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

Optimal. Leaf size=255 \[ -\frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-b \left (a B e^2+2 A c d e+B c d^2\right )+2 c \left (-a A e^2+2 a B d e+A c d^2\right )+A b^2 e^2\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^2}-\frac {\log \left (a+b x+c x^2\right ) \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )}{2 \left (a e^2-b d e+c d^2\right )^2}+\frac {B d-A e}{(d+e x) \left (a e^2-b d e+c d^2\right )}+\frac {\log (d+e x) \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )}{\left (a e^2-b d e+c d^2\right )^2} \]

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Rubi [A]  time = 0.44, antiderivative size = 255, 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} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-b \left (a B e^2+2 A c d e+B c d^2\right )+2 c \left (-a A e^2+2 a B d e+A c d^2\right )+A b^2 e^2\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^2}-\frac {\log \left (a+b x+c x^2\right ) \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )}{2 \left (a e^2-b d e+c d^2\right )^2}+\frac {B d-A e}{(d+e x) \left (a e^2-b d e+c d^2\right )}+\frac {\log (d+e x) \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )}{\left (a e^2-b d e+c d^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B*d - A*e)/((c*d^2 - b*d*e + a*e^2)*(d + e*x)) - ((A*b^2*e^2 + 2*c*(A*c*d^2 + 2*a*B*d*e - a*A*e^2) - b*(B*c*d
^2 + 2*A*c*d*e + a*B*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)^
2) + ((A*e*(2*c*d - b*e) - B*(c*d^2 - a*e^2))*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^2 - ((A*e*(2*c*d - b*e) -
B*(c*d^2 - a*e^2))*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^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 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)^2 \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)^2}+\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)}+\frac {a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )-c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) x}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {B d-A e}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {\left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^2}+\frac {\int \frac {a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )-c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) x}{a+b x+c x^2} \, dx}{\left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {B d-A e}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {\left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^2}-\frac {\left (A e (2 c d-b e)-B \left (c d^2-a e^2\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 )^2}+\frac {\left (A b^2 e^2+2 c \left (A c d^2+2 a B d e-a A e^2\right )-b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {B d-A e}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {\left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^2}-\frac {\left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (A b^2 e^2+2 c \left (A c d^2+2 a B d e-a A e^2\right )-b \left (B c d^2+2 A c d e+a B e^2\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 )^2}\\ &=\frac {B d-A e}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (A b^2 e^2+2 c \left (A c d^2+2 a B d e-a A e^2\right )-b \left (B c d^2+2 A c d e+a B e^2\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 )^2}+\frac {\left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^2}-\frac {\left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^2}\\ \end {align*}

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Mathematica [A]  time = 0.34, size = 219, normalized size = 0.86 \begin {gather*} \frac {\frac {2 \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right ) \left (-b \left (a B e^2+2 A c d e+B c d^2\right )+2 c \left (-a A e^2+2 a B d e+A c d^2\right )+A b^2 e^2\right )}{\sqrt {4 a c-b^2}}-2 \log (d+e x) \left (B \left (c d^2-a e^2\right )+A e (b e-2 c d)\right )+\log (a+x (b+c x)) \left (B \left (c d^2-a e^2\right )+A e (b e-2 c d)\right )+\frac {2 (B d-A e) \left (e (a e-b d)+c d^2\right )}{d+e x}}{2 \left (e (a e-b d)+c d^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 62.07, size = 1594, normalized size = 6.25

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*(B*b^2*c - 4*B*a*c^2)*d^3 - 2*(B*b^3 - 4*A*a*c^2 - (4*B*a*b - A*b^2)*c)*d^2*e + 2*(B*a*b^2 + A*b^3 - 4
*(B*a^2 + A*a*b)*c)*d*e^2 - 2*(A*a*b^2 - 4*A*a^2*c)*e^3 - (2*(2*B*a - A*b)*c*d^2*e - (B*b*c - 2*A*c^2)*d^3 - (
B*a*b - A*b^2 + 2*A*a*c)*d*e^2 + (2*(2*B*a - A*b)*c*d*e^2 - (B*b*c - 2*A*c^2)*d^2*e - (B*a*b - A*b^2 + 2*A*a*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)) + ((B*b^2*c - 4*B*a*c^2)*d^3 - 2*(A*b^2*c - 4*A*a*c^2)*d^2*e - (B*a*b^2 - A*b^3 - 4*(B*a^2 - A*a*b)*c
)*d*e^2 + ((B*b^2*c - 4*B*a*c^2)*d^2*e - 2*(A*b^2*c - 4*A*a*c^2)*d*e^2 - (B*a*b^2 - A*b^3 - 4*(B*a^2 - A*a*b)*
c)*e^3)*x)*log(c*x^2 + b*x + a) - 2*((B*b^2*c - 4*B*a*c^2)*d^3 - 2*(A*b^2*c - 4*A*a*c^2)*d^2*e - (B*a*b^2 - A*
b^3 - 4*(B*a^2 - A*a*b)*c)*d*e^2 + ((B*b^2*c - 4*B*a*c^2)*d^2*e - 2*(A*b^2*c - 4*A*a*c^2)*d*e^2 - (B*a*b^2 - A
*b^3 - 4*(B*a^2 - A*a*b)*c)*e^3)*x)*log(e*x + d))/((b^2*c^2 - 4*a*c^3)*d^5 - 2*(b^3*c - 4*a*b*c^2)*d^4*e + (b^
4 - 2*a*b^2*c - 8*a^2*c^2)*d^3*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d^2*e^3 + (a^2*b^2 - 4*a^3*c)*d*e^4 + ((b^2*c^2 - 4
*a*c^3)*d^4*e - 2*(b^3*c - 4*a*b*c^2)*d^3*e^2 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^3 - 2*(a*b^3 - 4*a^2*b*c)*
d*e^4 + (a^2*b^2 - 4*a^3*c)*e^5)*x), 1/2*(2*(B*b^2*c - 4*B*a*c^2)*d^3 - 2*(B*b^3 - 4*A*a*c^2 - (4*B*a*b - A*b^
2)*c)*d^2*e + 2*(B*a*b^2 + A*b^3 - 4*(B*a^2 + A*a*b)*c)*d*e^2 - 2*(A*a*b^2 - 4*A*a^2*c)*e^3 - 2*(2*(2*B*a - A*
b)*c*d^2*e - (B*b*c - 2*A*c^2)*d^3 - (B*a*b - A*b^2 + 2*A*a*c)*d*e^2 + (2*(2*B*a - A*b)*c*d*e^2 - (B*b*c - 2*A
