3.2370 \(\int \frac {A+B x}{(d+e x)^3 (a+b x+c x^2)} \, dx\)

Optimal. Leaf size=414 \[ \frac {\log \left (a+b x+c x^2\right ) \left (B \left (a b e^3-3 a c d e^2+c^2 d^3\right )-A e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac {\log (d+e x) \left (B \left (a b e^3-3 a c d e^2+c^2 d^3\right )-A e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-b^2 e^2 (a B e+3 A c d)+b c \left (-3 a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )-2 c \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )\right )+A b^3 e^3\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}+\frac {B d-A e}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {A e (2 c d-b e)-B \left (c d^2-a e^2\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.80, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {800, 634, 618, 206, 628} \[ \frac {\log \left (a+b x+c x^2\right ) \left (B \left (a b e^3-3 a c d e^2+c^2 d^3\right )-A e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac {\log (d+e x) \left (B \left (a b e^3-3 a c d e^2+c^2 d^3\right )-A e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-b^2 e^2 (a B e+3 A c d)+b c \left (-3 a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )-2 c \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )\right )+A b^3 e^3\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}+\frac {B d-A e}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {A e (2 c d-b e)-B \left (c d^2-a e^2\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 628

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 634

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 800

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

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Mathematica [A]  time = 0.47, size = 413, normalized size = 1.00 \[ -\frac {\log (d+e x) \left (A e \left (c e (a e+3 b d)-b^2 e^2-3 c^2 d^2\right )+B \left (a b e^3-3 a c d e^2+c^2 d^3\right )\right )}{\left (e (a e-b d)+c d^2\right )^3}+\frac {\log (a+x (b+c x)) \left (A e \left (c e (a e+3 b d)-b^2 e^2-3 c^2 d^2\right )+B \left (a b e^3-3 a c d e^2+c^2 d^3\right )\right )}{2 \left (e (a e-b d)+c d^2\right )^3}+\frac {\tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right ) \left (-b^2 e^2 (a B e+3 A c d)+b c \left (-3 