### 3.300 $$\int \frac{(3+2 x+5 x^2)^2 (2+x+3 x^2-5 x^3+4 x^4)}{d+e x} \, dx$$

Optimal. Leaf size=352 $\frac{x^6 \left (100 d^2+45 d e+111 e^2\right )}{6 e^3}-\frac{x^5 \left (45 d^2 e+100 d^3+111 d e^2+37 e^3\right )}{5 e^4}+\frac{x^4 \left (111 d^2 e^2+45 d^3 e+100 d^4+37 d e^3+148 e^4\right )}{4 e^5}-\frac{x^3 \left (111 d^3 e^2+37 d^2 e^3+45 d^4 e+100 d^5+148 d e^4-65 e^5\right )}{3 e^6}+\frac{x^2 \left (111 d^4 e^2+37 d^3 e^3+148 d^2 e^4+45 d^5 e+100 d^6-65 d e^5+107 e^6\right )}{2 e^7}-\frac{x \left (111 d^5 e^2+37 d^4 e^3+148 d^3 e^4-65 d^2 e^5+45 d^6 e+100 d^7+107 d e^6-33 e^7\right )}{e^8}+\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)}{e^9}-\frac{5 x^7 (20 d+9 e)}{7 e^2}+\frac{25 x^8}{2 e}$

[Out]

-(((100*d^7 + 45*d^6*e + 111*d^5*e^2 + 37*d^4*e^3 + 148*d^3*e^4 - 65*d^2*e^5 + 107*d*e^6 - 33*e^7)*x)/e^8) + (
(100*d^6 + 45*d^5*e + 111*d^4*e^2 + 37*d^3*e^3 + 148*d^2*e^4 - 65*d*e^5 + 107*e^6)*x^2)/(2*e^7) - ((100*d^5 +
45*d^4*e + 111*d^3*e^2 + 37*d^2*e^3 + 148*d*e^4 - 65*e^5)*x^3)/(3*e^6) + ((100*d^4 + 45*d^3*e + 111*d^2*e^2 +
37*d*e^3 + 148*e^4)*x^4)/(4*e^5) - ((100*d^3 + 45*d^2*e + 111*d*e^2 + 37*e^3)*x^5)/(5*e^4) + ((100*d^2 + 45*d*
e + 111*e^2)*x^6)/(6*e^3) - (5*(20*d + 9*e)*x^7)/(7*e^2) + (25*x^8)/(2*e) + ((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^9

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Rubi [A]  time = 0.316476, antiderivative size = 352, 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} $\frac{x^6 \left (100 d^2+45 d e+111 e^2\right )}{6 e^3}-\frac{x^5 \left (45 d^2 e+100 d^3+111 d e^2+37 e^3\right )}{5 e^4}+\frac{x^4 \left (111 d^2 e^2+45 d^3 e+100 d^4+37 d e^3+148 e^4\right )}{4 e^5}-\frac{x^3 \left (111 d^3 e^2+37 d^2 e^3+45 d^4 e+100 d^5+148 d e^4-65 e^5\right )}{3 e^6}+\frac{x^2 \left (111 d^4 e^2+37 d^3 e^3+148 d^2 e^4+45 d^5 e+100 d^6-65 d e^5+107 e^6\right )}{2 e^7}-\frac{x \left (111 d^5 e^2+37 d^4 e^3+148 d^3 e^4-65 d^2 e^5+45 d^6 e+100 d^7+107 d e^6-33 e^7\right )}{e^8}+\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)}{e^9}-\frac{5 x^7 (20 d+9 e)}{7 e^2}+\frac{25 x^8}{2 e}$

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),x]

[Out]