*c^2)*d^2*e - (B*a*b - A*b^2 + 2*A*a*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)) + ((B*b^2*c - 4*B*a*c^2)*d^3 - 2*(A*b^2*c - 4*A*a*c^2)*d^2*e - (B*a*b^2 - A*b^3 - 4*(B*a^2 - A*a*b)
*c)*d*e^2 + ((B*b^2*c - 4*B*a*c^2)*d^2*e - 2*(A*b^2*c - 4*A*a*c^2)*d*e^2 - (B*a*b^2 - A*b^3 - 4*(B*a^2 - A*a*b
)*c)*e^3)*x)*log(c*x^2 + b*x + a) - 2*((B*b^2*c - 4*B*a*c^2)*d^3 - 2*(A*b^2*c - 4*A*a*c^2)*d^2*e - (B*a*b^2 -
A*b^3 - 4*(B*a^2 - A*a*b)*c)*d*e^2 + ((B*b^2*c - 4*B*a*c^2)*d^2*e - 2*(A*b^2*c - 4*A*a*c^2)*d*e^2 - (B*a*b^2 -
A*b^3 - 4*(B*a^2 - A*a*b)*c)*e^3)*x)*log(e*x + d))/((b^2*c^2 - 4*a*c^3)*d^5 - 2*(b^3*c - 4*a*b*c^2)*d^4*e + (
b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^3*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d^2*e^3 + (a^2*b^2 - 4*a^3*c)*d*e^4 + ((b^2*c^2 -
4*a*c^3)*d^4*e - 2*(b^3*c - 4*a*b*c^2)*d^3*e^2 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^3 - 2*(a*b^3 - 4*a^2*b*c
)*d*e^4 + (a^2*b^2 - 4*a^3*c)*e^5)*x)]

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giac [A]  time = 0.17, size = 383, normalized size = 1.50 \begin {gather*} -\frac {{\left (B b c d^{2} e^{2} - 2 \, A c^{2} d^{2} e^{2} - 4 \, B a c d e^{3} + 2 \, A b c d e^{3} + B a b e^{4} - A b^{2} e^{4} + 2 \, A a c e^{4}\right )} \arctan \left (\frac {{\left (2 \, c d - \frac {2 \, c d^{2}}{x e + d} - b e + \frac {2 \, b d e}{x e + d} - \frac {2 \, a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt {-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\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 )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (B c d^{2} - 2 \, A c d e - B a e^{2} + A b e^{2}\right )} \log \left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\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 {\frac {B d e^{2}}{x e + d} - \frac {A e^{3}}{x e + d}}{c d^{2} e^{2} - b d e^{3} + a e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(B*b*c*d^2*e^2 - 2*A*c^2*d^2*e^2 - 4*B*a*c*d*e^3 + 2*A*b*c*d*e^3 + B*a*b*e^4 - A*b^2*e^4 + 2*A*a*c*e^4)*arcta
n((2*c*d - 2*c*d^2/(x*e + d) - b*e + 2*b*d*e/(x*e + d) - 2*a*e^2/(x*e + d))*e^(-1)/sqrt(-b^2 + 4*a*c))*e^(-2)/
((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)*sqrt(-b^2 + 4*a*c)) + 1/2*(B*c*
d^2 - 2*A*c*d*e - B*a*e^2 + A*b*e^2)*log(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e
+ d)^2 + a*e^2/(x*e + d)^2)/(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) + (B
*d*e^2/(x*e + d) - A*e^3/(x*e + d))/(c*d^2*e^2 - b*d*e^3 + a*e^4)

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maple [B]  time = 0.06, size = 729, normalized size = 2.86 \begin {gather*} -\frac {2 A a c \,e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {4 a c -b^{2}}}+\frac {A \,b^{2} e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {4 a c -b^{2}}}-\frac {2 A b c d e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {4 a c -b^{2}}}+\frac {2 A \,c^{2} d^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {4 a c -b^{2}}}-\frac {B a b \,e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {4 