### 3.303 $$\int \frac{(3+2 x+5 x^2)^2 (2+x+3 x^2-5 x^3+4 x^4)}{(d+e x)^4} \, dx$$

Optimal. Leaf size=360 $\frac{x^3 \left (1000 d^2+180 d e+111 e^2\right )}{3 e^6}-\frac{x^2 \left (450 d^2 e+2000 d^3+444 d e^2+37 e^3\right )}{2 e^7}+\frac{2 x \left (555 d^2 e^2+450 d^3 e+1750 d^4+74 d e^3+74 e^4\right )}{e^8}-\frac{1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4+945 d^5 e+2800 d^6-195 d e^5+107 e^6}{e^9 (d+e x)}+\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (88 d^3 e^2-4 d^2 e^3+127 d^4 e+160 d^5+64 d e^4-11 e^5\right )}{2 e^9 (d+e x)^2}-\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{3 e^9 (d+e x)^3}-\frac{\left (2220 d^3 e^2+370 d^2 e^3+1575 d^4 e+5600 d^5+592 d e^4-65 e^5\right ) \log (d+e x)}{e^9}-\frac{5 x^4 (80 d+9 e)}{4 e^5}+\frac{20 x^5}{e^4}$

[Out]

(2*(1750*d^4 + 450*d^3*e + 555*d^2*e^2 + 74*d*e^3 + 74*e^4)*x)/e^8 - ((2000*d^3 + 450*d^2*e + 444*d*e^2 + 37*e
^3)*x^2)/(2*e^7) + ((1000*d^2 + 180*d*e + 111*e^2)*x^3)/(3*e^6) - (5*(80*d + 9*e)*x^4)/(4*e^5) + (20*x^5)/e^4
- ((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))/(3*e^9*(d + e*x)^3) + ((5*d^2 - 2*
d*e + 3*e^2)*(160*d^5 + 127*d^4*e + 88*d^3*e^2 - 4*d^2*e^3 + 64*d*e^4 - 11*e^5))/(2*e^9*(d + e*x)^2) - (2800*d
^6 + 945*d^5*e + 1665*d^4*e^2 + 370*d^3*e^3 + 888*d^2*e^4 - 195*d*e^5 + 107*e^6)/(e^9*(d + e*x)) - ((5600*d^5
+ 1575*d^4*e + 2220*d^3*e^2 + 370*d^2*e^3 + 592*d*e^4 - 65*e^5)*Log[d + e*x])/e^9

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Rubi [A]  time = 0.358292, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 38, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.026, Rules used = {1628} $\frac{x^3 \left (1000 d^2+180 d e+111 e^2\right )}{3 e^6}-\frac{x^2 \left (450 d^2 e+2000 d^3+444 d e^2+37 e^3\right )}{2 e^7}+\frac{2 x \left (555 d^2 e^2+450 d^3 e+1750 d^4+74 d e^3+74 e^4\right )}{e^8}-\frac{1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4+945 d^5 e+2800 d^6-195 d e^5+107 e^6}{e^9 (d+e x)}+\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (88 d^3 e^2-4 d^2 e^3+127 d^4 e+160 d^5+64 d e^4-11 e^5\right )}{2 e^9 (d+e x)^2}-\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{3 e^9 (d+e x)^3}-\frac{\left (2220 d^3 e^2+370 d^2 e^3+1575 d^4 e+5600 d^5+592 d e^4-65 e^5\right ) \log (d+e x)}{e^9}-\frac{5 x^4 (80 d+9 e)}{4 e^5}+\frac{20 x^5}{e^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(1750*d^4 + 450*d^3*e + 555*d^2*e^2 + 74*d*e^3 + 74*e^4)*x)/e^8 - ((2000*d^3 + 450*d^2*e + 444*d*e^2 + 37*e
^3)*x^2)/(2*e^7) + ((1000*d^2 + 180*d*e + 111*e^2)*x^3)/(3*e^6) - (5*(80*d + 9*e)*x^4)/(4*e^5) + (20*x^5)/e^4
- ((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))/(3*e^9*(d + e*x)^3) + ((5*d^2 - 2*
d*e + 3*e^2)*(160*d^5 + 127*d^4*e + 88*d^3*e^2 - 4*d^2*e^3 + 64*d*e^4 - 11*e^5))/(2*e^9*(d + e*x)^2) - (2800*d
^6 + 945*d^5*e + 1665*d^4*e^2 + 370*d^3*e^3 + 888*d^2*e^4 - 195*d*e^5 + 107*e^6)/(e^9*(d + e*x)) - ((5600*d^5
+ 1575*d^4*e + 2220*d^3*e^2 + 370*d^2*e^3 + 592*d*e^4 - 65*e^5)*Log[d + e*x])/e^9