-(((100*d^7 + 45*d^6*e + 111*d^5*e^2 + 37*d^4*e^3 + 148*d^3*e^4 - 65*d^2*e^5 + 107*d*e^6 - 33*e^7)*x)/e^8) + (
(100*d^6 + 45*d^5*e + 111*d^4*e^2 + 37*d^3*e^3 + 148*d^2*e^4 - 65*d*e^5 + 107*e^6)*x^2)/(2*e^7) - ((100*d^5 +
45*d^4*e + 111*d^3*e^2 + 37*d^2*e^3 + 148*d*e^4 - 65*e^5)*x^3)/(3*e^6) + ((100*d^4 + 45*d^3*e + 111*d^2*e^2 +
37*d*e^3 + 148*e^4)*x^4)/(4*e^5) - ((100*d^3 + 45*d^2*e + 111*d*e^2 + 37*e^3)*x^5)/(5*e^4) + ((100*d^2 + 45*d*
e + 111*e^2)*x^6)/(6*e^3) - (5*(20*d + 9*e)*x^7)/(7*e^2) + (25*x^8)/(2*e) + ((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^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} \, dx &=\int \left (\frac{-100 d^7-45 d^6 e-111 d^5 e^2-37 d^4 e^3-148 d^3 e^4+65 d^2 e^5-107 d e^6+33 e^7}{e^8}+\frac{\left (100 d^6+45 d^5 e+111 d^4 e^2+37 d^3 e^3+148 d^2 e^4-65 d e^5+107 e^6\right ) x}{e^7}+\frac{\left (-100 d^5-45 d^4 e-111 d^3 e^2-37 d^2 e^3-148 d e^4+65 e^5\right ) x^2}{e^6}+\frac{\left (100 d^4+45 d^3 e+111 d^2 e^2+37 d e^3+148 e^4\right ) x^3}{e^5}-\frac{\left (100 d^3+45 d^2 e+111 d e^2+37 e^3\right ) x^4}{e^4}+\frac{\left (100 d^2+45 d e+111 e^2\right ) x^5}{e^3}-\frac{5 (20 d+9 e) x^6}{e^2}+\frac{100 x^7}{e}+\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)}\right ) \, dx\\ &=-\frac{\left (100 d^7+45 d^6 e+111 d^5 e^2+37 d^4 e^3+148 d^3 e^4-65 d^2 e^5+107 d e^6-33 e^7\right ) x}{e^8}+\frac{\left (100 d^6+45 d^5 e+111 d^4 e^2+37 d^3 e^3+148 d^2 e^4-65 d e^5+107 e^6\right ) x^2}{2 e^7}-\frac{\left (100 d^5+45 d^4 e+111 d^3 e^2+37 d^2 e^3+148 d e^4-65 e^5\right ) x^3}{3 e^6}+\frac{\left (100 d^4+45 d^3 e+111 d^2 e^2+37 d e^3+148 e^4\right ) x^4}{4 e^5}-\frac{\left (100 d^3+45 d^2 e+111 d e^2+37 e^3\right ) x^5}{5 e^4}+\frac{\left (100 d^2+45 d e+111 e^2\right ) x^6}{6 e^3}-\frac{5 (20 d+9 e) x^7}{7 e^2}+\frac{25 x^8}{2 e}+\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 ) \log (d+e x)}{e^9}\\ \end{align*}

Mathematica [A]  time = 0.1217, size = 262, normalized size = 0.74 $\frac{x \left (-70 d^5 e^2 \left (200 x^2-135 x+666\right )+210 d^4 e^3 \left (50 x^3-30 x^2+111 x-74\right )-105 d^3 e^4 \left (80 x^4-45 x^3+148 x^2-74 x+592\right )+35 d^2 e^5 \left (200 x^5-108 x^4+333 x^3-148 x^2+888 x+780\right )+2100 d^6 e (10 x-9)-42000 d^7-d e^6 \left (6000 x^6-3150 x^5+9324 x^4-3885 x^3+20720 x^2+13650 x+44940\right )+2 e^7 \left (2625 x^7-1350 x^6+3885 x^5-1554 x^4+7770 x^3+4550 x^2+11235 x+6930\right )\right )}{420 e^8}+\frac{\left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \left (5 d^2-2 d e+3 e^2\right )^2 \log (d+e x)}{e^9}$

