 2*d*e + 3*e^2)*x)

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Fricas [B]  time = 1.67982, size = 1191, normalized size = 5.32 \begin{align*} -\frac{95690 \, d^{3} e - 58786 \, d^{2} e^{2} + 65618 \, d e^{3} - 12306 \, e^{4} - \sqrt{14}{\left (19695 \, d^{3} e - 79269 \, d^{2} e^{2} + 33267 \, d e^{3} - 19869 \, e^{4} + 5 \,{\left (6565 \, d^{3} e - 26423 \, d^{2} e^{2} + 11089 \, d e^{3} - 6623 \, e^{4}\right )} x^{2} + 2 \,{\left (6565 \, d^{3} e - 26423 \, d^{2} e^{2} + 11089 \, d e^{3} - 6623 \, e^{4}\right )} x\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + 14 \,{\left (2115 \, d^{3} e - 7681 \, d^{2} e^{2} + 4003 \, d e^{3} - 4101 \, e^{4}\right )} x - 9800 \,{\left (12 \, d^{4} + 15 \, d^{3} e + 9 \, d^{2} e^{2} - 3 \, d e^{3} + 6 \, e^{4} + 5 \,{\left (4 \, d^{4} + 5 \, d^{3} e + 3 \, d^{2} e^{2} - d e^{3} + 2 \, e^{4}\right )} x^{2} + 2 \,{\left (4 \, d^{4} + 5 \, d^{3} e + 3 \, d^{2} e^{2} - d e^{3} + 2 \, e^{4}\right )} x\right )} \log \left (e x + d\right ) + 196 \,{\left (615 \, d^{3} e - 183 \, d^{2} e^{2} + 69 \, d e^{3} + 42 \, e^{4} + 5 \,{\left (205 \, d^{3} e - 61 \, d^{2} e^{2} + 23 \, d e^{3} + 14 \, e^{4}\right )} x^{2} + 2 \,{\left (205 \, d^{3} e - 61 \, d^{2} e^{2} + 23 \, d e^{3} + 14 \, e^{4}\right )} x\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{9800 \,{\left (75 \, d^{4} e - 60 \, d^{3} e^{2} + 102 \, d^{2} e^{3} - 36 \, d e^{4} + 27 \, e^{5} + 5 \,{\left (25 \, d^{4} e - 20 \, d^{3} e^{2} + 34 \, d^{2} e^{3} - 12 \, d e^{4} + 9 \, e^{5}\right )} x^{2} + 2 \,{\left (25 \, d^{4} e - 20 \, d^{3} e^{2} + 34 \, d^{2} e^{3} - 12 \, d e^{4} + 9 \, e^{5}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9800*(95690*d^3*e - 58786*d^2*e^2 + 65618*d*e^3 - 12306*e^4 - sqrt(14)*(19695*d^3*e - 79269*d^2*e^2 + 33267
*d*e^3 - 19869*e^4 + 5*(6565*d^3*e - 26423*d^2*e^2 + 11089*d*e^3 - 6623*e^4)*x^2 + 2*(6565*d^3*e - 26423*d^2*e
^2 + 11089*d*e^3 - 6623*e^4)*x)*arctan(1/14*sqrt(14)*(5*x + 1)) + 14*(2115*d^3*e - 7681*d^2*e^2 + 4003*d*e^3 -
4101*e^4)*x - 9800*(12*d^4 + 15*d^3*e + 9*d^2*e^2 - 3*d*e^3 + 6*e^4 + 5*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3
+ 2*e^4)*x^2 + 2*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*x)*log(e*x + d) + 196*(615*d^3*e - 183*d^2*e^2
+ 69*d*e^3 + 42*e^4 + 5*(205*d^3*e - 61*d^2*e^2 + 23*d*e^3 + 14*e^4)*x^2 + 2*(205*d^3*e - 61*d^2*e^2 + 23*d*e^
3 + 14*e^4)*x)*log(5*x^2 + 2*x + 3))/(75*d^4*e - 60*d^3*e^2 + 102*d^2*e^3 - 36*d*e^4 + 27*e^5 + 5*(25*d^4*e -
20*d^3*e^2 + 34*d^2*e^3 - 12*d*e^4 + 9*e^5)*x^2 + 2*(25*d^4*e - 20*d^3*e^2 + 34*d^2*e^3 - 12*d*e^4 + 9*e^5)*x)

________________________________________________________________________________________

Sympy [C]  time = 18.3775, size = 8322, normalized size = 37.15 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2
- 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))*log(x
+ (-6252890000000*d**12*e*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4
- 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 -
2*d*e + 3*e**2)**2))**2 + 1721036800000*d**12*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e*
*3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14
*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) - 33493264000000*d**11*e**2*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e +
11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*
d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 4402940080000*d**11*e*(-sqrt(14)*I*(6565
*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e*
*4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) + 305308416000*d**11 +
55032566400000*d**10*e**3*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 -
20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 -
2*d*e + 3*e**2)**2))**2 - 5332117966000*d**10*e**2*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 662
3*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2
+ 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) + 1028468958725*d**10*e - 141469554240000*d**9*e**4*(-sqrt(14)*I
*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3
+ 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 172629895704
00*d**9*e**3*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e +
34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**
2)**2)) + 95412070955*d**9*e**2 + 139354879664000*d**8*e**5*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e
**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 2
3*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 11862414903920*d**8*e**4*(-sqrt(14)*I*(6565*d**3 -
26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) -
