3.2471 \(\int \frac{A+B x}{(d+e x)^3 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=271 \[ \frac{\left (-4 b \left (a B e^2+2 A c d e+B c d^2\right )+4 c \left (-a A e^2+3 a B d e+2 A c d^2\right )+b^2 e (3 A e+B d)\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac{\sqrt{a+b x+c x^2} (B d-A e)}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{a+b x+c x^2} \left (3 A e (2 c d-b e)-B \left (e (b d-4 a e)+2 c d^2\right )\right )}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.378198, antiderivative size = 269, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {834, 806, 724, 206} \[ \frac{\left (-4 b \left (a B e^2+2 A c d e+B c d^2\right )+4 c \left (-a A e^2+3 a B d e+2 A c d^2\right )+b^2 e (3 A e+B d)\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac{\sqrt{a+b x+c x^2} (B d-A e)}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{a+b x+c x^2} \left (B e (b d-4 a e)-3 A e (2 c d-b e)+2 B c d^2\right )}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.624563, size = 264, normalized size = 0.97 \[ -\frac{\left (-4 b \left (a B e^2+2 A c d e+B c d^2\right )+4 c \left (-a A e^2+3 a B d e+2 A c d^2\right )+b^2 e (3 A e+B d)\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{8 \left (e (a e-b d)+c d^2\right )^{5/2}}+\frac{\sqrt{a+x (b+c x)} (B d-A e)}{2 (d+e x)^2 \left (e (a e-b d)+c d^2\right )}+\frac{\sqrt{a+x (b+c x)} \left (B e (b d-4 a e)+3 A e (b e-2 c d)+2 B c d^2\right )}{4 (d+e x) \left (e (a e-b d)+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.015, size = 2204, normalized size = 8.1 \begin{align*} \text{result too large to display} \end{align*}

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)^(1/2),x)

[Out]

-1/2/e/(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)*A+1/2/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)*B*d+3/4*e/(
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*A-3/4/(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*B*d-3/2/(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)*c*d*A+3/2/e/(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)*c*d^2*B-3/8*e/(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*A+3/8/(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*B*d+3/2
/(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*c
*d*A-3/2/e/(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*c*d^2*B-3/2/e/(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))*c^2*d^2*A+3/2/e^2/(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))*c^2*d^3*B+1/2/e*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))*A-3/2/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)*l
n((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*d-B/e/(a*e^2-b*d*e+c*d^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)+1/2*B/e/(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))*b

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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)^3/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 99.1968, size = 3688, normalized size = 13.61 \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)^3/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/16*((4*(B*b*c - 2*A*c^2)*d^4 - (B*b^2 + 4*(3*B*a - 2*A*b)*c)*d^3*e + (4*B*a*b - 3*A*b^2 + 4*A*a*c)*d^2*e^2
+ (4*(B*b*c - 2*A*c^2)*d^2*e^2 - (B*b^2 + 4*(3*B*a - 2*A*b)*c)*d*e^3 + (4*B*a*b - 3*A*b^2 + 4*A*a*c)*e^4)*x^2
+ 2*(4*(B*b*c - 2*A*c^2)*d^3*e - (B*b^2 + 4*(3*B*a - 2*A*b)*c)*d^2*e^2 + (4*B*a*b - 3*A*b^2 + 4*A*a*c)*d*e^3)*
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*(4*B*c^2*d^5 - 2*A*a^2*e^5 - (5*
B*b*c + 8*A*c^2)*d^4*e + (B*b^2 + (2*B*a + 13*A*b)*c)*d^3*e^2 + (B*a*b - 5*A*b^2 - 10*A*a*c)*d^2*e^3 - (2*B*a^
2 - 7*A*a*b)*d*e^4 + (2*B*c^2*d^4*e - (B*b*c + 6*A*c^2)*d^3*e^2 - (B*b^2 + (2*B*a - 9*A*b)*c)*d^2*e^3 + (5*B*a
*b - 3*A*b^2 - 6*A*a*c)*d*e^4 - (4*B*a^2 - 3*A*a*b)*e^5)*x)*sqrt(c*x^2 + b*x + a))/(c^3*d^8 - 3*b*c^2*d^7*e -
3*a^2*b*d^3*e^5 + a^3*d^2*e^6 + 3*(b^2*c + a*c^2)*d^6*e^2 - (b^3 + 6*a*b*c)*d^5*e^3 + 3*(a*b^2 + a^2*c)*d^4*e^
4 + (c^3*d^6*e^2 - 3*b*c^2*d^5*e^3 - 3*a^2*b*d*e^7 + a^3*e^8 + 3*(b^2*c + a*c^2)*d^4*e^4 - (b^3 + 6*a*b*c)*d^3
*e^5 + 3*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + 2*(c^3*d^7*e - 3*b*c^2*d^6*e^2 - 3*a^2*b*d^2*e^6 + a^3*d*e^7 + 3*(b^2*
c + a*c^2)*d^5*e^3 - (b^3 + 6*a*b*c)*d^4*e^4 + 3*(a*b^2 + a^2*c)*d^3*e^5)*x), -1/8*((4*(B*b*c - 2*A*c^2)*d^4 -
(B*b^2 + 4*(3*B*a - 2*A*b)*c)*d^3*e + (4*B*a*b - 3*A*b^2 + 4*A*a*c)*d^2*e^2 + (4*(B*b*c - 2*A*c^2)*d^2*e^2 -
(B*b^2 + 4*(3*B*a - 2*A*b)*c)*d*e^3 + (4*B*a*b - 3*A*b^2 + 4*A*a*c)*e^4)*x^2 + 2*(4*(B*b*c - 2*A*c^2)*d^3*e -
(B*b^2 + 4*(3*B*a - 2*A*b)*c)*d^2*e^2 + (4*B*a*b - 3*A*b^2 + 4*A*a*c)*d*e^3)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*a
rctan(-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*(4*B*c^2*d^5 - 2*A*a^
2*e^5 - (5*B*b*c + 8*A*c^2)*d^4*e + (B*b^2 + (2*B*a + 13*A*b)*c)*d^3*e^2 + (B*a*b - 5*A*b^2 - 10*A*a*c)*d^2*e^
3 - (2*B*a^2 - 7*A*a*b)*d*e^4 + (2*B*c^2*d^4*e - (B*b*c + 6*A*c^2)*d^3*e^2 - (B*b^2 + (2*B*a - 9*A*b)*c)*d^2*e
^3 + (5*B*a*b - 3*A*b^2 - 6*A*a*c)*d*e^4 - (4*B*a^2 - 3*A*a*b)*e^5)*x)*sqrt(c*x^2 + b*x + a))/(c^3*d^8 - 3*b*c
^2*d^7*e - 3*a^2*b*d^3*e^5 + a^3*d^2*e^6 + 3*(b^2*c + a*c^2)*d^6*e^2 - (b^3 + 6*a*b*c)*d^5*e^3 + 3*(a*b^2 + a^
2*c)*d^4*e^4 + (c^3*d^6*e^2 - 3*b*c^2*d^5*e^3 - 3*a^2*b*d*e^7 + a^3*e^8 + 3*(b^2*c + a*c^2)*d^4*e^4 - (b^3 + 6
*a*b*c)*d^3*e^5 + 3*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + 2*(c^3*d^7*e - 3*b*c^2*d^6*e^2 - 3*a^2*b*d^2*e^6 + a^3*d*e^
7 + 3*(b^2*c + a*c^2)*d^5*e^3 - (b^3 + 6*a*b*c)*d^4*e^4 + 3*(a*b^2 + a^2*c)*d^3*e^5)*x)]

