\(\int \frac {1}{(d+e x) (a+b x+c x^2)^2} \, dx\) [2196]

Optimal result

Integrand size = 20, antiderivative size = 224 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}+\frac {(2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2}+\frac {e^3 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^2}-\frac {e^3 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {754, 814, 648, 632, 212, 642} \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx=\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^2}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {e^3 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^2}+\frac {e^3 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^2} \]

[In]

Int[1/((d + e*x)*(a + b*x + c*x^2)^2),x]

[Out]

-((b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(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]])/((b^2 - 4*a*
c)^(3/2)*(c*d^2 - b*d*e + a*e^2)^2) + (e^3*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^2 - (e^3*Log[a + b*x + c*x^2]
)/(2*(c*d^2 - b*d*e + a*e^2)^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 754

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

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + 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] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx=\frac {-b^2 e+2 c (a e+c d x)+b c (d-e x)}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))}+\frac {(-2 c d+b e) \left (-2 c^2 d^2+b^2 e^2+2 c e (b d-3 a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2} \left (c d^2+e (-b d+a e)\right )^2}+\frac {e^3 \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^2}-\frac {e^3 \log (a+x (b+c x))}{2 \left (c d^2+e (-b d+a e)\right )^2} \]

[In]

Integrate[1/((d + e*x)*(a + b*x + c*x^2)^2),x]

[Out]

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

Maple [A] (verified)

Time = 20.93 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.57

method	result	size
default	\(-\frac {\frac {\frac {c \left (a b \,e^{3}-2 d \,e^{2} a c -b^{2} d \,e^{2}+3 b c e \,d^{2}-2 c^{2} d^{3}\right ) x}{4 a c -b^{2}}-\frac {2 a^{2} c \,e^{3}-b^{2} e^{3} a -a b c d \,e^{2}+2 a \,c^{2} d^{2} e +b^{3} d \,e^{2}-2 b^{2} c \,d^{2} e +d^{3} b \,c^{2}}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (4 a \,c^{2} e^{3}-b^{2} c \,e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (5 a b c \,e^{3}-6 c^{2} a d \,e^{2}-b^{3} e^{3}+3 b \,c^{2} d^{2} e -2 c^{3} d^{3}-\frac {\left (4 a \,c^{2} e^{3}-b^{2} c \,e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{\left (e^{2} a -b d e +c \,d^{2}\right )^{2}}+\frac {e^{3} \ln \left (e x +d \right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{2}}\)	\(351\)
risch	\(\text {Expression too large to display}\)	\(15758\)

[In]

int(1/(e*x+d)/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1054 vs. \(2 (218) = 436\).

Time = 19.06 (sec) , antiderivative size = 2128, normalized size of antiderivative = 9.50 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)/(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 - 4*(b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d^2*e + 2*(b^5 - 5*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 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d*e^2 - (a*b^3*c - 4*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) - 2*((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(e*x + d))/((a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^4 - 2*
(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d^3*e + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*d^2*e^2 - 2*(a^2*b^5 - 8*a
^3*b^3*c + 16*a^4*b*c^2)*d*e^3 + (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*e^4 + ((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c
^5)*d^4 - 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^3*e + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d^2*e^2 - 2*(a*b
^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d*e^3 + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*e^4)*x^2 + ((b^5*c^2 - 8
*a*b^3*c^3 + 16*a^2*b*c^4)*d^4 - 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^3*e + (b^7 - 6*a*b^5*c + 32*a^3*b*
c^3)*d^2*e^2 - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d*e^3 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^4)*x)
, -1/2*(2*(b^3*c^2 - 4*a*b*c^3)*d^3 - 4*(b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d^2*e + 2*(b^5 - 5*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)*arctan(-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 + (b^4*c - 2*
a*b^2*c^2 - 8*a^2*c^3)*d*e^2 - (a*b^3*c - 4*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) - 2*((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(e*x + d))/((a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^4 - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3
*b*c^3)*d^3*e + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*d^2*e^2 - 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d*e^3 +
(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*e^4 + ((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^4 - 2*(b^5*c^2 - 8*a*b^3*c^
3 + 16*a^2*b*c^4)*d^3*e + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d^2*e^2 - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c
^3)*d*e^3 + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*e^4)*x^2 + ((b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^4 -
2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^3*e + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d^2*e^2 - 2*(a*b^6 - 8*a^2*b
^4*c + 16*a^3*b^2*c^2)*d*e^3 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^4)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x+d)/(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (218) = 436\).

