\(\int \frac {(d+e x)^3}{(a+b x+c x^2)^2} \, dx\) [2192]

Optimal result

Integrand size = 20, antiderivative size = 173 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\frac {e^2 (2 c d-b e) x}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {(2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {e^3 \log \left (a+b x+c x^2\right )}{2 c^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {752, 787, 648, 632, 212, 642} \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {e^2 x (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e^3 \log \left (a+b x+c x^2\right )}{2 c^2} \]

[In]

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

[Out]

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

Rule 212

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

Rule 632

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 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

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

Rule 752

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[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(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] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 787

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*g*(x/c
), x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.16 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {2 \left (b^3 e^3 x+b^2 e^2 (a e-3 c d x)-2 c \left (a^2 e^3+c^2 d^3 x-3 a c d e (d+e x)\right )-b c \left (c d^2 (d-3 e x)+3 a e^2 (d+e x)\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 (-2 c d+b e) \left (-2 c^2 d^2+b^2 e^2+2 c e (b d-3 a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+e^3 \log (a+x (b+c x))}{2 c^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 39.73 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.66

method	result	size
default	\(\frac {\frac {\left (3 a b c \,e^{3}-6 c^{2} a d \,e^{2}-b^{3} e^{3}+3 b^{2} d \,e^{2} c -3 b \,c^{2} d^{2} e +2 c^{3} d^{3}\right ) x}{c^{2} \left (4 a c -b^{2}\right )}+\frac {2 a^{2} c \,e^{3}-b^{2} e^{3} a +3 a b c d \,e^{2}-6 a \,c^{2} d^{2} e +d^{3} b \,c^{2}}{c^{2} \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {\frac {\left (4 a c \,e^{3}-b^{2} e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-a b \,e^{3}+6 d \,e^{2} a c -3 b c e \,d^{2}+2 c^{2} d^{3}-\frac {\left (4 a c \,e^{3}-b^{2} e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c \left (4 a c -b^{2}\right )}\)	\(287\)
risch	\(\text {Expression too large to display}\)	\(2178\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 572 vs. \(2 (167) = 334\).

Time = 0.34 (sec) , antiderivative size = 1164, normalized size of antiderivative = 6.73 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/2*(2*(b^3*c^2 - 4*a*b*c^3)*d^3 - 12*(a*b^2*c^2 - 4*a^2*c^3)*d^2*e + 6*(a*b^3*c - 4*a^2*b*c^2)*d*e^2 - 2*(a
*b^4 - 6*a^2*b^2*c + 8*a^3*c^2)*e^3 - (4*a*c^3*d^3 - 6*a*b*c^2*d^2*e + 12*a^2*c^2*d*e^2 + (a*b^3 - 6*a^2*b*c)*
e^3 + (4*c^4*d^3 - 6*b*c^3*d^2*e + 12*a*c^3*d*e^2 + (b^3*c - 6*a*b*c^2)*e^3)*x^2 + (4*b*c^3*d^3 - 6*b^2*c^2*d^
2*e + 12*a*b*c^2*d*e^2 + (b^4 - 6*a*b^2*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)) + 2*(2*(b^2*c^3 - 4*a*c^4)*d^3 - 3*(b^3*c^2 - 4*a*b*c^3)*d^2
*e + 3*(b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d*e^2 - (b^5 - 7*a*b^3*c + 12*a^2*b*c^2)*e^3)*x - ((b^4*c - 8*a*b^2*c
^2 + 16*a^2*c^3)*e^3*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^3*x + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*e^3)*lo
g(c*x^2 + b*x + a))/(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^2 + (b^5*
c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x), -1/2*(2*(b^3*c^2 - 4*a*b*c^3)*d^3 - 12*(a*b^2*c^2 - 4*a^2*c^3)*d^2*e + 6
*(a*b^3*c - 4*a^2*b*c^2)*d*e^2 - 2*(a*b^4 - 6*a^2*b^2*c + 8*a^3*c^2)*e^3 - 2*(4*a*c^3*d^3 - 6*a*b*c^2*d^2*e +
12*a^2*c^2*d*e^2 + (a*b^3 - 6*a^2*b*c)*e^3 + (4*c^4*d^3 - 6*b*c^3*d^2*e + 12*a*c^3*d*e^2 + (b^3*c - 6*a*b*c^2)
*e^3)*x^2 + (4*b*c^3*d^3 - 6*b^2*c^2*d^2*e + 12*a*b*c^2*d*e^2 + (b^4 - 6*a*b^2*c)*e^3)*x)*sqrt(-b^2 + 4*a*c)*a
rctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*(2*(b^2*c^3 - 4*a*c^4)*d^3 - 3*(b^3*c^2 - 4*a*b*c^3)*
d^2*e + 3*(b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d*e^2 - (b^5 - 7*a*b^3*c + 12*a^2*b*c^2)*e^3)*x - ((b^4*c - 8*a*b^
2*c^2 + 16*a^2*c^3)*e^3*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^3*x + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*e^3)
*log(c*x^2 + b*x + a))/(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^2 + (b
^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1238 vs. \(2 (160) = 320\).

