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

Optimal. Leaf size=354 \[ \frac{3 x^4 \left (200 d^2+45 d e+37 e^2\right )}{4 e^5}-\frac{x^3 \left (270 d^2 e+1000 d^3+333 d e^2+37 e^3\right )}{3 e^6}+\frac{x^2 \left (666 d^2 e^2+450 d^3 e+1500 d^4+111 d e^3+148 e^4\right )}{2 e^7}-\frac{x \left (1110 d^3 e^2+222 d^2 e^3+675 d^4 e+2100 d^5+444 d e^4-65 e^5\right )}{e^8}+\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (88 d^3 e^2-4 d^2 e^3+127 d^4 e+160 d^5+64 d e^4-11 e^5\right )}{e^9 (d+e x)}-\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{2 e^9 (d+e x)^2}+\frac{\left (1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4+945 d^5 e+2800 d^6-195 d e^5+107 e^6\right ) \log (d+e x)}{e^9}-\frac{3 x^5 (20 d+3 e)}{e^4}+\frac{50 x^6}{3 e^3} \]

[Out]

-(((2100*d^5 + 675*d^4*e + 1110*d^3*e^2 + 222*d^2*e^3 + 444*d*e^4 - 65*e^5)*x)/e^8) + ((1500*d^4 + 450*d^3*e +
666*d^2*e^2 + 111*d*e^3 + 148*e^4)*x^2)/(2*e^7) - ((1000*d^3 + 270*d^2*e + 333*d*e^2 + 37*e^3)*x^3)/(3*e^6) +
(3*(200*d^2 + 45*d*e + 37*e^2)*x^4)/(4*e^5) - (3*(20*d + 3*e)*x^5)/e^4 + (50*x^6)/(3*e^3) - ((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))/(2*e^9*(d + e*x)^2) + ((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))/(e^9*(d + e*x)) + ((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)*Log[d + e*x])/e^9

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Rubi [A]  time = 0.343337, antiderivative size = 354, 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} \[ \frac{3 x^4 \left (200 d^2+45 d e+37 e^2\right )}{4 e^5}-\frac{x^3 \left (270 d^2 e+1000 d^3+333 d e^2+37 e^3\right )}{3 e^6}+\frac{x^2 \left (666 d^2 e^2+450 d^3 e+1500 d^4+111 d e^3+148 e^4\right )}{2 e^7}-\frac{x \left (1110 d^3 e^2+222 d^2 e^3+675 d^4 e+2100 d^5+444 d e^4-65 e^5\right )}{e^8}+\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (88 d^3 e^2-4 d^2 e^3+127 d^4 e+160 d^5+64 d e^4-11 e^5\right )}{e^9 (d+e x)}-\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{2 e^9 (d+e x)^2}+\frac{\left (1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4+945 d^5 e+2800 d^6-195 d e^5+107 e^6\right ) \log (d+e x)}{e^9}-\frac{3 x^5 (20 d+3 e)}{e^4}+\frac{50 x^6}{3 e^3} \]

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)^3,x]

[Out]

