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 \[ \frac {x^6 \left (100 d^2+45 d e+111 e^2\right )}{6 e^3}-\frac {x^5 \left (100 d^3+45 d^2 e+111 d e^2+37 e^3\right )}{5 e^4}+\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^9}+\frac {x^4 \left (100 d^4+45 d^3 e+111 d^2 e^2+37 d e^3+148 e^4\right )}{4 e^5}-\frac {x^3 \left (100 d^5+45 d^4 e+111 d^3 e^2+37 d^2 e^3+148 d e^4-65 e^5\right )}{3 e^6}+\frac {x^2 \left (100 d^6+45 d^5 e+111 d^4 e^2+37 d^3 e^3+148 d^2 e^4-65 d e^5+107 e^6\right )}{2 e^7}-\frac {x \left (100 d^7+45 d^6 e+111 d^5 e^2+37 d^4 e^3+148 d^3 e^4-65 d^2 e^5+107 d e^6-33 e^7\right )}{e^8}-\frac {5 x^7 (20 d+9 e)}{7 e^2}+\frac {25 x^8}{2 e} \]

[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+1/2*(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)*x^2/e^7-1/3*(100*d^5+45*d^4*e+111*d^3*e^2+37*d^2*e^3+148
*d*e^4-65*e^5)*x^3/e^6+1/4*(100*d^4+45*d^3*e+111*d^2*e^2+37*d*e^3+148*e^4)*x^4/e^5-1/5*(100*d^3+45*d^2*e+111*d
*e^2+37*e^3)*x^5/e^4+1/6*(100*d^2+45*d*e+111*e^2)*x^6/e^3-5/7*(20*d+9*e)*x^7/e^2+25/2*x^8/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)*ln(e*x+d)/e^9

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Rubi [A]  time = 0.32, antiderivative size = 352, 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} \[ \frac {x^6 \left (100 d^2+45 d e+111 e^2\right )}{6 e^3}-\frac {x^5 \left (45 d^2 e+100 d^3+111 d e^2+37 e^3\right )}{5 e^4}+\frac {x^4 \left (111 d^2 e^2+45 d^3 e+100 d^4+37 d e^3+148 e^4\right )}{4 e^5}-\frac {x^3 \left (111 d^3 e^2+37 d^2 e^3+45 d^4 e+100 d^5+148 d e^4-65 e^5\right )}{3 e^6}+\frac {x^2 \left (111 d^4 e^2+37 d^3 e^3+148 d^2 e^4+45 d^5 e+100 d^6-65 d e^5+107 e^6\right )}{2 e^7}-\frac {x \left (111 d^5 e^2+37 d^4 e^3+148 d^3 e^4-65 d^2 e^5+45 d^6 e+100 d^7+107 d e^6-33 e^7\right )}{e^8}+\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)}{e^9}-\frac {5 x^7 (20 d+9 e)}{7 e^2}+\frac {25 x^8}{2 e} \]

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*}

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Mathematica [A]  time = 0.12, size = 262, normalized size = 0.74 \[ \frac {\left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \left (5 d^2-2 d e+3 e^2\right )^2 \log (d+e x)}{e^9}+\frac {x \left (-42000 d^7+2100 d^6 e (10 x-9)-70 d^5 e^2 \left (200 x^2-135 x+666\right )+210 d^4 e^3 \left (50 x^3-30 x^2+111 x-74\right )-105 d^3 e^4 \left (80 x^4-45 x^3+148 x^2-74 x+592\right )+35 d^2 e^5 \left (200 x^5-108 x^4+333 x^3-148 x^2+888 x+780\right )-d e^6 \left (6000 x^6-3150 x^5+9324 x^4-3885 x^3+20720 x^2+13650 x+44940\right )+2 e^7 \left (2625 x^7-1350 x^6+3885 x^5-1554 x^4+7770 x^3+4550 x^2+11235 x+6930\right )\right )}{420 e^8} \]

