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

[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^8-1/2*(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/e^7+1/3*(500*d^4+180*d^3*e+333*d^2*e^2+74*d*e^3+148*e^4)*x^3/e^6-1/4*(4
00*d^3+135*d^2*e+222*d*e^2+37*e^3)*x^4/e^5+3/5*(100*d^2+30*d*e+37*e^2)*x^5/e^4-5/6*(40*d+9*e)*x^6/e^3+100/7*x^
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/(e*x+d)-(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)*ln(e*x+d)/e^9

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Rubi [A]  time = 0.33, antiderivative size = 353, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.14, size = 342, normalized size = 0.97 \[ \frac {252 e^5 x^5 \left (100 d^2+30 d e+37 e^2\right )-105 e^4 x^4 \left (400 d^3+135 d^2 e+222 d e^2+37 e^3\right )+140 e^3 x^3 \left (500 d^4+180 d^3 e+333 d^2 e^2+74 d e^3+148 e^4\right )-\frac {420 \left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{d+e x}-210 e^2 x^2 \left (600 d^5+225 d^4 e+444 d^3 e^2+111 d^2 e^3+296 d e^4-65 e^5\right )+420 e x \left (700 d^6+270 d^5 e+555 d^4 e^2+148 d^3 e^3+444 d^2 e^4-130 d e^5+107 e^6\right )-420 \left (800 d^7+315 d^6 e+666 d^5 e^2+185 d^4 e^3+592 d^3 e^4-195 d^2 e^5+214 d e^6-33 e^7\right ) \log (d+e x)-350 e^6 x^6 (40 d+9 e)+6000 e^7 x^7}{420 e^9} \]

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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fricas [A]  time = 1.00, size = 490, normalized size = 1.39 \[ \frac {6000 \, e^{8} x^{8} - 42000 \, d^{8} - 18900 \, d^{7} e - 46620 \, d^{6} e^{2} - 15540 \, d^{5} e^{3} - 62160 \, d^{4} e^{4} + 27300 \, d^{3} e^{5} - 44940 \, d^{2} e^{6} + 13860 \, d e^{7} - 7560 \, e^{8} - 50 \, {\left (160 \, d e^{7} + 63 \, e^{8}\right )} x^{7} + 14 \, {\left (800 \, d^{2} e^{6} + 315 \, d e^{7} + 666 \, e^{8}\right )} x^{6} - 21 \, {\left (800 \, d^{3} e^{5} + 315 \, d^{2} e^{6} + 666 \, d e^{7} + 185 \, e^{8}\right )} x^{5} + 35 \, {\left (800 \, d^{4} e^{4} + 315 \, d^{3} e^{5} + 666 \, d^{2} e^{6} + 185 \, d e^{7} + 592 \, e^{8}\right )} x^{4} - 70 \, {\left (800 \, d^{5} e^{3} + 315 \, d^{4} e^{4} + 666 \, d^{3} e^{5} + 185 \, d^{2} e^{6} + 592 \, d e^{7} - 195 \, e^{8}\right )} x^{3} + 210 \, {\left (800 \, d^{6} e^{2} + 315 \, d^{5} e^{3} + 666 \, d^{4} e^{4} + 185 \, d^{3} e^{5} + 592 \, d^{2} e^{6} - 195 \, d e^{7} + 214 \, e^{8}\right )} x^{2} + 420 \, {\left (700 \, d^{7} e + 270 \, d^{6} e^{2} + 555 \, d^{5} e^{3} + 148 \, d^{4} e^{4} + 444 \, d^{3} e^{5} - 130 \, d^{2} e^{6} + 107 \, d e^{7}\right )} x - 420 \, {\left (800 \, d^{8} + 315 \, d^{7} e + 666 \, d^{6} e^{2} + 185 \, d^{5} e^{3} + 592 \, d^{4} e^{4} - 195 \, d^{3} e^{5} + 214 \, d^{2} e^{6} - 33 \, d e^{7} + {\left (800 \, d^{7} e + 315 \, d^{6} e^{2} + 666 \, d^{5} e^{3} + 185 \, d^{4} e^{4} + 592 \, d^{3} e^{5} - 195 \, d^{2} e^{6} + 214 \, d e^{7} - 33 \, e^{8}\right )} x\right )} \log \left (e x + d\right )}{420 \, {\left (e^{10} x + d e^{9}\right )}} \]

