3.2479 \(\int \frac{A+B x}{(d+e x)^3 (a+b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=545 \[ \frac{3 e \left (A e \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )-B \left (-4 c d e (3 a e+b d)+b e^2 (4 a e+b d)+8 c^2 d^3\right )\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac{e \sqrt{a+b x+c x^2} \left (4 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-3 a A e^2+5 a B d e+2 A c d^2\right )+b^2 e (B d-5 A e)\right )}{2 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \sqrt{a+b x+c x^2} \left (-2 b^2 e \left (-6 a B e^2-19 A c d e+5 B c d^2\right )-4 b c \left (-13 a A e^3+9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+8 c \left (A c d \left (2 c d^2-13 a e^2\right )+a B e \left (11 c d^2-4 a e^2\right )\right )+3 b^3 e^2 (B d-5 A e)\right )}{4 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

[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)^2*Sqrt[a + b*x + c*x^2]) + (e*(b^2*e*(B*d - 5*A*e) - 4*c*(2*A*c*d^2 + 5*a*B
*d*e - 3*a*A*e^2) + 4*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[a + b*x + c*x^2])/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*
e + a*e^2)^2*(d + e*x)^2) - (e*(3*b^3*e^2*(B*d - 5*A*e) - 2*b^2*e*(5*B*c*d^2 - 19*A*c*d*e - 6*a*B*e^2) - 4*b*c
*(2*B*c*d^3 + 6*A*c*d^2*e + 9*a*B*d*e^2 - 13*a*A*e^3) + 8*c*(A*c*d*(2*c*d^2 - 13*a*e^2) + a*B*e*(11*c*d^2 - 4*
a*e^2)))*Sqrt[a + b*x + c*x^2])/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (3*e*(A*e*(16*c^2*d^2
+ 5*b^2*e^2 - 4*c*e*(4*b*d + a*e)) - B*(8*c^2*d^3 - 4*c*d*e*(b*d + 3*a*e) + b*e^2*(b*d + 4*a*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])])/(8*(c*d^2 - b*d*e + a*e^2)
^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.932847, antiderivative size = 545, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {822, 834, 806, 724, 206} \[ \frac{3 e \left (A e \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )-B \left (-4 c d e (3 a e+b d)+b e^2 (4 a e+b d)+8 c^2 d^3\right )\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac{e \sqrt{a+b x+c x^2} \left (4 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-3 a A e^2+5 a B d e+2 A c d^2\right )+b^2 e (B d-5 A e)\right )}{2 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \sqrt{a+b x+c x^2} \left (-2 b^2 e \left (-6 a B e^2-19 A c d e+5 B c d^2\right )-4 b c \left (-13 a A e^3+9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+8 c \left (A c d \left (2 c d^2-13 a e^2\right )+a B e \left (11 c d^2-4 a e^2\right )\right )+3 b^3 e^2 (B d-5 A e)\right )}{4 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^3*(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)^2*Sqrt[a + b*x + c*x^2]) + (e*(b^2*e*(B*d - 5*A*e) - 4*c*(2*A*c*d^2 + 5*a*B
*d*e - 3*a*A*e^2) + 4*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[a + b*x + c*x^2])/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*
e + a*e^2)^2*(d + e*x)^2) - (e*(3*b^3*e^2*(B*d - 5*A*e) - 2*b^2*e*(5*B*c*d^2 - 19*A*c*d*e - 6*a*B*e^2) - 4*b*c
*(2*B*c*d^3 + 6*A*c*d^2*e + 9*a*B*d*e^2 - 13*a*A*e^3) + 8*c*(A*c*d*(2*c*d^2 - 13*a*e^2) + a*B*e*(11*c*d^2 - 4*
a*e^2)))*Sqrt[a + b*x + c*x^2])/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (3*e*(A*e*(16*c^2*d^2
+ 5*b^2*e^2 - 4*c*e*(4*b*d + a*e)) - B*(8*c^2*d^3 - 4*c*d*e*(b*d + 3*a*e) + b*e^2*(b*d + 4*a*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])])/(8*(c*d^2 - b*d*e + a*e^2)
^(7/2))

Rule 822

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])