Time = 3.62 (sec) , antiderivative size = 1238, normalized size of antiderivative = 7.16 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\left (\frac {e^{3}}{2 c^{2}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) \log {\left (x + \frac {- 16 a^{2} c^{3} \left (\frac {e^{3}}{2 c^{2}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 8 a^{2} c e^{3} + 8 a b^{2} c^{2} \left (\frac {e^{3}}{2 c^{2}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) - a b^{2} e^{3} - 6 a b c d e^{2} - b^{4} c \left (\frac {e^{3}}{2 c^{2}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 3 b^{2} c d^{2} e - 2 b c^{2} d^{3}}{6 a b c e^{3} - 12 a c^{2} d e^{2} - b^{3} e^{3} + 6 b c^{2} d^{2} e - 4 c^{3} d^{3}} \right )} + \left (\frac {e^{3}}{2 c^{2}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) \log {\left (x + \frac {- 16 a^{2} c^{3} \left (\frac {e^{3}}{2 c^{2}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 8 a^{2} c e^{3} + 8 a b^{2} c^{2} \left (\frac {e^{3}}{2 c^{2}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) - a b^{2} e^{3} - 6 a b c d e^{2} - b^{4} c \left (\frac {e^{3}}{2 c^{2}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 3 b^{2} c d^{2} e - 2 b c^{2} d^{3}}{6 a b c e^{3} - 12 a c^{2} d e^{2} - b^{3} e^{3} + 6 b c^{2} d^{2} e - 4 c^{3} d^{3}} \right )} + \frac {2 a^{2} c e^{3} - a b^{2} e^{3} + 3 a b c d e^{2} - 6 a c^{2} d^{2} e + b c^{2} d^{3} + x \left (3 a b c e^{3} - 6 a c^{2} d e^{2} - b^{3} e^{3} + 3 b^{2} c d e^{2} - 3 b c^{2} d^{2} e + 2 c^{3} d^{3}\right )}{4 a^{2} c^{3} - a b^{2} c^{2} + x^{2} \cdot \left (4 a c^{4} - b^{2} c^{3}\right ) + x \left (4 a b c^{3} - b^{3} c^{2}\right )} \]

[In]

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

[Out]

(e**3/(2*c**2) - sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(2*
c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-16*a**2*c**3*(e**3/(2*c**2) - sqrt(-(
4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*c**3 - 48*
a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 8*a**2*c*e**3 + 8*a*b**2*c**2*(e**3/(2*c**2) - sqrt(-(4*a*c - b**2)**
3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2
+ 12*a*b**4*c - b**6))) - a*b**2*e**3 - 6*a*b*c*d*e**2 - b**4*c*(e**3/(2*c**2) - sqrt(-(4*a*c - b**2)**3)*(b*e
- 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*
b**4*c - b**6))) + 3*b**2*c*d**2*e - 2*b*c**2*d**3)/(6*a*b*c*e**3 - 12*a*c**2*d*e**2 - b**3*e**3 + 6*b*c**2*d*
*2*e - 4*c**3*d**3)) + (e**3/(2*c**2) + sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c
*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-16*a**2*c**3*(
e**3/(2*c**2) + sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(2*c
**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 8*a**2*c*e**3 + 8*a*b**2*c**2*(e**3/(2*c**2) +
sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*c*
*3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - a*b**2*e**3 - 6*a*b*c*d*e**2 - b**4*c*(e**3/(2*c**2) + sqrt(-
(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*c**3 - 48
*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 3*b**2*c*d**2*e - 2*b*c**2*d**3)/(6*a*b*c*e**3 - 12*a*c**2*d*e**2 -
b**3*e**3 + 6*b*c**2*d**2*e - 4*c**3*d**3)) + (2*a**2*c*e**3 - a*b**2*e**3 + 3*a*b*c*d*e**2 - 6*a*c**2*d**2*e
+ b*c**2*d**3 + x*(3*a*b*c*e**3 - 6*a*c**2*d*e**2 - b**3*e**3 + 3*b**2*c*d*e**2 - 3*b*c**2*d**2*e + 2*c**3*d**
3))/(4*a**2*c**3 - a*b**2*c**2 + x**2*(4*a*c**4 - b**2*c**3) + x*(4*a*b*c**3 - b**3*c**2))

