3.2472 \(\int \frac{A+B x}{(d+e x)^4 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=444 \[ \frac{\sqrt{a+b x+c x^2} \left (B \left (2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)+8 c^2 d^3\right )-A e \left (-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right )\right )}{24 (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{\left (-2 b^2 e \left (3 a B e^2+9 A c d e+2 B c d^2\right )+4 b c \left (-3 a A e^3+5 a B d e^2+6 A c d^2 e+2 B c d^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )+b^3 e^2 (5 A e+B d)\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac{\sqrt{a+b x+c x^2} (B d-A e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{a+b x+c x^2} \left (5 A e (2 c d-b e)-B \left (e (b d-6 a e)+4 c d^2\right )\right )}{12 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2} \]

[Out]

((B*d - A*e)*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - ((5*A*e*(2*c*d - b*e) - B*(4*c*d
^2 + e*(b*d - 6*a*e)))*Sqrt[a + b*x + c*x^2])/(12*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + ((B*(8*c^2*d^3 + 2*
c*d*e*(5*b*d - 26*a*e) - 3*b*e^2*(b*d - 6*a*e)) - A*e*(44*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(11*b*d + 4*a*e)))*Sqrt
[a + b*x + c*x^2])/(24*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) - ((b^3*e^2*(B*d + 5*A*e) - 2*b^2*e*(2*B*c*d^2 + 9
*A*c*d*e + 3*a*B*e^2) + 4*b*c*(2*B*c*d^3 + 6*A*c*d^2*e + 5*a*B*d*e^2 - 3*a*A*e^3) - 8*c*(A*c*d*(2*c*d^2 - 3*a*
e^2) + a*B*e*(4*c*d^2 - a*e^2)))*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])])/(16*(c*d^2 - b*d*e + a*e^2)^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.835473, antiderivative size = 442, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {834, 806, 724, 206} \[ \frac{\sqrt{a+b x+c x^2} \left (B \left (2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)+8 c^2 d^3\right )-A e \left (-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right )\right )}{24 (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{\left (-2 b^2 e \left (3 a B e^2+9 A c d e+2 B c d^2\right )+4 b c \left (-3 a A e^3+5 a B d e^2+6 A c d^2 e+2 B c d^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )+b^3 e^2 (5 A e+B d)\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac{\sqrt{a+b x+c x^2} (B d-A e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{a+b x+c x^2} \left (B e (b d-6 a e)-5 A e (2 c d-b e)+4 B c d^2\right )}{12 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]

[Out]

((B*d - A*e)*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + ((4*B*c*d^2 + B*e*(b*d - 6*a*e)
- 5*A*e*(2*c*d - b*e))*Sqrt[a + b*x + c*x^2])/(12*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + ((B*(8*c^2*d^3 + 2*
c*d*e*(5*b*d - 26*a*e) - 3*b*e^2*(b*d - 6*a*e)) - A*e*(44*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(11*b*d + 4*a*e)))*Sqrt
[a + b*x + c*x^2])/(24*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) - ((b^3*e^2*(B*d + 5*A*e) - 2*b^2*e*(2*B*c*d^2 + 9
*A*c*d*e + 3*a*B*e^2) + 4*b*c*(2*B*c*d^3 + 6*A*c*d^2*e + 5*a*B*d*e^2 - 3*a*A*e^3) - 8*c*(A*c*d*(2*c*d^2 - 3*a*
e^2) + a*B*e*(4*c*d^2 - a*e^2)))*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])])/(16*(c*d^2 - b*d*e + a*e^2)^(7/2))

Rule 834

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))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((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[Sim
plify[m + 2*p + 3], 0]

Rule 724

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 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])