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 \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)^2 \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{\frac{1}{2} e \left (b^2 (B d-5 A e)-12 a c (B d-A e)+4 b (A c d+a B e)\right )-2 c e (b B d-2 A c d+A b e-2 a B e) x}{(d+e x)^3 \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)^2 \sqrt{a+b x+c x^2}}+\frac{e \left (b^2 e (B d-5 A e)-4 c \left (2 A c d^2+5 a B d e-3 a A e^2\right )+4 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt{a+b x+c x^2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{\int \frac{\frac{1}{4} e \left (3 b^3 e (B d-5 A e)-4 b c \left (2 A c d^2+7 a B d e-13 a A e^2\right )-4 b^2 \left (2 B c d^2-7 A c d e-3 a B e^2\right )+16 a c \left (3 B c d^2-5 A c d e-2 a B e^2\right )\right )+\frac{1}{2} c e \left (b^2 e (B d-5 A e)-4 c \left (2 A c d^2+5 a B d e-3 a A e^2\right )+4 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) 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 )^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)^2 \sqrt{a+b x+c x^2}}+\frac{e \left (b^2 e (B d-5 A e)-4 c \left (2 A c d^2+5 a B d e-3 a A e^2\right )+4 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt{a+b x+c x^2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e \left (3 b^3 e^2 (B d-5 A e)-2 b^2 e \left (5 B c d^2-19 A c d e-6 a B e^2\right )-4 b c \left (2 B c d^3+6 A c d^2 e+9 a B d e^2-13 a A e^3\right )+8 c \left (A c d \left (2 c d^2-13 a e^2\right )+a B e \left (11 c d^2-4 a e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac{\left (3 e \left (A e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )-B \left (8 c^2 d^3-4 c d e (b d+3 a e)+b e^2 (b d+4 a e)\right )\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 )^3}\\ &=\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)^2 \sqrt{a+b x+c x^2}}+\frac{e \left (b^2 e (B d-5 A e)-4 c \left (2 A c d^2+5 a B d e-3 a A e^2\right )+4 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt{a+b x+c x^2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e \left (3 b^3 e^2 (B d-5 A e)-2 b^2 e \left (5 B c d^2-19 A c d e-6 a B e^2\right )-4 b c \left (2 B c d^3+6 A c d^2 e+9 a B d e^2-13 a A e^3\right )+8 c \left (A c d \left (2 c d^2-13 a e^2\right )+a B e \left (11 c d^2-4 a e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{\left (3 e \left (A e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )-B \left (8 c^2 d^3-4 c d e (b d+3 a e)+b e^2 (b d+4 a e)\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 )}{4 \left (c d^2-b d e+a e^2\right )^3}\\ &=\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)^2 \sqrt{a+b x+c x^2}}+\frac{e \left (b^2 e (B d-5 A e)-4 c \left (2 A c d^2+5 a B d e-3 a A e^2\right )+4 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt{a+b x+c x^2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e \left (3 b^3 e^2 (B d-5 A e)-2 b^2 e \left (5 B c d^2-19 A c d e-6 a B e^2\right )-4 b c \left (2 B c d^3+6 A c d^2 e+9 a B d e^2-13 a A e^3\right )+8 c \left (A c d \left (2 c d^2-13 a e^2\right )+a B e \left (11 c d^2-4 a e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac{3 e \left (A e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )-B \left (8 c^2 d^3-4 c d e (b d+3 a e)+b e^2 (b d+4 a e)\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 )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 4.11383, size = 527, normalized size = 0.97 \[ \frac{2 \left (\frac{3 e \left (b^2-4 a c\right ) \left (A e \left (4 c e (a e+4 b d)-5 b^2 e^2-16 c^2 d^2\right )+B \left (-4 c d e (3 a e+b d)+b e^2 (4 a e+b d)+8 c^2 d^3\right )\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{16 \left (e (a e-b d)+c d^2\right )^{5/2}}+\frac{e \sqrt{a+x (b+c x)} \left (4 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-3 a A e^2+5 a B d e+2 A c d^2\right )+b^2 e (B d-5 A e)\right )}{4 (d+e x)^2 \left (e (a e-b d)+c d^2\right )}-\frac{e \sqrt{a+x (b+c x)} \left (2 b^2 e \left (6 a B e^2+19 A c d e-5 B c d^2\right )-4 b c \left (-13 a A e^3+9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+8 c \left (A c d \left (2 c d^2-13 a e^2\right )+a B e \left (11 c d^2-4 a e^2\right )\right )+3 b^3 e^2 (B d-5 A e)\right )}{8 (d+e x) \left (e (a e-b d)+c d^2\right )^2}+\frac{-2 A c (a e+c d x)+a B (2 c (d-e x)-b e)+A b^2 e+A b c (e x-d)+b B c d x}{(d+e x)^2 \sqrt{a+x (b+c x)}}\right )}{\left (b^2-4 a c\right ) \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((e*(b^2*e*(B*d - 5*A*e) - 4*c*(2*A*c*d^2 + 5*a*B*d*e - 3*a*A*e^2) + 4*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*S
qrt[a + x*(b + c*x)])/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) - (e*(3*b^3*e^2*(B*d - 5*A*e) + 2*b^2*e*(-5*B
*c*d^2 + 19*A*c*d*e + 6*a*B*e^2) - 4*b*c*(2*B*c*d^3 + 6*A*c*d^2*e + 9*a*B*d*e^2 - 13*a*A*e^3) + 8*c*(A*c*d*(2*
c*d^2 - 13*a*e^2) + a*B*e*(11*c*d^2 - 4*a*e^2)))*Sqrt[a + x*(b + c*x)])/(8*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e
*x)) + (A*b^2*e + b*B*c*d*x - 2*A*c*(a*e + c*d*x) + A*b*c*(-d + e*x) + a*B*(-(b*e) + 2*c*(d - e*x)))/((d + e*x
)^2*Sqrt[a + x*(b + c*x)]) + (3*(b^2 - 4*a*c)*e*(A*e*(-16*c^2*d^2 - 5*b^2*e^2 + 4*c*e*(4*b*d + a*e)) + B*(8*c^
2*d^3 - 4*c*d*e*(b*d + 3*a*e) + b*e^2*(b*d + 4*a*e)))*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)])])/(16*(c*d^2 + e*(-(b*d) + a*e))^(5/2))))/((b^2 - 4*a*c)*(c*d^2 +
e*(-(b*d) + a*e)))

