### 3.295 $$\int \frac{(3+2 x+5 x^2) (2+x+3 x^2-5 x^3+4 x^4)}{(d+e x)^3} \, dx$$

Optimal. Leaf size=231 $\frac{x^2 \left (120 d^2+51 d e+17 e^2\right )}{2 e^5}-\frac{x \left (102 d^2 e+200 d^3+51 d e^2+4 e^3\right )}{e^6}+\frac{68 d^3 e^2+12 d^2 e^3+85 d^4 e+120 d^5+42 d e^4-7 e^5}{e^7 (d+e x)}-\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{2 e^7 (d+e x)^2}+\frac{\left (102 d^2 e^2+170 d^3 e+300 d^4+12 d e^3+21 e^4\right ) \log (d+e x)}{e^7}-\frac{x^3 (60 d+17 e)}{3 e^4}+\frac{5 x^4}{e^3}$

[Out]

-(((200*d^3 + 102*d^2*e + 51*d*e^2 + 4*e^3)*x)/e^6) + ((120*d^2 + 51*d*e + 17*e^2)*x^2)/(2*e^5) - ((60*d + 17*
e)*x^3)/(3*e^4) + (5*x^4)/e^3 - ((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))/(2*e^7
*(d + e*x)^2) + (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*(d + e*x)) + ((300*d^4
+ 170*d^3*e + 102*d^2*e^2 + 12*d*e^3 + 21*e^4)*Log[d + e*x])/e^7

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Rubi [A]  time = 0.203744, antiderivative size = 231, 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} $\frac{x^2 \left (120 d^2+51 d e+17 e^2\right )}{2 e^5}-\frac{x \left (102 d^2 e+200 d^3+51 d e^2+4 e^3\right )}{e^6}+\frac{68 d^3 e^2+12 d^2 e^3+85 d^4 e+120 d^5+42 d e^4-7 e^5}{e^7 (d+e x)}-\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{2 e^7 (d+e x)^2}+\frac{\left (102 d^2 e^2+170 d^3 e+300 d^4+12 d e^3+21 e^4\right ) \log (d+e x)}{e^7}-\frac{x^3 (60 d+17 e)}{3 e^4}+\frac{5 x^4}{e^3}$

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)^3,x]

[Out]

-(((200*d^3 + 102*d^2*e + 51*d*e^2 + 4*e^3)*x)/e^6) + ((120*d^2 + 51*d*e + 17*e^2)*x^2)/(2*e^5) - ((60*d + 17*
e)*x^3)/(3*e^4) + (5*x^4)/e^3 - ((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))/(2*e^7
*(d + e*x)^2) + (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*(d + e*x)) + ((300*d^4
+ 170*d^3*e + 102*d^2*e^2 + 12*d*e^3 + 21*e^4)*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)^3} \, dx &=\int \left (\frac{-200 d^3-102 d^2 e-51 d e^2-4 e^3}{e^6}+\frac{\left (120 d^2+51 d e+17 e^2\right ) x}{e^5}-\frac{(60 d+17 e) x^2}{e^4}+\frac{20 x^3}{e^3}+\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)^3}+\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)^2}+\frac{300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac{\left (200 d^3+102 d^2 e+51 d e^2+4 e^3\right ) x}{e^6}+\frac{\left (120 d^2+51 d e+17 e^2\right ) x^2}{2 e^5}-\frac{(60 d+17 e) x^3}{3 e^4}+\frac{5 x^4}{e^3}-\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 )}{2 e^7 (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^7 (d+e x)}+\frac{\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.0682934, size = 204, normalized size = 0.88 $\frac{-51 d^4 e^2 \left (40 x^2+2 x-7\right )-3 d^3 e^3 \left (200 x^3+357 x^2-34 x-20\right )+d^2 e^4 \left (150 x^4-340 x^3-561 x^2+48 x+189\right )+6 \left (102 d^2 e^2+170 d^3 e+300 d^4+12 d e^3+21 e^4\right ) (d+e x)^2 \log (d+e x)+d^5 e (459-480 x)+660 d^6-d e^5 \left (60 x^5-85 x^4+204 x^3+48 x^2-252 x+21\right )+e^6 \left (30 x^6-34 x^5+51 x^4-24 x^3-42 x-18\right )}{6 e^7 (d+e x)^2}$