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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giac [A]  time = 0.18, size = 459, normalized size = 1.30 \[ -\frac {1}{420} \, {\left (x e + d\right )}^{7} {\left (\frac {350 \, {\left (160 \, d e + 9 \, e^{2}\right )} e^{\left (-1\right )}}{x e + d} - \frac {84 \, {\left (2800 \, d^{2} e^{2} + 315 \, d e^{3} + 111 \, e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} + \frac {105 \, {\left (5600 \, d^{3} e^{3} + 945 \, d^{2} e^{4} + 666 \, d e^{5} + 37 \, e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} - \frac {140 \, {\left (7000 \, d^{4} e^{4} + 1575 \, d^{3} e^{5} + 1665 \, d^{2} e^{6} + 185 \, d e^{7} + 148 \, e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} + \frac {210 \, {\left (5600 \, d^{5} e^{5} + 1575 \, d^{4} e^{6} + 2220 \, d^{3} e^{7} + 370 \, d^{2} e^{8} + 592 \, d e^{9} - 65 \, e^{10}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}} - \frac {420 \, {\left (2800 \, d^{6} e^{6} + 945 \, d^{5} e^{7} + 1665 \, d^{4} e^{8} + 370 \, d^{3} e^{9} + 888 \, d^{2} e^{10} - 195 \, d e^{11} + 107 \, e^{12}\right )} e^{\left (-6\right )}}{{\left (x e + d\right )}^{6}} - 6000\right )} e^{\left (-9\right )} + {\left (800 \, d^{7} + 315 \, d^{6} e + 666 \, d^{5} e^{2} + 185 \, d^{4} e^{3} + 592 \, d^{3} e^{4} - 195 \, d^{2} e^{5} + 214 \, d e^{6} - 33 \, e^{7}\right )} e^{\left (-9\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {100 \, d^{8} e^{7}}{x e + d} + \frac {45 \, d^{7} e^{8}}{x e + d} + \frac {111 \, d^{6} e^{9}}{x e + d} + \frac {37 \, d^{5} e^{10}}{x e + d} + \frac {148 \, d^{4} e^{11}}{x e + d} - \frac {65 \, d^{3} e^{12}}{x e + d} + \frac {107 \, d^{2} e^{13}}{x e + d} - \frac {33 \, d e^{14}}{x e + d} + \frac {18 \, e^{15}}{x e + d}\right )} e^{\left (-16\right )} \]

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)