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Maple [B]  time = 0.019, size = 5528, normalized size = 10.1 \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)^3/(c*x^2+b*x+a)^(3/2),x)

[Out]

result too large to display

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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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

[Out]

Timed out

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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)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.60781, size = 5403, normalized size = 9.91 \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)^(3/2),x, algorithm="giac")

[Out]

2*((B*b*c^6*d^9 - 2*A*c^7*d^9 - 3*B*b^2*c^5*d^8*e - 6*B*a*c^6*d^8*e + 9*A*b*c^6*d^8*e + 3*B*b^3*c^4*d^7*e^2 +
24*B*a*b*c^5*d^7*e^2 - 18*A*b^2*c^5*d^7*e^2 - B*b^4*c^3*d^6*e^3 - 34*B*a*b^2*c^4*d^6*e^3 + 21*A*b^3*c^4*d^6*e^
3 - 16*B*a^2*c^5*d^6*e^3 + 21*B*a*b^3*c^3*d^5*e^4 - 15*A*b^4*c^3*d^5*e^4 + 42*B*a^2*b*c^4*d^5*e^4 - 6*A*a*b^2*
c^4*d^5*e^4 + 12*A*a^2*c^5*d^5*e^4 - 6*B*a*b^4*c^2*d^4*e^5 + 6*A*b^5*c^2*d^4*e^5 - 36*B*a^2*b^2*c^3*d^4*e^5 +
15*A*a*b^3*c^3*d^4*e^5 - 12*B*a^3*c^4*d^4*e^5 - 30*A*a^2*b*c^4*d^4*e^5 + B*a*b^5*c*d^3*e^6 - A*b^6*c*d^3*e^6 +
13*B*a^2*b^3*c^2*d^3*e^6 - 12*A*a*b^4*c^2*d^3*e^6 + 16*B*a^3*b*c^3*d^3*e^6 + 18*A*a^2*b^2*c^3*d^3*e^6 + 16*A*
a^3*c^4*d^3*e^6 - 3*B*a^2*b^4*c*d^2*e^7 + 3*A*a*b^5*c*d^2*e^7 - 6*B*a^3*b^2*c^2*d^2*e^7 + 3*A*a^2*b^3*c^2*d^2*
e^7 - 24*A*a^3*b*c^3*d^2*e^7 + 3*B*a^3*b^3*c*d*e^8 - 3*A*a^2*b^4*c*d*e^8 - 3*B*a^4*b*c^2*d*e^8 + 6*A*a^3*b^2*c
^2*d*e^8 + 6*A*a^4*c^3*d*e^8 - B*a^4*b^2*c*e^9 + A*a^3*b^3*c*e^9 + 2*B*a^5*c^2*e^9 - 3*A*a^4*b*c^2*e^9)*x/(b^2
*c^6*d^12 - 4*a*c^7*d^12 - 6*b^3*c^5*d^11*e + 24*a*b*c^6*d^11*e + 15*b^4*c^4*d^10*e^2 - 54*a*b^2*c^5*d^10*e^2
- 24*a^2*c^6*d^10*e^2 - 20*b^5*c^3*d^9*e^3 + 50*a*b^3*c^4*d^9*e^3 + 120*a^2*b*c^5*d^9*e^3 + 15*b^6*c^2*d^8*e^4
- 225*a^2*b^2*c^4*d^8*e^4 - 60*a^3*c^5*d^8*e^4 - 6*b^7*c*d^7*e^5 - 36*a*b^5*c^2*d^7*e^5 + 180*a^2*b^3*c^3*d^7
*e^5 + 240*a^3*b*c^4*d^7*e^5 + b^8*d^6*e^6 + 26*a*b^6*c*d^6*e^6 - 30*a^2*b^4*c^2*d^6*e^6 - 340*a^3*b^2*c^3*d^6
*e^6 - 80*a^4*c^4*d^6*e^6 - 6*a*b^7*d^5*e^7 - 36*a^2*b^5*c*d^5*e^7 + 180*a^3*b^3*c^2*d^5*e^7 + 240*a^4*b*c^3*d
^5*e^7 + 15*a^2*b^6*d^4*e^8 - 225*a^4*b^2*c^2*d^4*e^8 - 60*a^5*c^3*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 50*a^4*b^3*c
*d^3*e^9 + 120*a^5*b*c^2*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 54*a^5*b^2*c*d^2*e^10 - 24*a^6*c^2*d^2*e^10 - 6*a^5*b