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),x]

[Out]

(x*(-42000*d^7 + 2100*d^6*e*(-9 + 10*x) - 70*d^5*e^2*(666 - 135*x + 200*x^2) + 210*d^4*e^3*(-74 + 111*x - 30*x
^2 + 50*x^3) - 105*d^3*e^4*(592 - 74*x + 148*x^2 - 45*x^3 + 80*x^4) + 35*d^2*e^5*(780 + 888*x - 148*x^2 + 333*
x^3 - 108*x^4 + 200*x^5) - d*e^6*(44940 + 13650*x + 20720*x^2 - 3885*x^3 + 9324*x^4 - 3150*x^5 + 6000*x^6) + 2
*e^7*(6930 + 11235*x + 4550*x^2 + 7770*x^3 - 1554*x^4 + 3885*x^5 - 1350*x^6 + 2625*x^7)))/(420*e^8) + ((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^9

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Maple [A]  time = 0.052, size = 465, normalized size = 1.3 \begin{align*} -{\frac{45\,{x}^{7}}{7\,e}}+33\,{\frac{x}{e}}+18\,{\frac{\ln \left ( ex+d \right ) }{e}}+{\frac{65\,{x}^{3}}{3\,e}}+37\,{\frac{{x}^{4}}{e}}-{\frac{37\,{x}^{5}}{5\,e}}-{\frac{148\,d{x}^{3}}{3\,{e}^{2}}}-{\frac{37\,{x}^{3}{d}^{2}}{3\,{e}^{3}}}-{\frac{111\,{x}^{5}d}{5\,{e}^{2}}}+{\frac{111\,{x}^{4}{d}^{2}}{4\,{e}^{3}}}+{\frac{37\,d{x}^{4}}{4\,{e}^{2}}}+{\frac{37\,{x}^{2}{d}^{3}}{2\,{e}^{4}}}+65\,{\frac{{d}^{2}x}{{e}^{3}}}+{\frac{111\,{x}^{2}{d}^{4}}{2\,{e}^{5}}}-37\,{\frac{x{d}^{4}}{{e}^{5}}}-148\,{\frac{{d}^{3}x}{{e}^{4}}}-{\frac{65\,d{x}^{2}}{2\,{e}^{2}}}-111\,{\frac{{d}^{5}x}{{e}^{6}}}+100\,{\frac{\ln \left ( ex+d \right ){d}^{8}}{{e}^{9}}}-107\,{\frac{dx}{{e}^{2}}}+74\,{\frac{{x}^{2}{d}^{2}}{{e}^{3}}}+37\,{\frac{\ln \left ( ex+d \right ){d}^{5}}{{e}^{6}}}+148\,{\frac{\ln \left ( ex+d \right ){d}^{4}}{{e}^{5}}}+111\,{\frac{\ln \left ( ex+d \right ){d}^{6}}{{e}^{7}}}-65\,{\frac{\ln \left ( ex+d \right ){d}^{3}}{{e}^{4}}}+107\,{\frac{\ln \left ( ex+d \right ){d}^{2}}{{e}^{3}}}-33\,{\frac{\ln \left ( ex+d \right ) d}{{e}^{2}}}+25\,{\frac{{x}^{4}{d}^{4}}{{e}^{5}}}-15\,{\frac{{x}^{3}{d}^{4}}{{e}^{5}}}-100\,{\frac{{d}^{7}x}{{e}^{8}}}+{\frac{15\,d{x}^{6}}{2\,{e}^{2}}}+45\,{\frac{\ln \left ( ex+d \right ){d}^{7}}{{e}^{8}}}+{\frac{50\,{x}^{6}{d}^{2}}{3\,{e}^{3}}}-45\,{\frac{{d}^{6}x}{{e}^{7}}}+50\,{\frac{{x}^{2}{d}^{6}}{{e}^{7}}}-{\frac{100\,{x}^{3}{d}^{5}}{3\,{e}^{6}}}-20\,{\frac{{x}^{5}{d}^{3}}{{e}^{4}}}-37\,{\frac{{x}^{3}{d}^{3}}{{e}^{4}}}+{\frac{45\,{x}^{2}{d}^{5}}{2\,{e}^{6}}}-9\,{\frac{{x}^{5}{d}^{2}}{{e}^{3}}}-{\frac{100\,{x}^{7}d}{7\,{e}^{2}}}+{\frac{45\,{x}^{4}{d}^{3}}{4\,{e}^{4}}}+{\frac{107\,{x}^{2}}{2\,e}}+{\frac{25\,{x}^{8}}{2\,e}}+{\frac{37\,{x}^{6}}{2\,e}} \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),x)