a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )+2 c \left (A c d \left (3 a e^2-c d^2\right )+a B e \left (a e^2-3 c d^2\right )\right )+A b^3 e^3\right )}{\sqrt {4 a c-b^2} \left (e (b d-a e)-c d^2\right )^3}+\frac {B \left (c d^2-a e^2\right )+A e (b e-2 c d)}{(d+e x) \left (e (a e-b d)+c d^2\right )^2}+\frac {B d-A e}{2 (d+e x)^2 \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [A]  time = 0.19, size = 797, normalized size = 1.93 \[ \frac {{\left (B c^{2} d^{3} - 3 \, A c^{2} d^{2} e - 3 \, B a c d e^{2} + 3 \, A b c d e^{2} + B a b e^{3} - A b^{2} e^{3} + A a c e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )}} - \frac {{\left (B c^{2} d^{3} e - 3 \, A c^{2} d^{2} e^{2} - 3 \, B a c d e^{3} + 3 \, A b c d e^{3} + B a b e^{4} - A b^{2} e^{4} + A a c e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} + 3 \, a c^{2} d^{4} e^{3} - b^{3} d^{3} e^{4} - 6 \, a b c d^{3} e^{4} + 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} c d^{2} e^{5} - 3 \, a^{2} b d e^{6} + a^{3} e^{7}} - \frac {{\left (B b c^{2} d^{3} - 2 \, A c^{3} d^{3} - 6 \, B a c^{2} d^{2} e + 3 \, A b c^{2} d^{2} e + 3 \, B a b c d e^{2} - 3 \, A b^{2} c d e^{2} + 6 \, A a c^{2} d e^{2} - B a b^{2} e^{3} + A b^{3} e^{3} + 2 \, B a^{2} c e^{3} - 3 \, A a b c e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {3 \, B c^{2} d^{5} - 4 \, B b c d^{4} e - 5 \, A c^{2} d^{4} e + B b^{2} d^{3} e^{2} + 2 \, B a c d^{3} e^{2} + 8 \, A b c d^{3} e^{2} - 3 \, A b^{2} d^{2} e^{3} - 6 \, A a c d^{2} e^{3} - B a^{2} d e^{4} + 4 \, A a b d e^{4} - A a^{2} e^{5} + 2 \, {\left (B c^{2} d^{4} e - B b c d^{3} e^{2} - 2 \, A c^{2} d^{3} e^{2} + 3 \, A b c d^{2} e^{3} + B a b d e^{4} - A b^{2} d e^{4} - 2 \, A a c d e^{4} - B a^{2} e^{5} + A a b e^{5}\right )} x}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )}^{3} {\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(B*c^2*d^3 - 3*A*c^2*d^2*e - 3*B*a*c*d*e^2 + 3*A*b*c*d*e^2 + B*a*b*e^3 - A*b^2*e^3 + A*a*c*e^3)*log(c*x^2
+ b*x + a)/(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*
b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6) - (B*c^2*d^3*e - 3*A*c^2*d^2*e^2 - 3*B*a*c*d*e^3 + 3*
A*b*c*d*e^3 + B*a*b*e^4 - A*b^2*e^4 + A*a*c*e^4)*log(abs(x*e + d))/(c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*