a c -b^{2}}}+\frac {4 B a c d e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {4 a c -b^{2}}}-\frac {B b c \,d^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {4 a c -b^{2}}}-\frac {A b \,e^{2} \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {A b \,e^{2} \ln \left (c \,x^{2}+b x +a \right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {2 A c d e \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}-\frac {A c d e \ln \left (c \,x^{2}+b x +a \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {B a \,e^{2} \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}-\frac {B a \,e^{2} \ln \left (c \,x^{2}+b x +a \right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}-\frac {B c \,d^{2} \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {B c \,d^{2} \ln \left (c \,x^{2}+b x +a \right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}-\frac {A e}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )}+\frac {B d}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^2/(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 = 7.22, size = 2650, normalized size = 10.39

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(2*A*a*b^3*e^4 + A*b*c^3*d^4 + 6*B*a*c^3*d^4 + 6*B*a^3*c*e^4 + 2*A*b^4*e^4*x + 2*A*c^4*d^4*x - A*c^3*d^4*(
b^2 - 4*a*c)^(1/2) - 2*B*a^2*b^2*e^4 - 2*B*b^2*c^2*d^4 - 2*B*a*b^3*e^4*x - B*b*c^3*d^4*x + 2*A*a*b^2*e^4*(b^2
- 4*a*c)^(1/2) - A*a^2*c*e^4*(b^2 - 4*a*c)^(1/2) - 2*B*a^2*b*e^4*(b^2 - 4*a*c)^(1/2) + 2*B*b*c^2*d^4*(b^2 - 4*
a*c)^(1/2) + 2*A*b^3*e^4*x*(b^2 - 4*a*c)^(1/2) + 3*B*c^3*d^4*x*(b^2 - 4*a*c)^(1/2) + 16*A*a^2*c^2*d*e^3 + 2*A*
b^2*c^2*d^3*e - A*b^3*c*d^2*e^2 + 2*A*a^2*c^2*e^4*x - 20*B*a^2*c^2*d^2*e^2 - 7*A*a^2*b*c*e^4 - 16*A*a*c^3*d^3*
e + 10*A*b^2*c^2*d^2*e^2*x - 6*A*a*b^2*c*d*e^3 + 4*B*a*b*c^2*d^3*e + 4*B*a^2*b*c*d*e^3 - 8*A*a*b^2*c*e^4*x + 7
*B*a^2*b*c*e^4*x - 4*A*b*c^3*d^3*e*x - 8*A*b^3*c*d*e^3*x + 16*B*a*c^3*d^3*e*x - 2*A*b*c^2*d^3*e*(b^2 - 4*a*c)^
(1/2) - 8*B*a*c^2*d^3*e*(b^2 - 4*a*c)^(1/2) + 8*B*a^2*c*d*e^3*(b^2 - 4*a*c)^(1/2) - 2*B*a*b^2*e^4*x*(b^2 - 4*a
*c)^(1/2) + 3*B*a^2*c*e^4*x*(b^2 - 4*a*c)^(1/2) - 8*A*c^3*d^3*e*x*(b^2 - 4*a*c)^(1/2) + 10*A*a*b*c^2*d^2*e^2 +
2*B*a*b^2*c*d^2*e^2 - 28*A*a*c^3*d^2*e^2*x - 16*B*a^2*c^2*d*e^3*x - 2*B*b^2*c^2*d^3*e*x + B*b^3*c*d^2*e^2*x +
14*A*a*c^2*d^2*e^2*(b^2 - 4*a*c)^(1/2) + A*b^2*c*d^2*e^2*(b^2 - 4*a*c)^(1/2) + 8*A*a*c^2*d*e^3*x*(b^2 - 4*a*c
)^(1/2) - 8*A*b^2*c*d*e^3*x*(b^2 - 4*a*c)^(1/2) - 2*B*b*c^2*d^3*e*x*(b^2 - 4*a*c)^(1/2) - 10*B*a*b*c^2*d^2*e^2
*x + 12*A*b*c^2*d^2*e^2*x*(b^2 - 4*a*c)^(1/2) - 10*B*a*c^2*d^2*e^2*x*(b^2 - 4*a*c)^(1/2) + B*b^2*c*d^2*e^2*x*(
b^2 - 4*a*c)^(1/2) - 10*A*a*b*c*d*e^3*(b^2 - 4*a*c)^(1/2) - 4*A*a*b*c*e^4*x*(b^2 - 4*a*c)^(1/2) + 28*A*a*b*c^2
*d*e^3*x + 6*B*a*b^2*c*d*e^3*x + 6*B*a*b*c*d*e^3*x*(b^2 - 4*a*c)^(1/2))*(B*a*b^2*e^2 - A*b^3*e^2 + 4*B*a*c^2*d
^2 - 4*B*a^2*c*e^2 - B*b^2*c*d^2 - A*b^2*e^2*(b^2 - 4*a*c)^(1/2) - 2*A*c^2*d^2*(b^2 - 4*a*c)^(1/2) + 4*A*a*b*c
