3.4.2 \(\int \frac {(3+2 x+5 x^2)^2 (2+x+3 x^2-5 x^3+4 x^4)}{(d+e x)^3} \, dx\) [302]

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

[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+1/2*(1500*d^4+450*d^3*e+666*d^2*e^2+111*
d*e^3+148*e^4)*x^2/e^7-1/3*(1000*d^3+270*d^2*e+333*d*e^2+37*e^3)*x^3/e^6+3/4*(200*d^2+45*d*e+37*e^2)*x^4/e^5-3
*(20*d+3*e)*x^5/e^4+50/3*x^6/e^3-1/2*(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)^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/(e*x+d)+(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)*ln(e*x+d)/e^9

________________________________________________________________________________________

Rubi [A]
time = 0.22, antiderivative size = 354, 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 = {1642} \begin {gather*} \frac {3 x^4 \left (200 d^2+45 d e+37 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 {\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 {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 {\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 {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 (d+e x)}{e^9}-\frac {3 x^5 (20 d+3 e)}{e^4}+\frac {50 x^6}{3 e^3} \end {gather*}

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 1642

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 = 0.07, size = 311, normalized size = 0.88 \begin {gather*} \frac {9000 d^8-390 d^7 e (-9+40 x)-18 d^6 e^2 \left (-407+240 x+2300 x^2\right )-2 d^5 e^3 \left (-999+2664 x+6750 x^2+5600 x^3\right )+4 d^4 e^4 \left (1554-111 x-5661 x^2-945 x^3+700 x^4\right )-d^3 e^5 \left (1950-1776 x+4662 x^2+6660 x^3-945 x^4+1120 x^5\right )+d^2 e^6 \left (1926-1560 x-9768 x^2-1480 x^3+1665 x^4-378 x^5+560 x^6\right )+d e^7 \left (-198+2568 x+1560 x^2-3552 x^3+370 x^4-666 x^5+189 x^6-320 x^7\right )+e^8 \left (-108-396 x+780 x^3+888 x^4-148 x^5+333 x^6-108 x^7+200 x^8\right )+12 \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 ) (d+e x)^2 \log (d+e x)}{12 e^9 (d+e x)^2} \end {gather*}

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)