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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fricas [A]  time = 0.87, size = 368, normalized size = 1.05 \[ \frac {5250 \, e^{8} x^{8} - 300 \, {\left (20 \, d e^{7} + 9 \, e^{8}\right )} x^{7} + 70 \, {\left (100 \, d^{2} e^{6} + 45 \, d e^{7} + 111 \, e^{8}\right )} x^{6} - 84 \, {\left (100 \, d^{3} e^{5} + 45 \, d^{2} e^{6} + 111 \, d e^{7} + 37 \, e^{8}\right )} x^{5} + 105 \, {\left (100 \, d^{4} e^{4} + 45 \, d^{3} e^{5} + 111 \, d^{2} e^{6} + 37 \, d e^{7} + 148 \, e^{8}\right )} x^{4} - 140 \, {\left (100 \, d^{5} e^{3} + 45 \, d^{4} e^{4} + 111 \, d^{3} e^{5} + 37 \, d^{2} e^{6} + 148 \, d e^{7} - 65 \, e^{8}\right )} x^{3} + 210 \, {\left (100 \, d^{6} e^{2} + 45 \, d^{5} e^{3} + 111 \, d^{4} e^{4} + 37 \, d^{3} e^{5} + 148 \, d^{2} e^{6} - 65 \, d e^{7} + 107 \, e^{8}\right )} x^{2} - 420 \, {\left (100 \, d^{7} e + 45 \, d^{6} e^{2} + 111 \, d^{5} e^{3} + 37 \, d^{4} e^{4} + 148 \, d^{3} e^{5} - 65 \, d^{2} e^{6} + 107 \, d e^{7} - 33 \, e^{8}\right )} x + 420 \, {\left (100 \, d^{8} + 45 \, d^{7} e + 111 \, d^{6} e^{2} + 37 \, d^{5} e^{3} + 148 \, d^{4} e^{4} - 65 \, d^{3} e^{5} + 107 \, d^{2} e^{6} - 33 \, d e^{7} + 18 \, e^{8}\right )} \log \left (e x + d\right )}{420 \, e^{9}} \]

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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giac [A]  time = 0.17, size = 378, normalized size = 1.07 \[ {\left (100 \, d^{8} + 45 \, d^{7} e + 111 \, d^{6} e^{2} + 37 \, d^{5} e^{3} + 148 \, d^{4} e^{4} - 65 \, d^{3} e^{5} + 107 \, d^{2} e^{6} - 33 \, d e^{7} + 18 \, e^{8}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{420} \, {\left (5250 \, x^{8} e^{7} - 6000 \, d x^{7} e^{6} + 7000 \, d^{2} x^{6} e^{5} - 8400 \, d^{3} x^{5} e^{4} + 10500 \, d^{4} x^{4} e^{3} - 14000 \, d^{5} x^{3} e^{2} + 21000 \, d^{6} x^{2} e - 42000 \, d^{7} x - 2700 \, x^{7} e^{7} + 3150 \, d x^{6} e^{6} - 3780 \, d^{2} x^{5} e^{5} + 4725 \, d^{3} x^{4} e^{4} - 6300 \, d^{4} x^{3} e^{3} + 9450 \, d^{5} x^{2} e^{2} - 18900 \, d^{6} x e + 7770 \, x^{6} e^{7} - 9324 \, d x^{5} e^{6} + 11655 \, d^{2} x^{4} e^{5} - 15540 \, d^{3} x^{3} e^{4} + 23310 \, d^{4} x^{2} e^{3} - 46620 \, d^{5} x e^{2} - 3108 \, x^{5} e^{7} + 3885 \, d x^{4} e^{6} - 5180 \, d^{2} x^{3} e^{5} + 7770 \, d^{3} x^{2} e^{4} - 15540 \, d^{4} x e^{3} + 15540 \, x^{4} e^{7} - 20720 \, d x^{3} e^{6} + 31080 \, d^{2} x^{2} e^{5} - 62160 \, d^{3} x e^{4} + 9100 \, x^{3} e^{7} - 13650 \, d x^{2} e^{6} + 27300 \, d^{2} x e^{5} + 22470 \, x^{2} e^{7} - 44940 \, d x e^{6} + 13860 \, x e^{7}\right )} e^{\left (-8\right )} \]

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)

