### 3.301 $$\int \frac{(3+2 x+5 x^2)^2 (2+x+3 x^2-5 x^3+4 x^4)}{(d+e x)^2} \, dx$$

Optimal. Leaf size=353 $\frac{3 x^5 \left (100 d^2+30 d e+37 e^2\right )}{5 e^4}-\frac{x^4 \left (135 d^2 e+400 d^3+222 d e^2+37 e^3\right )}{4 e^5}+\frac{x^3 \left (333 d^2 e^2+180 d^3 e+500 d^4+74 d e^3+148 e^4\right )}{3 e^6}-\frac{x^2 \left (444 d^3 e^2+111 d^2 e^3+225 d^4 e+600 d^5+296 d e^4-65 e^5\right )}{2 e^7}+\frac{x \left (555 d^4 e^2+148 d^3 e^3+444 d^2 e^4+270 d^5 e+700 d^6-130 d e^5+107 e^6\right )}{e^8}-\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{e^9 (d+e x)}-\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (88 d^3 e^2-4 d^2 e^3+127 d^4 e+160 d^5+64 d e^4-11 e^5\right ) \log (d+e x)}{e^9}-\frac{5 x^6 (40 d+9 e)}{6 e^3}+\frac{100 x^7}{7 e^2}$

[Out]

((700*d^6 + 270*d^5*e + 555*d^4*e^2 + 148*d^3*e^3 + 444*d^2*e^4 - 130*d*e^5 + 107*e^6)*x)/e^8 - ((600*d^5 + 22
5*d^4*e + 444*d^3*e^2 + 111*d^2*e^3 + 296*d*e^4 - 65*e^5)*x^2)/(2*e^7) + ((500*d^4 + 180*d^3*e + 333*d^2*e^2 +
74*d*e^3 + 148*e^4)*x^3)/(3*e^6) - ((400*d^3 + 135*d^2*e + 222*d*e^2 + 37*e^3)*x^4)/(4*e^5) + (3*(100*d^2 + 3
0*d*e + 37*e^2)*x^5)/(5*e^4) - (5*(40*d + 9*e)*x^6)/(6*e^3) + (100*x^7)/(7*e^2) - ((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))/(e^9*(d + e*x)) - ((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)*Log[d + e*x])/e^9

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Rubi [A]  time = 0.32805, antiderivative size = 353, 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} $\frac{3 x^5 \left (100 d^2+30 d e+37 e^2\right )}{5 e^4}-\frac{x^4 \left (135 d^2 e+400 d^3+222 d e^2+37 e^3\right )}{4 e^5}+\frac{x^3 \left (333 d^2 e^2+180 d^3 e+500 d^4+74 d e^3+148 e^4\right )}{3 e^6}-\frac{x^2 \left (444 d^3 e^2+111 d^2 e^3+225 d^4 e+600 d^5+296 d e^4-65 e^5\right )}{2 e^7}+\frac{x \left (555 d^4 e^2+148 d^3 e^3+444 d^2 e^4+270 d^5 e+700 d^6-130 d e^5+107 e^6\right )}{e^8}-\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{e^9 (d+e x)}-\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (88 d^3 e^2-4 d^2 e^3+127 d^4 e+160 d^5+64 d e^4-11 e^5\right ) \log (d+e x)}{e^9}-\frac{5 x^6 (40 d+9 e)}{6 e^3}+\frac{100 x^7}{7 e^2}$

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)^2,x]

[Out]