________________________________________________________________________________________

Maple [A]
time = 0.11, size = 391, normalized size = 1.10

method	result	size
norman	\(\frac {\frac {\left (5600 d^{7}+1890 d^{6} e +3330 d^{5} e^{2}+740 d^{4} e^{3}+1776 d^{3} e^{4}-390 d^{2} e^{5}+214 d \,e^{6}-33 e^{7}\right ) x}{e^{8}}+\frac {50 x^{8}}{3 e}+\frac {8400 d^{8}+2835 d^{7} e +4995 d^{6} e^{2}+1110 d^{5} e^{3}+2664 d^{4} e^{4}-585 d^{3} e^{5}+321 d^{2} e^{6}-33 d \,e^{7}-18 e^{8}}{2 e^{9}}-\frac {\left (80 d +27 e \right ) x^{7}}{3 e^{2}}+\frac {\left (560 d^{2}+189 d e +333 e^{2}\right ) x^{6}}{12 e^{3}}-\frac {\left (560 d^{3}+189 d^{2} e +333 d \,e^{2}+74 e^{3}\right ) x^{5}}{6 e^{4}}+\frac {\left (2800 d^{4}+945 d^{3} e +1665 d^{2} e^{2}+370 d \,e^{3}+888 e^{4}\right ) x^{4}}{12 e^{5}}-\frac {\left (2800 d^{5}+945 d^{4} e +1665 d^{3} e^{2}+370 d^{2} e^{3}+888 d \,e^{4}-195 e^{5}\right ) x^{3}}{3 e^{6}}}{\left (e x +d \right )^{2}}+\frac {\left (2800 d^{6}+945 d^{5} e +1665 d^{4} e^{2}+370 d^{3} e^{3}+888 d^{2} e^{4}-195 d \,e^{5}+107 e^{6}\right ) \ln \left (e x +d \right )}{e^{9}}\)	\(359\)
default	\(\frac {\left (2800 d^{6}+945 d^{5} e +1665 d^{4} e^{2}+370 d^{3} e^{3}+888 d^{2} e^{4}-195 d \,e^{5}+107 e^{6}\right ) \ln \left (e x +d \right )}{e^{9}}-\frac {100 d^{8}+45 d^{7} e +111 d^{6} e^{2}+37 d^{5} e^{3}+148 d^{4} e^{4}-65 d^{3} e^{5}+107 d^{2} e^{6}-33 d \,e^{7}+18 e^{8}}{2 e^{9} \left (e x +d \right )^{2}}-\frac {-800 d^{7}-315 d^{6} e -666 d^{5} e^{2}-185 d^{4} e^{3}-592 d^{3} e^{4}+195 d^{2} e^{5}-214 d \,e^{6}+33 e^{7}}{e^{9} \left (e x +d \right )}-\frac {-\frac {50}{3} x^{6} e^{5}+60 x^{5} e^{4} d +9 x^{5} e^{5}-150 d^{2} e^{3} x^{4}-\frac {135}{4} x^{4} e^{4} d -\frac {111}{4} x^{4} e^{5}+\frac {1000}{3} d^{3} e^{2} x^{3}+90 d^{2} e^{3} x^{3}+111 d \,e^{4} x^{3}+\frac {37}{3} e^{5} x^{3}-750 d^{4} e \,x^{2}-225 d^{3} e^{2} x^{2}-333 d^{2} e^{3} x^{2}-\frac {111}{2} d \,e^{4} x^{2}-74 e^{5} x^{2}+2100 d^{5} x +675 d^{4} e x +1110 d^{3} e^{2} x +222 d^{2} e^{3} x +444 d \,e^{4} x -65 e^{5} x}{e^{8}}\)	\(391\)
risch	\(\frac {107 \ln \left (e x +d \right )}{e^{3}}+\frac {74 x^{2}}{e^{3}}+\frac {65 x}{e^{3}}-\frac {9 x^{5}}{e^{3}}-\frac {37 x^{3}}{3 e^{3}}+\frac {50 x^{6}}{3 e^{3}}+\frac {\left (800 d^{7}+315 d^{6} e +666 d^{5} e^{2}+185 d^{4} e^{3}+592 d^{3} e^{4}-195 d^{2} e^{5}+214 d \,e^{6}-33 e^{7}\right ) x +\frac {1500 d^{8}+585 d^{7} e +1221 d^{6} e^{2}+333 d^{5} e^{3}+1036 d^{4} e^{4}-325 d^{3} e^{5}+321 d^{2} e^{6}-33 d \,e^{7}-18 e^{8}}{2 e}}{e^{8} \left (e x +d \right )^{2}}+\frac {1665 \ln \left (e x +d \right ) d^{4}}{e^{7}}+\frac {370 \ln \left (e x +d \right ) d^{3}}{e^{6}}+\frac {888 \ln \left (e x +d \right ) d^{2}}{e^{5}}-\frac {195 \ln \left (e x +d \right ) d}{e^{4}}+\frac {111 d \,x^{2}}{2 e^{4}}-\frac {1110 d^{3} x}{e^{6}}-\frac {222 d^{2} x}{e^{5}}-\frac {444 d x}{e^{4}}-\frac {111 d \,x^{3}}{e^{4}}+\frac {333 d^{2} x^{2}}{e^{5}}-\frac {90 d^{2} x^{3}}{e^{5}}+\frac {750 d^{4} x^{2}}{e^{7}}+\frac {225 d^{3} x^{2}}{e^{6}}-\frac {2100 d^{5} x}{e^{8}}-\frac {675 d^{4} x}{e^{7}}+\frac {2800 \ln \left (e x +d \right ) d^{6}}{e^{9}}+\frac {945 \ln \left (e x +d \right ) d^{5}}{e^{8}}-\frac {60 x^{5} d}{e^{4}}+\frac {150 d^{2} x^{4}}{e^{5}}+\frac {135 x^{4} d}{4 e^{4}}-\frac {1000 d^{3} x^{3}}{3 e^{6}}+\frac {111 x^{4}}{4 e^{3}}\)	\(423\)