e^3 + 3*a*c^2*d^4*e^3 - b^3*d^3*e^4 - 6*a*b*c*d^3*e^4 + 3*a*b^2*d^2*e^5 + 3*a^2*c*d^2*e^5 - 3*a^2*b*d*e^6 + a^
3*e^7) - (B*b*c^2*d^3 - 2*A*c^3*d^3 - 6*B*a*c^2*d^2*e + 3*A*b*c^2*d^2*e + 3*B*a*b*c*d*e^2 - 3*A*b^2*c*d*e^2 +
6*A*a*c^2*d*e^2 - B*a*b^2*e^3 + A*b^3*e^3 + 2*B*a^2*c*e^3 - 3*A*a*b*c*e^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*
c))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^
2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)*sqrt(-b^2 + 4*a*c)) + 1/2*(3*B*c^2*d^5 - 4*B*b*c*d^4*e - 5*
A*c^2*d^4*e + B*b^2*d^3*e^2 + 2*B*a*c*d^3*e^2 + 8*A*b*c*d^3*e^2 - 3*A*b^2*d^2*e^3 - 6*A*a*c*d^2*e^3 - B*a^2*d*
e^4 + 4*A*a*b*d*e^4 - A*a^2*e^5 + 2*(B*c^2*d^4*e - B*b*c*d^3*e^2 - 2*A*c^2*d^3*e^2 + 3*A*b*c*d^2*e^3 + B*a*b*d
*e^4 - A*b^2*d*e^4 - 2*A*a*c*d*e^4 - B*a^2*e^5 + A*a*b*e^5)*x)/((c*d^2 - b*d*e + a*e^2)^3*(x*e + d)^2)

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maple [B]  time = 0.06, size = 1339, normalized size = 3.23 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((B*x+A)/(e*x+d)^3/(c*x^2+b*x+a),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?

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mupad [B]  time = 26.75, size = 7042, normalized size = 17.01 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(((A*b^4*e^3 - (3*A*b*e^3*(b^2 - 4*a*c)^(3/2))/4 + (B*a*e^3*(b^2 - 4*a*c)^(3/2))/2 - B*a*b^3*e^3 + 4*B*a*c
^3*d^3 - (A*b^3*e^3*(b^2 - 4*a*c)^(1/2))/4 + 2*A*c^3*d^3*(b^2 - 4*a*c)^(1/2) + 4*A*a^2*c^2*e^3 - B*b^2*c^2*d^3
+ (B*a*b^2*e^3*(b^2 - 4*a*c)^(1/2))/2 - B*b*c^2*d^3*(b^2 - 4*a*c)^(1/2) - (3*B*b^3*d*e^2*(b^2 - 4*a*c)^(1/2))
/4 + 3*A*b^2*c^2*d^2*e - 12*B*a^2*c^2*d*e^2 + (3*A*c*d*e^2*(b^2 - 4*a*c)^(3/2))/2 + (3*B*b*d*e^2*(b^2 - 4*a*c)
^(3/2))/4 - (3*B*c*d^2*e*(b^2 - 4*a*c)^(3/2))/2 - 5*A*a*b^2*c*e^3 + 4*B*a^2*b*c*e^3 - 12*A*a*c^3*d^2*e - 3*A*b
^3*c*d*e^2 + 12*A*a*b*c^2*d*e^2 + 3*B*a*b^2*c*d*e^2 - 3*A*b*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + (3*A*b^2*c*d*e^2*(
b^2 - 4*a*c)^(1/2))/2 + (3*B*b^2*c*d^2*e*(b^2 - 4*a*c)^(1/2))/2)*(4*B*d*e^5*(b^2 - 4*a*c)^(7/2) + 3*B*e^6*x*(b
^2 - 4*a*c)^(7/2) - 3*A*b^2*e^6*(b^2 - 4*a*c)^(5/2) + 2*A*b^4*e^6*(b^2 - 4*a*c)^(3/2) + A*b^6*e^6*(b^2 - 4*a*c
)^(1/2) + 128*B*a^4*c^3*e^6 - 32*A*b^6*c*e^6*x - 2*B*a*b^5*e^6*(b^2 - 4*a*c)^(1/2) + B*b^2*d*e^5*(b^2 - 4*a*c)