Time = 0.27 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.18 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx=\frac {e^{4} \log \left ({\left | e x + d \right |}\right )}{c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3} + 2 \, a c d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}} - \frac {e^{3} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )}} - \frac {{\left (4 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 12 \, a c^{2} d e^{2} + b^{3} e^{3} - 6 \, a b c e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {b c^{2} d^{3} - 2 \, b^{2} c d^{2} e + 2 \, a c^{2} d^{2} e + b^{3} d e^{2} - a b c d e^{2} - a b^{2} e^{3} + 2 \, a^{2} c e^{3} + {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{2} c d e^{2} + 2 \, a c^{2} d e^{2} - a b c e^{3}\right )} x}{{\left (c d^{2} - b d e + a e^{2}\right )}^{2} {\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )}} \]

[In]

integrate(1/(e*x+d)/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

e^4*log(abs(e*x + d))/(c^2*d^4*e - 2*b*c*d^3*e^2 + b^2*d^2*e^3 + 2*a*c*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5) - 1/2*
e^3*log(c*x^2 + b*x + a)/(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4) - (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*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^4*d^2*e^2 - 2*a*b^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 -
2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 - 4*a^3*c*e^4)*sqrt(-b^2 + 4*a*c)) - (b*c^2*d^3 - 2*b^2*c*d^2*e
+ 2*a*c^2*d^2*e + b^3*d*e^2 - 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 + b^2*c*d*e^
2 + 2*a*c^2*d*e^2 - a*b*c*e^3)*x)/((c*d^2 - b*d*e + a*e^2)^2*(c*x^2 + b*x + a)*(b^2 - 4*a*c))

Mupad [B] (verification not implemented)

Time = 14.04 (sec) , antiderivative size = 2953, normalized size of antiderivative = 13.18 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

int(1/((d + e*x)*(a + b*x + c*x^2)^2),x)

[Out]