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,method=_RETURNVERBOSE)

[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)*ln(e*x+d)/e^9-1/2*(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)/e^9/(e*x+d)^2-(-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/(e*x+d)-1/e^8*(-50/3*x^6*e^5+60*x^5*e^4
*d+9*x^5*e^5-150*d^2*e^3*x^4-135/4*x^4*e^4*d-111/4*x^4*e^5+1000/3*d^3*e^2*x^3+90*d^2*e^3*x^3+111*d*e^4*x^3+37/
3*e^5*x^3-750*d^4*e*x^2-225*d^3*e^2*x^2-333*d^2*e^3*x^2-111/2*d*e^4*x^2-74*e^5*x^2+2100*d^5*x+675*d^4*e*x+1110
*d^3*e^2*x+222*d^2*e^3*x+444*d*e^4*x-65*e^5*x)

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 342, normalized size = 0.97 \begin {gather*} {\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 (x e + d\right ) + \frac {1}{12} \, {\left (200 \, x^{6} e^{5} - 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\right )} e^{\left (-8\right )} + \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} + 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}}{2 \, {\left (x^{2} e^{11} + 2 \, d x e^{10} + d^{2} e^{9}\right )}} \end {gather*}

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]

(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(x*e + d) +
1/12*(200*x^6*e^5 - 36*(20*d*e^4 + 3*e^5)*x^5 + 9*(200*d^2*e^3 + 45*d*e^4 + 37*e^5)*x^4 - 4*(1000*d^3*e^2 + 27
0*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 - 1
2*(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) + 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)/(x^2*e^11 + 2*d
*x*e^10 + d^2*e^9)

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 501, normalized size = 1.42 \begin {gather*} \frac {9000 \, d^{8} + {\left (200 \, x^{8} - 108 \, x^{7} + 333 \, x^{6} - 148 \, x^{5} + 888 \, x^{4} + 780 \, x^{3} - 396 \, x - 108\right )} e^{8} - {\left (320 \, d x^{7} - 189 \, d x^{6} + 666 \, d x^{5} - 370 \, d x^{4} + 3552 \, d x^{3} - 1560 \, d x^{2} - 2568 \, d x + 198 \, d\right )} e^{7} + {\left (560 \, d^{2} x^{6} - 378 \, d^{2} x^{5} + 1665 \, d^{2} x^{4} - 1480 \, d^{2} x^{3} - 9768 \, d^{2} x^{2} - 1560 \, d^{2} x + 1926 \, d^{2}\right )} e^{6} - {\left (1120 \, d^{3} x^{5} - 945 \, d^{3} x^{4} + 6660 \, d^{3} x^{3} + 4662 \, d^{3} x^{2} - 1776 \, d^{3} x + 1950 \, d^{3}\right )} e^{5} + 4 \, {\left (700 \, d^{4} x^{4} - 945 \, d^{4} x^{3} - 5661 \, d^{4} x^{2} - 111 \, d^{4} x + 1554 \, d^{4}\right )} e^{4} - 2 \, {\left (5600 \, d^{5} x^{3} + 6750 \, d^{5} x^{2} + 2664 \, d^{5} x - 999 \, d^{5}\right )} e^{3} - 18 \, {\left (2300 \, d^{6} x^{2} + 240 \, d^{6} x - 407 \, d^{6}\right )} e^{2} - 390 \, {\left (40 \, d^{7} x - 9 \, d^{7}\right )} e + 12 \, {\left (2800 \, d^{8} + 107 \, x^{2} e^{8} - {\left (195 \, d x^{2} - 214 \, d x\right )} e^{7} + {\left (888 \, d^{2} x^{2} - 390 \, d^{2} x + 107 \, d^{2}\right )} e^{6} + {\left (370 \, d^{3} x^{2} + 1776 \, d^{3} x - 195 \, d^{3}\right )} e^{5} + 37 \, {\left (45 \, d^{4} x^{2} + 20 \, d^{4} x + 24 \, d^{4}\right )} e^{4} + 5 \, {\left (189 \, d^{5} x^{2} + 666 \, d^{5} x + 74 \, d^{5}\right )} e^{3} + 5 \, {\left (560 \, d^{6} x^{2} + 378 \, d^{6} x + 333 \, d^{6}\right )} e^{2} + 35 \, {\left (160 \, d^{7} x + 27 \, d^{7}\right )} e\right )} \log \left (x e + d\right )}{12 \, {\left (x^{2} e^{11} + 2 \, d x e^{10} + d^{2} e^{9}\right )}} \end {gather*}

