3.1509 \(\int \frac{(b+2 c x) (a+b x+c x^2)^2}{d+e x} \, dx\)

Optimal. Leaf size=229 \[ \frac{4 c (d+e x)^3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac{(d+e x)^2 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^6}+\frac{2 x \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^5}-\frac{(2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^6}-\frac{5 c^2 (d+e x)^4 (2 c d-b e)}{4 e^6}+\frac{2 c^3 (d+e x)^5}{5 e^6} \]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*x)/e^5 - ((2*c*d - b*e)*(10*c^2*d^2 + b^2
*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^2)/(2*e^6) + (4*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^
3)/(3*e^6) - (5*c^2*(2*c*d - b*e)*(d + e*x)^4)/(4*e^6) + (2*c^3*(d + e*x)^5)/(5*e^6) - ((2*c*d - b*e)*(c*d^2 -
b*d*e + a*e^2)^2*Log[d + e*x])/e^6

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Rubi [A]  time = 0.275187, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {771} \[ \frac{4 c (d+e x)^3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac{(d+e x)^2 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^6}+\frac{2 x \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^5}-\frac{(2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^6}-\frac{5 c^2 (d+e x)^4 (2 c d-b e)}{4 e^6}+\frac{2 c^3 (d+e x)^5}{5 e^6} \]

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*(a + b*x + c*x^2)^2)/(d + e*x),x]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*x)/e^5 - ((2*c*d - b*e)*(10*c^2*d^2 + b^2
*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^2)/(2*e^6) + (4*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^
3)/(3*e^6) - (5*c^2*(2*c*d - b*e)*(d + e*x)^4)/(4*e^6) + (2*c^3*(d + e*x)^5)/(5*e^6) - ((2*c*d - b*e)*(c*d^2 -
b*d*e + a*e^2)^2*Log[d + e*x])/e^6

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(b+2 c x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx &=\int \left (\frac{2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^5}+\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)}{e^5}+\frac{4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^2}{e^5}-\frac{5 c^2 (2 c d-b e) (d+e x)^3}{e^5}+\frac{2 c^3 (d+e x)^4}{e^5}\right ) \, dx\\ &=\frac{2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{e^5}-\frac{(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^2}{2 e^6}+\frac{4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^3}{3 e^6}-\frac{5 c^2 (2 c d-b e) (d+e x)^4}{4 e^6}+\frac{2 c^3 (d+e x)^5}{5 e^6}-\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 \log (d+e x)}{e^6}\\ \end{align*}

Mathematica [A]  time = 0.119278, size = 228, normalized size = 1. \[ \frac{e x \left (20 c e^2 \left (6 a^2 e^2+9 a b e (e x-2 d)+2 b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+30 b^2 e^3 (4 a e-2 b d+b e x)+5 c^2 e \left (8 a e \left (6 d^2-3 d e x+2 e^2 x^2\right )-5 b \left (-6 d^2 e x+12 d^3+4 d e^2 x^2-3 e^3 x^3\right )\right )+2 c^3 \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )\right )-60 (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^2}{60 e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^2)/(d + e*x),x]

[Out]

(e*x*(30*b^2*e^3*(-2*b*d + 4*a*e + b*e*x) + 2*c^3*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^
4*x^4) + 20*c*e^2*(6*a^2*e^2 + 9*a*b*e*(-2*d + e*x) + 2*b^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 5*c^2*e*(8*a*e*(6
*d^2 - 3*d*e*x + 2*e^2*x^2) - 5*b*(12*d^3 - 6*d^2*e*x + 4*d*e^2*x^2 - 3*e^3*x^3))) - 60*(2*c*d - b*e)*(c*d^2 +
e*(-(b*d) + a*e))^2*Log[d + e*x])/(60*e^6)