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]

Rubi steps

\begin{align*} \int \frac{\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^4} \, dx &=\int \left (\frac{2 \left (1750 d^4+450 d^3 e+555 d^2 e^2+74 d e^3+74 e^4\right )}{e^8}-\frac{\left (2000 d^3+450 d^2 e+444 d e^2+37 e^3\right ) x}{e^7}+\frac{\left (1000 d^2+180 d e+111 e^2\right ) x^2}{e^6}-\frac{5 (80 d+9 e) x^3}{e^5}+\frac{100 x^4}{e^4}+\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{e^8 (d+e x)^4}+\frac{-800 d^7-315 d^6 e-666 d^5 e^2-185 d^4 e^3-592 d^3 e^4+195 d^2 e^5-214 d e^6+33 e^7}{e^8 (d+e x)^3}+\frac{2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6}{e^8 (d+e x)^2}+\frac{-5600 d^5-1575 d^4 e-2220 d^3 e^2-370 d^2 e^3-592 d e^4+65 e^5}{e^8 (d+e x)}\right ) \, dx\\ &=\frac{2 \left (1750 d^4+450 d^3 e+555 d^2 e^2+74 d e^3+74 e^4\right ) x}{e^8}-\frac{\left (2000 d^3+450 d^2 e+444 d e^2+37 e^3\right ) x^2}{2 e^7}+\frac{\left (1000 d^2+180 d e+111 e^2\right ) x^3}{3 e^6}-\frac{5 (80 d+9 e) x^4}{4 e^5}+\frac{20 x^5}{e^4}-\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{3 e^9 (d+e x)^3}+\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (160 d^5+127 d^4 e+88 d^3 e^2-4 d^2 e^3+64 d e^4-11 e^5\right )}{2 e^9 (d+e x)^2}-\frac{2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6}{e^9 (d+e x)}-\frac{\left (5600 d^5+1575 d^4 e+2220 d^3 e^2+370 d^2 e^3+592 d e^4-65 e^5\right ) \log (d+e x)}{e^9}\\ \end{align*}