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*(9000*d^8 + (200*x^8 - 108*x^7 + 333*x^6 - 148*x^5 + 888*x^4 + 780*x^3 - 396*x - 108)*e^8 - (320*d*x^7 -
189*d*x^6 + 666*d*x^5 - 370*d*x^4 + 3552*d*x^3 - 1560*d*x^2 - 2568*d*x + 198*d)*e^7 + (560*d^2*x^6 - 378*d^2*x
^5 + 1665*d^2*x^4 - 1480*d^2*x^3 - 9768*d^2*x^2 - 1560*d^2*x + 1926*d^2)*e^6 - (1120*d^3*x^5 - 945*d^3*x^4 + 6
660*d^3*x^3 + 4662*d^3*x^2 - 1776*d^3*x + 1950*d^3)*e^5 + 4*(700*d^4*x^4 - 945*d^4*x^3 - 5661*d^4*x^2 - 111*d^
4*x + 1554*d^4)*e^4 - 2*(5600*d^5*x^3 + 6750*d^5*x^2 + 2664*d^5*x - 999*d^5)*e^3 - 18*(2300*d^6*x^2 + 240*d^6*
x - 407*d^6)*e^2 - 390*(40*d^7*x - 9*d^7)*e + 12*(2800*d^8 + 107*x^2*e^8 - (195*d*x^2 - 214*d*x)*e^7 + (888*d^
2*x^2 - 390*d^2*x + 107*d^2)*e^6 + (370*d^3*x^2 + 1776*d^3*x - 195*d^3)*e^5 + 37*(45*d^4*x^2 + 20*d^4*x + 24*d
^4)*e^4 + 5*(189*d^5*x^2 + 666*d^5*x + 74*d^5)*e^3 + 5*(560*d^6*x^2 + 378*d^6*x + 333*d^6)*e^2 + 35*(160*d^7*x
+ 27*d^7)*e)*log(x*e + d))/(x^2*e^11 + 2*d*x*e^10 + d^2*e^9)

________________________________________________________________________________________

Sympy [A]
time = 1.50, size = 394, normalized size = 1.11 \begin {gather*} x^{5} \left (- \frac {60 d}{e^{4}} - \frac {9}{e^{3}}\right ) + x^{4} \cdot \left (\frac {150 d^{2}}{e^{5}} + \frac {135 d}{4 e^{4}} + \frac {111}{4 e^{3}}\right ) + x^{3} \left (- \frac {1000 d^{3}}{3 e^{6}} - \frac {90 d^{2}}{e^{5}} - \frac {111 d}{e^{4}} - \frac {37}{3 e^{3}}\right ) + x^{2} \cdot \left (\frac {750 d^{4}}{e^{7}} + \frac {225 d^{3}}{e^{6}} + \frac {333 d^{2}}{e^{5}} + \frac {111 d}{2 e^{4}} + \frac {74}{e^{3}}\right ) + x \left (- \frac {2100 d^{5}}{e^{8}} - \frac {675 d^{4}}{e^{7}} - \frac {1110 d^{3}}{e^{6}} - \frac {222 d^{2}}{e^{5}} - \frac {444 d}{e^{4}} + \frac {65}{e^{3}}\right ) + \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 {\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 {gather*}