^(5/2) - 10*B*b^4*d*e^5*(b^2 - 4*a*c)^(3/2) + 5*B*b^6*d*e^5*(b^2 - 4*a*c)^(1/2) + B*b^2*e^6*x*(b^2 - 4*a*c)^(5
/2) - 3*B*b^4*e^6*x*(b^2 - 4*a*c)^(3/2) - B*b^6*e^6*x*(b^2 - 4*a*c)^(1/2) - 320*A*a^3*b*c^3*e^6 + 48*B*a^2*b^4
*c*e^6 + 640*A*a^3*c^4*d*e^5 - 32*A*b^2*c^5*d^5*e + 48*B*b^3*c^4*d^5*e - 48*A*c^2*d^2*e^4*(b^2 - 4*a*c)^(5/2)
- 64*A*c^4*d^4*e^2*(b^2 - 4*a*c)^(3/2) + 48*B*c^2*d^3*e^3*(b^2 - 4*a*c)^(5/2) + 272*A*a^2*b^3*c^2*e^6 - 224*B*
a^3*b^2*c^2*e^6 - 1280*A*a^2*c^5*d^3*e^3 - 48*A*b^3*c^4*d^4*e^2 + 96*A*b^4*c^3*d^3*e^3 - 64*A*b^5*c^2*d^2*e^4
+ 640*B*a^2*c^5*d^4*e^2 - 1280*B*a^3*c^4*d^2*e^4 - 16*B*b^4*c^3*d^4*e^2 + 2*B*a*b*e^6*(b^2 - 4*a*c)^(5/2) - 48
*A*a*b^5*c*e^6 + 128*A*a*c^6*d^5*e + 16*A*b^6*c*d*e^5 - 96*A*c^4*d^3*e^3*x*(b^2 - 4*a*c)^(3/2) + 40*B*c^2*d^2*
e^4*x*(b^2 - 4*a*c)^(5/2) + 48*B*c^4*d^4*e^2*x*(b^2 - 4*a*c)^(3/2) - 64*A*a*b^2*c^4*d^3*e^3 - 96*A*a*b^3*c^3*d
^2*e^4 + 1408*A*a^2*b*c^4*d^2*e^4 - 928*A*a^2*b^2*c^3*d*e^5 - 96*B*a*b^2*c^4*d^4*e^2 - 32*B*a*b^3*c^3*d^3*e^3
+ 64*B*a*b^4*c^2*d^2*e^4 + 128*B*a^2*b*c^4*d^3*e^3 - 144*B*a^2*b^3*c^2*d*e^5 - 256*A*a^2*b^2*c^3*e^6*x - 160*B
*a^2*b^3*c^2*e^6*x - 1024*A*a^2*c^5*d^2*e^4*x - 256*A*b^2*c^5*d^4*e^2*x + 512*A*b^3*c^4*d^3*e^3*x - 448*A*b^4*
c^3*d^2*e^4*x + 1536*B*a^2*c^5*d^3*e^3*x - 32*B*b^3*c^4*d^4*e^2*x + 30*A*b*c*d*e^5*(b^2 - 4*a*c)^(5/2) + 18*A*
b*c*e^6*x*(b^2 - 4*a*c)^(5/2) - 192*B*a*b*c^5*d^5*e - 16*B*a*b^5*c*d*e^5 - 24*A*b^2*c^2*d^2*e^4*(b^2 - 4*a*c)^
(3/2) + 80*A*b^2*c^4*d^4*e^2*(b^2 - 4*a*c)^(1/2) - 80*A*b^3*c^3*d^3*e^3*(b^2 - 4*a*c)^(1/2) + 40*A*b^4*c^2*d^2
*e^4*(b^2 - 4*a*c)^(1/2) - 88*B*b^2*c^2*d^3*e^3*(b^2 - 4*a*c)^(3/2) - 40*B*b^3*c^3*d^4*e^2*(b^2 - 4*a*c)^(1/2)
+ 40*B*b^4*c^2*d^3*e^3*(b^2 - 4*a*c)^(1/2) + 32*B*a*b^5*c*e^6*x - 256*B*a*c^6*d^5*e*x + 64*B*a^2*b^2*c^3*d^2*
e^4 - 32*A*b*c^5*d^5*e*(b^2 - 4*a*c)^(1/2) - 4*A*b^3*c*d*e^5*(b^2 - 4*a*c)^(3/2) - 10*A*b^5*c*d*e^5*(b^2 - 4*a
*c)^(1/2) - 28*B*b*c*d^2*e^4*(b^2 - 4*a*c)^(5/2) + 12*A*b^3*c*e^6*x*(b^2 - 4*a*c)^(3/2) + 2*A*b^5*c*e^6*x*(b^2
- 4*a*c)^(1/2) - 36*A*c^2*d*e^5*x*(b^2 - 4*a*c)^(5/2) - 64*A*c^6*d^5*e*x*(b^2 - 4*a*c)^(1/2) + 192*A*a*b*c^5*
d^4*e^2 + 128*A*a*b^4*c^2*d*e^5 + 832*B*a^3*b*c^3*d*e^5 + 192*A*a*b^4*c^2*e^6*x + 128*B*a^3*b*c^3*e^6*x + 1024