Rubi steps

\begin{align*} \int \frac{A+B x}{(d+e x)^4 \sqrt{a+b x+c x^2}} \, dx &=\frac{(B d-A e) \sqrt{a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{\int \frac{\frac{1}{2} (b B d-6 A c d+5 A b e-6 a B e)-2 c (B d-A e) x}{(d+e x)^3 \sqrt{a+b x+c x^2}} \, dx}{3 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{(B d-A e) \sqrt{a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{\left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) \sqrt{a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{\int \frac{\frac{1}{4} \left (3 b^2 e (B d+5 A e)+8 c \left (3 A c d^2+5 a B d e-2 a A e^2\right )-2 b \left (4 B c d^2+17 A c d e+9 a B e^2\right )\right )+\frac{1}{2} c \left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) x}{(d+e x)^2 \sqrt{a+b x+c x^2}} \, dx}{6 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{(B d-A e) \sqrt{a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{\left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) \sqrt{a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{\left (B \left (8 c^2 d^3+2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)\right )-A e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right )\right ) \sqrt{a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{\left (b^3 e^2 (B d+5 A e)-2 b^2 e \left (2 B c d^2+9 A c d e+3 a B e^2\right )+4 b c \left (2 B c d^3+6 A c d^2 e+5 a B d e^2-3 a A e^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{16 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{(B d-A e) \sqrt{a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{\left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) \sqrt{a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{\left (B \left (8 c^2 d^3+2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)\right )-A e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right )\right ) \sqrt{a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac{\left (b^3 e^2 (B d+5 A e)-2 b^2 e \left (2 B c d^2+9 A c d e+3 a B e^2\right )+4 b c \left (2 B c d^3+6 A c d^2 e+5 a B d e^2-3 a A e^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )\right ) \operatorname{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 )}{8 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{(B d-A e) \sqrt{a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{\left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) \sqrt{a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{\left (B \left (8 c^2 d^3+2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)\right )-A e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right )\right ) \sqrt{a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{\left (b^3 e^2 (B d+5 A e)-2 b^2 e \left (2 B c d^2+9 A c d e+3 a B e^2\right )+4 b c \left (2 B c d^3+6 A c d^2 e+5 a B d e^2-3 a A e^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )\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 )}{16 \left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 1.1698, size = 434, normalized size = 0.98 \[ \frac{\frac{\sqrt{a+x (b+c x)} \left (A e \left (4 c e (4 a e+11 b d)-15 b^2 e^2-44 c^2 d^2\right )+B \left (2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)+8 c^2 d^3\right )\right )}{4 (d+e x) \left (e (a e-b d)+c d^2\right )}+\frac{3 \left (-2 b^2 e \left (3 a B e^2+9 A c d e+2 B c d^2\right )+4 b c \left (-3 a A e^3+5 a B d e^2+6 A c d^2 e+2 B c d^3\right )+8 c \left (A c d \left (3 a e^2-2 c d^2\right )+a B e \left (a e^2-4 c d^2\right )\right )+b^3 e^2 (5 A e+B d)\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{8 \left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{2 \sqrt{a+x (b+c x)} (B d-A e) \left (e (a e-b d)+c d^2\right )}{(d+e x)^3}+\frac{\sqrt{a+x (b+c x)} \left (B e (b d-6 a e)+5 A e (b e-2 c d)+4 B c d^2\right )}{2 (d+e x)^2}}{6 \left (e (a e-b d)+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]

[Out]

((2*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)])/(d + e*x)^3 + ((4*B*c*d^2 + B*e*(b*d - 6*a*e
) + 5*A*e*(-2*c*d + b*e))*Sqrt[a + x*(b + c*x)])/(2*(d + e*x)^2) + ((B*(8*c^2*d^3 + 2*c*d*e*(5*b*d - 26*a*e) -
3*b*e^2*(b*d - 6*a*e)) + A*e*(-44*c^2*d^2 - 15*b^2*e^2 + 4*c*e*(11*b*d + 4*a*e)))*Sqrt[a + x*(b + c*x)])/(4*(
c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) + (3*(b^3*e^2*(B*d + 5*A*e) - 2*b^2*e*(2*B*c*d^2 + 9*A*c*d*e + 3*a*B*e^2)
+ 4*b*c*(2*B*c*d^3 + 6*A*c*d^2*e + 5*a*B*d*e^2 - 3*a*A*e^3) + 8*c*(a*B*e*(-4*c*d^2 + a*e^2) + A*c*d*(-2*c*d^2
+ 3*a*e^2)))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x
)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2)))/(6*(c*d^2 + e*(-(b*d) + a*e))^2)

