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

Optimal. Leaf size=228 \[ \frac{x^4 \left (20 d^2+17 d e+17 e^2\right )}{4 e^3}-\frac{x^3 \left (17 d^2 e+20 d^3+17 d e^2+4 e^3\right )}{3 e^4}+\frac{x^2 \left (17 d^2 e^2+17 d^3 e+20 d^4+4 d e^3+21 e^4\right )}{2 e^5}-\frac{x \left (17 d^3 e^2+4 d^2 e^3+17 d^4 e+20 d^5+21 d e^4-7 e^5\right )}{e^6}+\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)}{e^7}-\frac{x^5 (20 d+17 e)}{5 e^2}+\frac{10 x^6}{3 e} \]

[Out]

-(((20*d^5 + 17*d^4*e + 17*d^3*e^2 + 4*d^2*e^3 + 21*d*e^4 - 7*e^5)*x)/e^6) + ((20*d^4 + 17*d^3*e + 17*d^2*e^2
+ 4*d*e^3 + 21*e^4)*x^2)/(2*e^5) - ((20*d^3 + 17*d^2*e + 17*d*e^2 + 4*e^3)*x^3)/(3*e^4) + ((20*d^2 + 17*d*e +
17*e^2)*x^4)/(4*e^3) - ((20*d + 17*e)*x^5)/(5*e^2) + (10*x^6)/(3*e) + ((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)*Log[d + e*x])/e^7

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Rubi [A]  time = 0.193499, 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} \[ \frac{x^4 \left (20 d^2+17 d e+17 e^2\right )}{4 e^3}-\frac{x^3 \left (17 d^2 e+20 d^3+17 d e^2+4 e^3\right )}{3 e^4}+\frac{x^2 \left (17 d^2 e^2+17 d^3 e+20 d^4+4 d e^3+21 e^4\right )}{2 e^5}-\frac{x \left (17 d^3 e^2+4 d^2 e^3+17 d^4 e+20 d^5+21 d e^4-7 e^5\right )}{e^6}+\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)}{e^7}-\frac{x^5 (20 d+17 e)}{5 e^2}+\frac{10 x^6}{3 e} \]

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),x]

[Out]

-(((20*d^5 + 17*d^4*e + 17*d^3*e^2 + 4*d^2*e^3 + 21*d*e^4 - 7*e^5)*x)/e^6) + ((20*d^4 + 17*d^3*e + 17*d^2*e^2
+ 4*d*e^3 + 21*e^4)*x^2)/(2*e^5) - ((20*d^3 + 17*d^2*e + 17*d*e^2 + 4*e^3)*x^3)/(3*e^4) + ((20*d^2 + 17*d*e +
17*e^2)*x^4)/(4*e^3) - ((20*d + 17*e)*x^5)/(5*e^2) + (10*x^6)/(3*e) + ((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)*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} \, dx &=\int \left (\frac{-20 d^5-17 d^4 e-17 d^3 e^2-4 d^2 e^3-21 d e^4+7 e^5}{e^6}+\frac{\left (20 d^4+17 d^3 e+17 d^2 e^2+4 d e^3+21 e^4\right ) x}{e^5}-\frac{\left (20 d^3+17 d^2 e+17 d e^2+4 e^3\right ) x^2}{e^4}+\frac{\left (20 d^2+17 d e+17 e^2\right ) x^3}{e^3}-\frac{(20 d+17 e) x^4}{e^2}+\frac{20 x^5}{e}+\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)}\right ) \, dx\\ &=-\frac{\left (20 d^5+17 d^4 e+17 d^3 e^2+4 d^2 e^3+21 d e^4-7 e^5\right ) x}{e^6}+\frac{\left (20 d^4+17 d^3 e+17 d^2 e^2+4 d e^3+21 e^4\right ) x^2}{2 e^5}-\frac{\left (20 d^3+17 d^2 e+17 d e^2+4 e^3\right ) x^3}{3 e^4}+\frac{\left (20 d^2+17 d e+17 e^2\right ) x^4}{4 e^3}-\frac{(20 d+17 e) x^5}{5 e^2}+\frac{10 x^6}{3 e}+\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 ) \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.0585377, size = 179, normalized size = 0.79 \[ \frac{e x \left (-10 d^3 e^2 \left (40 x^2-51 x+102\right )+10 d^2 e^3 \left (30 x^3-34 x^2+51 x-24\right )+60 d^4 e (10 x-17)-1200 d^5-5 d e^4 \left (48 x^4-51 x^3+68 x^2-24 x+252\right )+e^5 \left (200 x^5-204 x^4+255 x^3-80 x^2+630 x+420\right )\right )+60 \left (17 d^4 e^2+4 d^3 e^3+21 d^2 e^4+17 d^5 e+20 d^6-7 d e^5+6 e^6\right ) \log (d+e x)}{60 e^7} \]

