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

Optimal. Leaf size=228 \[ \frac{x^3 \left (60 d^2+34 d e+17 e^2\right )}{3 e^4}-\frac{x^2 \left (51 d^2 e+80 d^3+34 d e^2+4 e^3\right )}{2 e^5}+\frac{x \left (51 d^2 e^2+68 d^3 e+100 d^4+8 d e^3+21 e^4\right )}{e^6}-\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{e^7 (d+e x)}-\frac{\left (68 d^3 e^2+12 d^2 e^3+85 d^4 e+120 d^5+42 d e^4-7 e^5\right ) \log (d+e x)}{e^7}-\frac{x^4 (40 d+17 e)}{4 e^3}+\frac{4 x^5}{e^2} \]

[Out]

((100*d^4 + 68*d^3*e + 51*d^2*e^2 + 8*d*e^3 + 21*e^4)*x)/e^6 - ((80*d^3 + 51*d^2*e + 34*d*e^2 + 4*e^3)*x^2)/(2
*e^5) + ((60*d^2 + 34*d*e + 17*e^2)*x^3)/(3*e^4) - ((40*d + 17*e)*x^4)/(4*e^3) + (4*x^5)/e^2 - ((5*d^2 - 2*d*e
+ 3*e^2)*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(e^7*(d + e*x)) - ((120*d^5 + 85*d^4*e + 68*d^3*e^2 +
12*d^2*e^3 + 42*d*e^4 - 7*e^5)*Log[d + e*x])/e^7

________________________________________________________________________________________

Rubi [A]  time = 0.191275, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.028, Rules used = {1628} \[ \frac{x^3 \left (60 d^2+34 d e+17 e^2\right )}{3 e^4}-\frac{x^2 \left (51 d^2 e+80 d^3+34 d e^2+4 e^3\right )}{2 e^5}+\frac{x \left (51 d^2 e^2+68 d^3 e+100 d^4+8 d e^3+21 e^4\right )}{e^6}-\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{e^7 (d+e x)}-\frac{\left (68 d^3 e^2+12 d^2 e^3+85 d^4 e+120 d^5+42 d e^4-7 e^5\right ) \log (d+e x)}{e^7}-\frac{x^4 (40 d+17 e)}{4 e^3}+\frac{4 x^5}{e^2} \]

Antiderivative was successfully verified.

[In]

Int[((3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(d + e*x)^2,x]

[Out]

((100*d^4 + 68*d^3*e + 51*d^2*e^2 + 8*d*e^3 + 21*e^4)*x)/e^6 - ((80*d^3 + 51*d^2*e + 34*d*e^2 + 4*e^3)*x^2)/(2
*e^5) + ((60*d^2 + 34*d*e + 17*e^2)*x^3)/(3*e^4) - ((40*d + 17*e)*x^4)/(4*e^3) + (4*x^5)/e^2 - ((5*d^2 - 2*d*e
+ 3*e^2)*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(e^7*(d + e*x)) - ((120*d^5 + 85*d^4*e + 68*d^3*e^2 +
12*d^2*e^3 + 42*d*e^4 - 7*e^5)*Log[d + e*x])/e^7