(205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) + 927554565402*d**8*e**3 - 160
769212620800*d**7*e**6*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20
*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d
*e + 3*e**2)**2))**2 + 13220300596608*d**7*e**5*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e
**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 1
4*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) - 1587450017342*d**7*e**4 + 92712805606400*d**6*e**7*(-sqrt(14)*I*(
6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 +
9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 376982672864*d
**6*e**6*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d
**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**
2)) + 1705352927600*d**6*e**5 - 61599603788800*d**5*e**8*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2
- 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d
*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 1766518292672*d**5*e**7*(-sqrt(14)*I*(6565*d**3 - 264
23*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205
*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) - 927094311444*d**5*e**6 + 1326767
3552000*d**4*e**9*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3
*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e +
3*e**2)**2))**2 + 6357651035680*d**4*e**8*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(
19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3
)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) + 5551412790*d**4*e**7 - 3617733504000*d**3*e**10*(-sqrt(14)*I*(6565*d**3
- 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4))
- (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 3730299722240*d**3*e**9
*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2
- 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) + 22
7625566062*d**3*e**8 - 3887664076800*d**2*e**11*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e
**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 1
4*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 2547991828368*d**2*e**10*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*
e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 -
61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) - 201172444677*d**2*e**9 + 1207100966400*d
*e**12*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**
2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)
)**2 - 703802088864*d*e**11*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4
- 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2
- 2*d*e + 3*e**2)**2)) + 62145783705*d*e**10 - 676838332800*e**13*(-sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 110
89*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2
*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 221086021968*e**12*(-sqrt(14)*I*(6565*d**3 -
26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) -
(205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) - 15706111904*e**11)/(11529190
4000*d**11 + 60548006400*d**10*e - 1205961319355*d**9*e**2 - 1979222576837*d**8*e**3 + 528572641642*d**7*e**4
- 1648297602686*d**6*e**5 + 151381570368*d**5*e**6 - 924616717780*d**4*e**7 + 478372778758*d**3*e**8 - 4786690
57938*d**2*e**9 + 139540516779*d*e**10 - 49409758967*e**11)) + (sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d
*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e +
23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))*log(x + (-6252890000000*d**12*e*(sqrt(14)*I*(6565*d**
3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4))
- (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 1721036800000*d**12*(s
qrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 1
2*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) - 334932
64000000*d**11*e**2*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**
3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e +
3*e**2)**2))**2 + 4402940080000*d**11*e*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19
600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/
(50*(5*d**2 - 2*d*e + 3*e**2)**2)) + 305308416000*d**11 + 55032566400000*d**10*e**3*(sqrt(14)*I*(6565*d**3 - 2
6423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (2
05*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 5332117966000*d**10*e**2*(s
qrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 1
2*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) + 102846
8958725*d**10*e - 141469554240000*d**9*e**4*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/
(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**