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),x]

[Out]

(e*x*(-1200*d^5 + 60*d^4*e*(-17 + 10*x) - 10*d^3*e^2*(102 - 51*x + 40*x^2) + 10*d^2*e^3*(-24 + 51*x - 34*x^2 +
30*x^3) - 5*d*e^4*(252 - 24*x + 68*x^2 - 51*x^3 + 48*x^4) + e^5*(420 + 630*x - 80*x^2 + 255*x^3 - 204*x^4 + 2
00*x^5)) + 60*(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)*Log[d + e*x])/(60*e^
7)

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Maple [A]  time = 0.05, size = 286, normalized size = 1.3 \begin{align*} 7\,{\frac{x}{e}}+6\,{\frac{\ln \left ( ex+d \right ) }{e}}-{\frac{4\,{x}^{3}}{3\,e}}+{\frac{17\,{x}^{4}}{4\,e}}-{\frac{17\,{x}^{5}}{5\,e}}-{\frac{17\,d{x}^{3}}{3\,{e}^{2}}}-{\frac{17\,{x}^{3}{d}^{2}}{3\,{e}^{3}}}-4\,{\frac{{x}^{5}d}{{e}^{2}}}+5\,{\frac{{x}^{4}{d}^{2}}{{e}^{3}}}+{\frac{17\,d{x}^{4}}{4\,{e}^{2}}}+{\frac{17\,{x}^{2}{d}^{3}}{2\,{e}^{4}}}-4\,{\frac{{d}^{2}x}{{e}^{3}}}+10\,{\frac{{x}^{2}{d}^{4}}{{e}^{5}}}-17\,{\frac{x{d}^{4}}{{e}^{5}}}-17\,{\frac{{d}^{3}x}{{e}^{4}}}+2\,{\frac{d{x}^{2}}{{e}^{2}}}-20\,{\frac{{d}^{5}x}{{e}^{6}}}-21\,{\frac{dx}{{e}^{2}}}+{\frac{17\,{x}^{2}{d}^{2}}{2\,{e}^{3}}}+17\,{\frac{\ln \left ( ex+d \right ){d}^{5}}{{e}^{6}}}+17\,{\frac{\ln \left ( ex+d \right ){d}^{4}}{{e}^{5}}}+20\,{\frac{\ln \left ( ex+d \right ){d}^{6}}{{e}^{7}}}+4\,{\frac{\ln \left ( ex+d \right ){d}^{3}}{{e}^{4}}}+21\,{\frac{\ln \left ( ex+d \right ){d}^{2}}{{e}^{3}}}-7\,{\frac{\ln \left ( ex+d \right ) d}{{e}^{2}}}-{\frac{20\,{x}^{3}{d}^{3}}{3\,{e}^{4}}}+{\frac{21\,{x}^{2}}{2\,e}}+{\frac{10\,{x}^{6}}{3\,e}} \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),x)

[Out]