-(((2100*d^5 + 675*d^4*e + 1110*d^3*e^2 + 222*d^2*e^3 + 444*d*e^4 - 65*e^5)*x)/e^8) + ((1500*d^4 + 450*d^3*e +
666*d^2*e^2 + 111*d*e^3 + 148*e^4)*x^2)/(2*e^7) - ((1000*d^3 + 270*d^2*e + 333*d*e^2 + 37*e^3)*x^3)/(3*e^6) +
(3*(200*d^2 + 45*d*e + 37*e^2)*x^4)/(4*e^5) - (3*(20*d + 3*e)*x^5)/e^4 + (50*x^6)/(3*e^3) - ((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))/(2*e^9*(d + e*x)^2) + ((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))/(e^9*(d + e*x)) + ((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)*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)^3} \, dx &=\int \left (\frac{-2100 d^5-675 d^4 e-1110 d^3 e^2-222 d^2 e^3-444 d e^4+65 e^5}{e^8}+\frac{\left (1500 d^4+450 d^3 e+666 d^2 e^2+111 d e^3+148 e^4\right ) x}{e^7}-\frac{\left (1000 d^3+270 d^2 e+333 d e^2+37 e^3\right ) x^2}{e^6}+\frac{3 \left (200 d^2+45 d e+37 e^2\right ) x^3}{e^5}-\frac{15 (20 d+3 e) x^4}{e^4}+\frac{100 x^5}{e^3}+\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)^3}+\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)^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^8 (d+e x)}\right ) \, dx\\ &=-\frac{\left (2100 d^5+675 d^4 e+1110 d^3 e^2+222 d^2 e^3+444 d e^4-65 e^5\right ) x}{e^8}+\frac{\left (1500 d^4+450 d^3 e+666 d^2 e^2+111 d e^3+148 e^4\right ) x^2}{2 e^7}-\frac{\left (1000 d^3+270 d^2 e+333 d e^2+37 e^3\right ) x^3}{3 e^6}+\frac{3 \left (200 d^2+45 d e+37 e^2\right ) x^4}{4 e^5}-\frac{3 (20 d+3 e) x^5}{e^4}+\frac{50 x^6}{3 e^3}-\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 )}{2 e^9 (d+e x)^2}+\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 )}{e^9 (d+e x)}+\frac{\left (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\right ) \log (d+e x)}{e^9}\\ \end{align*}

Mathematica [A]  time = 0.104478, size = 311, normalized size = 0.88 \[ \frac{-18 d^6 e^2 \left (2300 x^2+240 x-407\right )-2 d^5 e^3 \left (5600 x^3+6750 x^2+2664 x-999\right )+4 d^4 e^4 \left (700 x^4-945 x^3-5661 x^2-111 x+1554\right )-d^3 e^5 \left (1120 x^5-945 x^4+6660 x^3+4662 x^2-1776 x+1950\right )+d^2 e^6 \left (560 x^6-378 x^5+1665 x^4-1480 x^3-9768 x^2-1560 x+1926\right )+12 \left (1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4+945 d^5 e+2800 d^6-195 d e^5+107 e^6\right ) (d+e x)^2 \log (d+e x)-390 d^7 e (40 x-9)+9000 d^8+d e^7 \left (-320 x^7+189 x^6-666 x^5+370 x^4-3552 x^3+1560 x^2+2568 x-198\right )+e^8 \left (200 x^8-108 x^7+333 x^6-148 x^5+888 x^4+780 x^3-396 x-108\right )}{12 e^9 (d+e x)^2} \]

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)^3,x]

[Out]

(9000*d^8 - 390*d^7*e*(-9 + 40*x) - 18*d^6*e^2*(-407 + 240*x + 2300*x^2) - 2*d^5*e^3*(-999 + 2664*x + 6750*x^2
+ 5600*x^3) + 4*d^4*e^4*(1554 - 111*x - 5661*x^2 - 945*x^3 + 700*x^4) - d^3*e^5*(1950 - 1776*x + 4662*x^2 + 6
660*x^3 - 945*x^4 + 1120*x^5) + d^2*e^6*(1926 - 1560*x - 9768*x^2 - 1480*x^3 + 1665*x^4 - 378*x^5 + 560*x^6) +
d*e^7*(-198 + 2568*x + 1560*x^2 - 3552*x^3 + 370*x^4 - 666*x^5 + 189*x^6 - 320*x^7) + e^8*(-108 - 396*x + 780
*x^3 + 888*x^4 - 148*x^5 + 333*x^6 - 108*x^7 + 200*x^8) + 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)^2*Log[d + e*x])/(12*e^9*(d + e*x)^2)

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Maple [A]  time = 0.056, size = 531, normalized size = 1.5 \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)^3,x)

[Out]