Mathematica [A]  time = 0.121456, size = 344, normalized size = 0.96 $\frac{4 e^3 x^3 \left (1000 d^2+180 d e+111 e^2\right )-6 e^2 x^2 \left (450 d^2 e+2000 d^3+444 d e^2+37 e^3\right )+24 e x \left (555 d^2 e^2+450 d^3 e+1750 d^4+74 d e^3+74 e^4\right )-\frac{12 \left (1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4+945 d^5 e+2800 d^6-195 d e^5+107 e^6\right )}{d+e x}+\frac{6 \left (666 d^5 e^2+185 d^4 e^3+592 d^3 e^4-195 d^2 e^5+315 d^6 e+800 d^7+214 d e^6-33 e^7\right )}{(d+e x)^2}-\frac{4 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{(d+e x)^3}-12 \left (2220 d^3 e^2+370 d^2 e^3+1575 d^4 e+5600 d^5+592 d e^4-65 e^5\right ) \log (d+e x)-15 e^4 x^4 (80 d+9 e)+240 e^5 x^5}{12 e^9}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(24*e*(1750*d^4 + 450*d^3*e + 555*d^2*e^2 + 74*d*e^3 + 74*e^4)*x - 6*e^2*(2000*d^3 + 450*d^2*e + 444*d*e^2 + 3
7*e^3)*x^2 + 4*e^3*(1000*d^2 + 180*d*e + 111*e^2)*x^3 - 15*e^4*(80*d + 9*e)*x^4 + 240*e^5*x^5 - (4*(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))/(d + e*x)^3 + (6*(800*d^7 + 315*d^6*e + 666*d^5*
e^2 + 185*d^4*e^3 + 592*d^3*e^4 - 195*d^2*e^5 + 214*d*e^6 - 33*e^7))/(d + e*x)^2 - (12*(2800*d^6 + 945*d^5*e +
1665*d^4*e^2 + 370*d^3*e^3 + 888*d^2*e^4 - 195*d*e^5 + 107*e^6))/(d + e*x) - 12*(5600*d^5 + 1575*d^4*e + 2220
*d^3*e^2 + 370*d^2*e^3 + 592*d*e^4 - 65*e^5)*Log[d + e*x])/(12*e^9)

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Maple [A]  time = 0.056, size = 558, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

65/e^4*ln(e*x+d)-33/2/e^2/(e*x+d)^2-107/e^3/(e*x+d)-6/e/(e*x+d)^3+37/e^4*x^3-37/2/e^4*x^2+148/e^4*x-45/4/e^4*x
^4-148/3/e^5/(e*x+d)^3*d^4+65/3/e^4/(e*x+d)^3*d^3-107/3/e^3/(e*x+d)^3*d^2+185/2/e^6/(e*x+d)^2*d^4+296/e^5/(e*x
+d)^2*d^3-195/2/e^4/(e*x+d)^2*d^2+107/e^3/(e*x+d)^2*d+11/e^2/(e*x+d)^3*d-5600/e^9*ln(e*x+d)*d^5-1575/e^8*ln(e*
x+d)*d^4-2220/e^7*ln(e*x+d)*d^3-370/e^6*ln(e*x+d)*d^2-592/e^5*ln(e*x+d)*d-2800/e^9/(e*x+d)*d^6-945/e^8/(e*x+d)
*d^5-1665/e^7/(e*x+d)*d^4-100/e^5*x^4*d+1000/3/e^6*x^3*d^2+60/e^5*x^3*d-1000/e^7*x^2*d^3-225/e^6*x^2*d^2-222/e
^5*x^2*d+3500/e^8*d^4*x+900/e^7*x*d^3+1110/e^6*x*d^2+148/e^5*x*d-100/3/e^9/(e*x+d)^3*d^8-15/e^8/(e*x+d)^3*d^7-
37/e^7/(e*x+d)^3*d^6-37/3/e^6/(e*x+d)^3*d^5-370/e^6/(e*x+d)*d^3-888/e^5/(e*x+d)*d^2+195/e^4/(e*x+d)*d+400/e^9/
(e*x+d)^2*d^7+315/2/e^8/(e*x+d)^2*d^6+333/e^7/(e*x+d)^2*d^5+20*x^5/e^4