7/e*x+6/e*ln(e*x+d)-4/3/e*x^3+17/4*x^4/e-17/5/e*x^5-17/3/e^2*x^3*d-17/3/e^3*x^3*d^2-4/e^2*x^5*d+5/e^3*x^4*d^2+
17/4/e^2*x^4*d+17/2/e^4*x^2*d^3-4/e^3*x*d^2+10/e^5*x^2*d^4-17/e^5*x*d^4-17/e^4*x*d^3+2/e^2*x^2*d-20/e^6*d^5*x-
21/e^2*x*d+17/2/e^3*x^2*d^2+17/e^6*ln(e*x+d)*d^5+17/e^5*ln(e*x+d)*d^4+20/e^7*ln(e*x+d)*d^6+4/e^4*ln(e*x+d)*d^3
+21/e^3*ln(e*x+d)*d^2-7/e^2*ln(e*x+d)*d-20/3/e^4*x^3*d^3+21/2*x^2/e+10/3*x^6/e

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Maxima [A]  time = 1.00043, size = 308, normalized size = 1.35 \begin{align*} \frac{200 \, e^{5} x^{6} - 12 \,{\left (20 \, d e^{4} + 17 \, e^{5}\right )} x^{5} + 15 \,{\left (20 \, d^{2} e^{3} + 17 \, d e^{4} + 17 \, e^{5}\right )} x^{4} - 20 \,{\left (20 \, d^{3} e^{2} + 17 \, d^{2} e^{3} + 17 \, d e^{4} + 4 \, e^{5}\right )} x^{3} + 30 \,{\left (20 \, d^{4} e + 17 \, d^{3} e^{2} + 17 \, d^{2} e^{3} + 4 \, d e^{4} + 21 \, e^{5}\right )} x^{2} - 60 \,{\left (20 \, d^{5} + 17 \, d^{4} e + 17 \, d^{3} e^{2} + 4 \, d^{2} e^{3} + 21 \, d e^{4} - 7 \, e^{5}\right )} x}{60 \, e^{6}} + \frac{{\left (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}\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),x, algorithm="maxima")

[Out]

1/60*(200*e^5*x^6 - 12*(20*d*e^4 + 17*e^5)*x^5 + 15*(20*d^2*e^3 + 17*d*e^4 + 17*e^5)*x^4 - 20*(20*d^3*e^2 + 17
*d^2*e^3 + 17*d*e^4 + 4*e^5)*x^3 + 30*(20*d^4*e + 17*d^3*e^2 + 17*d^2*e^3 + 4*d*e^4 + 21*e^5)*x^2 - 60*(20*d^5
+ 17*d^4*e + 17*d^3*e^2 + 4*d^2*e^3 + 21*d*e^4 - 7*e^5)*x)/e^6 + (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)*log(e*x + d)/e^7

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Fricas [A]  time = 0.996529, size = 520, normalized size = 2.28 \begin{align*} \frac{200 \, e^{6} x^{6} - 12 \,{\left (20 \, d e^{5} + 17 \, e^{6}\right )} x^{5} + 15 \,{\left (20 \, d^{2} e^{4} + 17 \, d e^{5} + 17 \, e^{6}\right )} x^{4} - 20 \,{\left (20 \, d^{3} e^{3} + 17 \, d^{2} e^{4} + 17 \, d e^{5} + 4 \, e^{6}\right )} x^{3} + 30 \,{\left (20 \, d^{4} e^{2} + 17 \, d^{3} e^{3} + 17 \, d^{2} e^{4} + 4 \, d e^{5} + 21 \, e^{6}\right )} x^{2} - 60 \,{\left (20 \, d^{5} e + 17 \, d^{4} e^{2} + 17 \, d^{3} e^{3} + 4 \, d^{2} e^{4} + 21 \, d e^{5} - 7 \, e^{6}\right )} x + 60 \,{\left (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}\right )} \log \left (e x + d\right )}{60 \, 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),x, algorithm="fricas")

[Out]