((700*d^6 + 270*d^5*e + 555*d^4*e^2 + 148*d^3*e^3 + 444*d^2*e^4 - 130*d*e^5 + 107*e^6)*x)/e^8 - ((600*d^5 + 22
5*d^4*e + 444*d^3*e^2 + 111*d^2*e^3 + 296*d*e^4 - 65*e^5)*x^2)/(2*e^7) + ((500*d^4 + 180*d^3*e + 333*d^2*e^2 +
74*d*e^3 + 148*e^4)*x^3)/(3*e^6) - ((400*d^3 + 135*d^2*e + 222*d*e^2 + 37*e^3)*x^4)/(4*e^5) + (3*(100*d^2 + 3
0*d*e + 37*e^2)*x^5)/(5*e^4) - (5*(40*d + 9*e)*x^6)/(6*e^3) + (100*x^7)/(7*e^2) - ((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))/(e^9*(d + e*x)) - ((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)*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)^2} \, dx &=\int \left (\frac{700 d^6+270 d^5 e+555 d^4 e^2+148 d^3 e^3+444 d^2 e^4-130 d e^5+107 e^6}{e^8}+\frac{\left (-600 d^5-225 d^4 e-444 d^3 e^2-111 d^2 e^3-296 d e^4+65 e^5\right ) x}{e^7}+\frac{\left (500 d^4+180 d^3 e+333 d^2 e^2+74 d e^3+148 e^4\right ) x^2}{e^6}-\frac{\left (400 d^3+135 d^2 e+222 d e^2+37 e^3\right ) x^3}{e^5}+\frac{3 \left (100 d^2+30 d e+37 e^2\right ) x^4}{e^4}-\frac{5 (40 d+9 e) x^5}{e^3}+\frac{100 x^6}{e^2}+\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)^2}+\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)}\right ) \, dx\\ &=\frac{\left (700 d^6+270 d^5 e+555 d^4 e^2+148 d^3 e^3+444 d^2 e^4-130 d e^5+107 e^6\right ) x}{e^8}-\frac{\left (600 d^5+225 d^4 e+444 d^3 e^2+111 d^2 e^3+296 d e^4-65 e^5\right ) x^2}{2 e^7}+\frac{\left (500 d^4+180 d^3 e+333 d^2 e^2+74 d e^3+148 e^4\right ) x^3}{3 e^6}-\frac{\left (400 d^3+135 d^2 e+222 d e^2+37 e^3\right ) x^4}{4 e^5}+\frac{3 \left (100 d^2+30 d e+37 e^2\right ) x^5}{5 e^4}-\frac{5 (40 d+9 e) x^6}{6 e^3}+\frac{100 x^7}{7 e^2}-\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^9 (d+e x)}-\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 ) \log (d+e x)}{e^9}\\ \end{align*}

Mathematica [A]  time = 0.136448, size = 342, normalized size = 0.97 $\frac{252 e^5 x^5 \left (100 d^2+30 d e+37 e^2\right )-105 e^4 x^4 \left (135 d^2 e+400 d^3+222 d e^2+37 e^3\right )+140 e^3 x^3 \left (333 d^2 e^2+180 d^3 e+500 d^4+74 d e^3+148 e^4\right )-210 e^2 x^2 \left (444 d^3 e^2+111 d^2 e^3+225 d^4 e+600 d^5+296 d e^4-65 e^5\right )+420 e x \left (555 d^4 e^2+148 d^3 e^3+444 d^2 e^4+270 d^5 e+700 d^6-130 d e^5+107 e^6\right )-\frac{420 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{d+e x}-420 \left (666 d^5 e^2+185 d^4 e^3+592 d^3 e^4-195 d^2 e^5+315 d^6 e+800 d^7+214 d e^6-33 e^7\right ) \log (d+e x)-350 e^6 x^6 (40 d+9 e)+6000 e^7 x^7}{420 e^9}$

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)^2,x]

[Out]

(420*e*(700*d^6 + 270*d^5*e + 555*d^4*e^2 + 148*d^3*e^3 + 444*d^2*e^4 - 130*d*e^5 + 107*e^6)*x - 210*e^2*(600*
d^5 + 225*d^4*e + 444*d^3*e^2 + 111*d^2*e^3 + 296*d*e^4 - 65*e^5)*x^2 + 140*e^3*(500*d^4 + 180*d^3*e + 333*d^2
*e^2 + 74*d*e^3 + 148*e^4)*x^3 - 105*e^4*(400*d^3 + 135*d^2*e + 222*d*e^2 + 37*e^3)*x^4 + 252*e^5*(100*d^2 + 3
0*d*e + 37*e^2)*x^5 - 350*e^6*(40*d + 9*e)*x^6 + 6000*e^7*x^7 - (420*(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) - 420*(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)*Log[d + e*x])/(420*e^9)

