3.1541 \(\int \frac {(b+2 c x) (d+e x)^3}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=126 \[ \frac {6 e \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {3 e (d+e x) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(d+e x)^3}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {768, 722, 618, 206} \[ \frac {6 e \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {3 e (d+e x) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(d+e x)^3}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*(d + e*x)^3)/(a + b*x + c*x^2)^3,x]

[Out]

-(d + e*x)^3/(2*(a + b*x + c*x^2)^2) - (3*e*(d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b
*x + c*x^2)) + (6*e*(c*d^2 - b*d*e + a*e^2)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 722

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[(2*(2*p + 3)*(c*d
^2 - b*d*e + a*e^2))/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ
[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && LtQ[p, -1]

Rule 768

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(2*c*(p + 1)), x] - Dist[(e*g*m)/(2*c*(p + 1)), Int[(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[2*c*f - b*g, 0] && LtQ[p, -1]
&& GtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {(d+e x)^3}{2 \left (a+b x+c x^2\right )^2}+\frac {1}{2} (3 e) \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {(d+e x)^3}{2 \left (a+b x+c x^2\right )^2}-\frac {3 e (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (3 e \left (c d^2-b d e+a e^2\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{b^2-4 a c}\\ &=-\frac {(d+e x)^3}{2 \left (a+b x+c x^2\right )^2}-\frac {3 e (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (6 e \left (c d^2-b d e+a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{b^2-4 a c}\\ &=-\frac {(d+e x)^3}{2 \left (a+b x+c x^2\right )^2}-\frac {3 e (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {6 e \left (c d^2-b d e+a e^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.28, size = 216, normalized size = 1.71 \[ \frac {1}{2} \left (\frac {12 e \left (e (a e-b d)+c d^2\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac {e \left (b c \left (7 a e^2+3 c d (d-2 e x)\right )+2 c^2 \left (3 c d^2 x-a e (12 d+5 e x)\right )+b^3 \left (-e^2\right )+b^2 c e (3 d+4 e x)\right )}{c^2 \left (4 a c-b^2\right ) (a+x (b+c x))}+\frac {c e^2 (3 a d+a e x+3 b d x)-b e^3 (a+b x)-c^2 d^2 (d+3 e x)}{c^2 (a+x (b+c x))^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((b + 2*c*x)*(d + e*x)^3)/(a + b*x + c*x^2)^3,x]

[Out]

((-(b*e^3*(a + b*x)) - c^2*d^2*(d + 3*e*x) + c*e^2*(3*a*d + 3*b*d*x + a*e*x))/(c^2*(a + x*(b + c*x))^2) + (e*(
-(b^3*e^2) + b^2*c*e*(3*d + 4*e*x) + b*c*(7*a*e^2 + 3*c*d*(d - 2*e*x)) + 2*c^2*(3*c*d^2*x - a*e*(12*d + 5*e*x)
)))/(c^2*(-b^2 + 4*a*c)*(a + x*(b + c*x))) + (12*e*(c*d^2 + e*(-(b*d) + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4
*a*c]])/(-b^2 + 4*a*c)^(3/2))/2

________________________________________________________________________________________

fricas [B]  time = 0.95, size = 1455, normalized size = 11.55 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^3/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