________________________________________________________________________________________

Maple [B]  time = 0.017, size = 3898, normalized size = 8.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x)

[Out]

1/3/e^3/(a*e^2-b*d*e+c*d^2)/(x+d/e)^3*((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*d-5/
2/(a*e^2-b*d*e+c*d^2)^3/(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)*c^2*d^2*A-5/
8*e^2/(a*e^2-b*d*e+c*d^2)^3/(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)*b^2*A+5/
16*e^2/(a*e^2-b*d*e+c*d^2)^3/((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
))*b^3*A+1/2*B/e^2*c/(a*e^2-b*d*e+c*d^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))-3/4/(a*e^2-b*d*e+c*d^2)^2*c/((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))*b*A-3/8*B/(a*e^2-b*d*e+c*d^2)^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))*b^2+2/3/(a*e^2-b*d*e+c*d^2)^2*c/(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)*A-1/2*B/e^2/(a*e^2-b*d*e+c*d^2)/(x+d/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)+3/4*B/(a*e^2-b*d*e+c*d^2)^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)*b-1/3/e^2/(a*e^2-b*d*e+c*d^2)/(x+d/e)^3*((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)*A+5/12/(a*e^2-b*d*e+c*d^2)^2/(x+d/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*A-5/12/e/(a*e^2-b*d*e+c*d^2)^2/(x+d/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
*B*d-5/16*e/(a*e^2-b*d*e+c*d^2)^3/((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))*b^3*B*d+5/8*e/(a*e^2-b*d*e+c*d^2)^3/(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)*b^2*B*d+3/2/e/(a*e^2-b*d*e+c*d^2)^2*c^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))*d*A-3/e^2/(a*e^2-b*d*e+c*d^2)^2*c^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))*d^2*B-13/6/e/(a*e^2-b*d*e+c*d^2)^2*c/(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)*B*d+15/4/(a*e^2-b*d*e+c*d^2)^3/((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))*b*c^2*d^2*A+5/2/e^2/(a*e^2-b*d*e+c*d^2)^3/((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))*c^3*d^4*B-15/4/e/(a*e^2-b*d*e+c*d^2)^3/((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))*b*c^2*d^3*B-15/8*e/(a*e^2-b*d*e+c*d^2)^3/((
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))*b^2*c*d*A+5/2*e/(a*e^2-b*d*e
+c*d^2)^3/(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)*b*c*d*A+9/4/e/(a*e^2-b*d*e
+c*d^2)^2*c/((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))*b*B*d-5/6/e/(a
*e^2-b*d*e+c*d^2)^2/(x+d/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)*c*d*A+5/6/e^2/
(a*e^2-b*d*e+c*d^2)^2/(x+d/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)*c*d^2*B+5/2/
e/(a*e^2-b*d*e+c*d^2)^3/(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)*c^2*d^3*B-5/
2/(a*e^2-b*d*e+c*d^2)^3/(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)*b*c*d^2*B+15
/8/(a*e^2-b*d*e+c*d^2)^3/((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))*b
^2*c*d^2*B-5/2/e/(a*e^2-b*d*e+c*d^2)^3/((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))*c^3*d^3*A

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{\left (d + e x\right )^{4} \sqrt{a + b x + c x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)**4/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((A + B*x)/((d + e*x)**4*sqrt(a + b*x + c*x**2)), x)