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Maple [A]  time = 0.058, size = 500, normalized size = 1.4 \begin{align*} -666\,{\frac{\ln \left ( ex+d \right ){d}^{5}}{{e}^{7}}}-{\frac{15\,{x}^{6}}{2\,{e}^{2}}}-{\frac{37\,{x}^{4}}{4\,{e}^{2}}}+{\frac{148\,{x}^{3}}{3\,{e}^{2}}}+{\frac{65\,{x}^{2}}{2\,{e}^{2}}}+33\,{\frac{\ln \left ( ex+d \right ) }{{e}^{2}}}-18\,{\frac{1}{e \left ( ex+d \right ) }}-185\,{\frac{\ln \left ( ex+d \right ){d}^{4}}{{e}^{6}}}-592\,{\frac{\ln \left ( ex+d \right ){d}^{3}}{{e}^{5}}}+195\,{\frac{\ln \left ( ex+d \right ){d}^{2}}{{e}^{4}}}-214\,{\frac{\ln \left ( ex+d \right ) d}{{e}^{3}}}+555\,{\frac{{d}^{4}x}{{e}^{6}}}+148\,{\frac{{d}^{3}x}{{e}^{5}}}+444\,{\frac{{d}^{2}x}{{e}^{4}}}-130\,{\frac{dx}{{e}^{3}}}-{\frac{111\,d{x}^{4}}{2\,{e}^{3}}}+111\,{\frac{{x}^{3}{d}^{2}}{{e}^{4}}}+{\frac{74\,d{x}^{3}}{3\,{e}^{3}}}-222\,{\frac{{x}^{2}{d}^{3}}{{e}^{5}}}-{\frac{111\,{x}^{2}{d}^{2}}{2\,{e}^{4}}}-148\,{\frac{d{x}^{2}}{{e}^{3}}}-111\,{\frac{{d}^{6}}{{e}^{7} \left ( ex+d \right ) }}-37\,{\frac{{d}^{5}}{{e}^{6} \left ( ex+d \right ) }}+270\,{\frac{x{d}^{5}}{{e}^{7}}}+18\,{\frac{{x}^{5}d}{{e}^{3}}}-{\frac{135\,{x}^{4}{d}^{2}}{4\,{e}^{4}}}+60\,{\frac{{x}^{5}{d}^{2}}{{e}^{4}}}-{\frac{100\,d{x}^{6}}{3\,{e}^{3}}}+60\,{\frac{{x}^{3}{d}^{3}}{{e}^{5}}}-100\,{\frac{{x}^{4}{d}^{3}}{{e}^{5}}}-{\frac{225\,{x}^{2}{d}^{4}}{2\,{e}^{6}}}-300\,{\frac{{x}^{2}{d}^{5}}{{e}^{7}}}-100\,{\frac{{d}^{8}}{{e}^{9} \left ( ex+d \right ) }}-45\,{\frac{{d}^{7}}{{e}^{8} \left ( ex+d \right ) }}-800\,{\frac{\ln \left ( ex+d \right ){d}^{7}}{{e}^{9}}}-315\,{\frac{\ln \left ( ex+d \right ){d}^{6}}{{e}^{8}}}+{\frac{500\,{x}^{3}{d}^{4}}{3\,{e}^{6}}}+700\,{\frac{{d}^{6}x}{{e}^{8}}}-148\,{\frac{{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}+65\,{\frac{{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-107\,{\frac{{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+33\,{\frac{d}{{e}^{2} \left ( ex+d \right ) }}+107\,{\frac{x}{{e}^{2}}}+{\frac{100\,{x}^{7}}{7\,{e}^{2}}}+{\frac{111\,{x}^{5}}{5\,{e}^{2}}} \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)^2,x)

[Out]

-666/e^7*ln(e*x+d)*d^5-15/2/e^2*x^6-37/4/e^2*x^4+148/3/e^2*x^3+65/2/e^2*x^2+33/e^2*ln(e*x+d)-18/e/(e*x+d)-185/
e^6*ln(e*x+d)*d^4-592/e^5*ln(e*x+d)*d^3+195/e^4*ln(e*x+d)*d^2-214/e^3*ln(e*x+d)*d+555/e^6*d^4*x+148/e^5*x*d^3+
444/e^4*x*d^2-130/e^3*x*d-111/2/e^3*x^4*d+111/e^4*x^3*d^2+74/3/e^3*x^3*d-222/e^5*x^2*d^3-111/2/e^4*x^2*d^2-148
/e^3*x^2*d-111/e^7/(e*x+d)*d^6-37/e^6/(e*x+d)*d^5+270/e^7*x*d^5+18/e^3*x^5*d-135/4/e^4*x^4*d^2+60/e^4*x^5*d^2-
100/3/e^3*x^6*d+60/e^5*x^3*d^3-100/e^5*x^4*d^3-225/2/e^6*x^2*d^4-300/e^7*x^2*d^5-100/e^9/(e*x+d)*d^8-45/e^8/(e
*x+d)*d^7-800/e^9*ln(e*x+d)*d^7-315/e^8*ln(e*x+d)*d^6+500/3/e^6*x^3*d^4+700/e^8*d^6*x-148/e^5/(e*x+d)*d^4+65/e
^4/(e*x+d)*d^3-107/e^3/(e*x+d)*d^2+33/e^2/(e*x+d)*d+107*x/e^2+100/7*x^7/e^2+111/5*x^5/e^2