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]

x**5*(-60*d/e**4 - 9/e**3) + x**4*(150*d**2/e**5 + 135*d/(4*e**4) + 111/(4*e**3)) + x**3*(-1000*d**3/(3*e**6)
- 90*d**2/e**5 - 111*d/e**4 - 37/(3*e**3)) + x**2*(750*d**4/e**7 + 225*d**3/e**6 + 333*d**2/e**5 + 111*d/(2*e*
*4) + 74/e**3) + x*(-2100*d**5/e**8 - 675*d**4/e**7 - 1110*d**3/e**6 - 222*d**2/e**5 - 444*d/e**4 + 65/e**3) +
(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 - 3
3*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) + (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

________________________________________________________________________________________

Giac [A]
time = 4.82, size = 354, normalized size = 1.00 \begin {gather*} {\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 {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 0.13, size = 771, normalized size = 2.18 \begin {gather*} x^4\,\left (\frac {111}{4\,e^3}-\frac {75\,d^2}{e^5}+\frac {3\,d\,\left (\frac {300\,d}{e^4}+\frac {45}{e^3}\right )}{4\,e}\right )-x^3\,\left (\frac {37}{3\,e^3}+\frac {100\,d^3}{3\,e^6}+\frac {d\,\left (\frac {111}{e^3}-\frac {300\,d^2}{e^5}+\frac {3\,d\,\left (\frac {300\,d}{e^4}+\frac {45}{e^3}\right )}{e}\right )}{e}-\frac {d^2\,\left (\frac {300\,d}{e^4}+\frac {45}{e^3}\right )}{e^2}\right )-x^5\,\left (\frac {60\,d}{e^4}+\frac {9}{e^3}\right )+x\,\left (\frac {65}{e^3}-\frac {3\,d\,\left (\frac {148}{e^3}+\frac {3\,d\,\left (\frac {37}{e^3}+\frac {100\,d^3}{e^6}+\frac {3\,d\,\left (\frac {111}{e^3}-\frac {300\,d^2}{e^5}+\frac {3\,d\,\left (\frac {300\,d}{e^4}+\frac {45}{e^3}\right )}{e}\right )}{e}-\frac {3\,d^2\,\left (\frac {300\,d}{e^4}+\frac {45}{e^3}\right )}{e^2}\right )}{e}-\frac {3\,d^2\,\left (\frac {111}{e^3}-\frac {300\,d^2}{e^5}+\frac {3\,d\,\left (\frac {300\,d}{e^4}+\frac {45}{e^3}\right )}{e}\right )}{e^2}+\frac {d^3\,\left (\frac {300\,d}{e^4}+\frac {45}{e^3}\right )}{e^3}\right )}{e}+\frac {3\,d^2\,\left (\frac {37}{e^3}+\frac {100\,d^3}{e^6}+\frac {3\,d\,\left (\frac {111}{e^3}-\frac {300\,d^2}{e^5}+\frac {3\,d\,\left (\frac {300\,d}{e^4}+\frac {45}{e^3}\right )}{e}\right )}{e}-\frac {3\,d^2\,\left (\frac {300\,d}{e^4}+\frac {45}{e^3}\right )}{e^2}\right )}{e^2}-\frac {d^3\,\left (\frac {111}{e^3}-\frac {300\,d^2}{e^5}+\frac {3\,d\,\left (\frac {300\,d}{e^4}+\frac {45}{e^3}\right )}{e}\right )}{e^3}\right )+\frac {x\,\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 )+\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\,e}}{d^2\,e^8+2\,d\,e^9\,x+e^{10}\,x^2}+\frac {50\,x^6}{3\,e^3}+x^2\,\left (\frac {74}{e^3}+\frac {3\,d\,\left (\frac {37}{e^3}+\frac {100\,d^3}{e^6}+\frac {3\,d\,\left (\frac {111}{e^3}-\frac {300\,d^2}{e^5}+\frac {3\,d\,\left (\frac {300\,d}{e^4}+\frac {45}{e^3}\right )}{e}\right )}{e}-\frac {3\,d^2\,\left (\frac {300\,d}{e^4}+\frac {45}{e^3}\right )}{e^2}\right )}{2\,e}-\frac {3\,d^2\,\left (\frac {111}{e^3}-\frac {300\,d^2}{e^5}+\frac {3\,d\,\left (\frac {300\,d}{e^4}+\frac {45}{e^3}\right )}{e}\right )}{2\,e^2}+\frac {d^3\,\left (\frac {300\,d}{e^4}+\frac {45}{e^3}\right )}{2\,e^3}\right )+\frac {\ln \left (d+e\,x\right )\,\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^9} \end {gather*}