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.41 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\frac {e^{3} \log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} - \frac {{\left (4 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 12 \, a c^{2} d e^{2} + b^{3} e^{3} - 6 \, a b c e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {b c^{2} d^{3} - 6 \, a c^{2} d^{2} e + 3 \, a b c d e^{2} - a b^{2} e^{3} + 2 \, a^{2} c e^{3} + {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - 6 \, a c^{2} d e^{2} - b^{3} e^{3} + 3 \, a b c e^{3}\right )} x}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.77 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.79 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {2\,a^2\,c\,e^3-a\,b^2\,e^3+3\,a\,b\,c\,d\,e^2-6\,a\,c^2\,d^2\,e+b\,c^2\,d^3}{c^2\,\left (4\,a\,c-b^2\right )}-\frac {x\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-3\,a\,b\,c\,e^3-2\,c^3\,d^3+6\,a\,c^2\,d\,e^2\right )}{c^2\,\left (4\,a\,c-b^2\right )}}{c\,x^2+b\,x+a}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-64\,a^3\,c^3\,e^3+48\,a^2\,b^2\,c^2\,e^3-12\,a\,b^4\,c\,e^3+b^6\,e^3\right )}{2\,\left (64\,a^3\,c^5-48\,a^2\,b^2\,c^4+12\,a\,b^4\,c^3-b^6\,c^2\right )}+\frac {\mathrm {atan}\left (\frac {c^2\,\left (\frac {2\,x\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2+2\,b\,c\,d\,e-2\,c^2\,d^2-6\,a\,c\,e^2\right )}{c\,{\left (4\,a\,c-b^2\right )}^3}-\frac {\left (b\,e-2\,c\,d\right )\,\left (b^3\,c-4\,a\,b\,c^2\right )\,\left (b^2\,e^2+2\,b\,c\,d\,e-2\,c^2\,d^2-6\,a\,c\,e^2\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^4}\right )\,{\left (4\,a\,c-b^2\right )}^{5/2}}{b^3\,e^3-6\,b\,c^2\,d^2\,e-6\,a\,b\,c\,e^3+4\,c^3\,d^3+12\,a\,c^2\,d\,e^2}\right )\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2+2\,b\,c\,d\,e-2\,c^2\,d^2-6\,a\,c\,e^2\right )}{c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}} \]

[In]

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

[Out]

((b*c^2*d^3 - a*b^2*e^3 + 2*a^2*c*e^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2)/(c^2*(4*a*c - b^2)) - (x*(b^3*e^3 - 2*c
^3*d^3 - 3*a*b*c*e^3 + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2))/(c^2*(4*a*c - b^2)))/(a + b*x + c*x^2)
- (log(a + b*x + c*x^2)*(b^6*e^3 - 64*a^3*c^3*e^3 + 48*a^2*b^2*c^2*e^3 - 12*a*b^4*c*e^3))/(2*(64*a^3*c^5 - b^6
*c^2 + 12*a*b^4*c^3 - 48*a^2*b^2*c^4)) + (atan((c^2*((2*x*(b*e - 2*c*d)*(b^2*e^2 - 2*c^2*d^2 - 6*a*c*e^2 + 2*b
*c*d*e))/(c*(4*a*c - b^2)^3) - ((b*e - 2*c*d)*(b^3*c - 4*a*b*c^2)*(b^2*e^2 - 2*c^2*d^2 - 6*a*c*e^2 + 2*b*c*d*e
))/(c^3*(4*a*c - b^2)^4))*(4*a*c - b^2)^(5/2))/(b^3*e^3 + 4*c^3*d^3 - 6*a*b*c*e^3 + 12*a*c^2*d*e^2 - 6*b*c^2*d
^2*e))*(b*e - 2*c*d)*(b^2*e^2 - 2*c^2*d^2 - 6*a*c*e^2 + 2*b*c*d*e))/(c^2*(4*a*c - b^2)^(3/2))