3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 17262989570400*d**9*e**3*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11
089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**
2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) + 95412070955*d**9*e**2 + 139354879664000*d**8*e
**5*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e*
*2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2
- 11862414903920*d**8*e**4*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4
- 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 -
2*d*e + 3*e**2)**2)) + 927554565402*d**8*e**3 - 160769212620800*d**7*e**6*(sqrt(14)*I*(6565*d**3 - 26423*d**2
*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 -
61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 13220300596608*d**7*e**5*(sqrt(14)*I
*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3
+ 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) - 1587450017342*d
**7*e**4 + 92712805606400*d**6*e**7*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(
25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(
5*d**2 - 2*d*e + 3*e**2)**2))**2 - 376982672864*d**6*e**6*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2
- 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d
*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) + 1705352927600*d**6*e**5 - 61599603788800*d**5*e**8*(sqrt
(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d
*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 176651
8292672*d**5*e**7*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*
e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3
*e**2)**2)) - 927094311444*d**5*e**6 + 13267673552000*d**4*e**9*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*
d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e
+ 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 6357651035680*d**4*e**8*(sqrt(14)*I*(6565*d**3
- 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) -
(205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) + 5551412790*d**4*e**7 - 3617
733504000*d**3*e**10*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d*
*3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e
+ 3*e**2)**2))**2 - 3730299722240*d**3*e**9*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/
(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**
3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) + 227625566062*d**3*e**8 - 3887664076800*d**2*e**11*(sqrt(14)*I*(6565*d*
*3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)
) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 2547991828368*d**2*e*
*10*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e*
*2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) -
201172444677*d**2*e**9 + 1207100966400*d*e**12*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**
3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*
e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 - 703802088864*d*e**11*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 110
89*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2
*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2)) + 62145783705*d*e**10 - 676838332800*e**13*(sqrt(
14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34*d**2*e**2 - 12*d*
e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)**2))**2 + 2210860
21968*e**12*(sqrt(14)*I*(6565*d**3 - 26423*d**2*e + 11089*d*e**2 - 6623*e**3)/(19600*(25*d**4 - 20*d**3*e + 34
*d**2*e**2 - 12*d*e**3 + 9*e**4)) - (205*d**3 - 61*d**2*e + 23*d*e**2 + 14*e**3)/(50*(5*d**2 - 2*d*e + 3*e**2)
**2)) - 15706111904*e**11)/(115291904000*d**11 + 60548006400*d**10*e - 1205961319355*d**9*e**2 - 1979222576837
*d**8*e**3 + 528572641642*d**7*e**4 - 1648297602686*d**6*e**5 + 151381570368*d**5*e**6 - 924616717780*d**4*e**
7 + 478372778758*d**3*e**8 - 478669057938*d**2*e**9 + 139540516779*d*e**10 - 49409758967*e**11)) - (1367*d - 2
93*e + x*(423*d - 1367*e))/(10500*d**2 - 4200*d*e + 6300*e**2 + x**2*(17500*d**2 - 7000*d*e + 10500*e**2) + x*
(7000*d**2 - 2800*d*e + 4200*e**2)) + (4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)*log(x + (17210368000
00*d**12*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)/(e*(5*d**2 - 2*d*e + 3*e**2)**2) - 6252890000000*
d**12*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)**2/(e*(5*d**2 - 2*d*e + 3*e**2)**4) + 305308416000*d
**11 + 4402940080000*d**11*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)/(5*d**2 - 2*d*e + 3*e**2)**2 -
33493264000000*d**11*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**4 + 102
8468958725*d**10*e - 5332117966000*d**10*e*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)/(5*d**2 - 2*d*e