107/e^3*ln(e*x+d)-33/e^2/(e*x+d)-9/e/(e*x+d)^2+74/e^3*x^2+65/e^3*x-37/3/e^3*x^3+666/e^7/(e*x+d)*d^5+185/e^6/(e
*x+d)*d^4+1665/e^7*ln(e*x+d)*d^4+370/e^6*ln(e*x+d)*d^3+888/e^5*ln(e*x+d)*d^2+333/e^5*x^2*d^2+111/2/e^4*x^2*d-9
/e^3*x^5-60/e^4*x^5*d+150/e^5*x^4*d^2+135/4/e^4*x^4*d-1000/3/e^6*x^3*d^3-90/e^5*x^3*d^2+750/e^7*x^2*d^4+2800/e
^9*ln(e*x+d)*d^6+945/e^8*ln(e*x+d)*d^5+800/e^9/(e*x+d)*d^7+315/e^8/(e*x+d)*d^6-50/e^9/(e*x+d)^2*d^8-45/2/e^8/(
e*x+d)^2*d^7+225/e^6*x^2*d^3-2100/e^8*d^5*x-675/e^7*x*d^4+592/e^5/(e*x+d)*d^3-195/e^4/(e*x+d)*d^2+214/e^3/(e*x
+d)*d-111/2/e^7/(e*x+d)^2*d^6-37/2/e^6/(e*x+d)^2*d^5-74/e^5/(e*x+d)^2*d^4+65/2/e^4/(e*x+d)^2*d^3-107/2/e^3/(e*
x+d)^2*d^2+33/2/e^2/(e*x+d)^2*d-195/e^4*ln(e*x+d)*d-1110/e^6*d^3*x-222/e^5*x*d^2-444/e^4*x*d-111/e^4*x^3*d+50/
3*x^6/e^3+111/4*x^4/e^3

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Maxima [A]  time = 0.986461, size = 510, normalized size = 1.44 \begin{align*} \frac{1500 \, d^{8} + 585 \, d^{7} e + 1221 \, d^{6} e^{2} + 333 \, d^{5} e^{3} + 1036 \, d^{4} e^{4} - 325 \, d^{3} e^{5} + 321 \, d^{2} e^{6} - 33 \, d e^{7} - 18 \, e^{8} + 2 \,{\left (800 \, d^{7} e + 315 \, d^{6} e^{2} + 666 \, d^{5} e^{3} + 185 \, d^{4} e^{4} + 592 \, d^{3} e^{5} - 195 \, d^{2} e^{6} + 214 \, d e^{7} - 33 \, e^{8}\right )} x}{2 \,{\left (e^{11} x^{2} + 2 \, d e^{10} x + d^{2} e^{9}\right )}} + \frac{200 \, e^{5} x^{6} - 36 \,{\left (20 \, d e^{4} + 3 \, e^{5}\right )} x^{5} + 9 \,{\left (200 \, d^{2} e^{3} + 45 \, d e^{4} + 37 \, e^{5}\right )} x^{4} - 4 \,{\left (1000 \, d^{3} e^{2} + 270 \, d^{2} e^{3} + 333 \, d e^{4} + 37 \, e^{5}\right )} x^{3} + 6 \,{\left (1500 \, d^{4} e + 450 \, d^{3} e^{2} + 666 \, d^{2} e^{3} + 111 \, d e^{4} + 148 \, e^{5}\right )} x^{2} - 12 \,{\left (2100 \, d^{5} + 675 \, d^{4} e + 1110 \, d^{3} e^{2} + 222 \, d^{2} e^{3} + 444 \, d e^{4} - 65 \, e^{5}\right )} x}{12 \, e^{8}} + \frac{{\left (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}\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)^3,x, algorithm="maxima")

[Out]

