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*(a*e^2-b*d*e+c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*x/e^5-1/2*(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(-
3*a*e+5*b*d))*(e*x+d)^2/e^6+4/3*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^3/e^6-5/4*c^2*(-b*e+2*c*d)*(e*x
+d)^4/e^6+2/5*c^3*(e*x+d)^5/e^6-(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2*ln(e*x+d)/e^6

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Rubi [A]  time = 0.28, antiderivative size = 229, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.11, size = 228, normalized size = 1.00 \[ \frac {e x \left (20 c e^2 \left (6 a^2 e^2+9 a b e (e x-2 d)+2 b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+30 b^2 e^3 (4 a e-2 b d+b e x)+5 c^2 e \left (8 a e \left (6 d^2-3 d e x+2 e^2 x^2\right )-5 b \left (12 d^3-6 d^2 e x+4 d e^2 x^2-3 e^3 x^3\right )\right )+2 c^3 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )-60 (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^2}{60 e^6} \]

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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fricas [A]  time = 0.80, size = 308, normalized size = 1.34 \[ \frac {24 \, c^{3} e^{5} x^{5} - 15 \, {\left (2 \, c^{3} d e^{4} - 5 \, b c^{2} e^{5}\right )} x^{4} + 20 \, {\left (2 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} + 4 \, {\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} - 30 \, {\left (2 \, c^{3} d^{3} e^{2} - 5 \, b c^{2} d^{2} e^{3} + 4 \, {\left (b^{2} c + a c^{2}\right )} d e^{4} - {\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 60 \, {\left (2 \, c^{3} d^{4} e - 5 \, b c^{2} d^{3} e^{2} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d e^{4} + 2 \, {\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x - 60 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} \log \left (e x + d\right )}{60 \, e^{6}} \]

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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giac [A]  time = 0.16, size = 337, normalized size = 1.47 \[ -{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} - a^{2} b e^{5}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{60} \, {\left (24 \, c^{3} x^{5} e^{4} - 30 \, c^{3} d x^{4} e^{3} + 40 \, c^{3} d^{2} x^{3} e^{2} - 60 \, c^{3} d^{3} x^{2} e + 120 \, c^{3} d^{4} x + 75 \, b c^{2} x^{4} e^{4} - 100 \, b c^{2} d x^{3} e^{3} + 150 \, b c^{2} d^{2} x^{2} e^{2} - 300 \, b c^{2} d^{3} x e + 80 \, b^{2} c x^{3} e^{4} + 80 \, a c^{2} x^{3} e^{4} - 120 \, b^{2} c d x^{2} e^{3} - 120 \, a c^{2} d x^{2} e^{3} + 240 \, b^{2} c d^{2} x e^{2} + 240 \, a c^{2} d^{2} x e^{2} + 30 \, b^{3} x^{2} e^{4} + 180 \, a b c x^{2} e^{4} - 60 \, b^{3} d x e^{3} - 360 \, a b c d x e^{3} + 120 \, a b^{2} x e^{4} + 120 \, a^{2} c x e^{4}\right )} e^{\left (-5\right )} \]

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)

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maple [A]  time = 0.05, size = 406, normalized size = 1.77 \[ \frac {2 c^{3} x^{5}}{5 e}+\frac {5 b \,c^{2} x^{4}}{4 e}-\frac {c^{3} d \,x^{4}}{2 e^{2}}+\frac {4 a \,c^{2} x^{3}}{3 e}+\frac {4 b^{2} c \,x^{3}}{3 e}-\frac {5 b \,c^{2} d \,x^{3}}{3 e^{2}}+\frac {2 c^{3} d^{2} x^{3}}{3 e^{3}}+\frac {3 a b c \,x^{2}}{e}-\frac {2 a \,c^{2} d \,x^{2}}{e^{2}}+\frac {b^{3} x^{2}}{2 e}-\frac {2 b^{2} c d \,x^{2}}{e^{2}}+\frac {5 b \,c^{2} d^{2} x^{2}}{2 e^{3}}-\frac {c^{3} d^{3} x^{2}}{e^{4}}+\frac {a^{2} b \ln \left (e x +d \right )}{e}-\frac {2 a^{2} c d \ln \left (e x +d \right )}{e^{2}}+\frac {2 a^{2} c x}{e}-\frac {2 a \,b^{2} d \ln \left (e x +d \right )}{e^{2}}+\frac {2 a \,b^{2} x}{e}+\frac {6 a b c \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {6 a b c d x}{e^{2}}-\frac {4 a \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {4 a \,c^{2} d^{2} x}{e^{3}}+\frac {b^{3} d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {b^{3} d x}{e^{2}}-\frac {4 b^{2} c \,d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {4 b^{2} c \,d^{2} x}{e^{3}}+\frac {5 b \,c^{2} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {5 b \,c^{2} d^{3} x}{e^{4}}-\frac {2 c^{3} d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {2 c^{3} d^{4} x}{e^{5}} \]