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Maxima [A]  time = 1.0135, size = 527, normalized size = 1.46 \begin{align*} -\frac{14600 \, d^{8} + 4815 \, d^{7} e + 8214 \, d^{6} e^{2} + 1739 \, d^{5} e^{3} + 3848 \, d^{4} e^{4} - 715 \, d^{3} e^{5} + 214 \, d^{2} e^{6} + 33 \, d e^{7} + 36 \, e^{8} + 6 \,{\left (2800 \, d^{6} e^{2} + 945 \, d^{5} e^{3} + 1665 \, d^{4} e^{4} + 370 \, d^{3} e^{5} + 888 \, d^{2} e^{6} - 195 \, d e^{7} + 107 \, e^{8}\right )} x^{2} + 3 \,{\left (10400 \, d^{7} e + 3465 \, d^{6} e^{2} + 5994 \, d^{5} e^{3} + 1295 \, d^{4} e^{4} + 2960 \, d^{3} e^{5} - 585 \, d^{2} e^{6} + 214 \, d e^{7} + 33 \, e^{8}\right )} x}{6 \,{\left (e^{12} x^{3} + 3 \, d e^{11} x^{2} + 3 \, d^{2} e^{10} x + d^{3} e^{9}\right )}} + \frac{240 \, e^{4} x^{5} - 15 \,{\left (80 \, d e^{3} + 9 \, e^{4}\right )} x^{4} + 4 \,{\left (1000 \, d^{2} e^{2} + 180 \, d e^{3} + 111 \, e^{4}\right )} x^{3} - 6 \,{\left (2000 \, d^{3} e + 450 \, d^{2} e^{2} + 444 \, d e^{3} + 37 \, e^{4}\right )} x^{2} + 24 \,{\left (1750 \, d^{4} + 450 \, d^{3} e + 555 \, d^{2} e^{2} + 74 \, d e^{3} + 74 \, e^{4}\right )} x}{12 \, e^{8}} - \frac{{\left (5600 \, d^{5} + 1575 \, d^{4} e + 2220 \, d^{3} e^{2} + 370 \, d^{2} e^{3} + 592 \, d e^{4} - 65 \, e^{5}\right )} \log \left (e x + d\right )}{e^{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(14600*d^8 + 4815*d^7*e + 8214*d^6*e^2 + 1739*d^5*e^3 + 3848*d^4*e^4 - 715*d^3*e^5 + 214*d^2*e^6 + 33*d*e
^7 + 36*e^8 + 6*(2800*d^6*e^2 + 945*d^5*e^3 + 1665*d^4*e^4 + 370*d^3*e^5 + 888*d^2*e^6 - 195*d*e^7 + 107*e^8)*
x^2 + 3*(10400*d^7*e + 3465*d^6*e^2 + 5994*d^5*e^3 + 1295*d^4*e^4 + 2960*d^3*e^5 - 585*d^2*e^6 + 214*d*e^7 + 3
3*e^8)*x)/(e^12*x^3 + 3*d*e^11*x^2 + 3*d^2*e^10*x + d^3*e^9) + 1/12*(240*e^4*x^5 - 15*(80*d*e^3 + 9*e^4)*x^4 +
4*(1000*d^2*e^2 + 180*d*e^3 + 111*e^4)*x^3 - 6*(2000*d^3*e + 450*d^2*e^2 + 444*d*e^3 + 37*e^4)*x^2 + 24*(1750
*d^4 + 450*d^3*e + 555*d^2*e^2 + 74*d*e^3 + 74*e^4)*x)/e^8 - (5600*d^5 + 1575*d^4*e + 2220*d^3*e^2 + 370*d^2*e
^3 + 592*d*e^4 - 65*e^5)*log(e*x + d)/e^9

