\(\int \frac {A+B x}{(d+e x)^2 (a+b x+c x^2)^{3/2}} \, dx\) [2478]

Optimal result

Integrand size = 27, antiderivative size = 334 \[ \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (b^2 e (B d-3 A e)-4 c \left (A c d^2+3 a B d e-2 a A e^2\right )+2 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {e \left (3 A e (2 c d-b e)-B \left (4 c d^2-e (b d+2 a e)\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{5/2}} \]

[Out]

1/2*e*(3*A*e*(-b*e+2*c*d)-B*(4*c*d^2-e*(2*a*e+b*d)))*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2
)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(5/2)+2*(a*B*(-b*e+2*c*d)-A*(2*a*c*e-b^2*e+b*c*d)+c*(A*b*e-2*
A*c*d-2*B*a*e+B*b*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)/(c*x^2+b*x+a)^(1/2)+e*(b^2*e*(-3*A*e+B*d)-4*c
*(-2*A*a*e^2+A*c*d^2+3*B*a*d*e)+2*b*(2*A*c*d*e+B*a*e^2+B*c*d^2))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e
+c*d^2)^2/(e*x+d)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {836, 820, 738, 212} \[ \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {e \left (-B e (2 a e+b d)-3 A e (2 c d-b e)+4 B c d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac {e \sqrt {a+b x+c x^2} \left (2 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-2 a A e^2+3 a B d e+A c d^2\right )+b^2 e (B d-3 A e)\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^2}+\frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

[In]

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

[Out]

(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*x))/((b^2 - 4*a*c)
*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*Sqrt[a + b*x + c*x^2]) + (e*(b^2*e*(B*d - 3*A*e) - 4*c*(A*c*d^2 + 3*a*B*d*e
- 2*a*A*e^2) + 2*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*
e^2)^2*(d + e*x)) - (e*(4*B*c*d^2 - B*e*(b*d + 2*a*e) - 3*A*e*(2*c*d - b*e))*ArcTanh[(b*d - 2*a*e + (2*c*d - b
*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(2*(c*d^2 - b*d*e + a*e^2)^(5/2))

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 738

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

Rule 820

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

Rule 836

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {\frac {1}{2} e \left (b^2 (B d-3 A e)-8 a c (B d-A e)+2 b (A c d+a B e)\right )-c e (b B d-2 A c d+A b e-2 a B e) x}{(d+e x)^2 \sqrt {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 )} \\ & = \frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (b^2 e (B d-3 A e)-4 c \left (A c d^2+3 a B d e-2 a A e^2\right )+2 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {\left (e \left (4 B c d^2-B e (b d+2 a e)-3 A e (2 c d-b e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{2 \left (c d^2-b d e+a e^2\right )^2} \\ & = \frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (b^2 e (B d-3 A e)-4 c \left (A c d^2+3 a B d e-2 a A e^2\right )+2 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {\left (e \left (4 B c d^2-B e (b d+2 a e)-3 A e (2 c d-b e)\right )\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^2} \\ & = \frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (b^2 e (B d-3 A e)-4 c \left (A c d^2+3 a B d e-2 a A e^2\right )+2 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {e \left (4 B c d^2-B e (b d+2 a e)-3 A e (2 c d-b e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{5/2}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(866\) vs. \(2(334)=668\).