________________________________________________________________________________________

Giac [B]  time = 1.67182, size = 5736, normalized size = 12.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

-1/8*(8*B*b*c^2*d^3 - 16*A*c^3*d^3 - 4*B*b^2*c*d^2*e - 32*B*a*c^2*d^2*e + 24*A*b*c^2*d^2*e + B*b^3*d*e^2 + 20*
B*a*b*c*d*e^2 - 18*A*b^2*c*d*e^2 + 24*A*a*c^2*d*e^2 - 6*B*a*b^2*e^3 + 5*A*b^3*e^3 + 8*B*a^2*c*e^3 - 12*A*a*b*c
*e^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((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(-c*d^2 + b*d*e - a*e^2)) + 1/24*(64*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
3*B*c^4*d^6 - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b*c^3*d^5*e - 352*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^3*A*c^4*d^5*e + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*b*c^(7/2)*d^6 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^4*B*b*c^(5/2)*d^4*e^2 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*c^(7/2)*d^4*e^2 + 24*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^2*B*b^2*c^(5/2)*d^5*e - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a*c^(7/2)*d^5*e - 528
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b*c^(7/2)*d^5*e + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*b^2*c^3*d^
6 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*b*c^2*d^3*e^3 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*c^3*
d^3*e^3 + 168*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b^2*c^2*d^4*e^2 - 512*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^3*B*a*c^3*d^4*e^2 + 400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b*c^3*d^4*e^2 + 36*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*B*b^3*c^2*d^5*e - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a*b*c^3*d^5*e - 264*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))*A*b^2*c^3*d^5*e + 8*B*b^3*c^(5/2)*d^6 - 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*b^2*c^(3/2
)*d^3*e^3 - 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a*c^(5/2)*d^3*e^3 + 360*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^4*A*b*c^(5/2)*d^3*e^3 + 54*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*b^3*c^(3/2)*d^4*e^2 - 696*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2*B*a*b*c^(5/2)*d^4*e^2 + 756*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b^2*c^(5/2)*d^4
*e^2 + 816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a*c^(7/2)*d^4*e^2 + 10*B*b^4*c^(3/2)*d^5*e - 48*B*a*b^2*c^(
5/2)*d^5*e - 44*A*b^3*c^(5/2)*d^5*e - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*b^2*c*d^2*e^4 - 96*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^5*B*a*c^2*d^2*e^4 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*b*c^2*d^2*e^4 - 74*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b^3*c*d^3*e^3 - 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a*b*c^2*d^3*
e^3 - 204*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b^2*c^2*d^3*e^3 + 656*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*
A*a*c^3*d^3*e^3 - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*b^4*c*d^4*e^2 - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*B*a*b^2*c^2*d^4*e^2 + 336*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*b^3*c^2*d^4*e^2 + 96*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))*B*a^2*c^3*d^4*e^2 + 816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*b*c^3*d^4*e^2 + 15*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^4*B*b^3*sqrt(c)*d^2*e^4 + 300*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a*b*c^(3/2)*d^2*
e^4 - 270*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*b^2*c^(3/2)*d^2*e^4 + 360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^4*A*a*c^(5/2)*d^2*e^4 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*b^4*sqrt(c)*d^3*e^3 + 252*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^2*B*a*b^2*c^(3/2)*d^3*e^3 - 498*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b^3*c^(3/2)*d^3*e
^3 + 816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^2*c^(5/2)*d^3*e^3 - 648*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^2*A*a*b*c^(5/2)*d^3*e^3 - 3*B*b^5*sqrt(c)*d^4*e^2 - 70*B*a*b^3*c^(3/2)*d^4*e^2 + 44*A*b^4*c^(3/2)*d^4*e^2 + 4
8*B*a^2*b*c^(5/2)*d^4*e^2 + 204*A*a*b^2*c^(5/2)*d^4*e^2 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*b^3*d*e^5
+ 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a*b*c*d*e^5 - 54*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*b^2*c*d*