*A*a*c^6*d^4*e^2*x + 192*A*b^5*c^2*d*e^5*x - 256*B*a^3*c^4*d*e^5*x + 64*B*b^2*c^5*d^5*e*x + 80*A*b*c^3*d^3*e^3
*(b^2 - 4*a*c)^(3/2) + 56*B*b*c^3*d^4*e^2*(b^2 - 4*a*c)^(3/2) + 16*B*b^2*c^4*d^5*e*(b^2 - 4*a*c)^(1/2) + 48*B*
b^3*c*d^2*e^4*(b^2 - 4*a*c)^(3/2) - 20*B*b^5*c*d^2*e^4*(b^2 - 4*a*c)^(1/2) + 32*B*b*c^5*d^5*e*x*(b^2 - 4*a*c)^
(1/2) + 12*B*b^3*c*d*e^5*x*(b^2 - 4*a*c)^(3/2) + 10*B*b^5*c*d*e^5*x*(b^2 - 4*a*c)^(1/2) - 2048*A*a*b*c^5*d^3*e
^3*x - 1024*A*a*b^3*c^3*d*e^5*x + 1024*A*a^2*b*c^4*d*e^5*x + 128*B*a*b*c^5*d^4*e^2*x - 192*B*a*b^4*c^2*d*e^5*x
+ 144*A*b*c^3*d^2*e^4*x*(b^2 - 4*a*c)^(3/2) + 160*A*b*c^5*d^4*e^2*x*(b^2 - 4*a*c)^(1/2) - 72*A*b^2*c^2*d*e^5*
x*(b^2 - 4*a*c)^(3/2) - 20*A*b^4*c^2*d*e^5*x*(b^2 - 4*a*c)^(1/2) - 48*B*b*c^3*d^3*e^3*x*(b^2 - 4*a*c)^(3/2) +
2048*A*a*b^2*c^4*d^2*e^4*x - 384*B*a*b^2*c^4*d^3*e^3*x + 448*B*a*b^3*c^3*d^2*e^4*x - 1792*B*a^2*b*c^4*d^2*e^4*
x + 832*B*a^2*b^2*c^3*d*e^5*x - 22*B*b*c*d*e^5*x*(b^2 - 4*a*c)^(5/2) - 160*A*b^2*c^4*d^3*e^3*x*(b^2 - 4*a*c)^(
1/2) + 80*A*b^3*c^3*d^2*e^4*x*(b^2 - 4*a*c)^(1/2) - 80*B*b^2*c^4*d^4*e^2*x*(b^2 - 4*a*c)^(1/2) + 80*B*b^3*c^3*
d^3*e^3*x*(b^2 - 4*a*c)^(1/2) - 40*B*b^4*c^2*d^2*e^4*x*(b^2 - 4*a*c)^(1/2)))/(64*(4*a*c - b^2)^2*(a*e^2 + c*d^
2 - b*d*e)^6) + (c^2*e*(A^2*b^3*e^4 + A*B*c^3*d^4 + B^2*a^2*b*e^4 - 2*A^2*c^3*d^3*e + 2*A^2*a*c^2*d*e^3 - 4*A^
2*b^2*c*d*e^3 + 2*B^2*a*c^2*d^3*e - 2*B^2*a^2*c*d*e^3 + 5*A^2*b*c^2*d^2*e^2 - 2*A*B*a*b^2*e^4 + A*B*a^2*c*e^4
- A^2*a*b*c*e^4 - 2*A*B*b*c^2*d^3*e - 6*A*B*a*c^2*d^2*e^2 + A*B*b^2*c*d^2*e^2 - B^2*a*b*c*d^2*e^2 + 6*A*B*a*b*
c*d*e^3))/(a*e^2 + c*d^2 - b*d*e)^4 + (c^3*e*x*(A*b*e^2 - B*a*e^2 + B*c*d^2 - 2*A*c*d*e)^2)/(a*e^2 + c*d^2 - b
*d*e)^4)*((A*e^3*(4*a*c - b^2)^2)/4 - (3*A*b*e^3*(b^2 - 4*a*c)^(3/2))/4 + (B*a*e^3*(b^2 - 4*a*c)^(3/2))/2 - (3
*A*b^2*e^3*(4*a*c - b^2))/4 - (A*b^3*e^3*(b^2 - 4*a*c)^(1/2))/4 + 2*A*c^3*d^3*(b^2 - 4*a*c)^(1/2) + B*c^2*d^3*
(4*a*c - b^2) - (3*B*d*e^2*(4*a*c - b^2)^2)/4 + (B*a*b^2*e^3*(b^2 - 4*a*c)^(1/2))/2 - B*b*c^2*d^3*(b^2 - 4*a*c
)^(1/2) - 3*A*c^2*d^2*e*(4*a*c - b^2) - (3*B*b^2*d*e^2*(4*a*c - b^2))/4 - (3*B*b^3*d*e^2*(b^2 - 4*a*c)^(1/2))/
4 + B*a*b*e^3*(4*a*c - b^2) + (3*A*c*d*e^2*(b^2 - 4*a*c)^(3/2))/2 + (3*B*b*d*e^2*(b^2 - 4*a*c)^(3/2))/4 - (3*B