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maple [A]  time = 0.01, size = 500, normalized size = 1.42 \[ \frac {100 x^{7}}{7 e^{2}}-\frac {100 d \,x^{6}}{3 e^{3}}-\frac {15 x^{6}}{2 e^{2}}+\frac {60 d^{2} x^{5}}{e^{4}}+\frac {18 d \,x^{5}}{e^{3}}+\frac {111 x^{5}}{5 e^{2}}-\frac {100 d^{3} x^{4}}{e^{5}}-\frac {135 d^{2} x^{4}}{4 e^{4}}-\frac {111 d \,x^{4}}{2 e^{3}}-\frac {37 x^{4}}{4 e^{2}}+\frac {500 d^{4} x^{3}}{3 e^{6}}+\frac {60 d^{3} x^{3}}{e^{5}}+\frac {111 d^{2} x^{3}}{e^{4}}+\frac {74 d \,x^{3}}{3 e^{3}}+\frac {148 x^{3}}{3 e^{2}}-\frac {300 d^{5} x^{2}}{e^{7}}-\frac {225 d^{4} x^{2}}{2 e^{6}}-\frac {222 d^{3} x^{2}}{e^{5}}-\frac {111 d^{2} x^{2}}{2 e^{4}}-\frac {148 d \,x^{2}}{e^{3}}+\frac {65 x^{2}}{2 e^{2}}-\frac {100 d^{8}}{\left (e x +d \right ) e^{9}}-\frac {45 d^{7}}{\left (e x +d \right ) e^{8}}-\frac {800 d^{7} \ln \left (e x +d \right )}{e^{9}}-\frac {111 d^{6}}{\left (e x +d \right ) e^{7}}+\frac {700 d^{6} x}{e^{8}}-\frac {315 d^{6} \ln \left (e x +d \right )}{e^{8}}-\frac {37 d^{5}}{\left (e x +d \right ) e^{6}}+\frac {270 d^{5} x}{e^{7}}-\frac {666 d^{5} \ln \left (e x +d \right )}{e^{7}}-\frac {148 d^{4}}{\left (e x +d \right ) e^{5}}+\frac {555 d^{4} x}{e^{6}}-\frac {185 d^{4} \ln \left (e x +d \right )}{e^{6}}+\frac {65 d^{3}}{\left (e x +d \right ) e^{4}}+\frac {148 d^{3} x}{e^{5}}-\frac {592 d^{3} \ln \left (e x +d \right )}{e^{5}}-\frac {107 d^{2}}{\left (e x +d \right ) e^{3}}+\frac {444 d^{2} x}{e^{4}}+\frac {195 d^{2} \ln \left (e x +d \right )}{e^{4}}+\frac {33 d}{\left (e x +d \right ) e^{2}}-\frac {130 d x}{e^{3}}-\frac {214 d \ln \left (e x +d \right )}{e^{3}}-\frac {18}{\left (e x +d \right ) e}+\frac {107 x}{e^{2}}+\frac {33 \ln \left (e x +d \right )}{e^{2}} \]

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]

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

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maxima [A]  time = 0.44, size = 372, normalized size = 1.05 \[ -\frac {100 \, d^{8} + 45 \, d^{7} e + 111 \, d^{6} e^{2} + 37 \, d^{5} e^{3} + 148 \, d^{4} e^{4} - 65 \, d^{3} e^{5} + 107 \, d^{2} e^{6} - 33 \, d e^{7} + 18 \, e^{8}}{e^{10} x + d e^{9}} + \frac {6000 \, e^{6} x^{7} - 350 \, {\left (40 \, d e^{5} + 9 \, e^{6}\right )} x^{6} + 252 \, {\left (100 \, d^{2} e^{4} + 30 \, d e^{5} + 37 \, e^{6}\right )} x^{5} - 105 \, {\left (400 \, d^{3} e^{3} + 135 \, d^{2} e^{4} + 222 \, d e^{5} + 37 \, e^{6}\right )} x^{4} + 140 \, {\left (500 \, d^{4} e^{2} + 180 \, d^{3} e^{3} + 333 \, d^{2} e^{4} + 74 \, d e^{5} + 148 \, e^{6}\right )} x^{3} - 210 \, {\left (600 \, d^{5} e + 225 \, d^{4} e^{2} + 444 \, d^{3} e^{3} + 111 \, d^{2} e^{4} + 296 \, d e^{5} - 65 \, e^{6}\right )} x^{2} + 420 \, {\left (700 \, d^{6} + 270 \, d^{5} e + 555 \, d^{4} e^{2} + 148 \, d^{3} e^{3} + 444 \, d^{2} e^{4} - 130 \, d e^{5} + 107 \, e^{6}\right )} x}{420 \, e^{8}} - \frac {{\left (800 \, d^{7} + 315 \, d^{6} e + 666 \, d^{5} e^{2} + 185 \, d^{4} e^{3} + 592 \, d^{3} e^{4} - 195 \, d^{2} e^{5} + 214 \, d e^{6} - 33 \, e^{7}\right )} \log \left (e x + d\right )}{e^{9}} \]