1/2*(1500*d^8 + 585*d^7*e + 1221*d^6*e^2 + 333*d^5*e^3 + 1036*d^4*e^4 - 325*d^3*e^5 + 321*d^2*e^6 - 33*d*e^7 -
18*e^8 + 2*(800*d^7*e + 315*d^6*e^2 + 666*d^5*e^3 + 185*d^4*e^4 + 592*d^3*e^5 - 195*d^2*e^6 + 214*d*e^7 - 33*
e^8)*x)/(e^11*x^2 + 2*d*e^10*x + d^2*e^9) + 1/12*(200*e^5*x^6 - 36*(20*d*e^4 + 3*e^5)*x^5 + 9*(200*d^2*e^3 + 4
5*d*e^4 + 37*e^5)*x^4 - 4*(1000*d^3*e^2 + 270*d^2*e^3 + 333*d*e^4 + 37*e^5)*x^3 + 6*(1500*d^4*e + 450*d^3*e^2
+ 666*d^2*e^3 + 111*d*e^4 + 148*e^5)*x^2 - 12*(2100*d^5 + 675*d^4*e + 1110*d^3*e^2 + 222*d^2*e^3 + 444*d*e^4 -
65*e^5)*x)/e^8 + (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)*log(
e*x + d)/e^9

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Fricas [A]  time = 1.00041, size = 1323, normalized size = 3.74 \begin{align*} \frac{200 \, e^{8} x^{8} + 9000 \, d^{8} + 3510 \, d^{7} e + 7326 \, d^{6} e^{2} + 1998 \, d^{5} e^{3} + 6216 \, d^{4} e^{4} - 1950 \, d^{3} e^{5} + 1926 \, d^{2} e^{6} - 198 \, d e^{7} - 108 \, e^{8} - 4 \,{\left (80 \, d e^{7} + 27 \, e^{8}\right )} x^{7} +{\left (560 \, d^{2} e^{6} + 189 \, d e^{7} + 333 \, e^{8}\right )} x^{6} - 2 \,{\left (560 \, d^{3} e^{5} + 189 \, d^{2} e^{6} + 333 \, d e^{7} + 74 \, e^{8}\right )} x^{5} +{\left (2800 \, d^{4} e^{4} + 945 \, d^{3} e^{5} + 1665 \, d^{2} e^{6} + 370 \, d e^{7} + 888 \, e^{8}\right )} x^{4} - 4 \,{\left (2800 \, d^{5} e^{3} + 945 \, d^{4} e^{4} + 1665 \, d^{3} e^{5} + 370 \, d^{2} e^{6} + 888 \, d e^{7} - 195 \, e^{8}\right )} x^{3} - 6 \,{\left (6900 \, d^{6} e^{2} + 2250 \, d^{5} e^{3} + 3774 \, d^{4} e^{4} + 777 \, d^{3} e^{5} + 1628 \, d^{2} e^{6} - 260 \, d e^{7}\right )} x^{2} - 12 \,{\left (1300 \, d^{7} e + 360 \, d^{6} e^{2} + 444 \, d^{5} e^{3} + 37 \, d^{4} e^{4} - 148 \, d^{3} e^{5} + 130 \, d^{2} e^{6} - 214 \, d e^{7} + 33 \, e^{8}\right )} x + 12 \,{\left (2800 \, d^{8} + 945 \, d^{7} e + 1665 \, d^{6} e^{2} + 370 \, d^{5} e^{3} + 888 \, d^{4} e^{4} - 195 \, d^{3} e^{5} + 107 \, d^{2} e^{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} + 2 \,{\left (2800 \, d^{7} e + 945 \, d^{6} e^{2} + 1665 \, d^{5} e^{3} + 370 \, d^{4} e^{4} + 888 \, d^{3} e^{5} - 195 \, d^{2} e^{6} + 107 \, d e^{7}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{11} x^{2} + 2 \, d e^{10} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