Time = 13.30 (sec) , antiderivative size = 866, normalized size of antiderivative = 2.59 \[ \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-\frac {A \left (b^3 e^2 (2 d+3 e x)+b^2 e \left (a e^2+c \left (-4 d^2+2 d e x+15 e^2 x^2\right )-\sqrt {c} e (5 d+9 e x) \sqrt {a+x (b+c x)}\right )+2 b \left (a c e^2 (-d+5 e x)+c^2 \left (d^3-9 d^2 e x-12 d e^2 x^2+6 e^3 x^3\right )-2 a \sqrt {c} e^3 \sqrt {a+x (b+c x)}+6 c^{3/2} e \left (d^2+d e x-e^2 x^2\right ) \sqrt {a+x (b+c x)}\right )+4 c \left (a^2 e^3-a c d e (2 d+5 e x)+2 a \sqrt {c} d e^2 \sqrt {a+x (b+c x)}+c^{3/2} d \left (d^2-3 d e x-6 e^2 x^2\right ) \left (\sqrt {c} x-\sqrt {a+x (b+c x)}\right )\right )\right )-B \left (4 a^2 c e^2 (2 d+e x)+a \left (b^2 e^2 (3 d+2 e x)+2 b c e \left (-4 d^2+7 d e x+5 e^2 x^2\right )-4 c^2 \left (d^3+5 d^2 e x-d e^2 x^2-2 e^3 x^3\right )-2 b \sqrt {c} e^2 (5 d+3 e x) \sqrt {a+x (b+c x)}-4 c^{3/2} e \left (-3 d^2+d e x+2 e^2 x^2\right ) \sqrt {a+x (b+c x)}\right )+d \left (b^3 e^2 x+b^2 c e x (-4 d+5 e x)+2 b c^2 x \left (-3 d^2-9 d e x+2 e^2 x^2\right )+b^2 \sqrt {c} e (d-3 e x) \sqrt {a+x (b+c x)}+2 b c^{3/2} \left (d^2+5 d e x-2 e^2 x^2\right ) \sqrt {a+x (b+c x)}+8 c^{5/2} d x (d+2 e x) \left (-\sqrt {c} x+\sqrt {a+x (b+c x)}\right )\right )\right )}{(d+e x) \left (b^2 \left (-4 \sqrt {c} x+\sqrt {a+x (b+c x)}\right )+8 c^2 x^2 \left (-\sqrt {c} x+\sqrt {a+x (b+c x)}\right )+4 b c x \left (-3 \sqrt {c} x+2 \sqrt {a+x (b+c x)}\right )-4 a \sqrt {c} \left (b+2 c x-\sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}+\frac {e \left (-4 B c d^2+B e (b d+2 a e)-3 A e (-2 c d+b e)\right ) \arctan \left (\frac {-\sqrt {c} (d+e x)+e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{\sqrt {-c d^2+e (b d-a e)}}}{\left (c d^2+e (-b d+a e)\right )^2} \]

[In]

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

[Out]

(-((A*(b^3*e^2*(2*d + 3*e*x) + b^2*e*(a*e^2 + c*(-4*d^2 + 2*d*e*x + 15*e^2*x^2) - Sqrt[c]*e*(5*d + 9*e*x)*Sqrt
[a + x*(b + c*x)]) + 2*b*(a*c*e^2*(-d + 5*e*x) + c^2*(d^3 - 9*d^2*e*x - 12*d*e^2*x^2 + 6*e^3*x^3) - 2*a*Sqrt[c
]*e^3*Sqrt[a + x*(b + c*x)] + 6*c^(3/2)*e*(d^2 + d*e*x - e^2*x^2)*Sqrt[a + x*(b + c*x)]) + 4*c*(a^2*e^3 - a*c*
d*e*(2*d + 5*e*x) + 2*a*Sqrt[c]*d*e^2*Sqrt[a + x*(b + c*x)] + c^(3/2)*d*(d^2 - 3*d*e*x - 6*e^2*x^2)*(Sqrt[c]*x
- Sqrt[a + x*(b + c*x)]))) - B*(4*a^2*c*e^2*(2*d + e*x) + a*(b^2*e^2*(3*d + 2*e*x) + 2*b*c*e*(-4*d^2 + 7*d*e*
x + 5*e^2*x^2) - 4*c^2*(d^3 + 5*d^2*e*x - d*e^2*x^2 - 2*e^3*x^3) - 2*b*Sqrt[c]*e^2*(5*d + 3*e*x)*Sqrt[a + x*(b
+ c*x)] - 4*c^(3/2)*e*(-3*d^2 + d*e*x + 2*e^2*x^2)*Sqrt[a + x*(b + c*x)]) + d*(b^3*e^2*x + b^2*c*e*x*(-4*d +
5*e*x) + 2*b*c^2*x*(-3*d^2 - 9*d*e*x + 2*e^2*x^2) + b^2*Sqrt[c]*e*(d - 3*e*x)*Sqrt[a + x*(b + c*x)] + 2*b*c^(3
/2)*(d^2 + 5*d*e*x - 2*e^2*x^2)*Sqrt[a + x*(b + c*x)] + 8*c^(5/2)*d*x*(d + 2*e*x)*(-(Sqrt[c]*x) + Sqrt[a + x*(
b + c*x)]))))/((d + e*x)*(b^2*(-4*Sqrt[c]*x + Sqrt[a + x*(b + c*x)]) + 8*c^2*x^2*(-(Sqrt[c]*x) + Sqrt[a + x*(b
+ c*x)]) + 4*b*c*x*(-3*Sqrt[c]*x + 2*Sqrt[a + x*(b + c*x)]) - 4*a*Sqrt[c]*(b + 2*c*x - Sqrt[c]*Sqrt[a + x*(b
+ c*x)])))) + (e*(-4*B*c*d^2 + B*e*(b*d + 2*a*e) - 3*A*e*(-2*c*d + b*e))*ArcTan[(-(Sqrt[c]*(d + e*x)) + e*Sqrt
[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]])/Sqrt[-(c*d^2) + e*(b*d - a*e)])/(c*d^2 + e*(-(b*d) + a*e))
^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1063\) vs. \(2(320)=640\).