*c*d^2*e*(b^2 - 4*a*c)^(3/2))/2 + 3*A*b*c*d*e^2*(4*a*c - b^2) - 3*A*b*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + (3*A*b^2
*c*d*e^2*(b^2 - 4*a*c)^(1/2))/2 + (3*B*b^2*c*d^2*e*(b^2 - 4*a*c)^(1/2))/2))/((4*a*c - b^2)*((4*a*c - b^2)*((3*
a*d^2*e^4)/2 - 3*b*d^3*e^3 + (3*c*d^4*e^2)/2) + 2*a^3*e^6 + 2*c^3*d^6 - 5*b^3*d^3*e^3 + (15*a*b^2*d^2*e^4)/2 +
(15*b^2*c*d^4*e^2)/2 - 6*a^2*b*d*e^5 - 6*b*c^2*d^5*e)) - (log(d + e*x)*(e^2*(3*A*b*c*d - 3*B*a*c*d) + e^3*(A*
a*c - A*b^2 + B*a*b) + B*c^2*d^3 - 3*A*c^2*d^2*e))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^
2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3) - ((A*a*e^3 -
3*B*c*d^3 - 3*A*b*d*e^2 + B*a*d*e^2 + 5*A*c*d^2*e + B*b*d^2*e)/(2*(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)) - (x*(A*b*e^3 - B*a*e^3 - 2*A*c*d*e^2 + B*c*d^2*e))/(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))/(d^2 + e^2*x^2 + 2*d*e*x) - (log((c^2*e*(A^2*b^3*e^4
+ A*B*c^3*d^4 + B^2*a^2*b*e^4 - 2*A^2*c^3*d^3*e + 2*A^2*a*c^2*d*e^3 - 4*A^2*b^2*c*d*e^3 + 2*B^2*a*c^2*d^3*e -
2*B^2*a^2*c*d*e^3 + 5*A^2*b*c^2*d^2*e^2 - 2*A*B*a*b^2*e^4 + A*B*a^2*c*e^4 - A^2*a*b*c*e^4 - 2*A*B*b*c^2*d^3*e
- 6*A*B*a*c^2*d^2*e^2 + A*B*b^2*c*d^2*e^2 - B^2*a*b*c*d^2*e^2 + 6*A*B*a*b*c*d*e^3))/(a*e^2 + c*d^2 - b*d*e)^4
- ((A*b^4*e^3 + (3*A*b*e^3*(b^2 - 4*a*c)^(3/2))/4 - (B*a*e^3*(b^2 - 4*a*c)^(3/2))/2 - B*a*b^3*e^3 + 4*B*a*c^3*
d^3 + (A*b^3*e^3*(b^2 - 4*a*c)^(1/2))/4 - 2*A*c^3*d^3*(b^2 - 4*a*c)^(1/2) + 4*A*a^2*c^2*e^3 - B*b^2*c^2*d^3 -
(B*a*b^2*e^3*(b^2 - 4*a*c)^(1/2))/2 + B*b*c^2*d^3*(b^2 - 4*a*c)^(1/2) + (3*B*b^3*d*e^2*(b^2 - 4*a*c)^(1/2))/4
+ 3*A*b^2*c^2*d^2*e - 12*B*a^2*c^2*d*e^2 - (3*A*c*d*e^2*(b^2 - 4*a*c)^(3/2))/2 - (3*B*b*d*e^2*(b^2 - 4*a*c)^(3
/2))/4 + (3*B*c*d^2*e*(b^2 - 4*a*c)^(3/2))/2 - 5*A*a*b^2*c*e^3 + 4*B*a^2*b*c*e^3 - 12*A*a*c^3*d^2*e - 3*A*b^3*
c*d*e^2 + 12*A*a*b*c^2*d*e^2 + 3*B*a*b^2*c*d*e^2 + 3*A*b*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - (3*A*b^2*c*d*e^2*(b^2
- 4*a*c)^(1/2))/2 - (3*B*b^2*c*d^2*e*(b^2 - 4*a*c)^(1/2))/2)*(4*B*d*e^5*(b^2 - 4*a*c)^(7/2) + 3*B*e^6*x*(b^2
- 4*a*c)^(7/2) - 3*A*b^2*e^6*(b^2 - 4*a*c)^(5/2) + 2*A*b^4*e^6*(b^2 - 4*a*c)^(3/2) + A*b^6*e^6*(b^2 - 4*a*c)^(
1/2) - 128*B*a^4*c^3*e^6 + 32*A*b^6*c*e^6*x - 2*B*a*b^5*e^6*(b^2 - 4*a*c)^(1/2) + B*b^2*d*e^5*(b^2 - 4*a*c)^(5
/2) - 10*B*b^4*d*e^5*(b^2 - 4*a*c)^(3/2) + 5*B*b^6*d*e^5*(b^2 - 4*a*c)^(1/2) + B*b^2*e^6*x*(b^2 - 4*a*c)^(5/2)