1/12*(200*e^8*x^8 + 9000*d^8 + 3510*d^7*e + 7326*d^6*e^2 + 1998*d^5*e^3 + 6216*d^4*e^4 - 1950*d^3*e^5 + 1926*d
^2*e^6 - 198*d*e^7 - 108*e^8 - 4*(80*d*e^7 + 27*e^8)*x^7 + (560*d^2*e^6 + 189*d*e^7 + 333*e^8)*x^6 - 2*(560*d^
3*e^5 + 189*d^2*e^6 + 333*d*e^7 + 74*e^8)*x^5 + (2800*d^4*e^4 + 945*d^3*e^5 + 1665*d^2*e^6 + 370*d*e^7 + 888*e
^8)*x^4 - 4*(2800*d^5*e^3 + 945*d^4*e^4 + 1665*d^3*e^5 + 370*d^2*e^6 + 888*d*e^7 - 195*e^8)*x^3 - 6*(6900*d^6*
e^2 + 2250*d^5*e^3 + 3774*d^4*e^4 + 777*d^3*e^5 + 1628*d^2*e^6 - 260*d*e^7)*x^2 - 12*(1300*d^7*e + 360*d^6*e^2
+ 444*d^5*e^3 + 37*d^4*e^4 - 148*d^3*e^5 + 130*d^2*e^6 - 214*d*e^7 + 33*e^8)*x + 12*(2800*d^8 + 945*d^7*e + 1
665*d^6*e^2 + 370*d^5*e^3 + 888*d^4*e^4 - 195*d^3*e^5 + 107*d^2*e^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 + 2*(2800*d^7*e + 945*d^6*e^2 + 1665*d^5*e^3 + 370*d
^4*e^4 + 888*d^3*e^5 - 195*d^2*e^6 + 107*d*e^7)*x)*log(e*x + d))/(e^11*x^2 + 2*d*e^10*x + d^2*e^9)

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Sympy [A]  time = 2.82858, size = 379, normalized size = 1.07 \begin{align*} \frac{1500 d^{8} + 585 d^{7} e + 1221 d^{6} e^{2} + 333 d^{5} e^{3} + 1036 d^{4} e^{4} - 325 d^{3} e^{5} + 321 d^{2} e^{6} - 33 d e^{7} - 18 e^{8} + x \left (1600 d^{7} e + 630 d^{6} e^{2} + 1332 d^{5} e^{3} + 370 d^{4} e^{4} + 1184 d^{3} e^{5} - 390 d^{2} e^{6} + 428 d e^{7} - 66 e^{8}\right )}{2 d^{2} e^{9} + 4 d e^{10} x + 2 e^{11} x^{2}} + \frac{50 x^{6}}{3 e^{3}} - \frac{x^{5} \left (60 d + 9 e\right )}{e^{4}} + \frac{x^{4} \left (600 d^{2} + 135 d e + 111 e^{2}\right )}{4 e^{5}} - \frac{x^{3} \left (1000 d^{3} + 270 d^{2} e + 333 d e^{2} + 37 e^{3}\right )}{3 e^{6}} + \frac{x^{2} \left (1500 d^{4} + 450 d^{3} e + 666 d^{2} e^{2} + 111 d e^{3} + 148 e^{4}\right )}{2 e^{7}} - \frac{x \left (2100 d^{5} + 675 d^{4} e + 1110 d^{3} e^{2} + 222 d^{2} e^{3} + 444 d e^{4} - 65 e^{5}\right )}{e^{8}} + \frac{\left (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}\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)**3,x)

[Out]