[Out]

-45/7/e*x^7+33/e*x+18/e*ln(e*x+d)+65/3/e*x^3+37*x^4/e-37/5/e*x^5-148/3/e^2*x^3*d-37/3/e^3*x^3*d^2-111/5/e^2*x^
5*d+111/4/e^3*x^4*d^2+37/4/e^2*x^4*d+37/2/e^4*x^2*d^3+65/e^3*x*d^2+111/2/e^5*x^2*d^4-37/e^5*x*d^4-148/e^4*x*d^
3-65/2/e^2*x^2*d-111/e^6*d^5*x+100/e^9*ln(e*x+d)*d^8-107/e^2*x*d+74/e^3*x^2*d^2+37/e^6*ln(e*x+d)*d^5+148/e^5*l
n(e*x+d)*d^4+111/e^7*ln(e*x+d)*d^6-65/e^4*ln(e*x+d)*d^3+107/e^3*ln(e*x+d)*d^2-33/e^2*ln(e*x+d)*d+25/e^5*x^4*d^
4-15/e^5*x^3*d^4-100/e^8*d^7*x+15/2/e^2*x^6*d+45/e^8*ln(e*x+d)*d^7+50/3/e^3*x^6*d^2-45/e^7*x*d^6+50/e^7*x^2*d^
6-100/3/e^6*x^3*d^5-20/e^4*x^5*d^3-37/e^4*x^3*d^3+45/2/e^6*x^2*d^5-9/e^3*x^5*d^2-100/7/e^2*x^7*d+45/4/e^4*x^4*
d^3+107/2*x^2/e+25/2*x^8/e+37/2*x^6/e

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Maxima [A]  time = 1.01593, size = 494, normalized size = 1.4 \begin{align*} \frac{5250 \, e^{7} x^{8} - 300 \,{\left (20 \, d e^{6} + 9 \, e^{7}\right )} x^{7} + 70 \,{\left (100 \, d^{2} e^{5} + 45 \, d e^{6} + 111 \, e^{7}\right )} x^{6} - 84 \,{\left (100 \, d^{3} e^{4} + 45 \, d^{2} e^{5} + 111 \, d e^{6} + 37 \, e^{7}\right )} x^{5} + 105 \,{\left (100 \, d^{4} e^{3} + 45 \, d^{3} e^{4} + 111 \, d^{2} e^{5} + 37 \, d e^{6} + 148 \, e^{7}\right )} x^{4} - 140 \,{\left (100 \, d^{5} e^{2} + 45 \, d^{4} e^{3} + 111 \, d^{3} e^{4} + 37 \, d^{2} e^{5} + 148 \, d e^{6} - 65 \, e^{7}\right )} x^{3} + 210 \,{\left (100 \, d^{6} e + 45 \, d^{5} e^{2} + 111 \, d^{4} e^{3} + 37 \, d^{3} e^{4} + 148 \, d^{2} e^{5} - 65 \, d e^{6} + 107 \, e^{7}\right )} x^{2} - 420 \,{\left (100 \, d^{7} + 45 \, d^{6} e + 111 \, d^{5} e^{2} + 37 \, d^{4} e^{3} + 148 \, d^{3} e^{4} - 65 \, d^{2} e^{5} + 107 \, d e^{6} - 33 \, e^{7}\right )} x}{420 \, e^{8}} + \frac{{\left (100 \, d^{8} + 45 \, d^{7} e + 111 \, d^{6} e^{2} + 37 \, d^{5} e^{3} + 148 \, d^{4} e^{4} - 65 \, d^{3} e^{5} + 107 \, d^{2} e^{6} - 33 \, d e^{7} + 18 \, e^{8}\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),x, algorithm="maxima")