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 ) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^2} \, dx &=\int \left (\frac{100 d^4+68 d^3 e+51 d^2 e^2+8 d e^3+21 e^4}{e^6}-\frac{\left (80 d^3+51 d^2 e+34 d e^2+4 e^3\right ) x}{e^5}+\frac{\left (60 d^2+34 d e+17 e^2\right ) x^2}{e^4}-\frac{(40 d+17 e) x^3}{e^3}+\frac{20 x^4}{e^2}+\frac{20 d^6+17 d^5 e+17 d^4 e^2+4 d^3 e^3+21 d^2 e^4-7 d e^5+6 e^6}{e^6 (d+e x)^2}+\frac{-120 d^5-85 d^4 e-68 d^3 e^2-12 d^2 e^3-42 d e^4+7 e^5}{e^6 (d+e x)}\right ) \, dx\\ &=\frac{\left (100 d^4+68 d^3 e+51 d^2 e^2+8 d e^3+21 e^4\right ) x}{e^6}-\frac{\left (80 d^3+51 d^2 e+34 d e^2+4 e^3\right ) x^2}{2 e^5}+\frac{\left (60 d^2+34 d e+17 e^2\right ) x^3}{3 e^4}-\frac{(40 d+17 e) x^4}{4 e^3}+\frac{4 x^5}{e^2}-\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{e^7 (d+e x)}-\frac{\left (120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right ) \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.0886713, size = 223, normalized size = 0.98 \[ \frac{4 e^3 x^3 \left (60 d^2+34 d e+17 e^2\right )-6 e^2 x^2 \left (51 d^2 e+80 d^3+34 d e^2+4 e^3\right )+12 e x \left (51 d^2 e^2+68 d^3 e+100 d^4+8 d e^3+21 e^4\right )-\frac{12 \left (17 d^4 e^2+4 d^3 e^3+21 d^2 e^4+17 d^5 e+20 d^6-7 d e^5+6 e^6\right )}{d+e x}-12 \left (68 d^3 e^2+12 d^2 e^3+85 d^4 e+120 d^5+42 d e^4-7 e^5\right ) \log (d+e x)-3 e^4 x^4 (40 d+17 e)+48 e^5 x^5}{12 e^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(d + e*x)^2,x]

[Out]

(12*e*(100*d^4 + 68*d^3*e + 51*d^2*e^2 + 8*d*e^3 + 21*e^4)*x - 6*e^2*(80*d^3 + 51*d^2*e + 34*d*e^2 + 4*e^3)*x^
2 + 4*e^3*(60*d^2 + 34*d*e + 17*e^2)*x^3 - 3*e^4*(40*d + 17*e)*x^4 + 48*e^5*x^5 - (12*(20*d^6 + 17*d^5*e + 17*
d^4*e^2 + 4*d^3*e^3 + 21*d^2*e^4 - 7*d*e^5 + 6*e^6))/(d + e*x) - 12*(120*d^5 + 85*d^4*e + 68*d^3*e^2 + 12*d^2*
e^3 + 42*d*e^4 - 7*e^5)*Log[d + e*x])/(12*e^7)