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Maple [A]  time = 0.006, size = 406, normalized size = 1.8 \begin{align*}{\frac{{x}^{2}{b}^{3}}{2\,e}}+{\frac{2\,{c}^{3}{x}^{5}}{5\,e}}-5\,{\frac{b{d}^{3}{c}^{2}x}{{e}^{4}}}+4\,{\frac{{b}^{2}{d}^{2}cx}{{e}^{3}}}-6\,{\frac{abcdx}{{e}^{2}}}+6\,{\frac{\ln \left ( ex+d \right ) abc{d}^{2}}{{e}^{3}}}+{\frac{5\,{x}^{4}b{c}^{2}}{4\,e}}-{\frac{{x}^{4}{c}^{3}d}{2\,{e}^{2}}}+2\,{\frac{{c}^{3}{d}^{4}x}{{e}^{5}}}-{\frac{{b}^{3}dx}{{e}^{2}}}+{\frac{4\,{x}^{3}{b}^{2}c}{3\,e}}+{\frac{2\,{x}^{3}{c}^{3}{d}^{2}}{3\,{e}^{3}}}-{\frac{{x}^{2}{c}^{3}{d}^{3}}{{e}^{4}}}+2\,{\frac{c{a}^{2}x}{e}}+2\,{\frac{{b}^{2}ax}{e}}+{\frac{4\,{x}^{3}a{c}^{2}}{3\,e}}-2\,{\frac{\ln \left ( ex+d \right ){c}^{3}{d}^{5}}{{e}^{6}}}+{\frac{\ln \left ( ex+d \right ){a}^{2}b}{e}}+{\frac{\ln \left ( ex+d \right ){b}^{3}{d}^{2}}{{e}^{3}}}+4\,{\frac{a{c}^{2}{d}^{2}x}{{e}^{3}}}-2\,{\frac{a{x}^{2}{c}^{2}d}{{e}^{2}}}-2\,{\frac{{b}^{2}{x}^{2}cd}{{e}^{2}}}+{\frac{5\,b{x}^{2}{c}^{2}{d}^{2}}{2\,{e}^{3}}}-{\frac{5\,b{x}^{3}{c}^{2}d}{3\,{e}^{2}}}+3\,{\frac{a{x}^{2}bc}{e}}+5\,{\frac{\ln \left ( ex+d \right ) b{c}^{2}{d}^{4}}{{e}^{5}}}-2\,{\frac{\ln \left ( ex+d \right ){a}^{2}cd}{{e}^{2}}}-2\,{\frac{\ln \left ( ex+d \right ) a{b}^{2}d}{{e}^{2}}}-4\,{\frac{\ln \left ( ex+d \right ) a{c}^{2}{d}^{3}}{{e}^{4}}}-4\,{\frac{\ln \left ( ex+d \right ){b}^{2}c{d}^{3}}{{e}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a)^2/(e*x+d),x)

[Out]

1/2/e*x^2*b^3+2/5/e*c^3*x^5-5/e^4*b*c^2*d^3*x+4/e^3*b^2*c*d^2*x-6/e^2*a*b*c*d*x+6/e^3*ln(e*x+d)*a*b*c*d^2+5/4/
e*x^4*b*c^2-1/2/e^2*x^4*c^3*d+2/e^5*c^3*d^4*x-1/e^2*b^3*d*x+4/3/e*x^3*b^2*c+2/3/e^3*x^3*c^3*d^2-1/e^4*x^2*c^3*
d^3+2/e*c*a^2*x+2/e*a*b^2*x+4/3/e*x^3*a*c^2-2/e^6*ln(e*x+d)*c^3*d^5+1/e*ln(e*x+d)*a^2*b+1/e^3*ln(e*x+d)*b^3*d^
2+4/e^3*a*c^2*d^2*x-2/e^2*x^2*a*c^2*d-2/e^2*x^2*b^2*c*d+5/2/e^3*x^2*b*c^2*d^2-5/3/e^2*x^3*b*c^2*d+3/e*x^2*a*b*
c+5/e^5*ln(e*x+d)*b*c^2*d^4-2/e^2*ln(e*x+d)*a^2*c*d-2/e^2*ln(e*x+d)*a*b^2*d-4/e^4*ln(e*x+d)*a*c^2*d^3-4/e^4*ln
(e*x+d)*b^2*c*d^3