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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mupad [B]  time = 4.22, size = 939, normalized size = 2.66 \[ x^2\,\left (\frac {65}{2\,e^2}-\frac {d\,\left (\frac {148}{e^2}+\frac {2\,d\,\left (\frac {37}{e^2}+\frac {2\,d\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e}-\frac {d^2\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e^2}\right )}{e}+\frac {d^2\,\left (\frac {37}{e^2}+\frac {2\,d\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e}-\frac {d^2\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e^2}\right )}{2\,e^2}\right )+x^3\,\left (\frac {148}{3\,e^2}+\frac {2\,d\,\left (\frac {37}{e^2}+\frac {2\,d\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e}-\frac {d^2\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e^2}\right )}{3\,e}-\frac {d^2\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{3\,e^2}\right )-x^4\,\left (\frac {37}{4\,e^2}+\frac {d\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{2\,e}-\frac {d^2\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{4\,e^2}\right )+x^5\,\left (\frac {111}{5\,e^2}-\frac {20\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{5\,e}\right )-x^6\,\left (\frac {100\,d}{3\,e^3}+\frac {15}{2\,e^2}\right )-x\,\left (\frac {2\,d\,\left (\frac {65}{e^2}-\frac {2\,d\,\left (\frac {148}{e^2}+\frac {2\,d\,\left (\frac {37}{e^2}+\frac {2\,d\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e}-\frac {d^2\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e^2}\right )}{e}+\frac {d^2\,\left (\frac {37}{e^2}+\frac {2\,d\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e}-\frac {d^2\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e^2}\right )}{e^2}\right )}{e}-\frac {107}{e^2}+\frac {d^2\,\left (\frac {148}{e^2}+\frac {2\,d\,\left (\frac {37}{e^2}+\frac {2\,d\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e}-\frac {d^2\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e^2}\right )}{e^2}\right )+\frac {100\,x^7}{7\,e^2}-\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\,\left (x\,e^9+d\,e^8\right )}-\frac {\ln \left (d+e\,x\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^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*(65/(2*e^2) - (d*(148/e^2 + (2*d*(37/e^2 + (2*d*(111/e^2 - (100*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2))/e)
)/e - (d^2*((200*d)/e^3 + 45/e^2))/e^2))/e - (d^2*(111/e^2 - (100*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2))/e))/
e^2))/e + (d^2*(37/e^2 + (2*d*(111/e^2 - (100*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2))/e))/e - (d^2*((200*d)/e^
3 + 45/e^2))/e^2))/(2*e^2)) + x^3*(148/(3*e^2) + (2*d*(37/e^2 + (2*d*(111/e^2 - (100*d^2)/e^4 + (2*d*((200*d)/
e^3 + 45/e^2))/e))/e - (d^2*((200*d)/e^3 + 45/e^2))/e^2))/(3*e) - (d^2*(111/e^2 - (100*d^2)/e^4 + (2*d*((200*d
)/e^3 + 45/e^2))/e))/(3*e^2)) - x^4*(37/(4*e^2) + (d*(111/e^2 - (100*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2))/e
))/(2*e) - (d^2*((200*d)/e^3 + 45/e^2))/(4*e^2)) + x^5*(111/(5*e^2) - (20*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^
2))/(5*e)) - x^6*((100*d)/(3*e^3) + 15/(2*e^2)) - x*((2*d*(65/e^2 - (2*d*(148/e^2 + (2*d*(37/e^2 + (2*d*(111/e
^2 - (100*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2))/e))/e - (d^2*((200*d)/e^3 + 45/e^2))/e^2))/e - (d^2*(111/e^2
- (100*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2))/e))/e^2))/e + (d^2*(37/e^2 + (2*d*(111/e^2 - (100*d^2)/e^4 + (
2*d*((200*d)/e^3 + 45/e^2))/e))/e - (d^2*((200*d)/e^3 + 45/e^2))/e^2))/e^2))/e - 107/e^2 + (d^2*(148/e^2 + (2*
d*(37/e^2 + (2*d*(111/e^2 - (100*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2))/e))/e - (d^2*((200*d)/e^3 + 45/e^2))/
e^2))/e - (d^2*(111/e^2 - (100*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2))/e))/e^2))/e^2) + (100*x^7)/(7*e^2) - (4
5*d^7*e - 33*d*e^7 + 100*d^8 + 18*e^8 + 107*d^2*e^6 - 65*d^3*e^5 + 148*d^4*e^4 + 37*d^5*e^3 + 111*d^6*e^2)/(e*
(d*e^8 + e^9*x)) - (log(d + e*x)*(214*d*e^6 + 315*d^6*e + 800*d^7 - 33*e^7 - 195*d^2*e^5 + 592*d^3*e^4 + 185*d
^4*e^3 + 666*d^5*e^2))/e^9