________________________________________________________________________________________

Maple [A]  time = 0.057, size = 313, normalized size = 1.4 \begin{align*} -120\,{\frac{\ln \left ( ex+d \right ){d}^{5}}{{e}^{7}}}-{\frac{17\,{x}^{4}}{4\,{e}^{2}}}+{\frac{17\,{x}^{3}}{3\,{e}^{2}}}-2\,{\frac{{x}^{2}}{{e}^{2}}}+7\,{\frac{\ln \left ( ex+d \right ) }{{e}^{2}}}-6\,{\frac{1}{e \left ( ex+d \right ) }}-85\,{\frac{\ln \left ( ex+d \right ){d}^{4}}{{e}^{6}}}-68\,{\frac{\ln \left ( ex+d \right ){d}^{3}}{{e}^{5}}}-12\,{\frac{\ln \left ( ex+d \right ){d}^{2}}{{e}^{4}}}-42\,{\frac{\ln \left ( ex+d \right ) d}{{e}^{3}}}+100\,{\frac{{d}^{4}x}{{e}^{6}}}+68\,{\frac{{d}^{3}x}{{e}^{5}}}+51\,{\frac{{d}^{2}x}{{e}^{4}}}+8\,{\frac{dx}{{e}^{3}}}-10\,{\frac{d{x}^{4}}{{e}^{3}}}+20\,{\frac{{x}^{3}{d}^{2}}{{e}^{4}}}+{\frac{34\,d{x}^{3}}{3\,{e}^{3}}}-40\,{\frac{{x}^{2}{d}^{3}}{{e}^{5}}}-{\frac{51\,{x}^{2}{d}^{2}}{2\,{e}^{4}}}-17\,{\frac{d{x}^{2}}{{e}^{3}}}-20\,{\frac{{d}^{6}}{{e}^{7} \left ( ex+d \right ) }}-17\,{\frac{{d}^{5}}{{e}^{6} \left ( ex+d \right ) }}-17\,{\frac{{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}-4\,{\frac{{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-21\,{\frac{{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+7\,{\frac{d}{{e}^{2} \left ( ex+d \right ) }}+21\,{\frac{x}{{e}^{2}}}+4\,{\frac{{x}^{5}}{{e}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-120/e^7*ln(e*x+d)*d^5-17/4/e^2*x^4+17/3/e^2*x^3-2/e^2*x^2+7/e^2*ln(e*x+d)-6/e/(e*x+d)-85/e^6*ln(e*x+d)*d^4-68
/e^5*ln(e*x+d)*d^3-12/e^4*ln(e*x+d)*d^2-42/e^3*ln(e*x+d)*d+100/e^6*d^4*x+68/e^5*x*d^3+51/e^4*x*d^2+8/e^3*x*d-1
0/e^3*x^4*d+20/e^4*x^3*d^2+34/3/e^3*x^3*d-40/e^5*x^2*d^3-51/2/e^4*x^2*d^2-17/e^3*x^2*d-20/e^7/(e*x+d)*d^6-17/e
^6/(e*x+d)*d^5-17/e^5/(e*x+d)*d^4-4/e^4/(e*x+d)*d^3-21/e^3/(e*x+d)*d^2+7/e^2/(e*x+d)*d+21*x/e^2+4*x^5/e^2

________________________________________________________________________________________

Maxima [A]  time = 0.994265, size = 316, normalized size = 1.39 \begin{align*} -\frac{20 \, d^{6} + 17 \, d^{5} e + 17 \, d^{4} e^{2} + 4 \, d^{3} e^{3} + 21 \, d^{2} e^{4} - 7 \, d e^{5} + 6 \, e^{6}}{e^{8} x + d e^{7}} + \frac{48 \, e^{4} x^{5} - 3 \,{\left (40 \, d e^{3} + 17 \, e^{4}\right )} x^{4} + 4 \,{\left (60 \, d^{2} e^{2} + 34 \, d e^{3} + 17 \, e^{4}\right )} x^{3} - 6 \,{\left (80 \, d^{3} e + 51 \, d^{2} e^{2} + 34 \, d e^{3} + 4 \, e^{4}\right )} x^{2} + 12 \,{\left (100 \, d^{4} + 68 \, d^{3} e + 51 \, d^{2} e^{2} + 8 \, d e^{3} + 21 \, e^{4}\right )} x}{12 \, e^{6}} - \frac{{\left (120 \, d^{5} + 85 \, d^{4} e + 68 \, d^{3} e^{2} + 12 \, d^{2} e^{3} + 42 \, d e^{4} - 7 \, e^{5}\right )} \log \left (e x + d\right )}{e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d)^2,x, algorithm="maxima")

[Out]

-(20*d^6 + 17*d^5*e + 17*d^4*e^2 + 4*d^3*e^3 + 21*d^2*e^4 - 7*d*e^5 + 6*e^6)/(e^8*x + d*e^7) + 1/12*(48*e^4*x^
5 - 3*(40*d*e^3 + 17*e^4)*x^4 + 4*(60*d^2*e^2 + 34*d*e^3 + 17*e^4)*x^3 - 6*(80*d^3*e + 51*d^2*e^2 + 34*d*e^3 +
4*e^4)*x^2 + 12*(100*d^4 + 68*d^3*e + 51*d^2*e^2 + 8*d*e^3 + 21*e^4)*x)/e^6 - (120*d^5 + 85*d^4*e + 68*d^3*e^
2 + 12*d^2*e^3 + 42*d*e^4 - 7*e^5)*log(e*x + d)/e^7