[-1/2*((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3 + 3*(a*b^3*c - 4*a^2*b*c^2)*d^2*e - 12*(a^2*b^2*c - 4*a^3*c^2)*d
*e^2 + 3*(a^2*b^3 - 4*a^3*b*c)*e^3 + 2*(3*(b^2*c^3 - 4*a*c^4)*d^2*e - 3*(b^3*c^2 - 4*a*b*c^3)*d*e^2 + (2*b^4*c
- 13*a*b^2*c^2 + 20*a^2*c^3)*e^3)*x^3 + 3*(3*(b^3*c^2 - 4*a*b*c^3)*d^2*e - (b^4*c + 4*a*b^2*c^2 - 32*a^2*c^3)
*d*e^2 + (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e^3)*x^2 + 6*(a^2*c^2*d^2*e - a^2*b*c*d*e^2 + a^3*c*e^3 + (c^4*d^2*e
- b*c^3*d*e^2 + a*c^3*e^3)*x^4 + 2*(b*c^3*d^2*e - b^2*c^2*d*e^2 + a*b*c^2*e^3)*x^3 + ((b^2*c^2 + 2*a*c^3)*d^2*
e - (b^3*c + 2*a*b*c^2)*d*e^2 + (a*b^2*c + 2*a^2*c^2)*e^3)*x^2 + 2*(a*b*c^2*d^2*e - a*b^2*c*d*e^2 + a^2*b*c*e^
3)*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x +
a)) + 6*((b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d^2*e - 3*(a*b^3*c - 4*a^2*b*c^2)*d*e^2 + (a*b^4 - 5*a^2*b^2*c + 4
*a^3*c^2)*e^3)*x)/(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + 2*(b^5*
c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 +
16*a^3*b*c^3)*x), -1/2*((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3 + 3*(a*b^3*c - 4*a^2*b*c^2)*d^2*e - 12*(a^2*b^2
*c - 4*a^3*c^2)*d*e^2 + 3*(a^2*b^3 - 4*a^3*b*c)*e^3 + 2*(3*(b^2*c^3 - 4*a*c^4)*d^2*e - 3*(b^3*c^2 - 4*a*b*c^3)
*d*e^2 + (2*b^4*c - 13*a*b^2*c^2 + 20*a^2*c^3)*e^3)*x^3 + 3*(3*(b^3*c^2 - 4*a*b*c^3)*d^2*e - (b^4*c + 4*a*b^2*
c^2 - 32*a^2*c^3)*d*e^2 + (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e^3)*x^2 - 12*(a^2*c^2*d^2*e - a^2*b*c*d*e^2 + a^3*c
*e^3 + (c^4*d^2*e - b*c^3*d*e^2 + a*c^3*e^3)*x^4 + 2*(b*c^3*d^2*e - b^2*c^2*d*e^2 + a*b*c^2*e^3)*x^3 + ((b^2*c
^2 + 2*a*c^3)*d^2*e - (b^3*c + 2*a*b*c^2)*d*e^2 + (a*b^2*c + 2*a^2*c^2)*e^3)*x^2 + 2*(a*b*c^2*d^2*e - a*b^2*c*
d*e^2 + a^2*b*c*e^3)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 6*((b^4*c -
5*a*b^2*c^2 + 4*a^2*c^3)*d^2*e - 3*(a*b^3*c - 4*a^2*b*c^2)*d*e^2 + (a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*e^3)*x)/
(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + 2*(b^5*c^2 - 8*a*b^3*c^3
+ 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x)]

________________________________________________________________________________________

giac [B]  time = 0.18, size = 292, normalized size = 2.32 \[ -\frac {6 \, {\left (c d^{2} e - b d e^{2} + a e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {6 \, c^{3} d^{2} x^{3} e - 6 \, b c^{2} d x^{3} e^{2} + 9 \, b c^{2} d^{2} x^{2} e + 4 \, b^{2} c x^{3} e^{3} - 10 \, a c^{2} x^{3} e^{3} - 3 \, b^{2} c d x^{2} e^{2} - 24 \, a c^{2} d x^{2} e^{2} + 6 \, b^{2} c d^{2} x e - 6 \, a c^{2} d^{2} x e + b^{2} c d^{3} - 4 \, a c^{2} d^{3} + 3 \, b^{3} x^{2} e^{3} - 3 \, a b c x^{2} e^{3} - 18 \, a b c d x e^{2} + 3 \, a b c d^{2} e + 6 \, a b^{2} x e^{3} - 6 \, a^{2} c x e^{3} - 12 \, a^{2} c d e^{2} + 3 \, a^{2} b e^{3}}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} {\left (c x^{2} + b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^3/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

-6*(c*d^2*e - b*d*e^2 + a*e^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2 - 4*a*c)*sqrt(-b^2 + 4*a*c)) - 1/2
*(6*c^3*d^2*x^3*e - 6*b*c^2*d*x^3*e^2 + 9*b*c^2*d^2*x^2*e + 4*b^2*c*x^3*e^3 - 10*a*c^2*x^3*e^3 - 3*b^2*c*d*x^2
*e^2 - 24*a*c^2*d*x^2*e^2 + 6*b^2*c*d^2*x*e - 6*a*c^2*d^2*x*e + b^2*c*d^3 - 4*a*c^2*d^3 + 3*b^3*x^2*e^3 - 3*a*
b*c*x^2*e^3 - 18*a*b*c*d*x*e^2 + 3*a*b*c*d^2*e + 6*a*b^2*x*e^3 - 6*a^2*c*x*e^3 - 12*a^2*c*d*e^2 + 3*a^2*b*e^3)
/((b^2*c - 4*a*c^2)*(c*x^2 + b*x + a)^2)