- 3*B*b^4*e^6*x*(b^2 - 4*a*c)^(3/2) - B*b^6*e^6*x*(b^2 - 4*a*c)^(1/2) + 320*A*a^3*b*c^3*e^6 - 48*B*a^2*b^4*c*
e^6 - 640*A*a^3*c^4*d*e^5 + 32*A*b^2*c^5*d^5*e - 48*B*b^3*c^4*d^5*e - 48*A*c^2*d^2*e^4*(b^2 - 4*a*c)^(5/2) - 6
4*A*c^4*d^4*e^2*(b^2 - 4*a*c)^(3/2) + 48*B*c^2*d^3*e^3*(b^2 - 4*a*c)^(5/2) - 272*A*a^2*b^3*c^2*e^6 + 224*B*a^3
*b^2*c^2*e^6 + 1280*A*a^2*c^5*d^3*e^3 + 48*A*b^3*c^4*d^4*e^2 - 96*A*b^4*c^3*d^3*e^3 + 64*A*b^5*c^2*d^2*e^4 - 6
40*B*a^2*c^5*d^4*e^2 + 1280*B*a^3*c^4*d^2*e^4 + 16*B*b^4*c^3*d^4*e^2 + 2*B*a*b*e^6*(b^2 - 4*a*c)^(5/2) + 48*A*
a*b^5*c*e^6 - 128*A*a*c^6*d^5*e - 16*A*b^6*c*d*e^5 - 96*A*c^4*d^3*e^3*x*(b^2 - 4*a*c)^(3/2) + 40*B*c^2*d^2*e^4
*x*(b^2 - 4*a*c)^(5/2) + 48*B*c^4*d^4*e^2*x*(b^2 - 4*a*c)^(3/2) + 64*A*a*b^2*c^4*d^3*e^3 + 96*A*a*b^3*c^3*d^2*
e^4 - 1408*A*a^2*b*c^4*d^2*e^4 + 928*A*a^2*b^2*c^3*d*e^5 + 96*B*a*b^2*c^4*d^4*e^2 + 32*B*a*b^3*c^3*d^3*e^3 - 6
4*B*a*b^4*c^2*d^2*e^4 - 128*B*a^2*b*c^4*d^3*e^3 + 144*B*a^2*b^3*c^2*d*e^5 + 256*A*a^2*b^2*c^3*e^6*x + 160*B*a^
2*b^3*c^2*e^6*x + 1024*A*a^2*c^5*d^2*e^4*x + 256*A*b^2*c^5*d^4*e^2*x - 512*A*b^3*c^4*d^3*e^3*x + 448*A*b^4*c^3
*d^2*e^4*x - 1536*B*a^2*c^5*d^3*e^3*x + 32*B*b^3*c^4*d^4*e^2*x + 30*A*b*c*d*e^5*(b^2 - 4*a*c)^(5/2) + 18*A*b*c
*e^6*x*(b^2 - 4*a*c)^(5/2) + 192*B*a*b*c^5*d^5*e + 16*B*a*b^5*c*d*e^5 - 24*A*b^2*c^2*d^2*e^4*(b^2 - 4*a*c)^(3/
2) + 80*A*b^2*c^4*d^4*e^2*(b^2 - 4*a*c)^(1/2) - 80*A*b^3*c^3*d^3*e^3*(b^2 - 4*a*c)^(1/2) + 40*A*b^4*c^2*d^2*e^
4*(b^2 - 4*a*c)^(1/2) - 88*B*b^2*c^2*d^3*e^3*(b^2 - 4*a*c)^(3/2) - 40*B*b^3*c^3*d^4*e^2*(b^2 - 4*a*c)^(1/2) +
40*B*b^4*c^2*d^3*e^3*(b^2 - 4*a*c)^(1/2) - 32*B*a*b^5*c*e^6*x + 256*B*a*c^6*d^5*e*x - 64*B*a^2*b^2*c^3*d^2*e^4
- 32*A*b*c^5*d^5*e*(b^2 - 4*a*c)^(1/2) - 4*A*b^3*c*d*e^5*(b^2 - 4*a*c)^(3/2) - 10*A*b^5*c*d*e^5*(b^2 - 4*a*c)
^(1/2) - 28*B*b*c*d^2*e^4*(b^2 - 4*a*c)^(5/2) + 12*A*b^3*c*e^6*x*(b^2 - 4*a*c)^(3/2) + 2*A*b^5*c*e^6*x*(b^2 -
4*a*c)^(1/2) - 36*A*c^2*d*e^5*x*(b^2 - 4*a*c)^(5/2) - 64*A*c^6*d^5*e*x*(b^2 - 4*a*c)^(1/2) - 192*A*a*b*c^5*d^4
*e^2 - 128*A*a*b^4*c^2*d*e^5 - 832*B*a^3*b*c^3*d*e^5 - 192*A*a*b^4*c^2*e^6*x - 128*B*a^3*b*c^3*e^6*x - 1024*A*
a*c^6*d^4*e^2*x - 192*A*b^5*c^2*d*e^5*x + 256*B*a^3*c^4*d*e^5*x - 64*B*b^2*c^5*d^5*e*x + 80*A*b*c^3*d^3*e^3*(b