________________________________________________________________________________________

Fricas [A]  time = 0.951872, size = 728, normalized size = 3.19 \begin{align*} \frac{48 \, e^{6} x^{6} - 240 \, d^{6} - 204 \, d^{5} e - 204 \, d^{4} e^{2} - 48 \, d^{3} e^{3} - 252 \, d^{2} e^{4} + 84 \, d e^{5} - 72 \, e^{6} - 3 \,{\left (24 \, d e^{5} + 17 \, e^{6}\right )} x^{5} +{\left (120 \, d^{2} e^{4} + 85 \, d e^{5} + 68 \, e^{6}\right )} x^{4} - 2 \,{\left (120 \, d^{3} e^{3} + 85 \, d^{2} e^{4} + 68 \, d e^{5} + 12 \, e^{6}\right )} x^{3} + 6 \,{\left (120 \, d^{4} e^{2} + 85 \, d^{3} e^{3} + 68 \, d^{2} e^{4} + 12 \, d e^{5} + 42 \, e^{6}\right )} x^{2} + 12 \,{\left (100 \, d^{5} e + 68 \, d^{4} e^{2} + 51 \, d^{3} e^{3} + 8 \, d^{2} e^{4} + 21 \, d e^{5}\right )} x - 12 \,{\left (120 \, d^{6} + 85 \, d^{5} e + 68 \, d^{4} e^{2} + 12 \, d^{3} e^{3} + 42 \, d^{2} e^{4} - 7 \, d e^{5} +{\left (120 \, d^{5} e + 85 \, d^{4} e^{2} + 68 \, d^{3} e^{3} + 12 \, d^{2} e^{4} + 42 \, d e^{5} - 7 \, e^{6}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{8} x + d e^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d)^2,x, algorithm="fricas")

[Out]

1/12*(48*e^6*x^6 - 240*d^6 - 204*d^5*e - 204*d^4*e^2 - 48*d^3*e^3 - 252*d^2*e^4 + 84*d*e^5 - 72*e^6 - 3*(24*d*
e^5 + 17*e^6)*x^5 + (120*d^2*e^4 + 85*d*e^5 + 68*e^6)*x^4 - 2*(120*d^3*e^3 + 85*d^2*e^4 + 68*d*e^5 + 12*e^6)*x
^3 + 6*(120*d^4*e^2 + 85*d^3*e^3 + 68*d^2*e^4 + 12*d*e^5 + 42*e^6)*x^2 + 12*(100*d^5*e + 68*d^4*e^2 + 51*d^3*e
^3 + 8*d^2*e^4 + 21*d*e^5)*x - 12*(120*d^6 + 85*d^5*e + 68*d^4*e^2 + 12*d^3*e^3 + 42*d^2*e^4 - 7*d*e^5 + (120*
d^5*e + 85*d^4*e^2 + 68*d^3*e^3 + 12*d^2*e^4 + 42*d*e^5 - 7*e^6)*x)*log(e*x + d))/(e^8*x + d*e^7)