Time = 0.54 (sec) , antiderivative size = 1064, normalized size of antiderivative = 3.19

method	result	size
default	\(\text {Expression too large to display}\)	\(1064\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2040 vs. \(2 (320) = 640\).

Time = 11.32 (sec) , antiderivative size = 4122, normalized size of antiderivative = 12.34 \[ \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/4*((4*(B*a*b^2*c - 4*B*a^2*c^2)*d^3*e - (B*a*b^3 - 24*A*a^2*c^2 - 2*(2*B*a^2*b - 3*A*a*b^2)*c)*d^2*e^2 - (2
*B*a^2*b^2 - 3*A*a*b^3 - 4*(2*B*a^3 - 3*A*a^2*b)*c)*d*e^3 + (4*(B*b^2*c^2 - 4*B*a*c^3)*d^2*e^2 - (B*b^3*c - 24
*A*a*c^3 - 2*(2*B*a*b - 3*A*b^2)*c^2)*d*e^3 + (4*(2*B*a^2 - 3*A*a*b)*c^2 - (2*B*a*b^2 - 3*A*b^3)*c)*e^4)*x^3 +
(4*(B*b^2*c^2 - 4*B*a*c^3)*d^3*e + 3*(B*b^3*c + 8*A*a*c^3 - 2*(2*B*a*b + A*b^2)*c^2)*d^2*e^2 - (B*b^4 - 4*(2*
B*a^2 + 3*A*a*b)*c^2 - (2*B*a*b^2 - 3*A*b^3)*c)*d*e^3 - (2*B*a*b^3 - 3*A*b^4 - 4*(2*B*a^2*b - 3*A*a*b^2)*c)*e^
4)*x^2 + (4*(B*b^3*c - 4*B*a*b*c^2)*d^3*e - (B*b^4 + 8*(2*B*a^2 - 3*A*a*b)*c^2 - 2*(4*B*a*b^2 - 3*A*b^3)*c)*d^
2*e^2 - 3*(B*a*b^3 - A*b^4 - 8*A*a^2*c^2 - 2*(2*B*a^2*b - 3*A*a*b^2)*c)*d*e^3 - (2*B*a^2*b^2 - 3*A*a*b^3 - 4*(
2*B*a^3 - 3*A*a^2*b)*c)*e^4)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (
8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*
a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) + 4*(2*
(2*B*a - A*b)*c^3*d^5 - 2*(4*A*a*c^3 + (4*B*a*b - 3*A*b^2)*c^2)*d^4*e - (4*(B*a^2 - 3*A*a*b)*c^2 - (7*B*a*b^2
- 6*A*b^3)*c)*d^3*e^2 - (3*B*a*b^3 - 2*A*b^4 + 4*A*a^2*c^2 - (4*B*a^2*b - 3*A*a*b^2)*c)*d^2*e^3 + (3*B*a^2*b^2
- A*a*b^3 - 2*(4*B*a^3 - A*a^2*b)*c)*d*e^4 - (A*a^2*b^2 - 4*A*a^3*c)*e^5 + (2*(B*b*c^3 - 2*A*c^4)*d^4*e - (B*
b^2*c^2 + 4*(3*B*a - 2*A*b)*c^3)*d^3*e^2 - (B*b^3*c - 4*A*a*c^3 - (16*B*a*b - 7*A*b^2)*c^2)*d^2*e^3 - (4*(3*B*
a^2 + A*a*b)*c^2 + (B*a*b^2 - 3*A*b^3)*c)*d*e^4 + (8*A*a^2*c^2 + (2*B*a^2*b - 3*A*a*b^2)*c)*e^5)*x^2 + (2*(B*b
*c^3 - 2*A*c^4)*d^5 - 2*(B*b^2*c^2 + (2*B*a - 3*A*b)*c^3)*d^4*e + (B*b^3*c - 8*A*a*c^3)*d^3*e^2 - (B*b^4 + 8*(
B*a^2 - 2*A*a*b)*c^2 - (8*B*a*b^2 - 5*A*b^3)*c)*d^2*e^3 - (B*a*b^3 - 3*A*b^4 + 4*A*a^2*c^2 + 2*(B*a^2*b + 4*A*
a*b^2)*c)*d*e^4 + (2*B*a^2*b^2 - 3*A*a*b^3 - 2*(2*B*a^3 - 5*A*a^2*b)*c)*e^5)*x)*sqrt(c*x^2 + b*x + a))/((a*b^2
*c^3 - 4*a^2*c^4)*d^7 - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^6*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^5*e^2 -