1/60*(200*e^6*x^6 - 12*(20*d*e^5 + 17*e^6)*x^5 + 15*(20*d^2*e^4 + 17*d*e^5 + 17*e^6)*x^4 - 20*(20*d^3*e^3 + 17
*d^2*e^4 + 17*d*e^5 + 4*e^6)*x^3 + 30*(20*d^4*e^2 + 17*d^3*e^3 + 17*d^2*e^4 + 4*d*e^5 + 21*e^6)*x^2 - 60*(20*d
^5*e + 17*d^4*e^2 + 17*d^3*e^3 + 4*d^2*e^4 + 21*d*e^5 - 7*e^6)*x + 60*(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)*log(e*x + d))/e^7

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Sympy [A]  time = 0.573867, size = 221, normalized size = 0.97 \begin{align*} \frac{10 x^{6}}{3 e} - \frac{x^{5} \left (20 d + 17 e\right )}{5 e^{2}} + \frac{x^{4} \left (20 d^{2} + 17 d e + 17 e^{2}\right )}{4 e^{3}} - \frac{x^{3} \left (20 d^{3} + 17 d^{2} e + 17 d e^{2} + 4 e^{3}\right )}{3 e^{4}} + \frac{x^{2} \left (20 d^{4} + 17 d^{3} e + 17 d^{2} e^{2} + 4 d e^{3} + 21 e^{4}\right )}{2 e^{5}} - \frac{x \left (20 d^{5} + 17 d^{4} e + 17 d^{3} e^{2} + 4 d^{2} e^{3} + 21 d e^{4} - 7 e^{5}\right )}{e^{6}} + \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 ) \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),x)

[Out]

10*x**6/(3*e) - x**5*(20*d + 17*e)/(5*e**2) + x**4*(20*d**2 + 17*d*e + 17*e**2)/(4*e**3) - x**3*(20*d**3 + 17*
d**2*e + 17*d*e**2 + 4*e**3)/(3*e**4) + x**2*(20*d**4 + 17*d**3*e + 17*d**2*e**2 + 4*d*e**3 + 21*e**4)/(2*e**5
) - x*(20*d**5 + 17*d**4*e + 17*d**3*e**2 + 4*d**2*e**3 + 21*d*e**4 - 7*e**5)/e**6 + (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)*log(d + e*x)/e**7

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Giac [A]  time = 1.16144, size = 308, normalized size = 1.35 \begin{align*}{\left (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}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (200 \, x^{6} e^{5} - 240 \, d x^{5} e^{4} + 300 \, d^{2} x^{4} e^{3} - 400 \, d^{3} x^{3} e^{2} + 600 \, d^{4} x^{2} e - 1200 \, d^{5} x - 204 \, x^{5} e^{5} + 255 \, d x^{4} e^{4} - 340 \, d^{2} x^{3} e^{3} + 510 \, d^{3} x^{2} e^{2} - 1020 \, d^{4} x e + 255 \, x^{4} e^{5} - 340 \, d x^{3} e^{4} + 510 \, d^{2} x^{2} e^{3} - 1020 \, d^{3} x e^{2} - 80 \, x^{3} e^{5} + 120 \, d x^{2} e^{4} - 240 \, d^{2} x e^{3} + 630 \, x^{2} e^{5} - 1260 \, d x e^{4} + 420 \, x e^{5}\right )} e^{\left (-6\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),x, algorithm="giac")

[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^(-7)*log(abs(x*e + d)) + 1/60*(2
00*x^6*e^5 - 240*d*x^5*e^4 + 300*d^2*x^4*e^3 - 400*d^3*x^3*e^2 + 600*d^4*x^2*e - 1200*d^5*x - 204*x^5*e^5 + 25
5*d*x^4*e^4 - 340*d^2*x^3*e^3 + 510*d^3*x^2*e^2 - 1020*d^4*x*e + 255*x^4*e^5 - 340*d*x^3*e^4 + 510*d^2*x^2*e^3
- 1020*d^3*x*e^2 - 80*x^3*e^5 + 120*d*x^2*e^4 - 240*d^2*x*e^3 + 630*x^2*e^5 - 1260*d*x*e^4 + 420*x*e^5)*e^(-6
)