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Maxima [A]  time = 0.996031, size = 502, normalized size = 1.42 \begin{align*} -\frac{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^{10} x + d e^{9}} + \frac{6000 \, e^{6} x^{7} - 350 \,{\left (40 \, d e^{5} + 9 \, e^{6}\right )} x^{6} + 252 \,{\left (100 \, d^{2} e^{4} + 30 \, d e^{5} + 37 \, e^{6}\right )} x^{5} - 105 \,{\left (400 \, d^{3} e^{3} + 135 \, d^{2} e^{4} + 222 \, d e^{5} + 37 \, e^{6}\right )} x^{4} + 140 \,{\left (500 \, d^{4} e^{2} + 180 \, d^{3} e^{3} + 333 \, d^{2} e^{4} + 74 \, d e^{5} + 148 \, e^{6}\right )} x^{3} - 210 \,{\left (600 \, d^{5} e + 225 \, d^{4} e^{2} + 444 \, d^{3} e^{3} + 111 \, d^{2} e^{4} + 296 \, d e^{5} - 65 \, e^{6}\right )} x^{2} + 420 \,{\left (700 \, d^{6} + 270 \, d^{5} e + 555 \, d^{4} e^{2} + 148 \, d^{3} e^{3} + 444 \, d^{2} e^{4} - 130 \, d e^{5} + 107 \, e^{6}\right )} x}{420 \, e^{8}} - \frac{{\left (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}\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)^2,x, algorithm="maxima")

[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^10*x + d*e^9) + 1/420*(6000*e^6*x^7 - 350*(40*d*e^5 + 9*e^6)*x^6 + 252*(100*d^2*e^4 + 30*d*e^5 + 37*e^6)*x^
5 - 105*(400*d^3*e^3 + 135*d^2*e^4 + 222*d*e^5 + 37*e^6)*x^4 + 140*(500*d^4*e^2 + 180*d^3*e^3 + 333*d^2*e^4 +
74*d*e^5 + 148*e^6)*x^3 - 210*(600*d^5*e + 225*d^4*e^2 + 444*d^3*e^3 + 111*d^2*e^4 + 296*d*e^5 - 65*e^6)*x^2 +
420*(700*d^6 + 270*d^5*e + 555*d^4*e^2 + 148*d^3*e^3 + 444*d^2*e^4 - 130*d*e^5 + 107*e^6)*x)/e^8 - (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)*log(e*x + d)/e^9

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Fricas [A]  time = 0.984773, size = 1211, normalized size = 3.43 \begin{align*} \frac{6000 \, e^{8} x^{8} - 42000 \, d^{8} - 18900 \, d^{7} e - 46620 \, d^{6} e^{2} - 15540 \, d^{5} e^{3} - 62160 \, d^{4} e^{4} + 27300 \, d^{3} e^{5} - 44940 \, d^{2} e^{6} + 13860 \, d e^{7} - 7560 \, e^{8} - 50 \,{\left (160 \, d e^{7} + 63 \, e^{8}\right )} x^{7} + 14 \,{\left (800 \, d^{2} e^{6} + 315 \, d e^{7} + 666 \, e^{8}\right )} x^{6} - 21 \,{\left (800 \, d^{3} e^{5} + 315 \, d^{2} e^{6} + 666 \, d e^{7} + 185 \, e^{8}\right )} x^{5} + 35 \,{\left (800 \, d^{4} e^{4} + 315 \, d^{3} e^{5} + 666 \, d^{2} e^{6} + 185 \, d e^{7} + 592 \, e^{8}\right )} x^{4} - 70 \,{\left (800 \, d^{5} e^{3} + 315 \, d^{4} e^{4} + 666 \, d^{3} e^{5} + 185 \, d^{2} e^{6} + 592 \, d e^{7} - 195 \, e^{8}\right )} x^{3} + 210 \,{\left (800 \, d^{6} e^{2} + 315 \, d^{5} e^{3} + 666 \, d^{4} e^{4} + 185 \, d^{3} e^{5} + 592 \, d^{2} e^{6} - 195 \, d e^{7} + 214 \, e^{8}\right )} x^{2} + 420 \,{\left (700 \, d^{7} e + 270 \, d^{6} e^{2} + 555 \, d^{5} e^{3} + 148 \, d^{4} e^{4} + 444 \, d^{3} e^{5} - 130 \, d^{2} e^{6} + 107 \, d e^{7}\right )} x - 420 \,{\left (800 \, d^{8} + 315 \, d^{7} e + 666 \, d^{6} e^{2} + 185 \, d^{5} e^{3} + 592 \, d^{4} e^{4} - 195 \, d^{3} e^{5} + 214 \, d^{2} e^{6} - 33 \, d e^{7} +{\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\right )} \log \left (e x + d\right )}{420 \,{\left (e^{10} x + d 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)^2,x, algorithm="fricas")