*e^2 - 8*A*a*c^2*d*e + 2*A*b^2*c*d*e + 2*A*a*c*e^2*(b^2 - 4*a*c)^(1/2) + B*a*b*e^2*(b^2 - 4*a*c)^(1/2) + B*b*c
*d^2*(b^2 - 4*a*c)^(1/2) + 2*A*b*c*d*e*(b^2 - 4*a*c)^(1/2) - 4*B*a*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*(4*a*c^3*d^4
+ 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^2*d^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e^3 + 2*b^3*c*d^3*e -
8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 2*a*b^2*c*d^2*e^2)) - (log(d + e*x)*(e^2*(A*b - B*a) + B*c*d^2 - 2*A*c*d*
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) - (log(2*A*a*b^3*e^4 + A*b*c
^3*d^4 + 6*B*a*c^3*d^4 + 6*B*a^3*c*e^4 + 2*A*b^4*e^4*x + 2*A*c^4*d^4*x + A*c^3*d^4*(b^2 - 4*a*c)^(1/2) - 2*B*a
^2*b^2*e^4 - 2*B*b^2*c^2*d^4 - 2*B*a*b^3*e^4*x - B*b*c^3*d^4*x - 2*A*a*b^2*e^4*(b^2 - 4*a*c)^(1/2) + A*a^2*c*e
^4*(b^2 - 4*a*c)^(1/2) + 2*B*a^2*b*e^4*(b^2 - 4*a*c)^(1/2) - 2*B*b*c^2*d^4*(b^2 - 4*a*c)^(1/2) - 2*A*b^3*e^4*x
*(b^2 - 4*a*c)^(1/2) - 3*B*c^3*d^4*x*(b^2 - 4*a*c)^(1/2) + 16*A*a^2*c^2*d*e^3 + 2*A*b^2*c^2*d^3*e - A*b^3*c*d^
2*e^2 + 2*A*a^2*c^2*e^4*x - 20*B*a^2*c^2*d^2*e^2 - 7*A*a^2*b*c*e^4 - 16*A*a*c^3*d^3*e + 10*A*b^2*c^2*d^2*e^2*x
- 6*A*a*b^2*c*d*e^3 + 4*B*a*b*c^2*d^3*e + 4*B*a^2*b*c*d*e^3 - 8*A*a*b^2*c*e^4*x + 7*B*a^2*b*c*e^4*x - 4*A*b*c
^3*d^3*e*x - 8*A*b^3*c*d*e^3*x + 16*B*a*c^3*d^3*e*x + 2*A*b*c^2*d^3*e*(b^2 - 4*a*c)^(1/2) + 8*B*a*c^2*d^3*e*(b
^2 - 4*a*c)^(1/2) - 8*B*a^2*c*d*e^3*(b^2 - 4*a*c)^(1/2) + 2*B*a*b^2*e^4*x*(b^2 - 4*a*c)^(1/2) - 3*B*a^2*c*e^4*
x*(b^2 - 4*a*c)^(1/2) + 8*A*c^3*d^3*e*x*(b^2 - 4*a*c)^(1/2) + 10*A*a*b*c^2*d^2*e^2 + 2*B*a*b^2*c*d^2*e^2 - 28*
A*a*c^3*d^2*e^2*x - 16*B*a^2*c^2*d*e^3*x - 2*B*b^2*c^2*d^3*e*x + B*b^3*c*d^2*e^2*x - 14*A*a*c^2*d^2*e^2*(b^2 -
4*a*c)^(1/2) - A*b^2*c*d^2*e^2*(b^2 - 4*a*c)^(1/2) - 8*A*a*c^2*d*e^3*x*(b^2 - 4*a*c)^(1/2) + 8*A*b^2*c*d*e^3*
x*(b^2 - 4*a*c)^(1/2) + 2*B*b*c^2*d^3*e*x*(b^2 - 4*a*c)^(1/2) - 10*B*a*b*c^2*d^2*e^2*x - 12*A*b*c^2*d^2*e^2*x*
(b^2 - 4*a*c)^(1/2) + 10*B*a*c^2*d^2*e^2*x*(b^2 - 4*a*c)^(1/2) - B*b^2*c*d^2*e^2*x*(b^2 - 4*a*c)^(1/2) + 10*A*
a*b*c*d*e^3*(b^2 - 4*a*c)^(1/2) + 4*A*a*b*c*e^4*x*(b^2 - 4*a*c)^(1/2) + 28*A*a*b*c^2*d*e^3*x + 6*B*a*b^2*c*d*e
^3*x - 6*B*a*b*c*d*e^3*x*(b^2 - 4*a*c)^(1/2))*(A*b^3*e^2 - B*a*b^2*e^2 - 4*B*a*c^2*d^2 + 4*B*a^2*c*e^2 + B*b^2
*c*d^2 - A*b^2*e^2*(b^2 - 4*a*c)^(1/2) - 2*A*c^2*d^2*(b^2 - 4*a*c)^(1/2) - 4*A*a*b*c*e^2 + 8*A*a*c^2*d*e - 2*A
*b^2*c*d*e + 2*A*a*c*e^2*(b^2 - 4*a*c)^(1/2) + B*a*b*e^2*(b^2 - 4*a*c)^(1/2) + B*b*c*d^2*(b^2 - 4*a*c)^(1/2) +
2*A*b*c*d*e*(b^2 - 4*a*c)^(1/2) - 4*B*a*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*(4*a*c^3*d^4 + 4*a^3*c*e^4 - a^2*b^2*e
^4 - b^2*c^2*d^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e^3 + 2*b^3*c*d^3*e - 8*a*b*c^2*d^3*e - 8*a^2*b
*c*d*e^3 + 2*a*b^2*c*d^2*e^2)) - (A*e - B*d)/((d + e*x)*(a*e^2 + c*d^2 - b*d*e))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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