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Sympy [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)**3/(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.29644, size = 1993, normalized size = 7.35 \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)^3/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

-1/4*(4*B*b*c*d^2 - 8*A*c^2*d^2 - B*b^2*d*e - 12*B*a*c*d*e + 8*A*b*c*d*e + 4*B*a*b*e^2 - 3*A*b^2*e^2 + 4*A*a*c
*e^2)*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^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(-c*d^2 + b*d*e - a*e^2)) + 1/4*(8*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^2*B*c^(5/2)*d^4 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*b*c^(3/2)*d^3*e - 24
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*c^(5/2)*d^3*e + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*b*c^2*d^4 + 4
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b*c*d^2*e^2 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*c^2*d^2*e^2 -
16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a*c^2*d^3*e - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*b*c^2*d^3*e +
2*B*b^2*c^(3/2)*d^4 + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*b^2*sqrt(c)*d^2*e^2 - 20*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^2*B*a*c^(3/2)*d^2*e^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b*c^(3/2)*d^2*e^2 + B*b^3*s
qrt(c)*d^3*e - 8*B*a*b*c^(3/2)*d^3*e - 6*A*b^2*c^(3/2)*d^3*e - (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b^2*d*e
^3 - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a*c*d*e^3 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b*c*d*e^
3 + (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*b^3*d^2*e^2 - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a*b*c*d^2*e^2
+ 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*b^2*c*d^2*e^2 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*c^2*d^2
*e^2 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a*b*sqrt(c)*d*e^3 - 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A
*b^2*sqrt(c)*d*e^3 + 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a*c^(3/2)*d*e^3 - 5*B*a*b^2*sqrt(c)*d^2*e^2 +
3*A*b^3*sqrt(c)*d^2*e^2 + 4*B*a^2*c^(3/2)*d^2*e^2 + 20*A*a*b*c^(3/2)*d^2*e^2 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^3*B*a*b*e^4 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b^2*e^4 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^3*A*a*c*e^4 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a*b^2*d*e^3 - 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*b
^3*d*e^3 + 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^2*c*d*e^3 - 28*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*b
*c*d*e^3 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^2*sqrt(c)*e^4 + 12*B*a^2*b*sqrt(c)*d*e^3 - 11*A*a*b^2*s
qrt(c)*d*e^3 - 12*A*a^2*c^(3/2)*d*e^3 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^2*b*e^4 + 5*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))*A*a*b^2*e^4 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*c*e^4 - 8*B*a^3*sqrt(c)*e^4 + 8*
A*a^2*b*sqrt(c)*e^4)/((c^2*d^4*e - 2*b*c*d^3*e^2 + b^2*d^2*e^3 + 2*a*c*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5)*((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)^2)