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Maxima [A]  time = 1.00159, size = 414, normalized size = 1.81 \begin{align*} \frac{24 \, c^{3} e^{4} x^{5} - 15 \,{\left (2 \, c^{3} d e^{3} - 5 \, b c^{2} e^{4}\right )} x^{4} + 20 \,{\left (2 \, c^{3} d^{2} e^{2} - 5 \, b c^{2} d e^{3} + 4 \,{\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{3} - 30 \,{\left (2 \, c^{3} d^{3} e - 5 \, b c^{2} d^{2} e^{2} + 4 \,{\left (b^{2} c + a c^{2}\right )} d e^{3} -{\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{2} + 60 \,{\left (2 \, c^{3} d^{4} - 5 \, b c^{2} d^{3} e + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x}{60 \, e^{5}} - \frac{{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} \log \left (e x + d\right )}{e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^2/(e*x+d),x, algorithm="maxima")

[Out]

1/60*(24*c^3*e^4*x^5 - 15*(2*c^3*d*e^3 - 5*b*c^2*e^4)*x^4 + 20*(2*c^3*d^2*e^2 - 5*b*c^2*d*e^3 + 4*(b^2*c + a*c
^2)*e^4)*x^3 - 30*(2*c^3*d^3*e - 5*b*c^2*d^2*e^2 + 4*(b^2*c + a*c^2)*d*e^3 - (b^3 + 6*a*b*c)*e^4)*x^2 + 60*(2*
c^3*d^4 - 5*b*c^2*d^3*e + 4*(b^2*c + a*c^2)*d^2*e^2 - (b^3 + 6*a*b*c)*d*e^3 + 2*(a*b^2 + a^2*c)*e^4)*x)/e^5 -
(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*
c)*d*e^4)*log(e*x + d)/e^6

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Fricas [A]  time = 1.42433, size = 639, normalized size = 2.79 \begin{align*} \frac{24 \, c^{3} e^{5} x^{5} - 15 \,{\left (2 \, c^{3} d e^{4} - 5 \, b c^{2} e^{5}\right )} x^{4} + 20 \,{\left (2 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} + 4 \,{\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} - 30 \,{\left (2 \, c^{3} d^{3} e^{2} - 5 \, b c^{2} d^{2} e^{3} + 4 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} -{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 60 \,{\left (2 \, c^{3} d^{4} e - 5 \, b c^{2} d^{3} e^{2} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d e^{4} + 2 \,{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x - 60 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} \log \left (e x + d\right )}{60 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^2/(e*x+d),x, algorithm="fricas")

[Out]

1/60*(24*c^3*e^5*x^5 - 15*(2*c^3*d*e^4 - 5*b*c^2*e^5)*x^4 + 20*(2*c^3*d^2*e^3 - 5*b*c^2*d*e^4 + 4*(b^2*c + a*c
^2)*e^5)*x^3 - 30*(2*c^3*d^3*e^2 - 5*b*c^2*d^2*e^3 + 4*(b^2*c + a*c^2)*d*e^4 - (b^3 + 6*a*b*c)*e^5)*x^2 + 60*(
2*c^3*d^4*e - 5*b*c^2*d^3*e^2 + 4*(b^2*c + a*c^2)*d^2*e^3 - (b^3 + 6*a*b*c)*d*e^4 + 2*(a*b^2 + a^2*c)*e^5)*x -
60*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 +
a^2*c)*d*e^4)*log(e*x + d))/e^6

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Sympy [A]  time = 0.959153, size = 280, normalized size = 1.22 \begin{align*} \frac{2 c^{3} x^{5}}{5 e} + \frac{x^{4} \left (5 b c^{2} e - 2 c^{3} d\right )}{4 e^{2}} + \frac{x^{3} \left (4 a c^{2} e^{2} + 4 b^{2} c e^{2} - 5 b c^{2} d e + 2 c^{3} d^{2}\right )}{3 e^{3}} + \frac{x^{2} \left (6 a b c e^{3} - 4 a c^{2} d e^{2} + b^{3} e^{3} - 4 b^{2} c d e^{2} + 5 b c^{2} d^{2} e - 2 c^{3} d^{3}\right )}{2 e^{4}} + \frac{x \left (2 a^{2} c e^{4} + 2 a b^{2} e^{4} - 6 a b c d e^{3} + 4 a c^{2} d^{2} e^{2} - b^{3} d e^{3} + 4 b^{2} c d^{2} e^{2} - 5 b c^{2} d^{3} e + 2 c^{3} d^{4}\right )}{e^{5}} + \frac{\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2} \log{\left (d + e x \right )}}{e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x**2+b*x+a)**2/(e*x+d),x)