________________________________________________________________________________________

maple [B]  time = 0.06, size = 365, normalized size = 2.90 \[ \frac {6 a \,e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}-\frac {6 b d \,e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}+\frac {6 c \,d^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}+\frac {-\frac {\left (5 a c \,e^{2}-2 b^{2} e^{2}+3 b c d e -3 c^{2} d^{2}\right ) e \,x^{3}}{4 a c -b^{2}}-\frac {3 \left (a b c \,e^{2}+8 a \,c^{2} d e -b^{3} e^{2}+b^{2} c d e -3 b \,c^{2} d^{2}\right ) e \,x^{2}}{2 \left (4 a c -b^{2}\right ) c}-\frac {3 \left (a^{2} c \,e^{2}-a \,b^{2} e^{2}+3 a b c d e +a \,c^{2} d^{2}-b^{2} c \,d^{2}\right ) e x}{\left (4 a c -b^{2}\right ) c}+\frac {3 a^{2} b \,e^{3}-12 a^{2} c d \,e^{2}+3 a b c \,d^{2} e -4 a \,c^{2} d^{3}+b^{2} c \,d^{3}}{2 \left (4 a c -b^{2}\right ) c}}{\left (c \,x^{2}+b x +a \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^3/(c*x^2+b*x+a)^3,x)

[Out]

(-e*(5*a*c*e^2-2*b^2*e^2+3*b*c*d*e-3*c^2*d^2)/(4*a*c-b^2)*x^3-3/2*e*(a*b*c*e^2+8*a*c^2*d*e-b^3*e^2+b^2*c*d*e-3
*b*c^2*d^2)/(4*a*c-b^2)/c*x^2-3/c*e*(a^2*c*e^2-a*b^2*e^2+3*a*b*c*d*e+a*c^2*d^2-b^2*c*d^2)/(4*a*c-b^2)*x+1/2*(3
*a^2*b*e^3-12*a^2*c*d*e^2+3*a*b*c*d^2*e-4*a*c^2*d^3+b^2*c*d^3)/(4*a*c-b^2)/c)/(c*x^2+b*x+a)^2+6*e^3/(4*a*c-b^2
)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a-6*e^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*d+
6*e/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c*d^2

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^3/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 2.13, size = 412, normalized size = 3.27 \[ \frac {\frac {3\,a^2\,b\,e^3-12\,a^2\,c\,d\,e^2+3\,a\,b\,c\,d^2\,e-4\,a\,c^2\,d^3+b^2\,c\,d^3}{2\,c\,\left (4\,a\,c-b^2\right )}+\frac {e\,x^3\,\left (2\,b^2\,e^2-3\,b\,c\,d\,e+3\,c^2\,d^2-5\,a\,c\,e^2\right )}{4\,a\,c-b^2}-\frac {3\,e\,x\,\left (a^2\,c\,e^2-a\,b^2\,e^2+3\,a\,b\,c\,d\,e+a\,c^2\,d^2-b^2\,c\,d^2\right )}{c\,\left (4\,a\,c-b^2\right )}-\frac {3\,e\,x^2\,\left (-b^3\,e^2+b^2\,c\,d\,e-3\,b\,c^2\,d^2+a\,b\,c\,e^2+8\,a\,c^2\,d\,e\right )}{2\,c\,\left (4\,a\,c-b^2\right )}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}-\frac {6\,e\,\mathrm {atan}\left (\frac {\left (\frac {3\,e\,\left (b^3-4\,a\,b\,c\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {6\,c\,e\,x\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (4\,a\,c-b^2\right )}{3\,c\,d^2\,e-3\,b\,d\,e^2+3\,a\,e^3}\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b + 2*c*x)*(d + e*x)^3)/(a + b*x + c*x^2)^3,x)

[Out]

((3*a^2*b*e^3 - 4*a*c^2*d^3 + b^2*c*d^3 - 12*a^2*c*d*e^2 + 3*a*b*c*d^2*e)/(2*c*(4*a*c - b^2)) + (e*x^3*(2*b^2*
e^2 + 3*c^2*d^2 - 5*a*c*e^2 - 3*b*c*d*e))/(4*a*c - b^2) - (3*e*x*(a*c^2*d^2 - a*b^2*e^2 + a^2*c*e^2 - b^2*c*d^
2 + 3*a*b*c*d*e))/(c*(4*a*c - b^2)) - (3*e*x^2*(a*b*c*e^2 - 3*b*c^2*d^2 - b^3*e^2 + 8*a*c^2*d*e + b^2*c*d*e))/
(2*c*(4*a*c - b^2)))/(x^2*(2*a*c + b^2) + a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*x^3) - (6*e*atan((((3*e*(b^3 - 4*a*b
*c)*(a*e^2 + c*d^2 - b*d*e))/(4*a*c - b^2)^(5/2) - (6*c*e*x*(a*e^2 + c*d^2 - b*d*e))/(4*a*c - b^2)^(3/2))*(4*a
*c - b^2))/(3*a*e^3 - 3*b*d*e^2 + 3*c*d^2*e))*(a*e^2 + c*d^2 - b*d*e))/(4*a*c - b^2)^(3/2)