[Out]

1/420*(5250*e^7*x^8 - 300*(20*d*e^6 + 9*e^7)*x^7 + 70*(100*d^2*e^5 + 45*d*e^6 + 111*e^7)*x^6 - 84*(100*d^3*e^4
+ 45*d^2*e^5 + 111*d*e^6 + 37*e^7)*x^5 + 105*(100*d^4*e^3 + 45*d^3*e^4 + 111*d^2*e^5 + 37*d*e^6 + 148*e^7)*x^
4 - 140*(100*d^5*e^2 + 45*d^4*e^3 + 111*d^3*e^4 + 37*d^2*e^5 + 148*d*e^6 - 65*e^7)*x^3 + 210*(100*d^6*e + 45*d
^5*e^2 + 111*d^4*e^3 + 37*d^3*e^4 + 148*d^2*e^5 - 65*d*e^6 + 107*e^7)*x^2 - 420*(100*d^7 + 45*d^6*e + 111*d^5*
e^2 + 37*d^4*e^3 + 148*d^3*e^4 - 65*d^2*e^5 + 107*d*e^6 - 33*e^7)*x)/e^8 + (100*d^8 + 45*d^7*e + 111*d^6*e^2 +
37*d^5*e^3 + 148*d^4*e^4 - 65*d^3*e^5 + 107*d^2*e^6 - 33*d*e^7 + 18*e^8)*log(e*x + d)/e^9

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Fricas [A]  time = 1.05444, size = 875, normalized size = 2.49 \begin{align*} \frac{5250 \, e^{8} x^{8} - 300 \,{\left (20 \, d e^{7} + 9 \, e^{8}\right )} x^{7} + 70 \,{\left (100 \, d^{2} e^{6} + 45 \, d e^{7} + 111 \, e^{8}\right )} x^{6} - 84 \,{\left (100 \, d^{3} e^{5} + 45 \, d^{2} e^{6} + 111 \, d e^{7} + 37 \, e^{8}\right )} x^{5} + 105 \,{\left (100 \, d^{4} e^{4} + 45 \, d^{3} e^{5} + 111 \, d^{2} e^{6} + 37 \, d e^{7} + 148 \, e^{8}\right )} x^{4} - 140 \,{\left (100 \, d^{5} e^{3} + 45 \, d^{4} e^{4} + 111 \, d^{3} e^{5} + 37 \, d^{2} e^{6} + 148 \, d e^{7} - 65 \, e^{8}\right )} x^{3} + 210 \,{\left (100 \, d^{6} e^{2} + 45 \, d^{5} e^{3} + 111 \, d^{4} e^{4} + 37 \, d^{3} e^{5} + 148 \, d^{2} e^{6} - 65 \, d e^{7} + 107 \, e^{8}\right )} x^{2} - 420 \,{\left (100 \, d^{7} e + 45 \, d^{6} e^{2} + 111 \, d^{5} e^{3} + 37 \, d^{4} e^{4} + 148 \, d^{3} e^{5} - 65 \, d^{2} e^{6} + 107 \, d e^{7} - 33 \, e^{8}\right )} x + 420 \,{\left (100 \, d^{8} + 45 \, d^{7} e + 111 \, d^{6} e^{2} + 37 \, d^{5} e^{3} + 148 \, d^{4} e^{4} - 65 \, d^{3} e^{5} + 107 \, d^{2} e^{6} - 33 \, d e^{7} + 18 \, e^{8}\right )} \log \left (e x + d\right )}{420 \, 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),x, algorithm="fricas")