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maple [A]  time = 0.01, size = 465, normalized size = 1.32 \[ \frac {25 x^{8}}{2 e}-\frac {100 d \,x^{7}}{7 e^{2}}-\frac {45 x^{7}}{7 e}+\frac {50 d^{2} x^{6}}{3 e^{3}}+\frac {15 d \,x^{6}}{2 e^{2}}+\frac {37 x^{6}}{2 e}-\frac {20 d^{3} x^{5}}{e^{4}}-\frac {9 d^{2} x^{5}}{e^{3}}-\frac {111 d \,x^{5}}{5 e^{2}}-\frac {37 x^{5}}{5 e}+\frac {25 d^{4} x^{4}}{e^{5}}+\frac {45 d^{3} x^{4}}{4 e^{4}}+\frac {111 d^{2} x^{4}}{4 e^{3}}+\frac {37 d \,x^{4}}{4 e^{2}}+\frac {37 x^{4}}{e}-\frac {100 d^{5} x^{3}}{3 e^{6}}-\frac {15 d^{4} x^{3}}{e^{5}}-\frac {37 d^{3} x^{3}}{e^{4}}-\frac {37 d^{2} x^{3}}{3 e^{3}}-\frac {148 d \,x^{3}}{3 e^{2}}+\frac {65 x^{3}}{3 e}+\frac {50 d^{6} x^{2}}{e^{7}}+\frac {45 d^{5} x^{2}}{2 e^{6}}+\frac {111 d^{4} x^{2}}{2 e^{5}}+\frac {37 d^{3} x^{2}}{2 e^{4}}+\frac {74 d^{2} x^{2}}{e^{3}}-\frac {65 d \,x^{2}}{2 e^{2}}+\frac {107 x^{2}}{2 e}+\frac {100 d^{8} \ln \left (e x +d \right )}{e^{9}}-\frac {100 d^{7} x}{e^{8}}+\frac {45 d^{7} \ln \left (e x +d \right )}{e^{8}}-\frac {45 d^{6} x}{e^{7}}+\frac {111 d^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {111 d^{5} x}{e^{6}}+\frac {37 d^{5} \ln \left (e x +d \right )}{e^{6}}-\frac {37 d^{4} x}{e^{5}}+\frac {148 d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {148 d^{3} x}{e^{4}}-\frac {65 d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {65 d^{2} x}{e^{3}}+\frac {107 d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {107 d x}{e^{2}}-\frac {33 d \ln \left (e x +d \right )}{e^{2}}+\frac {33 x}{e}+\frac {18 \ln \left (e x +d \right )}{e} \]

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]

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

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maxima [A]  time = 0.44, size = 366, normalized size = 1.04 \[ \frac {5250 \, e^{7} x^{8} - 300 \, {\left (20 \, d e^{6} + 9 \, e^{7}\right )} x^{7} + 70 \, {\left (100 \, d^{2} e^{5} + 45 \, d e^{6} + 111 \, e^{7}\right )} x^{6} - 84 \, {\left (100 \, d^{3} e^{4} + 45 \, d^{2} e^{5} + 111 \, d e^{6} + 37 \, e^{7}\right )} x^{5} + 105 \, {\left (100 \, d^{4} e^{3} + 45 \, d^{3} e^{4} + 111 \, d^{2} e^{5} + 37 \, d e^{6} + 148 \, e^{7}\right )} x^{4} - 140 \, {\left (100 \, d^{5} e^{2} + 45 \, d^{4} e^{3} + 111 \, d^{3} e^{4} + 37 \, d^{2} e^{5} + 148 \, d e^{6} - 65 \, e^{7}\right )} x^{3} + 210 \, {\left (100 \, d^{6} e + 45 \, d^{5} e^{2} + 111 \, d^{4} e^{3} + 37 \, d^{3} e^{4} + 148 \, d^{2} e^{5} - 65 \, d e^{6} + 107 \, e^{7}\right )} x^{2} - 420 \, {\left (100 \, d^{7} + 45 \, d^{6} e + 111 \, d^{5} e^{2} + 37 \, d^{4} e^{3} + 148 \, d^{3} e^{4} - 65 \, d^{2} e^{5} + 107 \, d e^{6} - 33 \, e^{7}\right )} x}{420 \, e^{8}} + \frac {{\left (100 \, d^{8} + 45 \, d^{7} e + 111 \, d^{6} e^{2} + 37 \, d^{5} e^{3} + 148 \, d^{4} e^{4} - 65 \, d^{3} e^{5} + 107 \, d^{2} e^{6} - 33 \, d e^{7} + 18 \, e^{8}\right )} \log \left (e x + d\right )}{e^{9}} \]