________________________________________________________________________________________

Sympy [A]  time = 1.04, size = 226, normalized size = 0.99 \begin{align*} - \frac{20 d^{6} + 17 d^{5} e + 17 d^{4} e^{2} + 4 d^{3} e^{3} + 21 d^{2} e^{4} - 7 d e^{5} + 6 e^{6}}{d e^{7} + e^{8} x} + \frac{4 x^{5}}{e^{2}} - \frac{x^{4} \left (40 d + 17 e\right )}{4 e^{3}} + \frac{x^{3} \left (60 d^{2} + 34 d e + 17 e^{2}\right )}{3 e^{4}} - \frac{x^{2} \left (80 d^{3} + 51 d^{2} e + 34 d e^{2} + 4 e^{3}\right )}{2 e^{5}} + \frac{x \left (100 d^{4} + 68 d^{3} e + 51 d^{2} e^{2} + 8 d e^{3} + 21 e^{4}\right )}{e^{6}} - \frac{\left (120 d^{5} + 85 d^{4} e + 68 d^{3} e^{2} + 12 d^{2} e^{3} + 42 d e^{4} - 7 e^{5}\right ) \log{\left (d + e x \right )}}{e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(20*d**6 + 17*d**5*e + 17*d**4*e**2 + 4*d**3*e**3 + 21*d**2*e**4 - 7*d*e**5 + 6*e**6)/(d*e**7 + e**8*x) + 4*x
**5/e**2 - x**4*(40*d + 17*e)/(4*e**3) + x**3*(60*d**2 + 34*d*e + 17*e**2)/(3*e**4) - x**2*(80*d**3 + 51*d**2*
e + 34*d*e**2 + 4*e**3)/(2*e**5) + x*(100*d**4 + 68*d**3*e + 51*d**2*e**2 + 8*d*e**3 + 21*e**4)/e**6 - (120*d*
*5 + 85*d**4*e + 68*d**3*e**2 + 12*d**2*e**3 + 42*d*e**4 - 7*e**5)*log(d + e*x)/e**7

________________________________________________________________________________________

Giac [A]  time = 1.16966, size = 416, normalized size = 1.82 \begin{align*} -\frac{1}{12} \,{\left (x e + d\right )}^{5}{\left (\frac{3 \,{\left (120 \, d e + 17 \, e^{2}\right )} e^{\left (-1\right )}}{x e + d} - \frac{4 \,{\left (300 \, d^{2} e^{2} + 85 \, d e^{3} + 17 \, e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} + \frac{12 \,{\left (200 \, d^{3} e^{3} + 85 \, d^{2} e^{4} + 34 \, d e^{5} + 2 \, e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} - \frac{12 \,{\left (300 \, d^{4} e^{4} + 170 \, d^{3} e^{5} + 102 \, d^{2} e^{6} + 12 \, d e^{7} + 21 \, e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} - 48\right )} e^{\left (-7\right )} +{\left (120 \, d^{5} + 85 \, d^{4} e + 68 \, d^{3} e^{2} + 12 \, d^{2} e^{3} + 42 \, d e^{4} - 7 \, e^{5}\right )} e^{\left (-7\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{20 \, d^{6} e^{5}}{x e + d} + \frac{17 \, d^{5} e^{6}}{x e + d} + \frac{17 \, d^{4} e^{7}}{x e + d} + \frac{4 \, d^{3} e^{8}}{x e + d} + \frac{21 \, d^{2} e^{9}}{x e + d} - \frac{7 \, d e^{10}}{x e + d} + \frac{6 \, e^{11}}{x e + d}\right )} e^{\left (-12\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d)^2,x, algorithm="giac")

[Out]

-1/12*(x*e + d)^5*(3*(120*d*e + 17*e^2)*e^(-1)/(x*e + d) - 4*(300*d^2*e^2 + 85*d*e^3 + 17*e^4)*e^(-2)/(x*e + d
)^2 + 12*(200*d^3*e^3 + 85*d^2*e^4 + 34*d*e^5 + 2*e^6)*e^(-3)/(x*e + d)^3 - 12*(300*d^4*e^4 + 170*d^3*e^5 + 10
2*d^2*e^6 + 12*d*e^7 + 21*e^8)*e^(-4)/(x*e + d)^4 - 48)*e^(-7) + (120*d^5 + 85*d^4*e + 68*d^3*e^2 + 12*d^2*e^3
+ 42*d*e^4 - 7*e^5)*e^(-7)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - (20*d^6*e^5/(x*e + d) + 17*d^5*e^6/(x*e + d
) + 17*d^4*e^7/(x*e + d) + 4*d^3*e^8/(x*e + d) + 21*d^2*e^9/(x*e + d) - 7*d*e^10/(x*e + d) + 6*e^11/(x*e + d))
*e^(-12)