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)^3,x)

[Out]

x^4*(111/(4*e^3) - (75*d^2)/e^5 + (3*d*((300*d)/e^4 + 45/e^3))/(4*e)) - x^3*(37/(3*e^3) + (100*d^3)/(3*e^6) +
(d*(111/e^3 - (300*d^2)/e^5 + (3*d*((300*d)/e^4 + 45/e^3))/e))/e - (d^2*((300*d)/e^4 + 45/e^3))/e^2) - x^5*((6
0*d)/e^4 + 9/e^3) + x*(65/e^3 - (3*d*(148/e^3 + (3*d*(37/e^3 + (100*d^3)/e^6 + (3*d*(111/e^3 - (300*d^2)/e^5 +
(3*d*((300*d)/e^4 + 45/e^3))/e))/e - (3*d^2*((300*d)/e^4 + 45/e^3))/e^2))/e - (3*d^2*(111/e^3 - (300*d^2)/e^5
+ (3*d*((300*d)/e^4 + 45/e^3))/e))/e^2 + (d^3*((300*d)/e^4 + 45/e^3))/e^3))/e + (3*d^2*(37/e^3 + (100*d^3)/e^
6 + (3*d*(111/e^3 - (300*d^2)/e^5 + (3*d*((300*d)/e^4 + 45/e^3))/e))/e - (3*d^2*((300*d)/e^4 + 45/e^3))/e^2))/
e^2 - (d^3*(111/e^3 - (300*d^2)/e^5 + (3*d*((300*d)/e^4 + 45/e^3))/e))/e^3) + (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) + (585*d^7*e - 33*d*e^7 + 1500*d^8 - 18*
e^8 + 321*d^2*e^6 - 325*d^3*e^5 + 1036*d^4*e^4 + 333*d^5*e^3 + 1221*d^6*e^2)/(2*e))/(d^2*e^8 + e^10*x^2 + 2*d*
e^9*x) + (50*x^6)/(3*e^3) + x^2*(74/e^3 + (3*d*(37/e^3 + (100*d^3)/e^6 + (3*d*(111/e^3 - (300*d^2)/e^5 + (3*d*
((300*d)/e^4 + 45/e^3))/e))/e - (3*d^2*((300*d)/e^4 + 45/e^3))/e^2))/(2*e) - (3*d^2*(111/e^3 - (300*d^2)/e^5 +
(3*d*((300*d)/e^4 + 45/e^3))/e))/(2*e^2) + (d^3*((300*d)/e^4 + 45/e^3))/(2*e^3)) + (log(d + e*x)*(945*d^5*e -
195*d*e^5 + 2800*d^6 + 107*e^6 + 888*d^2*e^4 + 370*d^3*e^3 + 1665*d^4*e^2))/e^9

________________________________________________________________________________________