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)^3,x]

[Out]

(660*d^6 + d^5*e*(459 - 480*x) - 51*d^4*e^2*(-7 + 2*x + 40*x^2) - 3*d^3*e^3*(-20 - 34*x + 357*x^2 + 200*x^3) +
d^2*e^4*(189 + 48*x - 561*x^2 - 340*x^3 + 150*x^4) - d*e^5*(21 - 252*x + 48*x^2 + 204*x^3 - 85*x^4 + 60*x^5)
+ e^6*(-18 - 42*x - 24*x^3 + 51*x^4 - 34*x^5 + 30*x^6) + 6*(300*d^4 + 170*d^3*e + 102*d^2*e^2 + 12*d*e^3 + 21*
e^4)*(d + e*x)^2*Log[d + e*x])/(6*e^7*(d + e*x)^2)

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Maple [A]  time = 0.051, size = 336, normalized size = 1.5 \begin{align*} 21\,{\frac{\ln \left ( ex+d \right ) }{{e}^{3}}}-7\,{\frac{1}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{1}{e \left ( ex+d \right ) ^{2}}}+{\frac{17\,{x}^{2}}{2\,{e}^{3}}}-4\,{\frac{x}{{e}^{3}}}-{\frac{17\,{x}^{3}}{3\,{e}^{3}}}+120\,{\frac{{d}^{5}}{{e}^{7} \left ( ex+d \right ) }}+85\,{\frac{{d}^{4}}{{e}^{6} \left ( ex+d \right ) }}+300\,{\frac{\ln \left ( ex+d \right ){d}^{4}}{{e}^{7}}}+170\,{\frac{\ln \left ( ex+d \right ){d}^{3}}{{e}^{6}}}+102\,{\frac{\ln \left ( ex+d \right ){d}^{2}}{{e}^{5}}}+60\,{\frac{{x}^{2}{d}^{2}}{{e}^{5}}}+{\frac{51\,d{x}^{2}}{2\,{e}^{4}}}+68\,{\frac{{d}^{3}}{{e}^{5} \left ( ex+d \right ) }}+12\,{\frac{{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}+42\,{\frac{d}{{e}^{3} \left ( ex+d \right ) }}-10\,{\frac{{d}^{6}}{{e}^{7} \left ( ex+d \right ) ^{2}}}-{\frac{17\,{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{17\,{d}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-2\,{\frac{{d}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{21\,{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{7\,d}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+12\,{\frac{\ln \left ( ex+d \right ) d}{{e}^{4}}}-200\,{\frac{{d}^{3}x}{{e}^{6}}}-102\,{\frac{{d}^{2}x}{{e}^{5}}}-51\,{\frac{dx}{{e}^{4}}}-20\,{\frac{d{x}^{3}}{{e}^{4}}}+5\,{\frac{{x}^{4}}{{e}^{3}}} \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)^3,x)

[Out]

21/e^3*ln(e*x+d)-7/e^2/(e*x+d)-3/e/(e*x+d)^2+17/2/e^3*x^2-4/e^3*x-17/3/e^3*x^3+120/e^7/(e*x+d)*d^5+85/e^6/(e*x
+d)*d^4+300/e^7*ln(e*x+d)*d^4+170/e^6*ln(e*x+d)*d^3+102/e^5*ln(e*x+d)*d^2+60/e^5*x^2*d^2+51/2/e^4*x^2*d+68/e^5
/(e*x+d)*d^3+12/e^4/(e*x+d)*d^2+42/e^3/(e*x+d)*d-10/e^7/(e*x+d)^2*d^6-17/2/e^6/(e*x+d)^2*d^5-17/2/e^5/(e*x+d)^
2*d^4-2/e^4/(e*x+d)^2*d^3-21/2/e^3/(e*x+d)^2*d^2+7/2/e^2/(e*x+d)^2*d+12/e^4*ln(e*x+d)*d-200/e^6*d^3*x-102/e^5*
x*d^2-51/e^4*x*d-20/e^4*x^3*d+5*x^4/e^3