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]

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

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maxima [A]  time = 0.50, size = 307, normalized size = 1.34 \[ \frac {24 \, c^{3} e^{4} x^{5} - 15 \, {\left (2 \, c^{3} d e^{3} - 5 \, b c^{2} e^{4}\right )} x^{4} + 20 \, {\left (2 \, c^{3} d^{2} e^{2} - 5 \, b c^{2} d e^{3} + 4 \, {\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{3} - 30 \, {\left (2 \, c^{3} d^{3} e - 5 \, b c^{2} d^{2} e^{2} + 4 \, {\left (b^{2} c + a c^{2}\right )} d e^{3} - {\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{2} + 60 \, {\left (2 \, c^{3} d^{4} - 5 \, b c^{2} d^{3} e + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x}{60 \, e^{5}} - \frac {{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} \log \left (e x + d\right )}{e^{6}} \]

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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mupad [B]  time = 1.81, size = 326, normalized size = 1.42 \[ x^4\,\left (\frac {5\,b\,c^2}{4\,e}-\frac {c^3\,d}{2\,e^2}\right )+x^2\,\left (\frac {b^3+6\,a\,c\,b}{2\,e}+\frac {d\,\left (\frac {d\,\left (\frac {5\,b\,c^2}{e}-\frac {2\,c^3\,d}{e^2}\right )}{e}-\frac {4\,c\,\left (b^2+a\,c\right )}{e}\right )}{2\,e}\right )-x^3\,\left (\frac {d\,\left (\frac {5\,b\,c^2}{e}-\frac {2\,c^3\,d}{e^2}\right )}{3\,e}-\frac {4\,c\,\left (b^2+a\,c\right )}{3\,e}\right )+x\,\left (\frac {2\,a\,\left (b^2+a\,c\right )}{e}-\frac {d\,\left (\frac {b^3+6\,a\,c\,b}{e}+\frac {d\,\left (\frac {d\,\left (\frac {5\,b\,c^2}{e}-\frac {2\,c^3\,d}{e^2}\right )}{e}-\frac {4\,c\,\left (b^2+a\,c\right )}{e}\right )}{e}\right )}{e}\right )+\frac {2\,c^3\,x^5}{5\,e}-\frac {\ln \left (d+e\,x\right )\,\left (-a^2\,b\,e^5+2\,a^2\,c\,d\,e^4+2\,a\,b^2\,d\,e^4-6\,a\,b\,c\,d^2\,e^3+4\,a\,c^2\,d^3\,e^2-b^3\,d^2\,e^3+4\,b^2\,c\,d^3\,e^2-5\,b\,c^2\,d^4\,e+2\,c^3\,d^5\right )}{e^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.75, size = 280, normalized size = 1.22 \[ \frac {2 c^{3} x^{5}}{5 e} + x^{4} \left (\frac {5 b c^{2}}{4 e} - \frac {c^{3} d}{2 e^{2}}\right ) + x^{3} \left (\frac {4 a c^{2}}{3 e} + \frac {4 b^{2} c}{3 e} - \frac {5 b c^{2} d}{3 e^{2}} + \frac {2 c^{3} d^{2}}{3 e^{3}}\right ) + x^{2} \left (\frac {3 a b c}{e} - \frac {2 a c^{2} d}{e^{2}} + \frac {b^{3}}{2 e} - \frac {2 b^{2} c d}{e^{2}} + \frac {5 b c^{2} d^{2}}{2 e^{3}} - \frac {c^{3} d^{3}}{e^{4}}\right ) + x \left (\frac {2 a^{2} c}{e} + \frac {2 a b^{2}}{e} - \frac {6 a b c d}{e^{2}} + \frac {4 a c^{2} d^{2}}{e^{3}} - \frac {b^{3} d}{e^{2}} + \frac {4 b^{2} c d^{2}}{e^{3}} - \frac {5 b c^{2} d^{3}}{e^{4}} + \frac {2 c^{3} d^{4}}{e^{5}}\right ) + \frac {\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2} \log {\left (d + e x \right )}}{e^{6}} \]

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