________________________________________________________________________________________

sympy [B]  time = 40.98, size = 762, normalized size = 6.05 \[ - 3 e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (x + \frac {- 48 a^{2} c^{2} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 24 a b^{2} c e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 3 a b e^{3} - 3 b^{4} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 3 b^{2} d e^{2} + 3 b c d^{2} e}{6 a c e^{3} - 6 b c d e^{2} + 6 c^{2} d^{2} e} \right )} + 3 e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (x + \frac {48 a^{2} c^{2} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 24 a b^{2} c e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 3 a b e^{3} + 3 b^{4} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 3 b^{2} d e^{2} + 3 b c d^{2} e}{6 a c e^{3} - 6 b c d e^{2} + 6 c^{2} d^{2} e} \right )} + \frac {3 a^{2} b e^{3} - 12 a^{2} c d e^{2} + 3 a b c d^{2} e - 4 a c^{2} d^{3} + b^{2} c d^{3} + x^{3} \left (- 10 a c^{2} e^{3} + 4 b^{2} c e^{3} - 6 b c^{2} d e^{2} + 6 c^{3} d^{2} e\right ) + x^{2} \left (- 3 a b c e^{3} - 24 a c^{2} d e^{2} + 3 b^{3} e^{3} - 3 b^{2} c d e^{2} + 9 b c^{2} d^{2} e\right ) + x \left (- 6 a^{2} c e^{3} + 6 a b^{2} e^{3} - 18 a b c d e^{2} - 6 a c^{2} d^{2} e + 6 b^{2} c d^{2} e\right )}{8 a^{3} c^{2} - 2 a^{2} b^{2} c + x^{4} \left (8 a c^{4} - 2 b^{2} c^{3}\right ) + x^{3} \left (16 a b c^{3} - 4 b^{3} c^{2}\right ) + x^{2} \left (16 a^{2} c^{3} + 4 a b^{2} c^{2} - 2 b^{4} c\right ) + x \left (16 a^{2} b c^{2} - 4 a b^{3} c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)**3/(c*x**2+b*x+a)**3,x)

[Out]

-3*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2)*log(x + (-48*a**2*c**2*e*sqrt(-1/(4*a*c - b**2)**3)*
(a*e**2 - b*d*e + c*d**2) + 24*a*b**2*c*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) + 3*a*b*e**3 -
3*b**4*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) - 3*b**2*d*e**2 + 3*b*c*d**2*e)/(6*a*c*e**3 - 6*
b*c*d*e**2 + 6*c**2*d**2*e)) + 3*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2)*log(x + (48*a**2*c**2*
e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) - 24*a*b**2*c*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*
d*e + c*d**2) + 3*a*b*e**3 + 3*b**4*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) - 3*b**2*d*e**2 + 3
*b*c*d**2*e)/(6*a*c*e**3 - 6*b*c*d*e**2 + 6*c**2*d**2*e)) + (3*a**2*b*e**3 - 12*a**2*c*d*e**2 + 3*a*b*c*d**2*e
- 4*a*c**2*d**3 + b**2*c*d**3 + x**3*(-10*a*c**2*e**3 + 4*b**2*c*e**3 - 6*b*c**2*d*e**2 + 6*c**3*d**2*e) + x*
*2*(-3*a*b*c*e**3 - 24*a*c**2*d*e**2 + 3*b**3*e**3 - 3*b**2*c*d*e**2 + 9*b*c**2*d**2*e) + x*(-6*a**2*c*e**3 +
6*a*b**2*e**3 - 18*a*b*c*d*e**2 - 6*a*c**2*d**2*e + 6*b**2*c*d**2*e))/(8*a**3*c**2 - 2*a**2*b**2*c + x**4*(8*a
*c**4 - 2*b**2*c**3) + x**3*(16*a*b*c**3 - 4*b**3*c**2) + x**2*(16*a**2*c**3 + 4*a*b**2*c**2 - 2*b**4*c) + x*(
16*a**2*b*c**2 - 4*a*b**3*c))

________________________________________________________________________________________