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Maxima [A]  time = 0.99386, size = 324, normalized size = 1.4 \begin{align*} \frac{220 \, d^{6} + 153 \, d^{5} e + 119 \, d^{4} e^{2} + 20 \, d^{3} e^{3} + 63 \, d^{2} e^{4} - 7 \, d e^{5} - 6 \, e^{6} + 2 \,{\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}{2 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac{30 \, e^{3} x^{4} - 2 \,{\left (60 \, d e^{2} + 17 \, e^{3}\right )} x^{3} + 3 \,{\left (120 \, d^{2} e + 51 \, d e^{2} + 17 \, e^{3}\right )} x^{2} - 6 \,{\left (200 \, d^{3} + 102 \, d^{2} e + 51 \, d e^{2} + 4 \, e^{3}\right )} x}{6 \, e^{6}} + \frac{{\left (300 \, d^{4} + 170 \, d^{3} e + 102 \, d^{2} e^{2} + 12 \, d e^{3} + 21 \, e^{4}\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)^3,x, algorithm="maxima")

[Out]

1/2*(220*d^6 + 153*d^5*e + 119*d^4*e^2 + 20*d^3*e^3 + 63*d^2*e^4 - 7*d*e^5 - 6*e^6 + 2*(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)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7) + 1/6*(30*e^3*x^4 - 2*(60*d*
e^2 + 17*e^3)*x^3 + 3*(120*d^2*e + 51*d*e^2 + 17*e^3)*x^2 - 6*(200*d^3 + 102*d^2*e + 51*d*e^2 + 4*e^3)*x)/e^6
+ (300*d^4 + 170*d^3*e + 102*d^2*e^2 + 12*d*e^3 + 21*e^4)*log(e*x + d)/e^7

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Fricas [A]  time = 0.923654, size = 819, normalized size = 3.55 \begin{align*} \frac{30 \, e^{6} x^{6} + 660 \, d^{6} + 459 \, d^{5} e + 357 \, d^{4} e^{2} + 60 \, d^{3} e^{3} + 189 \, d^{2} e^{4} - 21 \, d e^{5} - 18 \, e^{6} - 2 \,{\left (30 \, d e^{5} + 17 \, e^{6}\right )} x^{5} +{\left (150 \, d^{2} e^{4} + 85 \, d e^{5} + 51 \, e^{6}\right )} x^{4} - 4 \,{\left (150 \, d^{3} e^{3} + 85 \, d^{2} e^{4} + 51 \, d e^{5} + 6 \, e^{6}\right )} x^{3} - 3 \,{\left (680 \, d^{4} e^{2} + 357 \, d^{3} e^{3} + 187 \, d^{2} e^{4} + 16 \, d e^{5}\right )} x^{2} - 6 \,{\left (80 \, d^{5} e + 17 \, d^{4} e^{2} - 17 \, d^{3} e^{3} - 8 \, d^{2} e^{4} - 42 \, d e^{5} + 7 \, e^{6}\right )} x + 6 \,{\left (300 \, d^{6} + 170 \, d^{5} e + 102 \, d^{4} e^{2} + 12 \, d^{3} e^{3} + 21 \, d^{2} e^{4} +{\left (300 \, d^{4} e^{2} + 170 \, d^{3} e^{3} + 102 \, d^{2} e^{4} + 12 \, d e^{5} + 21 \, e^{6}\right )} x^{2} + 2 \,{\left (300 \, d^{5} e + 170 \, d^{4} e^{2} + 102 \, d^{3} e^{3} + 12 \, d^{2} e^{4} + 21 \, d e^{5}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

1/6*(30*e^6*x^6 + 660*d^6 + 459*d^5*e + 357*d^4*e^2 + 60*d^3*e^3 + 189*d^2*e^4 - 21*d*e^5 - 18*e^6 - 2*(30*d*e
^5 + 17*e^6)*x^5 + (150*d^2*e^4 + 85*d*e^5 + 51*e^6)*x^4 - 4*(150*d^3*e^3 + 85*d^2*e^4 + 51*d*e^5 + 6*e^6)*x^3
- 3*(680*d^4*e^2 + 357*d^3*e^3 + 187*d^2*e^4 + 16*d*e^5)*x^2 - 6*(80*d^5*e + 17*d^4*e^2 - 17*d^3*e^3 - 8*d^2*
e^4 - 42*d*e^5 + 7*e^6)*x + 6*(300*d^6 + 170*d^5*e + 102*d^4*e^2 + 12*d^3*e^3 + 21*d^2*e^4 + (300*d^4*e^2 + 17
0*d^3*e^3 + 102*d^2*e^4 + 12*d*e^5 + 21*e^6)*x^2 + 2*(300*d^5*e + 170*d^4*e^2 + 102*d^3*e^3 + 12*d^2*e^4 + 21*
d*e^5)*x)*log(e*x + d))/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