(e^3*log(d + e*x))/(a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2) - ((2*a*c*e -
b^2*e + b*c*d)/(a*b^2*e^2 - 4*a*c^2*d^2 - 4*a^2*c*e^2 + b^2*c*d^2 - b^3*d*e + 4*a*b*c*d*e) + (x*(2*c^2*d - b*
c*e))/(a*b^2*e^2 - 4*a*c^2*d^2 - 4*a^2*c*e^2 + b^2*c*d^2 - b^3*d*e + 4*a*b*c*d*e))/(a + b*x + c*x^2) + (log(96
*a^4*c^3*e^5 - 2*b^7*e^5*x - 2*a*b^6*e^5 - 2*b^3*c^4*d^5 - 2*c^4*d^5*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b^3*e^5*(-
(4*a*c - b^2)^3)^(1/2) + 23*a^2*b^4*c*e^5 - 32*a^2*c^5*d^4*e + 3*b^4*c^3*d^4*e + b^6*c*d^2*e^3 - 4*b^2*c^5*d^5
*x + 2*b^4*e^5*x*(-(4*a*c - b^2)^3)^(1/2) - 84*a^3*b^2*c^2*e^5 - 192*a^3*c^4*d^2*e^3 + 8*a*b*c^5*d^5 + 16*a*c^
6*d^5*x + 2*a*b^5*c*d*e^4 + 24*a*b^5*c*e^5*x + 4*b^6*c*d*e^4*x + 72*a^2*b^2*c^3*d^2*e^3 - 9*a^2*b*c*e^5*(-(4*a
*c - b^2)^3)^(1/2) - 4*a*b^2*c^4*d^4*e + 72*a^3*b*c^3*d*e^4 + 3*b*c^3*d^4*e*(-(4*a*c - b^2)^3)^(1/2) + 120*a^3
*b*c^3*e^5*x - 240*a^3*c^4*d*e^4*x + 10*b^3*c^4*d^4*e*x - 4*c^4*d^4*e*x*(-(4*a*c - b^2)^3)^(1/2) - 20*a*b^3*c^
3*d^3*e^2 - 10*a*b^4*c^2*d^2*e^3 + 80*a^2*b*c^4*d^3*e^2 - 26*a^2*b^3*c^2*d*e^4 - 4*a*c^3*d^3*e^2*(-(4*a*c - b^
2)^3)^(1/2) + 30*a^2*c^2*d*e^4*(-(4*a*c - b^2)^3)^(1/2) + b^3*c*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 94*a^2*b^3*
c^2*e^5*x + 12*a^2*c^2*e^5*x*(-(4*a*c - b^2)^3)^(1/2) + 32*a^2*c^5*d^3*e^2*x - 8*b^4*c^3*d^3*e^2*x + 2*b^5*c^2
*d^2*e^3*x - 6*a*b*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 24*a*b^2*c^4*d^3*e^2*x + 4*a*b^3*c^3*d^2*e^3*x - 48*
a^2*b*c^4*d^2*e^3*x + 204*a^2*b^2*c^3*d*e^4*x - 24*a*c^3*d^2*e^3*x*(-(4*a*c - b^2)^3)^(1/2) + 8*b*c^3*d^3*e^2*
x*(-(4*a*c - b^2)^3)^(1/2) - 40*a*b*c^5*d^4*e*x - 6*a*b^2*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^2*c*e^5*x*
(-(4*a*c - b^2)^3)^(1/2) - 52*a*b^4*c^2*d*e^4*x - 4*b^3*c*d*e^4*x*(-(4*a*c - b^2)^3)^(1/2) + 24*a*b*c^2*d*e^4*
x*(-(4*a*c - b^2)^3)^(1/2))*(c*(6*a*b^4*e^3 - 3*a*b*e^3*(-(4*a*c - b^2)^3)^(1/2)) + c^3*(32*a^3*e^3 + 2*d^3*(-
(4*a*c - b^2)^3)^(1/2)) - c^2*(24*a^2*b^2*e^3 - 6*a*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 3*b*d^2*e*(-(4*a*c - b^2)
^3)^(1/2)) - (b^6*e^3)/2 + (b^3*e^3*(-(4*a*c - b^2)^3)^(1/2))/2))/(a^2*b^6*e^4 - 64*a^3*c^5*d^4 - 64*a^5*c^3*e
^4 + b^6*c^2*d^4 + b^8*d^2*e^2 - 12*a*b^4*c^3*d^4 - 12*a^3*b^4*c*e^4 + 48*a^2*b^2*c^4*d^4 + 48*a^4*b^2*c^2*e^4
- 128*a^4*c^4*d^2*e^2 - 2*a*b^7*d*e^3 - 2*b^7*c*d^3*e + 24*a^2*b^4*c^2*d^2*e^2 + 32*a^3*b^2*c^3*d^2*e^2 + 24*