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sympy [A]  time = 2.31, size = 393, normalized size = 1.11 \[ x^{6} \left (- \frac {100 d}{3 e^{3}} - \frac {15}{2 e^{2}}\right ) + x^{5} \left (\frac {60 d^{2}}{e^{4}} + \frac {18 d}{e^{3}} + \frac {111}{5 e^{2}}\right ) + x^{4} \left (- \frac {100 d^{3}}{e^{5}} - \frac {135 d^{2}}{4 e^{4}} - \frac {111 d}{2 e^{3}} - \frac {37}{4 e^{2}}\right ) + x^{3} \left (\frac {500 d^{4}}{3 e^{6}} + \frac {60 d^{3}}{e^{5}} + \frac {111 d^{2}}{e^{4}} + \frac {74 d}{3 e^{3}} + \frac {148}{3 e^{2}}\right ) + x^{2} \left (- \frac {300 d^{5}}{e^{7}} - \frac {225 d^{4}}{2 e^{6}} - \frac {222 d^{3}}{e^{5}} - \frac {111 d^{2}}{2 e^{4}} - \frac {148 d}{e^{3}} + \frac {65}{2 e^{2}}\right ) + x \left (\frac {700 d^{6}}{e^{8}} + \frac {270 d^{5}}{e^{7}} + \frac {555 d^{4}}{e^{6}} + \frac {148 d^{3}}{e^{5}} + \frac {444 d^{2}}{e^{4}} - \frac {130 d}{e^{3}} + \frac {107}{e^{2}}\right ) + \frac {- 100 d^{8} - 45 d^{7} e - 111 d^{6} e^{2} - 37 d^{5} e^{3} - 148 d^{4} e^{4} + 65 d^{3} e^{5} - 107 d^{2} e^{6} + 33 d e^{7} - 18 e^{8}}{d e^{9} + e^{10} x} + \frac {100 x^{7}}{7 e^{2}} - \frac {\left (5 d^{2} - 2 d e + 3 e^{2}\right ) \left (160 d^{5} + 127 d^{4} e + 88 d^{3} e^{2} - 4 d^{2} e^{3} + 64 d e^{4} - 11 e^{5}\right ) \log {\left (d + e x \right )}}{e^{9}} \]

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]

x**6*(-100*d/(3*e**3) - 15/(2*e**2)) + x**5*(60*d**2/e**4 + 18*d/e**3 + 111/(5*e**2)) + x**4*(-100*d**3/e**5 -
135*d**2/(4*e**4) - 111*d/(2*e**3) - 37/(4*e**2)) + x**3*(500*d**4/(3*e**6) + 60*d**3/e**5 + 111*d**2/e**4 +
74*d/(3*e**3) + 148/(3*e**2)) + x**2*(-300*d**5/e**7 - 225*d**4/(2*e**6) - 222*d**3/e**5 - 111*d**2/(2*e**4) -
148*d/e**3 + 65/(2*e**2)) + x*(700*d**6/e**8 + 270*d**5/e**7 + 555*d**4/e**6 + 148*d**3/e**5 + 444*d**2/e**4
- 130*d/e**3 + 107/e**2) + (-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) - (5*d**2 - 2*d*e + 3*e**2)*(1
60*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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