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Fricas [A]  time = 1.02185, size = 1434, normalized size = 3.98 \begin{align*} \frac{240 \, e^{8} x^{8} - 29200 \, d^{8} - 9630 \, d^{7} e - 16428 \, d^{6} e^{2} - 3478 \, d^{5} e^{3} - 7696 \, d^{4} e^{4} + 1430 \, d^{3} e^{5} - 428 \, d^{2} e^{6} - 66 \, d e^{7} - 72 \, e^{8} - 15 \,{\left (32 \, d e^{7} + 9 \, e^{8}\right )} x^{7} +{\left (1120 \, d^{2} e^{6} + 315 \, d e^{7} + 444 \, e^{8}\right )} x^{6} - 3 \,{\left (1120 \, d^{3} e^{5} + 315 \, d^{2} e^{6} + 444 \, d e^{7} + 74 \, e^{8}\right )} x^{5} + 3 \,{\left (5600 \, d^{4} e^{4} + 1575 \, d^{3} e^{5} + 2220 \, d^{2} e^{6} + 370 \, d e^{7} + 592 \, e^{8}\right )} x^{4} + 2 \,{\left (47000 \, d^{5} e^{3} + 12510 \, d^{4} e^{4} + 16206 \, d^{3} e^{5} + 2331 \, d^{2} e^{6} + 2664 \, d e^{7}\right )} x^{3} + 6 \,{\left (13400 \, d^{6} e^{2} + 3060 \, d^{5} e^{3} + 2886 \, d^{4} e^{4} + 111 \, d^{3} e^{5} - 888 \, d^{2} e^{6} + 390 \, d e^{7} - 214 \, e^{8}\right )} x^{2} - 6 \,{\left (3400 \, d^{7} e + 1665 \, d^{6} e^{2} + 3774 \, d^{5} e^{3} + 999 \, d^{4} e^{4} + 2664 \, d^{3} e^{5} - 585 \, d^{2} e^{6} + 214 \, d e^{7} + 33 \, e^{8}\right )} x - 12 \,{\left (5600 \, d^{8} + 1575 \, d^{7} e + 2220 \, d^{6} e^{2} + 370 \, d^{5} e^{3} + 592 \, d^{4} e^{4} - 65 \, d^{3} e^{5} +{\left (5600 \, d^{5} e^{3} + 1575 \, d^{4} e^{4} + 2220 \, d^{3} e^{5} + 370 \, d^{2} e^{6} + 592 \, d e^{7} - 65 \, e^{8}\right )} x^{3} + 3 \,{\left (5600 \, d^{6} e^{2} + 1575 \, d^{5} e^{3} + 2220 \, d^{4} e^{4} + 370 \, d^{3} e^{5} + 592 \, d^{2} e^{6} - 65 \, d e^{7}\right )} x^{2} + 3 \,{\left (5600 \, d^{7} e + 1575 \, d^{6} e^{2} + 2220 \, d^{5} e^{3} + 370 \, d^{4} e^{4} + 592 \, d^{3} e^{5} - 65 \, d^{2} e^{6}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{12} x^{3} + 3 \, d e^{11} x^{2} + 3 \, d^{2} e^{10} x + d^{3} e^{9}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(240*e^8*x^8 - 29200*d^8 - 9630*d^7*e - 16428*d^6*e^2 - 3478*d^5*e^3 - 7696*d^4*e^4 + 1430*d^3*e^5 - 428*
d^2*e^6 - 66*d*e^7 - 72*e^8 - 15*(32*d*e^7 + 9*e^8)*x^7 + (1120*d^2*e^6 + 315*d*e^7 + 444*e^8)*x^6 - 3*(1120*d
^3*e^5 + 315*d^2*e^6 + 444*d*e^7 + 74*e^8)*x^5 + 3*(5600*d^4*e^4 + 1575*d^3*e^5 + 2220*d^2*e^6 + 370*d*e^7 + 5
92*e^8)*x^4 + 2*(47000*d^5*e^3 + 12510*d^4*e^4 + 16206*d^3*e^5 + 2331*d^2*e^6 + 2664*d*e^7)*x^3 + 6*(13400*d^6
*e^2 + 3060*d^5*e^3 + 2886*d^4*e^4 + 111*d^3*e^5 - 888*d^2*e^6 + 390*d*e^7 - 214*e^8)*x^2 - 6*(3400*d^7*e + 16
65*d^6*e^2 + 3774*d^5*e^3 + 999*d^4*e^4 + 2664*d^3*e^5 - 585*d^2*e^6 + 214*d*e^7 + 33*e^8)*x - 12*(5600*d^8 +
1575*d^7*e + 2220*d^6*e^2 + 370*d^5*e^3 + 592*d^4*e^4 - 65*d^3*e^5 + (5600*d^5*e^3 + 1575*d^4*e^4 + 2220*d^3*e
^5 + 370*d^2*e^6 + 592*d*e^7 - 65*e^8)*x^3 + 3*(5600*d^6*e^2 + 1575*d^5*e^3 + 2220*d^4*e^4 + 370*d^3*e^5 + 592
*d^2*e^6 - 65*d*e^7)*x^2 + 3*(5600*d^7*e + 1575*d^6*e^2 + 2220*d^5*e^3 + 370*d^4*e^4 + 592*d^3*e^5 - 65*d^2*e^
6)*x)*log(e*x + d))/(e^12*x^3 + 3*d*e^11*x^2 + 3*d^2*e^10*x + d^3*e^9)