(1500*d**8 + 585*d**7*e + 1221*d**6*e**2 + 333*d**5*e**3 + 1036*d**4*e**4 - 325*d**3*e**5 + 321*d**2*e**6 - 33
*d*e**7 - 18*e**8 + x*(1600*d**7*e + 630*d**6*e**2 + 1332*d**5*e**3 + 370*d**4*e**4 + 1184*d**3*e**5 - 390*d**
2*e**6 + 428*d*e**7 - 66*e**8))/(2*d**2*e**9 + 4*d*e**10*x + 2*e**11*x**2) + 50*x**6/(3*e**3) - x**5*(60*d + 9
*e)/e**4 + x**4*(600*d**2 + 135*d*e + 111*e**2)/(4*e**5) - x**3*(1000*d**3 + 270*d**2*e + 333*d*e**2 + 37*e**3
)/(3*e**6) + x**2*(1500*d**4 + 450*d**3*e + 666*d**2*e**2 + 111*d*e**3 + 148*e**4)/(2*e**7) - x*(2100*d**5 + 6
75*d**4*e + 1110*d**3*e**2 + 222*d**2*e**3 + 444*d*e**4 - 65*e**5)/e**8 + (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)*log(d + e*x)/e**9

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Giac [A]  time = 1.1506, size = 478, normalized size = 1.35 \begin{align*}{\left (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}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (200 \, x^{6} e^{15} - 720 \, d x^{5} e^{14} + 1800 \, d^{2} x^{4} e^{13} - 4000 \, d^{3} x^{3} e^{12} + 9000 \, d^{4} x^{2} e^{11} - 25200 \, d^{5} x e^{10} - 108 \, x^{5} e^{15} + 405 \, d x^{4} e^{14} - 1080 \, d^{2} x^{3} e^{13} + 2700 \, d^{3} x^{2} e^{12} - 8100 \, d^{4} x e^{11} + 333 \, x^{4} e^{15} - 1332 \, d x^{3} e^{14} + 3996 \, d^{2} x^{2} e^{13} - 13320 \, d^{3} x e^{12} - 148 \, x^{3} e^{15} + 666 \, d x^{2} e^{14} - 2664 \, d^{2} x e^{13} + 888 \, x^{2} e^{15} - 5328 \, d x e^{14} + 780 \, x e^{15}\right )} e^{\left (-18\right )} + \frac{{\left (1500 \, d^{8} + 585 \, d^{7} e + 1221 \, d^{6} e^{2} + 333 \, d^{5} e^{3} + 1036 \, d^{4} e^{4} - 325 \, d^{3} e^{5} + 321 \, d^{2} e^{6} + 2 \,{\left (800 \, d^{7} e + 315 \, d^{6} e^{2} + 666 \, d^{5} e^{3} + 185 \, d^{4} e^{4} + 592 \, d^{3} e^{5} - 195 \, d^{2} e^{6} + 214 \, d e^{7} - 33 \, e^{8}\right )} x - 33 \, d e^{7} - 18 \, e^{8}\right )} e^{\left (-9\right )}}{2 \,{\left (x e + d\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

(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)*log(abs(x*e + d
)) + 1/12*(200*x^6*e^15 - 720*d*x^5*e^14 + 1800*d^2*x^4*e^13 - 4000*d^3*x^3*e^12 + 9000*d^4*x^2*e^11 - 25200*d
^5*x*e^10 - 108*x^5*e^15 + 405*d*x^4*e^14 - 1080*d^2*x^3*e^13 + 2700*d^3*x^2*e^12 - 8100*d^4*x*e^11 + 333*x^4*
e^15 - 1332*d*x^3*e^14 + 3996*d^2*x^2*e^13 - 13320*d^3*x*e^12 - 148*x^3*e^15 + 666*d*x^2*e^14 - 2664*d^2*x*e^1
3 + 888*x^2*e^15 - 5328*d*x*e^14 + 780*x*e^15)*e^(-18) + 1/2*(1500*d^8 + 585*d^7*e + 1221*d^6*e^2 + 333*d^5*e^
3 + 1036*d^4*e^4 - 325*d^3*e^5 + 321*d^2*e^6 + 2*(800*d^7*e + 315*d^6*e^2 + 666*d^5*e^3 + 185*d^4*e^4 + 592*d^
3*e^5 - 195*d^2*e^6 + 214*d*e^7 - 33*e^8)*x - 33*d*e^7 - 18*e^8)*e^(-9)/(x*e + d)^2