[Out]

1/420*(6000*e^8*x^8 - 42000*d^8 - 18900*d^7*e - 46620*d^6*e^2 - 15540*d^5*e^3 - 62160*d^4*e^4 + 27300*d^3*e^5
- 44940*d^2*e^6 + 13860*d*e^7 - 7560*e^8 - 50*(160*d*e^7 + 63*e^8)*x^7 + 14*(800*d^2*e^6 + 315*d*e^7 + 666*e^8
)*x^6 - 21*(800*d^3*e^5 + 315*d^2*e^6 + 666*d*e^7 + 185*e^8)*x^5 + 35*(800*d^4*e^4 + 315*d^3*e^5 + 666*d^2*e^6
+ 185*d*e^7 + 592*e^8)*x^4 - 70*(800*d^5*e^3 + 315*d^4*e^4 + 666*d^3*e^5 + 185*d^2*e^6 + 592*d*e^7 - 195*e^8)
*x^3 + 210*(800*d^6*e^2 + 315*d^5*e^3 + 666*d^4*e^4 + 185*d^3*e^5 + 592*d^2*e^6 - 195*d*e^7 + 214*e^8)*x^2 + 4
20*(700*d^7*e + 270*d^6*e^2 + 555*d^5*e^3 + 148*d^4*e^4 + 444*d^3*e^5 - 130*d^2*e^6 + 107*d*e^7)*x - 420*(800*
d^8 + 315*d^7*e + 666*d^6*e^2 + 185*d^5*e^3 + 592*d^4*e^4 - 195*d^3*e^5 + 214*d^2*e^6 - 33*d*e^7 + (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)*log(e*x + d))/(
e^10*x + d*e^9)

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Sympy [A]  time = 1.45614, size = 367, normalized size = 1.04 \begin{align*} - \frac{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}}{d e^{9} + e^{10} x} + \frac{100 x^{7}}{7 e^{2}} - \frac{x^{6} \left (200 d + 45 e\right )}{6 e^{3}} + \frac{x^{5} \left (300 d^{2} + 90 d e + 111 e^{2}\right )}{5 e^{4}} - \frac{x^{4} \left (400 d^{3} + 135 d^{2} e + 222 d e^{2} + 37 e^{3}\right )}{4 e^{5}} + \frac{x^{3} \left (500 d^{4} + 180 d^{3} e + 333 d^{2} e^{2} + 74 d e^{3} + 148 e^{4}\right )}{3 e^{6}} - \frac{x^{2} \left (600 d^{5} + 225 d^{4} e + 444 d^{3} e^{2} + 111 d^{2} e^{3} + 296 d e^{4} - 65 e^{5}\right )}{2 e^{7}} + \frac{x \left (700 d^{6} + 270 d^{5} e + 555 d^{4} e^{2} + 148 d^{3} e^{3} + 444 d^{2} e^{4} - 130 d e^{5} + 107 e^{6}\right )}{e^{8}} - \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 ) \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)**2,x)