^3*d*e^11 + 24*a^6*b*c*d*e^11 + a^6*b^2*e^12 - 4*a^7*c*e^12) + (2*B*a*c^6*d^9 - A*b*c^6*d^9 - 9*B*a*b*c^5*d^8*
e + 6*A*b^2*c^5*d^8*e - 6*A*a*c^6*d^8*e + 18*B*a*b^2*c^4*d^7*e^2 - 15*A*b^3*c^4*d^7*e^2 + 24*A*a*b*c^5*d^7*e^2
- 21*B*a*b^3*c^3*d^6*e^3 + 20*A*b^4*c^3*d^6*e^3 - 34*A*a*b^2*c^4*d^6*e^3 - 16*A*a^2*c^5*d^6*e^3 + 15*B*a*b^4*
c^2*d^5*e^4 - 15*A*b^5*c^2*d^5*e^4 + 6*B*a^2*b^2*c^3*d^5*e^4 + 15*A*a*b^3*c^3*d^5*e^4 - 12*B*a^3*c^4*d^5*e^4 +
54*A*a^2*b*c^4*d^5*e^4 - 6*B*a*b^5*c*d^4*e^5 + 6*A*b^6*c*d^4*e^5 - 15*B*a^2*b^3*c^2*d^4*e^5 + 9*A*a*b^4*c^2*d
^4*e^5 + 30*B*a^3*b*c^3*d^4*e^5 - 66*A*a^2*b^2*c^3*d^4*e^5 - 12*A*a^3*c^4*d^4*e^5 + B*a*b^6*d^3*e^6 - A*b^7*d^
3*e^6 + 12*B*a^2*b^4*c*d^3*e^6 - 11*A*a*b^5*c*d^3*e^6 - 18*B*a^3*b^2*c^2*d^3*e^6 + 31*A*a^2*b^3*c^2*d^3*e^6 -
16*B*a^4*c^3*d^3*e^6 + 32*A*a^3*b*c^3*d^3*e^6 - 3*B*a^2*b^5*d^2*e^7 + 3*A*a*b^6*d^2*e^7 - 3*B*a^3*b^3*c*d^2*e^
7 + 24*B*a^4*b*c^2*d^2*e^7 - 30*A*a^3*b^2*c^2*d^2*e^7 + 3*B*a^3*b^4*d*e^8 - 3*A*a^2*b^5*d*e^8 - 6*B*a^4*b^2*c*
d*e^8 + 9*A*a^3*b^3*c*d*e^8 - 6*B*a^5*c^2*d*e^8 + 3*A*a^4*b*c^2*d*e^8 - B*a^4*b^3*e^9 + A*a^3*b^4*e^9 + 3*B*a^
5*b*c*e^9 - 4*A*a^4*b^2*c*e^9 + 2*A*a^5*c^2*e^9)/(b^2*c^6*d^12 - 4*a*c^7*d^12 - 6*b^3*c^5*d^11*e + 24*a*b*c^6*
d^11*e + 15*b^4*c^4*d^10*e^2 - 54*a*b^2*c^5*d^10*e^2 - 24*a^2*c^6*d^10*e^2 - 20*b^5*c^3*d^9*e^3 + 50*a*b^3*c^4
*d^9*e^3 + 120*a^2*b*c^5*d^9*e^3 + 15*b^6*c^2*d^8*e^4 - 225*a^2*b^2*c^4*d^8*e^4 - 60*a^3*c^5*d^8*e^4 - 6*b^7*c
*d^7*e^5 - 36*a*b^5*c^2*d^7*e^5 + 180*a^2*b^3*c^3*d^7*e^5 + 240*a^3*b*c^4*d^7*e^5 + b^8*d^6*e^6 + 26*a*b^6*c*d
^6*e^6 - 30*a^2*b^4*c^2*d^6*e^6 - 340*a^3*b^2*c^3*d^6*e^6 - 80*a^4*c^4*d^6*e^6 - 6*a*b^7*d^5*e^7 - 36*a^2*b^5*
c*d^5*e^7 + 180*a^3*b^3*c^2*d^5*e^7 + 240*a^4*b*c^3*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 225*a^4*b^2*c^2*d^4*e^8 - 6
0*a^5*c^3*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 50*a^4*b^3*c*d^3*e^9 + 120*a^5*b*c^2*d^3*e^9 + 15*a^4*b^4*d^2*e^10 -
54*a^5*b^2*c*d^2*e^10 - 24*a^6*c^2*d^2*e^10 - 6*a^5*b^3*d*e^11 + 24*a^6*b*c*d*e^11 + a^6*b^2*e^12 - 4*a^7*c*e^
12))/sqrt(c*x^2 + b*x + a) - 3/4*(8*B*c^2*d^3*e - 4*B*b*c*d^2*e^2 - 16*A*c^2*d^2*e^2 + B*b^2*d*e^3 - 12*B*a*c*
d*e^3 + 16*A*b*c*d*e^3 + 4*B*a*b*e^4 - 5*A*b^2*e^4 + 4*A*a*c*e^4)*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)) + 1/4*(40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*c^(5/2)*d^4*e + 16*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^3*B*c^2*d^3*e^2 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*b*c^2*d^4*e - 28*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^2*B*b*c^(3/2)*d^3*e^2 - 56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*c^(5/2)*d^3*e^2 + 10*B*b^2*c^(3/2