[Out]

1/420*(5250*e^8*x^8 - 300*(20*d*e^7 + 9*e^8)*x^7 + 70*(100*d^2*e^6 + 45*d*e^7 + 111*e^8)*x^6 - 84*(100*d^3*e^5
+ 45*d^2*e^6 + 111*d*e^7 + 37*e^8)*x^5 + 105*(100*d^4*e^4 + 45*d^3*e^5 + 111*d^2*e^6 + 37*d*e^7 + 148*e^8)*x^
4 - 140*(100*d^5*e^3 + 45*d^4*e^4 + 111*d^3*e^5 + 37*d^2*e^6 + 148*d*e^7 - 65*e^8)*x^3 + 210*(100*d^6*e^2 + 45
*d^5*e^3 + 111*d^4*e^4 + 37*d^3*e^5 + 148*d^2*e^6 - 65*d*e^7 + 107*e^8)*x^2 - 420*(100*d^7*e + 45*d^6*e^2 + 11
1*d^5*e^3 + 37*d^4*e^4 + 148*d^3*e^5 - 65*d^2*e^6 + 107*d*e^7 - 33*e^8)*x + 420*(100*d^8 + 45*d^7*e + 111*d^6*
e^2 + 37*d^5*e^3 + 148*d^4*e^4 - 65*d^3*e^5 + 107*d^2*e^6 - 33*d*e^7 + 18*e^8)*log(e*x + d))/e^9

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Sympy [A]  time = 0.717569, size = 347, normalized size = 0.99 \begin{align*} \frac{25 x^{8}}{2 e} - \frac{x^{7} \left (100 d + 45 e\right )}{7 e^{2}} + \frac{x^{6} \left (100 d^{2} + 45 d e + 111 e^{2}\right )}{6 e^{3}} - \frac{x^{5} \left (100 d^{3} + 45 d^{2} e + 111 d e^{2} + 37 e^{3}\right )}{5 e^{4}} + \frac{x^{4} \left (100 d^{4} + 45 d^{3} e + 111 d^{2} e^{2} + 37 d e^{3} + 148 e^{4}\right )}{4 e^{5}} - \frac{x^{3} \left (100 d^{5} + 45 d^{4} e + 111 d^{3} e^{2} + 37 d^{2} e^{3} + 148 d e^{4} - 65 e^{5}\right )}{3 e^{6}} + \frac{x^{2} \left (100 d^{6} + 45 d^{5} e + 111 d^{4} e^{2} + 37 d^{3} e^{3} + 148 d^{2} e^{4} - 65 d e^{5} + 107 e^{6}\right )}{2 e^{7}} - \frac{x \left (100 d^{7} + 45 d^{6} e + 111 d^{5} e^{2} + 37 d^{4} e^{3} + 148 d^{3} e^{4} - 65 d^{2} e^{5} + 107 d e^{6} - 33 e^{7}\right )}{e^{8}} + \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 ) \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),x)

[Out]

25*x**8/(2*e) - x**7*(100*d + 45*e)/(7*e**2) + x**6*(100*d**2 + 45*d*e + 111*e**2)/(6*e**3) - x**5*(100*d**3 +
45*d**2*e + 111*d*e**2 + 37*e**3)/(5*e**4) + x**4*(100*d**4 + 45*d**3*e + 111*d**2*e**2 + 37*d*e**3 + 148*e**
4)/(4*e**5) - x**3*(100*d**5 + 45*d**4*e + 111*d**3*e**2 + 37*d**2*e**3 + 148*d*e**4 - 65*e**5)/(3*e**6) + x**
2*(100*d**6 + 45*d**5*e + 111*d**4*e**2 + 37*d**3*e**3 + 148*d**2*e**4 - 65*d*e**5 + 107*e**6)/(2*e**7) - x*(1
00*d**7 + 45*d**6*e + 111*d**5*e**2 + 37*d**4*e**3 + 148*d**3*e**4 - 65*d**2*e**5 + 107*d*e**6 - 33*e**7)/e**8
+ (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**9