[Out]

2*c**3*x**5/(5*e) + x**4*(5*b*c**2*e - 2*c**3*d)/(4*e**2) + x**3*(4*a*c**2*e**2 + 4*b**2*c*e**2 - 5*b*c**2*d*e
+ 2*c**3*d**2)/(3*e**3) + x**2*(6*a*b*c*e**3 - 4*a*c**2*d*e**2 + b**3*e**3 - 4*b**2*c*d*e**2 + 5*b*c**2*d**2*
e - 2*c**3*d**3)/(2*e**4) + x*(2*a**2*c*e**4 + 2*a*b**2*e**4 - 6*a*b*c*d*e**3 + 4*a*c**2*d**2*e**2 - b**3*d*e*
*3 + 4*b**2*c*d**2*e**2 - 5*b*c**2*d**3*e + 2*c**3*d**4)/e**5 + (b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**2*log
(d + e*x)/e**6

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Giac [A]  time = 1.17294, size = 455, normalized size = 1.99 \begin{align*} -{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} - a^{2} b e^{5}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (24 \, c^{3} x^{5} e^{4} - 30 \, c^{3} d x^{4} e^{3} + 40 \, c^{3} d^{2} x^{3} e^{2} - 60 \, c^{3} d^{3} x^{2} e + 120 \, c^{3} d^{4} x + 75 \, b c^{2} x^{4} e^{4} - 100 \, b c^{2} d x^{3} e^{3} + 150 \, b c^{2} d^{2} x^{2} e^{2} - 300 \, b c^{2} d^{3} x e + 80 \, b^{2} c x^{3} e^{4} + 80 \, a c^{2} x^{3} e^{4} - 120 \, b^{2} c d x^{2} e^{3} - 120 \, a c^{2} d x^{2} e^{3} + 240 \, b^{2} c d^{2} x e^{2} + 240 \, a c^{2} d^{2} x e^{2} + 30 \, b^{3} x^{2} e^{4} + 180 \, a b c x^{2} e^{4} - 60 \, b^{3} d x e^{3} - 360 \, a b c d x e^{3} + 120 \, a b^{2} x e^{4} + 120 \, a^{2} c x e^{4}\right )} e^{\left (-5\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^2/(e*x+d),x, algorithm="giac")

[Out]

-(2*c^3*d^5 - 5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 + 4*a*c^2*d^3*e^2 - b^3*d^2*e^3 - 6*a*b*c*d^2*e^3 + 2*a*b^2*d*e^
4 + 2*a^2*c*d*e^4 - a^2*b*e^5)*e^(-6)*log(abs(x*e + d)) + 1/60*(24*c^3*x^5*e^4 - 30*c^3*d*x^4*e^3 + 40*c^3*d^2
*x^3*e^2 - 60*c^3*d^3*x^2*e + 120*c^3*d^4*x + 75*b*c^2*x^4*e^4 - 100*b*c^2*d*x^3*e^3 + 150*b*c^2*d^2*x^2*e^2 -
300*b*c^2*d^3*x*e + 80*b^2*c*x^3*e^4 + 80*a*c^2*x^3*e^4 - 120*b^2*c*d*x^2*e^3 - 120*a*c^2*d*x^2*e^3 + 240*b^2
*c*d^2*x*e^2 + 240*a*c^2*d^2*x*e^2 + 30*b^3*x^2*e^4 + 180*a*b*c*x^2*e^4 - 60*b^3*d*x*e^3 - 360*a*b*c*d*x*e^3 +
120*a*b^2*x*e^4 + 120*a^2*c*x*e^4)*e^(-5)