[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)/(d*e**9 + e**10*x) + 100*x**7/(7*e**2) - x**6*(200*d + 45*e)/(6*e**3) + x**5*(300*d**2 + 90*d*e
+ 111*e**2)/(5*e**4) - x**4*(400*d**3 + 135*d**2*e + 222*d*e**2 + 37*e**3)/(4*e**5) + x**3*(500*d**4 + 180*d**
3*e + 333*d**2*e**2 + 74*d*e**3 + 148*e**4)/(3*e**6) - x**2*(600*d**5 + 225*d**4*e + 444*d**3*e**2 + 111*d**2*
e**3 + 296*d*e**4 - 65*e**5)/(2*e**7) + x*(700*d**6 + 270*d**5*e + 555*d**4*e**2 + 148*d**3*e**3 + 444*d**2*e*
*4 - 130*d*e**5 + 107*e**6)/e**8 - (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)*log(d + e*x)/e**9

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Giac [A]  time = 1.16612, size = 620, normalized size = 1.76 \begin{align*} -\frac{1}{420} \,{\left (x e + d\right )}^{7}{\left (\frac{350 \,{\left (160 \, d e + 9 \, e^{2}\right )} e^{\left (-1\right )}}{x e + d} - \frac{84 \,{\left (2800 \, d^{2} e^{2} + 315 \, d e^{3} + 111 \, e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} + \frac{105 \,{\left (5600 \, d^{3} e^{3} + 945 \, d^{2} e^{4} + 666 \, d e^{5} + 37 \, e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} - \frac{140 \,{\left (7000 \, d^{4} e^{4} + 1575 \, d^{3} e^{5} + 1665 \, d^{2} e^{6} + 185 \, d e^{7} + 148 \, e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} + \frac{210 \,{\left (5600 \, d^{5} e^{5} + 1575 \, d^{4} e^{6} + 2220 \, d^{3} e^{7} + 370 \, d^{2} e^{8} + 592 \, d e^{9} - 65 \, e^{10}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}} - \frac{420 \,{\left (2800 \, d^{6} e^{6} + 945 \, d^{5} e^{7} + 1665 \, d^{4} e^{8} + 370 \, d^{3} e^{9} + 888 \, d^{2} e^{10} - 195 \, d e^{11} + 107 \, e^{12}\right )} e^{\left (-6\right )}}{{\left (x e + d\right )}^{6}} - 6000\right )} e^{\left (-9\right )} +{\left (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}\right )} e^{\left (-9\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{100 \, d^{8} e^{7}}{x e + d} + \frac{45 \, d^{7} e^{8}}{x e + d} + \frac{111 \, d^{6} e^{9}}{x e + d} + \frac{37 \, d^{5} e^{10}}{x e + d} + \frac{148 \, d^{4} e^{11}}{x e + d} - \frac{65 \, d^{3} e^{12}}{x e + d} + \frac{107 \, d^{2} e^{13}}{x e + d} - \frac{33 \, d e^{14}}{x e + d} + \frac{18 \, e^{15}}{x e + d}\right )} e^{\left (-16\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)^2,x, algorithm="giac")

[Out]

-1/420*(x*e + d)^7*(350*(160*d*e + 9*e^2)*e^(-1)/(x*e + d) - 84*(2800*d^2*e^2 + 315*d*e^3 + 111*e^4)*e^(-2)/(x
*e + d)^2 + 105*(5600*d^3*e^3 + 945*d^2*e^4 + 666*d*e^5 + 37*e^6)*e^(-3)/(x*e + d)^3 - 140*(7000*d^4*e^4 + 157
5*d^3*e^5 + 1665*d^2*e^6 + 185*d*e^7 + 148*e^8)*e^(-4)/(x*e + d)^4 + 210*(5600*d^5*e^5 + 1575*d^4*e^6 + 2220*d
^3*e^7 + 370*d^2*e^8 + 592*d*e^9 - 65*e^10)*e^(-5)/(x*e + d)^5 - 420*(2800*d^6*e^6 + 945*d^5*e^7 + 1665*d^4*e^
8 + 370*d^3*e^9 + 888*d^2*e^10 - 195*d*e^11 + 107*e^12)*e^(-6)/(x*e + d)^6 - 6000)*e^(-9) + (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^(-9)*log(abs(x*e + d)*e^(-1
)/(x*e + d)^2) - (100*d^8*e^7/(x*e + d) + 45*d^7*e^8/(x*e + d) + 111*d^6*e^9/(x*e + d) + 37*d^5*e^10/(x*e + d)
+ 148*d^4*e^11/(x*e + d) - 65*d^3*e^12/(x*e + d) + 107*d^2*e^13/(x*e + d) - 33*d*e^14/(x*e + d) + 18*e^15/(x*
e + d))*e^(-16)