^2 - 4*a*c)^(3/2) + 56*B*b*c^3*d^4*e^2*(b^2 - 4*a*c)^(3/2) + 16*B*b^2*c^4*d^5*e*(b^2 - 4*a*c)^(1/2) + 48*B*b^3
*c*d^2*e^4*(b^2 - 4*a*c)^(3/2) - 20*B*b^5*c*d^2*e^4*(b^2 - 4*a*c)^(1/2) + 32*B*b*c^5*d^5*e*x*(b^2 - 4*a*c)^(1/
2) + 12*B*b^3*c*d*e^5*x*(b^2 - 4*a*c)^(3/2) + 10*B*b^5*c*d*e^5*x*(b^2 - 4*a*c)^(1/2) + 2048*A*a*b*c^5*d^3*e^3*
x + 1024*A*a*b^3*c^3*d*e^5*x - 1024*A*a^2*b*c^4*d*e^5*x - 128*B*a*b*c^5*d^4*e^2*x + 192*B*a*b^4*c^2*d*e^5*x +
144*A*b*c^3*d^2*e^4*x*(b^2 - 4*a*c)^(3/2) + 160*A*b*c^5*d^4*e^2*x*(b^2 - 4*a*c)^(1/2) - 72*A*b^2*c^2*d*e^5*x*(
b^2 - 4*a*c)^(3/2) - 20*A*b^4*c^2*d*e^5*x*(b^2 - 4*a*c)^(1/2) - 48*B*b*c^3*d^3*e^3*x*(b^2 - 4*a*c)^(3/2) - 204
8*A*a*b^2*c^4*d^2*e^4*x + 384*B*a*b^2*c^4*d^3*e^3*x - 448*B*a*b^3*c^3*d^2*e^4*x + 1792*B*a^2*b*c^4*d^2*e^4*x -
832*B*a^2*b^2*c^3*d*e^5*x - 22*B*b*c*d*e^5*x*(b^2 - 4*a*c)^(5/2) - 160*A*b^2*c^4*d^3*e^3*x*(b^2 - 4*a*c)^(1/2
) + 80*A*b^3*c^3*d^2*e^4*x*(b^2 - 4*a*c)^(1/2) - 80*B*b^2*c^4*d^4*e^2*x*(b^2 - 4*a*c)^(1/2) + 80*B*b^3*c^3*d^3
*e^3*x*(b^2 - 4*a*c)^(1/2) - 40*B*b^4*c^2*d^2*e^4*x*(b^2 - 4*a*c)^(1/2)))/(64*(4*a*c - b^2)^2*(a*e^2 + c*d^2 -
b*d*e)^6) + (c^3*e*x*(A*b*e^2 - B*a*e^2 + B*c*d^2 - 2*A*c*d*e)^2)/(a*e^2 + c*d^2 - b*d*e)^4)*((B*a*e^3*(b^2 -
4*a*c)^(3/2))/2 - (3*A*b*e^3*(b^2 - 4*a*c)^(3/2))/4 - (A*e^3*(4*a*c - b^2)^2)/4 + (3*A*b^2*e^3*(4*a*c - b^2))
/4 - (A*b^3*e^3*(b^2 - 4*a*c)^(1/2))/4 + 2*A*c^3*d^3*(b^2 - 4*a*c)^(1/2) - B*c^2*d^3*(4*a*c - b^2) + (3*B*d*e^
2*(4*a*c - b^2)^2)/4 + (B*a*b^2*e^3*(b^2 - 4*a*c)^(1/2))/2 - B*b*c^2*d^3*(b^2 - 4*a*c)^(1/2) + 3*A*c^2*d^2*e*(
4*a*c - b^2) + (3*B*b^2*d*e^2*(4*a*c - b^2))/4 - (3*B*b^3*d*e^2*(b^2 - 4*a*c)^(1/2))/4 - B*a*b*e^3*(4*a*c - b^
2) + (3*A*c*d*e^2*(b^2 - 4*a*c)^(3/2))/2 + (3*B*b*d*e^2*(b^2 - 4*a*c)^(3/2))/4 - (3*B*c*d^2*e*(b^2 - 4*a*c)^(3
/2))/2 - 3*A*b*c*d*e^2*(4*a*c - b^2) - 3*A*b*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + (3*A*b^2*c*d*e^2*(b^2 - 4*a*c)^(1
/2))/2 + (3*B*b^2*c*d^2*e*(b^2 - 4*a*c)^(1/2))/2))/((4*a*c - b^2)*((4*a*c - b^2)*((3*a*d^2*e^4)/2 - 3*b*d^3*e^
3 + (3*c*d^4*e^2)/2) + 2*a^3*e^6 + 2*c^3*d^6 - 5*b^3*d^3*e^3 + (15*a*b^2*d^2*e^4)/2 + (15*b^2*c*d^4*e^2)/2 - 6
*a^2*b*d*e^5 - 6*b*c^2*d^5*e))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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