(a*b^5 + 2*a^2*b^3*c - 24*a^3*b*c^2)*d^4*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^3*e^4 - 3*(a^3*b^3 - 4*
a^4*b*c)*d^2*e^5 + (a^4*b^2 - 4*a^5*c)*d*e^6 + ((b^2*c^4 - 4*a*c^5)*d^6*e - 3*(b^3*c^3 - 4*a*b*c^4)*d^5*e^2 +
3*(b^4*c^2 - 3*a*b^2*c^3 - 4*a^2*c^4)*d^4*e^3 - (b^5*c + 2*a*b^3*c^2 - 24*a^2*b*c^3)*d^3*e^4 + 3*(a*b^4*c - 3*
a^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^5 - 3*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^6 + (a^3*b^2*c - 4*a^4*c^2)*e^7)*x^3 + ((b^
2*c^4 - 4*a*c^5)*d^7 - 2*(b^3*c^3 - 4*a*b*c^4)*d^6*e + 3*(a*b^2*c^3 - 4*a^2*c^4)*d^5*e^2 + (2*b^5*c - 11*a*b^3
*c^2 + 12*a^2*b*c^3)*d^4*e^3 - (b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^3*e^4 + 3*(a*b^5 - 4*a^2*b^3*c)
*d^2*e^5 - (3*a^2*b^4 - 13*a^3*b^2*c + 4*a^4*c^2)*d*e^6 + (a^3*b^3 - 4*a^4*b*c)*e^7)*x^2 + ((b^3*c^3 - 4*a*b*c
^4)*d^7 - (3*b^4*c^2 - 13*a*b^2*c^3 + 4*a^2*c^4)*d^6*e + 3*(b^5*c - 4*a*b^3*c^2)*d^5*e^2 - (b^6 - a*b^4*c - 15
*a^2*b^2*c^2 + 12*a^3*c^3)*d^4*e^3 + (2*a*b^5 - 11*a^2*b^3*c + 12*a^3*b*c^2)*d^3*e^4 + 3*(a^3*b^2*c - 4*a^4*c^
2)*d^2*e^5 - 2*(a^3*b^3 - 4*a^4*b*c)*d*e^6 + (a^4*b^2 - 4*a^5*c)*e^7)*x), -1/2*((4*(B*a*b^2*c - 4*B*a^2*c^2)*d
^3*e - (B*a*b^3 - 24*A*a^2*c^2 - 2*(2*B*a^2*b - 3*A*a*b^2)*c)*d^2*e^2 - (2*B*a^2*b^2 - 3*A*a*b^3 - 4*(2*B*a^3
- 3*A*a^2*b)*c)*d*e^3 + (4*(B*b^2*c^2 - 4*B*a*c^3)*d^2*e^2 - (B*b^3*c - 24*A*a*c^3 - 2*(2*B*a*b - 3*A*b^2)*c^2
)*d*e^3 + (4*(2*B*a^2 - 3*A*a*b)*c^2 - (2*B*a*b^2 - 3*A*b^3)*c)*e^4)*x^3 + (4*(B*b^2*c^2 - 4*B*a*c^3)*d^3*e +
3*(B*b^3*c + 8*A*a*c^3 - 2*(2*B*a*b + A*b^2)*c^2)*d^2*e^2 - (B*b^4 - 4*(2*B*a^2 + 3*A*a*b)*c^2 - (2*B*a*b^2 -
3*A*b^3)*c)*d*e^3 - (2*B*a*b^3 - 3*A*b^4 - 4*(2*B*a^2*b - 3*A*a*b^2)*c)*e^4)*x^2 + (4*(B*b^3*c - 4*B*a*b*c^2)*
d^3*e - (B*b^4 + 8*(2*B*a^2 - 3*A*a*b)*c^2 - 2*(4*B*a*b^2 - 3*A*b^3)*c)*d^2*e^2 - 3*(B*a*b^3 - A*b^4 - 8*A*a^2
*c^2 - 2*(2*B*a^2*b - 3*A*a*b^2)*c)*d*e^3 - (2*B*a^2*b^2 - 3*A*a*b^3 - 4*(2*B*a^3 - 3*A*a^2*b)*c)*e^4)*x)*sqrt
(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d
- b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x))
- 2*(2*(2*B*a - A*b)*c^3*d^5 - 2*(4*A*a*c^3 + (4*B*a*b - 3*A*b^2)*c^2)*d^4*e - (4*(B*a^2 - 3*A*a*b)*c^2 - (7*
B*a*b^2 - 6*A*b^3)*c)*d^3*e^2 - (3*B*a*b^3 - 2*A*b^4 + 4*A*a^2*c^2 - (4*B*a^2*b - 3*A*a*b^2)*c)*d^2*e^3 + (3*B