e^5 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*c^2*d*e^5 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b^4*d
^2*e^4 + 252*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a*b^2*c*d^2*e^4 - 34*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
3*A*b^3*c*d^2*e^4 + 624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^2*c^2*d^2*e^4 - 264*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*A*a*b*c^2*d^2*e^4 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*b^5*d^3*e^3 + 186*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))*B*a*b^3*c*d^3*e^3 - 180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*b^4*c*d^3*e^3 + 864*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))*B*a^2*b*c^2*d^3*e^3 - 900*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*b^2*c^2*d^3*e^3 -
480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*c^3*d^3*e^3 - 90*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a*b^2*s
qrt(c)*d*e^5 + 75*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*b^3*sqrt(c)*d*e^5 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^4*B*a^2*c^(3/2)*d*e^5 - 180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a*b*c^(3/2)*d*e^5 - 24*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2*B*a*b^3*sqrt(c)*d^2*e^4 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b^4*sqrt(c)*d
^2*e^4 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^2*b*c^(3/2)*d^2*e^4 + 432*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*A*a*b^2*c^(3/2)*d^2*e^4 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^2*c^(5/2)*d^2*e^4 + 24*B*a*b^
4*sqrt(c)*d^3*e^3 - 15*A*b^5*sqrt(c)*d^3*e^3 + 240*B*a^2*b^2*c^(3/2)*d^3*e^3 - 206*A*a*b^3*c^(3/2)*d^3*e^3 - 1
6*B*a^3*c^(5/2)*d^3*e^3 - 240*A*a^2*b*c^(5/2)*d^3*e^3 - 18*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a*b^2*e^6 +
15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*b^3*e^6 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^2*c*e^6 - 3
6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*b*c*e^6 - 56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a*b^3*d*e^5 +
40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b^4*d*e^5 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^2*b*c*d*
e^5 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a*b^2*c*d*e^5 - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a
^2*c^2*d*e^5 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a*b^4*d^2*e^4 + 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*A*b^5*d^2*e^4 - 432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^2*b^2*c*d^2*e^4 + 450*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))*A*a*b^3*c*d^2*e^4 - 528*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^3*c^2*d^2*e^4 + 432*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))*A*a^2*b*c^2*d^2*e^4 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a*b^3*sqrt(c)*d*e^5 - 192
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^3*c^(3/2)*d*e^5 - 87*B*a^2*b^3*sqrt(c)*d^2*e^4 + 78*A*a*b^4*sqrt(c)
*d^2*e^4 - 284*B*a^3*b*c^(3/2)*d^2*e^4 + 222*A*a^2*b^2*c^(3/2)*d^2*e^4 + 88*A*a^3*c^(5/2)*d^2*e^4 + 48*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^3*B*a^2*b^2*e^6 - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a*b^3*e^6 + 96*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^2*b*c*e^6 + 57*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^2*b^3*d*e^5 - 66*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*b^4*d*e^5 + 276*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^3*b*c*d*e^5 - 3
06*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*b^2*c*d*e^5 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^3*c^2*d
*e^5 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^3*b*sqrt(c)*e^6 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2
*A*a^3*c^(3/2)*e^6 + 114*B*a^3*b^2*sqrt(c)*d*e^5 - 111*A*a^2*b^3*sqrt(c)*d*e^5 + 104*B*a^4*c^(3/2)*d*e^5 - 28*
A*a^3*b*c^(3/2)*d*e^5 - 30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^3*b^2*e^6 + 33*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))*A*a^2*b^3*e^6 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^4*c*e^6 + 36*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*A*a^3*b*c*e^6 - 48*B*a^4*b*sqrt(c)*e^6 + 48*A*a^3*b^2*sqrt(c)*e^6 - 32*A*a^4*c^(3/2)*e^6)/((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)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))*sqrt(c)*d + b*d - a*e)^3)