+ 3*e**2)**2 + 55032566400000*d**10*e*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)**2/(5*d**2 - 2*d*e
+ 3*e**2)**4 + 95412070955*d**9*e**2 + 17262989570400*d**9*e**2*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*
e**4)/(5*d**2 - 2*d*e + 3*e**2)**2 - 141469554240000*d**9*e**2*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e
**4)**2/(5*d**2 - 2*d*e + 3*e**2)**4 + 927554565402*d**8*e**3 - 11862414903920*d**8*e**3*(4*d**4 + 5*d**3*e +
3*d**2*e**2 - d*e**3 + 2*e**4)/(5*d**2 - 2*d*e + 3*e**2)**2 + 139354879664000*d**8*e**3*(4*d**4 + 5*d**3*e + 3
*d**2*e**2 - d*e**3 + 2*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**4 - 1587450017342*d**7*e**4 + 13220300596608*d**7*
e**4*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)/(5*d**2 - 2*d*e + 3*e**2)**2 - 160769212620800*d**7*e
**4*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**4 + 1705352927600*d**6*e
**5 - 376982672864*d**6*e**5*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)/(5*d**2 - 2*d*e + 3*e**2)**2
+ 92712805606400*d**6*e**5*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**4
- 927094311444*d**5*e**6 - 1766518292672*d**5*e**6*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)/(5*d**
2 - 2*d*e + 3*e**2)**2 - 61599603788800*d**5*e**6*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)**2/(5*d*
*2 - 2*d*e + 3*e**2)**4 + 5551412790*d**4*e**7 + 6357651035680*d**4*e**7*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*
e**3 + 2*e**4)/(5*d**2 - 2*d*e + 3*e**2)**2 + 13267673552000*d**4*e**7*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e*
*3 + 2*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**4 + 227625566062*d**3*e**8 - 3730299722240*d**3*e**8*(4*d**4 + 5*d*
*3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)/(5*d**2 - 2*d*e + 3*e**2)**2 - 3617733504000*d**3*e**8*(4*d**4 + 5*d**3*
e + 3*d**2*e**2 - d*e**3 + 2*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**4 - 201172444677*d**2*e**9 + 2547991828368*d*
*2*e**9*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)/(5*d**2 - 2*d*e + 3*e**2)**2 - 3887664076800*d**2*
e**9*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**4 + 62145783705*d*e**10
- 703802088864*d*e**10*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)/(5*d**2 - 2*d*e + 3*e**2)**2 + 120
7100966400*d*e**10*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**4 - 15706
111904*e**11 + 221086021968*e**11*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)/(5*d**2 - 2*d*e + 3*e**2
)**2 - 676838332800*e**11*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)**2/(5*d**2 - 2*d*e + 3*e**2)**4)
/(115291904000*d**11 + 60548006400*d**10*e - 1205961319355*d**9*e**2 - 1979222576837*d**8*e**3 + 528572641642*
d**7*e**4 - 1648297602686*d**6*e**5 + 151381570368*d**5*e**6 - 924616717780*d**4*e**7 + 478372778758*d**3*e**8
- 478669057938*d**2*e**9 + 139540516779*d*e**10 - 49409758967*e**11))/(e*(5*d**2 - 2*d*e + 3*e**2)**2)

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Giac [A]  time = 1.17548, size = 383, normalized size = 1.71 \begin{align*} \frac{\sqrt{14}{\left (6565 \, d^{3} - 26423 \, d^{2} e + 11089 \, d e^{2} - 6623 \, e^{3}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right )}{9800 \,{\left (25 \, d^{4} - 20 \, d^{3} e + 34 \, d^{2} e^{2} - 12 \, d e^{3} + 9 \, e^{4}\right )}} - \frac{{\left (205 \, d^{3} - 61 \, d^{2} e + 23 \, d e^{2} + 14 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{50 \,{\left (25 \, d^{4} - 20 \, d^{3} e + 34 \, d^{2} e^{2} - 12 \, d e^{3} + 9 \, e^{4}\right )}} + \frac{{\left (4 \, d^{4} + 5 \, d^{3} e + 3 \, d^{2} e^{2} - d e^{3} + 2 \, e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{25 \, d^{4} e - 20 \, d^{3} e^{2} + 34 \, d^{2} e^{3} - 12 \, d e^{4} + 9 \, e^{5}} - \frac{6835 \, d^{3} - 4199 \, d^{2} e +{\left (2115 \, d^{3} - 7681 \, d^{2} e + 4003 \, d e^{2} - 4101 \, e^{3}\right )} x + 4687 \, d e^{2} - 879 \, e^{3}}{700 \,{\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}^{2}{\left (5 \, x^{2} + 2 \, x + 3\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9800*sqrt(14)*(6565*d^3 - 26423*d^2*e + 11089*d*e^2 - 6623*e^3)*arctan(1/14*sqrt(14)*(5*x + 1))/(25*d^4 - 20
*d^3*e + 34*d^2*e^2 - 12*d*e^3 + 9*e^4) - 1/50*(205*d^3 - 61*d^2*e + 23*d*e^2 + 14*e^3)*log(5*x^2 + 2*x + 3)/(
25*d^4 - 20*d^3*e + 34*d^2*e^2 - 12*d*e^3 + 9*e^4) + (4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*log(abs(x*e
+ d))/(25*d^4*e - 20*d^3*e^2 + 34*d^2*e^3 - 12*d*e^4 + 9*e^5) - 1/700*(6835*d^3 - 4199*d^2*e + (2115*d^3 - 76
81*d^2*e + 4003*d*e^2 - 4101*e^3)*x + 4687*d*e^2 - 879*e^3)/((5*d^2 - 2*d*e + 3*e^2)^2*(5*x^2 + 2*x + 3))