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Giac [A]  time = 1.14073, size = 510, normalized size = 1.45 \begin{align*}{\left (100 \, d^{8} + 45 \, d^{7} e + 111 \, d^{6} e^{2} + 37 \, d^{5} e^{3} + 148 \, d^{4} e^{4} - 65 \, d^{3} e^{5} + 107 \, d^{2} e^{6} - 33 \, d e^{7} + 18 \, e^{8}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{420} \,{\left (5250 \, x^{8} e^{7} - 6000 \, d x^{7} e^{6} + 7000 \, d^{2} x^{6} e^{5} - 8400 \, d^{3} x^{5} e^{4} + 10500 \, d^{4} x^{4} e^{3} - 14000 \, d^{5} x^{3} e^{2} + 21000 \, d^{6} x^{2} e - 42000 \, d^{7} x - 2700 \, x^{7} e^{7} + 3150 \, d x^{6} e^{6} - 3780 \, d^{2} x^{5} e^{5} + 4725 \, d^{3} x^{4} e^{4} - 6300 \, d^{4} x^{3} e^{3} + 9450 \, d^{5} x^{2} e^{2} - 18900 \, d^{6} x e + 7770 \, x^{6} e^{7} - 9324 \, d x^{5} e^{6} + 11655 \, d^{2} x^{4} e^{5} - 15540 \, d^{3} x^{3} e^{4} + 23310 \, d^{4} x^{2} e^{3} - 46620 \, d^{5} x e^{2} - 3108 \, x^{5} e^{7} + 3885 \, d x^{4} e^{6} - 5180 \, d^{2} x^{3} e^{5} + 7770 \, d^{3} x^{2} e^{4} - 15540 \, d^{4} x e^{3} + 15540 \, x^{4} e^{7} - 20720 \, d x^{3} e^{6} + 31080 \, d^{2} x^{2} e^{5} - 62160 \, d^{3} x e^{4} + 9100 \, x^{3} e^{7} - 13650 \, d x^{2} e^{6} + 27300 \, d^{2} x e^{5} + 22470 \, x^{2} e^{7} - 44940 \, d x e^{6} + 13860 \, x e^{7}\right )} e^{\left (-8\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),x, algorithm="giac")

[Out]

(100*d^8 + 45*d^7*e + 111*d^6*e^2 + 37*d^5*e^3 + 148*d^4*e^4 - 65*d^3*e^5 + 107*d^2*e^6 - 33*d*e^7 + 18*e^8)*e
^(-9)*log(abs(x*e + d)) + 1/420*(5250*x^8*e^7 - 6000*d*x^7*e^6 + 7000*d^2*x^6*e^5 - 8400*d^3*x^5*e^4 + 10500*d
^4*x^4*e^3 - 14000*d^5*x^3*e^2 + 21000*d^6*x^2*e - 42000*d^7*x - 2700*x^7*e^7 + 3150*d*x^6*e^6 - 3780*d^2*x^5*
e^5 + 4725*d^3*x^4*e^4 - 6300*d^4*x^3*e^3 + 9450*d^5*x^2*e^2 - 18900*d^6*x*e + 7770*x^6*e^7 - 9324*d*x^5*e^6 +
11655*d^2*x^4*e^5 - 15540*d^3*x^3*e^4 + 23310*d^4*x^2*e^3 - 46620*d^5*x*e^2 - 3108*x^5*e^7 + 3885*d*x^4*e^6 -
5180*d^2*x^3*e^5 + 7770*d^3*x^2*e^4 - 15540*d^4*x*e^3 + 15540*x^4*e^7 - 20720*d*x^3*e^6 + 31080*d^2*x^2*e^5 -
62160*d^3*x*e^4 + 9100*x^3*e^7 - 13650*d*x^2*e^6 + 27300*d^2*x*e^5 + 22470*x^2*e^7 - 44940*d*x*e^6 + 13860*x*
e^7)*e^(-8)