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Sympy [A]  time = 5.23281, size = 391, normalized size = 1.09 \begin{align*} - \frac{14600 d^{8} + 4815 d^{7} e + 8214 d^{6} e^{2} + 1739 d^{5} e^{3} + 3848 d^{4} e^{4} - 715 d^{3} e^{5} + 214 d^{2} e^{6} + 33 d e^{7} + 36 e^{8} + x^{2} \left (16800 d^{6} e^{2} + 5670 d^{5} e^{3} + 9990 d^{4} e^{4} + 2220 d^{3} e^{5} + 5328 d^{2} e^{6} - 1170 d e^{7} + 642 e^{8}\right ) + x \left (31200 d^{7} e + 10395 d^{6} e^{2} + 17982 d^{5} e^{3} + 3885 d^{4} e^{4} + 8880 d^{3} e^{5} - 1755 d^{2} e^{6} + 642 d e^{7} + 99 e^{8}\right )}{6 d^{3} e^{9} + 18 d^{2} e^{10} x + 18 d e^{11} x^{2} + 6 e^{12} x^{3}} + \frac{20 x^{5}}{e^{4}} - \frac{x^{4} \left (400 d + 45 e\right )}{4 e^{5}} + \frac{x^{3} \left (1000 d^{2} + 180 d e + 111 e^{2}\right )}{3 e^{6}} - \frac{x^{2} \left (2000 d^{3} + 450 d^{2} e + 444 d e^{2} + 37 e^{3}\right )}{2 e^{7}} + \frac{x \left (3500 d^{4} + 900 d^{3} e + 1110 d^{2} e^{2} + 148 d e^{3} + 148 e^{4}\right )}{e^{8}} - \frac{\left (5600 d^{5} + 1575 d^{4} e + 2220 d^{3} e^{2} + 370 d^{2} e^{3} + 592 d e^{4} - 65 e^{5}\right ) \log{\left (d + e x \right )}}{e^{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(14600*d**8 + 4815*d**7*e + 8214*d**6*e**2 + 1739*d**5*e**3 + 3848*d**4*e**4 - 715*d**3*e**5 + 214*d**2*e**6
+ 33*d*e**7 + 36*e**8 + x**2*(16800*d**6*e**2 + 5670*d**5*e**3 + 9990*d**4*e**4 + 2220*d**3*e**5 + 5328*d**2*e
**6 - 1170*d*e**7 + 642*e**8) + x*(31200*d**7*e + 10395*d**6*e**2 + 17982*d**5*e**3 + 3885*d**4*e**4 + 8880*d*
*3*e**5 - 1755*d**2*e**6 + 642*d*e**7 + 99*e**8))/(6*d**3*e**9 + 18*d**2*e**10*x + 18*d*e**11*x**2 + 6*e**12*x
**3) + 20*x**5/e**4 - x**4*(400*d + 45*e)/(4*e**5) + x**3*(1000*d**2 + 180*d*e + 111*e**2)/(3*e**6) - x**2*(20
00*d**3 + 450*d**2*e + 444*d*e**2 + 37*e**3)/(2*e**7) + x*(3500*d**4 + 900*d**3*e + 1110*d**2*e**2 + 148*d*e**
3 + 148*e**4)/e**8 - (5600*d**5 + 1575*d**4*e + 2220*d**3*e**2 + 370*d**2*e**3 + 592*d*e**4 - 65*e**5)*log(d +
e*x)/e**9