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Sympy [A]  time = 1.82195, size = 238, normalized size = 1.03 \begin{align*} \frac{220 d^{6} + 153 d^{5} e + 119 d^{4} e^{2} + 20 d^{3} e^{3} + 63 d^{2} e^{4} - 7 d e^{5} - 6 e^{6} + x \left (240 d^{5} e + 170 d^{4} e^{2} + 136 d^{3} e^{3} + 24 d^{2} e^{4} + 84 d e^{5} - 14 e^{6}\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac{5 x^{4}}{e^{3}} - \frac{x^{3} \left (60 d + 17 e\right )}{3 e^{4}} + \frac{x^{2} \left (120 d^{2} + 51 d e + 17 e^{2}\right )}{2 e^{5}} - \frac{x \left (200 d^{3} + 102 d^{2} e + 51 d e^{2} + 4 e^{3}\right )}{e^{6}} + \frac{\left (300 d^{4} + 170 d^{3} e + 102 d^{2} e^{2} + 12 d e^{3} + 21 e^{4}\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)**3,x)

[Out]

(220*d**6 + 153*d**5*e + 119*d**4*e**2 + 20*d**3*e**3 + 63*d**2*e**4 - 7*d*e**5 - 6*e**6 + x*(240*d**5*e + 170
*d**4*e**2 + 136*d**3*e**3 + 24*d**2*e**4 + 84*d*e**5 - 14*e**6))/(2*d**2*e**7 + 4*d*e**8*x + 2*e**9*x**2) + 5
*x**4/e**3 - x**3*(60*d + 17*e)/(3*e**4) + x**2*(120*d**2 + 51*d*e + 17*e**2)/(2*e**5) - x*(200*d**3 + 102*d**
2*e + 51*d*e**2 + 4*e**3)/e**6 + (300*d**4 + 170*d**3*e + 102*d**2*e**2 + 12*d*e**3 + 21*e**4)*log(d + e*x)/e*
*7

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Giac [A]  time = 1.16683, size = 292, normalized size = 1.26 \begin{align*}{\left (300 \, d^{4} + 170 \, d^{3} e + 102 \, d^{2} e^{2} + 12 \, d e^{3} + 21 \, e^{4}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (30 \, x^{4} e^{9} - 120 \, d x^{3} e^{8} + 360 \, d^{2} x^{2} e^{7} - 1200 \, d^{3} x e^{6} - 34 \, x^{3} e^{9} + 153 \, d x^{2} e^{8} - 612 \, d^{2} x e^{7} + 51 \, x^{2} e^{9} - 306 \, d x e^{8} - 24 \, x e^{9}\right )} e^{\left (-12\right )} + \frac{{\left (220 \, d^{6} + 153 \, d^{5} e + 119 \, d^{4} e^{2} + 20 \, d^{3} e^{3} + 63 \, d^{2} e^{4} + 2 \,{\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 - 7 \, d e^{5} - 6 \, e^{6}\right )} e^{\left (-7\right )}}{2 \,{\left (x e + d\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

(300*d^4 + 170*d^3*e + 102*d^2*e^2 + 12*d*e^3 + 21*e^4)*e^(-7)*log(abs(x*e + d)) + 1/6*(30*x^4*e^9 - 120*d*x^3
*e^8 + 360*d^2*x^2*e^7 - 1200*d^3*x*e^6 - 34*x^3*e^9 + 153*d*x^2*e^8 - 612*d^2*x*e^7 + 51*x^2*e^9 - 306*d*x*e^
8 - 24*x*e^9)*e^(-12) + 1/2*(220*d^6 + 153*d^5*e + 119*d^4*e^2 + 20*d^3*e^3 + 63*d^2*e^4 + 2*(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 - 7*d*e^5 - 6*e^6)*e^(-7)/(x*e + d)^2