a*b^5*c^2*d^3*e - 10*a*b^6*c*d^2*e^2 + 24*a^2*b^5*c*d*e^3 + 128*a^3*b*c^4*d^3*e + 128*a^4*b*c^3*d*e^3 - 96*a^2
*b^3*c^3*d^3*e - 96*a^3*b^3*c^2*d*e^3) - (log(96*a^4*c^3*e^5 - 2*b^7*e^5*x - 2*a*b^6*e^5 - 2*b^3*c^4*d^5 + 2*c
^4*d^5*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b^3*e^5*(-(4*a*c - b^2)^3)^(1/2) + 23*a^2*b^4*c*e^5 - 32*a^2*c^5*d^4*e +
3*b^4*c^3*d^4*e + b^6*c*d^2*e^3 - 4*b^2*c^5*d^5*x - 2*b^4*e^5*x*(-(4*a*c - b^2)^3)^(1/2) - 84*a^3*b^2*c^2*e^5
- 192*a^3*c^4*d^2*e^3 + 8*a*b*c^5*d^5 + 16*a*c^6*d^5*x + 2*a*b^5*c*d*e^4 + 24*a*b^5*c*e^5*x + 4*b^6*c*d*e^4*x
+ 72*a^2*b^2*c^3*d^2*e^3 + 9*a^2*b*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^2*c^4*d^4*e + 72*a^3*b*c^3*d*e^4 -
3*b*c^3*d^4*e*(-(4*a*c - b^2)^3)^(1/2) + 120*a^3*b*c^3*e^5*x - 240*a^3*c^4*d*e^4*x + 10*b^3*c^4*d^4*e*x + 4*c^
4*d^4*e*x*(-(4*a*c - b^2)^3)^(1/2) - 20*a*b^3*c^3*d^3*e^2 - 10*a*b^4*c^2*d^2*e^3 + 80*a^2*b*c^4*d^3*e^2 - 26*a
^2*b^3*c^2*d*e^4 + 4*a*c^3*d^3*e^2*(-(4*a*c - b^2)^3)^(1/2) - 30*a^2*c^2*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - b^3*
c*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 94*a^2*b^3*c^2*e^5*x - 12*a^2*c^2*e^5*x*(-(4*a*c - b^2)^3)^(1/2) + 32*a^2
*c^5*d^3*e^2*x - 8*b^4*c^3*d^3*e^2*x + 2*b^5*c^2*d^2*e^3*x + 6*a*b*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 24*a
*b^2*c^4*d^3*e^2*x + 4*a*b^3*c^3*d^2*e^3*x - 48*a^2*b*c^4*d^2*e^3*x + 204*a^2*b^2*c^3*d*e^4*x + 24*a*c^3*d^2*e
^3*x*(-(4*a*c - b^2)^3)^(1/2) - 8*b*c^3*d^3*e^2*x*(-(4*a*c - b^2)^3)^(1/2) - 40*a*b*c^5*d^4*e*x + 6*a*b^2*c*d*
e^4*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b^2*c*e^5*x*(-(4*a*c - b^2)^3)^(1/2) - 52*a*b^4*c^2*d*e^4*x + 4*b^3*c*d*e^
4*x*(-(4*a*c - b^2)^3)^(1/2) - 24*a*b*c^2*d*e^4*x*(-(4*a*c - b^2)^3)^(1/2))*(c^2*(24*a^2*b^2*e^3 + 6*a*d*e^2*(
-(4*a*c - b^2)^3)^(1/2) - 3*b*d^2*e*(-(4*a*c - b^2)^3)^(1/2)) - c^3*(32*a^3*e^3 - 2*d^3*(-(4*a*c - b^2)^3)^(1/
2)) - c*(6*a*b^4*e^3 + 3*a*b*e^3*(-(4*a*c - b^2)^3)^(1/2)) + (b^6*e^3)/2 + (b^3*e^3*(-(4*a*c - b^2)^3)^(1/2))/
2))/(a^2*b^6*e^4 - 64*a^3*c^5*d^4 - 64*a^5*c^3*e^4 + b^6*c^2*d^4 + b^8*d^2*e^2 - 12*a*b^4*c^3*d^4 - 12*a^3*b^4
*c*e^4 + 48*a^2*b^2*c^4*d^4 + 48*a^4*b^2*c^2*e^4 - 128*a^4*c^4*d^2*e^2 - 2*a*b^7*d*e^3 - 2*b^7*c*d^3*e + 24*a^
2*b^4*c^2*d^2*e^2 + 32*a^3*b^2*c^3*d^2*e^2 + 24*a*b^5*c^2*d^3*e - 10*a*b^6*c*d^2*e^2 + 24*a^2*b^5*c*d*e^3 + 12
8*a^3*b*c^4*d^3*e + 128*a^4*b*c^3*d*e^3 - 96*a^2*b^3*c^3*d^3*e - 96*a^3*b^3*c^2*d*e^3)