*a^2*b^2 - A*a*b^3 - 2*(4*B*a^3 - A*a^2*b)*c)*d*e^4 - (A*a^2*b^2 - 4*A*a^3*c)*e^5 + (2*(B*b*c^3 - 2*A*c^4)*d^4
*e - (B*b^2*c^2 + 4*(3*B*a - 2*A*b)*c^3)*d^3*e^2 - (B*b^3*c - 4*A*a*c^3 - (16*B*a*b - 7*A*b^2)*c^2)*d^2*e^3 -
(4*(3*B*a^2 + A*a*b)*c^2 + (B*a*b^2 - 3*A*b^3)*c)*d*e^4 + (8*A*a^2*c^2 + (2*B*a^2*b - 3*A*a*b^2)*c)*e^5)*x^2 +
(2*(B*b*c^3 - 2*A*c^4)*d^5 - 2*(B*b^2*c^2 + (2*B*a - 3*A*b)*c^3)*d^4*e + (B*b^3*c - 8*A*a*c^3)*d^3*e^2 - (B*b
^4 + 8*(B*a^2 - 2*A*a*b)*c^2 - (8*B*a*b^2 - 5*A*b^3)*c)*d^2*e^3 - (B*a*b^3 - 3*A*b^4 + 4*A*a^2*c^2 + 2*(B*a^2*
b + 4*A*a*b^2)*c)*d*e^4 + (2*B*a^2*b^2 - 3*A*a*b^3 - 2*(2*B*a^3 - 5*A*a^2*b)*c)*e^5)*x)*sqrt(c*x^2 + b*x + a))
/((a*b^2*c^3 - 4*a^2*c^4)*d^7 - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^6*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^
5*e^2 - (a*b^5 + 2*a^2*b^3*c - 24*a^3*b*c^2)*d^4*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^3*e^4 - 3*(a^3*
b^3 - 4*a^4*b*c)*d^2*e^5 + (a^4*b^2 - 4*a^5*c)*d*e^6 + ((b^2*c^4 - 4*a*c^5)*d^6*e - 3*(b^3*c^3 - 4*a*b*c^4)*d^
5*e^2 + 3*(b^4*c^2 - 3*a*b^2*c^3 - 4*a^2*c^4)*d^4*e^3 - (b^5*c + 2*a*b^3*c^2 - 24*a^2*b*c^3)*d^3*e^4 + 3*(a*b^
4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^5 - 3*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^6 + (a^3*b^2*c - 4*a^4*c^2)*e^7)*x^
3 + ((b^2*c^4 - 4*a*c^5)*d^7 - 2*(b^3*c^3 - 4*a*b*c^4)*d^6*e + 3*(a*b^2*c^3 - 4*a^2*c^4)*d^5*e^2 + (2*b^5*c -
11*a*b^3*c^2 + 12*a^2*b*c^3)*d^4*e^3 - (b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^3*e^4 + 3*(a*b^5 - 4*a^
2*b^3*c)*d^2*e^5 - (3*a^2*b^4 - 13*a^3*b^2*c + 4*a^4*c^2)*d*e^6 + (a^3*b^3 - 4*a^4*b*c)*e^7)*x^2 + ((b^3*c^3 -
4*a*b*c^4)*d^7 - (3*b^4*c^2 - 13*a*b^2*c^3 + 4*a^2*c^4)*d^6*e + 3*(b^5*c - 4*a*b^3*c^2)*d^5*e^2 - (b^6 - a*b^
4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^4*e^3 + (2*a*b^5 - 11*a^2*b^3*c + 12*a^3*b*c^2)*d^3*e^4 + 3*(a^3*b^2*c -
4*a^4*c^2)*d^2*e^5 - 2*(a^3*b^3 - 4*a^4*b*c)*d*e^6 + (a^4*b^2 - 4*a^5*c)*e^7)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate((B*x+A)/(e*x+d)^2/(c*x^2+b*x+a)^(3/2),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(a*e^2-b*d*e>0)', see `assume?`
for more de

Giac [F]

\[ \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{{\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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