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Giac [A]  time = 1.1417, size = 466, normalized size = 1.29 \begin{align*} -{\left (5600 \, d^{5} + 1575 \, d^{4} e + 2220 \, d^{3} e^{2} + 370 \, d^{2} e^{3} + 592 \, d e^{4} - 65 \, e^{5}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (240 \, x^{5} e^{16} - 1200 \, d x^{4} e^{15} + 4000 \, d^{2} x^{3} e^{14} - 12000 \, d^{3} x^{2} e^{13} + 42000 \, d^{4} x e^{12} - 135 \, x^{4} e^{16} + 720 \, d x^{3} e^{15} - 2700 \, d^{2} x^{2} e^{14} + 10800 \, d^{3} x e^{13} + 444 \, x^{3} e^{16} - 2664 \, d x^{2} e^{15} + 13320 \, d^{2} x e^{14} - 222 \, x^{2} e^{16} + 1776 \, d x e^{15} + 1776 \, x e^{16}\right )} e^{\left (-20\right )} - \frac{{\left (14600 \, d^{8} + 4815 \, d^{7} e + 8214 \, d^{6} e^{2} + 1739 \, d^{5} e^{3} + 3848 \, d^{4} e^{4} - 715 \, d^{3} e^{5} + 6 \,{\left (2800 \, d^{6} e^{2} + 945 \, d^{5} e^{3} + 1665 \, d^{4} e^{4} + 370 \, d^{3} e^{5} + 888 \, d^{2} e^{6} - 195 \, d e^{7} + 107 \, e^{8}\right )} x^{2} + 214 \, d^{2} e^{6} + 3 \,{\left (10400 \, d^{7} e + 3465 \, d^{6} e^{2} + 5994 \, d^{5} e^{3} + 1295 \, d^{4} e^{4} + 2960 \, d^{3} e^{5} - 585 \, d^{2} e^{6} + 214 \, d e^{7} + 33 \, e^{8}\right )} x + 33 \, d e^{7} + 36 \, e^{8}\right )} e^{\left (-9\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(5600*d^5 + 1575*d^4*e + 2220*d^3*e^2 + 370*d^2*e^3 + 592*d*e^4 - 65*e^5)*e^(-9)*log(abs(x*e + d)) + 1/12*(24
0*x^5*e^16 - 1200*d*x^4*e^15 + 4000*d^2*x^3*e^14 - 12000*d^3*x^2*e^13 + 42000*d^4*x*e^12 - 135*x^4*e^16 + 720*
d*x^3*e^15 - 2700*d^2*x^2*e^14 + 10800*d^3*x*e^13 + 444*x^3*e^16 - 2664*d*x^2*e^15 + 13320*d^2*x*e^14 - 222*x^
2*e^16 + 1776*d*x*e^15 + 1776*x*e^16)*e^(-20) - 1/6*(14600*d^8 + 4815*d^7*e + 8214*d^6*e^2 + 1739*d^5*e^3 + 38
48*d^4*e^4 - 715*d^3*e^5 + 6*(2800*d^6*e^2 + 945*d^5*e^3 + 1665*d^4*e^4 + 370*d^3*e^5 + 888*d^2*e^6 - 195*d*e^
7 + 107*e^8)*x^2 + 214*d^2*e^6 + 3*(10400*d^7*e + 3465*d^6*e^2 + 5994*d^5*e^3 + 1295*d^4*e^4 + 2960*d^3*e^5 -
585*d^2*e^6 + 214*d*e^7 + 33*e^8)*x + 33*d*e^7 + 36*e^8)*e^(-9)/(x*e + d)^3