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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mupad [B]  time = 0.08, size = 434, normalized size = 1.23 \[ x\,\left (\frac {33}{e}-\frac {d\,\left (\frac {107}{e}-\frac {d\,\left (\frac {65}{e}-\frac {d\,\left (\frac {148}{e}+\frac {d\,\left (\frac {37}{e}+\frac {d\,\left (\frac {111}{e}+\frac {d\,\left (\frac {100\,d}{e^2}+\frac {45}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )-x^7\,\left (\frac {100\,d}{7\,e^2}+\frac {45}{7\,e}\right )+x^6\,\left (\frac {37}{2\,e}+\frac {d\,\left (\frac {100\,d}{e^2}+\frac {45}{e}\right )}{6\,e}\right )-x^5\,\left (\frac {37}{5\,e}+\frac {d\,\left (\frac {111}{e}+\frac {d\,\left (\frac {100\,d}{e^2}+\frac {45}{e}\right )}{e}\right )}{5\,e}\right )+x^4\,\left (\frac {37}{e}+\frac {d\,\left (\frac {37}{e}+\frac {d\,\left (\frac {111}{e}+\frac {d\,\left (\frac {100\,d}{e^2}+\frac {45}{e}\right )}{e}\right )}{e}\right )}{4\,e}\right )+x^3\,\left (\frac {65}{3\,e}-\frac {d\,\left (\frac {148}{e}+\frac {d\,\left (\frac {37}{e}+\frac {d\,\left (\frac {111}{e}+\frac {d\,\left (\frac {100\,d}{e^2}+\frac {45}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{3\,e}\right )+x^2\,\left (\frac {107}{2\,e}-\frac {d\,\left (\frac {65}{e}-\frac {d\,\left (\frac {148}{e}+\frac {d\,\left (\frac {37}{e}+\frac {d\,\left (\frac {111}{e}+\frac {d\,\left (\frac {100\,d}{e^2}+\frac {45}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{2\,e}\right )+\frac {25\,x^8}{2\,e}+\frac {\ln \left (d+e\,x\right )\,\left (100\,d^8+45\,d^7\,e+111\,d^6\,e^2+37\,d^5\,e^3+148\,d^4\,e^4-65\,d^3\,e^5+107\,d^2\,e^6-33\,d\,e^7+18\,e^8\right )}{e^9} \]

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),x)

[Out]

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

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sympy [A]  time = 1.00, size = 372, normalized size = 1.06 \[ x^{7} \left (- \frac {100 d}{7 e^{2}} - \frac {45}{7 e}\right ) + x^{6} \left (\frac {50 d^{2}}{3 e^{3}} + \frac {15 d}{2 e^{2}} + \frac {37}{2 e}\right ) + x^{5} \left (- \frac {20 d^{3}}{e^{4}} - \frac {9 d^{2}}{e^{3}} - \frac {111 d}{5 e^{2}} - \frac {37}{5 e}\right ) + x^{4} \left (\frac {25 d^{4}}{e^{5}} + \frac {45 d^{3}}{4 e^{4}} + \frac {111 d^{2}}{4 e^{3}} + \frac {37 d}{4 e^{2}} + \frac {37}{e}\right ) + x^{3} \left (- \frac {100 d^{5}}{3 e^{6}} - \frac {15 d^{4}}{e^{5}} - \frac {37 d^{3}}{e^{4}} - \frac {37 d^{2}}{3 e^{3}} - \frac {148 d}{3 e^{2}} + \frac {65}{3 e}\right ) + x^{2} \left (\frac {50 d^{6}}{e^{7}} + \frac {45 d^{5}}{2 e^{6}} + \frac {111 d^{4}}{2 e^{5}} + \frac {37 d^{3}}{2 e^{4}} + \frac {74 d^{2}}{e^{3}} - \frac {65 d}{2 e^{2}} + \frac {107}{2 e}\right ) + x \left (- \frac {100 d^{7}}{e^{8}} - \frac {45 d^{6}}{e^{7}} - \frac {111 d^{5}}{e^{6}} - \frac {37 d^{4}}{e^{5}} - \frac {148 d^{3}}{e^{4}} + \frac {65 d^{2}}{e^{3}} - \frac {107 d}{e^{2}} + \frac {33}{e}\right ) + \frac {25 x^{8}}{2 e} + \frac {\left (5 d^{2} - 2 d e + 3 e^{2}\right )^{2} \left (4 d^{4} + 5 d^{3} e + 3 d^{2} e^{2} - d e^{3} + 2 e^{4}\right ) \log {\left (d + e x \right )}}{e^{9}} \]

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),x)

[Out]

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