)*d^4*e - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b*c*d^2*e^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*
c^2*d^2*e^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*b^2*c*d^3*e^2 - 64*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*
B*a*c^2*d^3*e^2 - 56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*b*c^2*d^3*e^2 + 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^2*B*b^2*sqrt(c)*d^2*e^3 - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a*c^(3/2)*d^2*e^3 + 48*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^2*A*b*c^(3/2)*d^2*e^3 - 3*B*b^3*sqrt(c)*d^3*e^2 - 32*B*a*b*c^(3/2)*d^3*e^2 - 14*A*b^2*c^(
3/2)*d^3*e^2 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b^2*d*e^4 - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*
B*a*c*d*e^4 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b*c*d*e^4 + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*b
^3*d^2*e^3 + 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a*b*c*d^2*e^3 + 44*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A
*b^2*c*d^2*e^3 + 88*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*c^2*d^2*e^3 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^2*B*a*b*sqrt(c)*d*e^4 - 13*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b^2*sqrt(c)*d*e^4 + 28*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^2*A*a*c^(3/2)*d*e^4 + 7*B*a*b^2*sqrt(c)*d^2*e^3 + 7*A*b^3*sqrt(c)*d^2*e^3 + 20*B*a^2*c^(3/2)
*d^2*e^3 + 44*A*a*b*c^(3/2)*d^2*e^3 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a*b*e^5 - 7*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^3*A*b^2*e^5 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a*c*e^5 - (sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*B*a*b^2*d*e^4 - 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*b^3*d*e^4 + 20*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*B*a^2*c*d*e^4 - 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*b*c*d*e^4 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*B*a^2*sqrt(c)*e^5 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a*b*sqrt(c)*e^5 + 4*B*a^2*b*sqrt(c)*d*e^
4 - 23*A*a*b^2*sqrt(c)*d*e^4 - 28*A*a^2*c^(3/2)*d*e^4 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^2*b*e^5 + 9*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*b^2*e^5 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*c*e^5 - 8*B*a^3*
sqrt(c)*e^5 + 16*A*a^2*b*sqrt(c)*e^5)/((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)*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)