3.1205 \(\int \frac{A+B x}{(d+e x)^3 (b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=374 \[ -\frac{e \sqrt{b x+c x^2} \left (-2 b^2 c d e (5 B d-19 A e)+3 b^3 e^2 (B d-5 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )}{4 b^2 d^3 (d+e x) (c d-b e)^3}-\frac{3 e \left (B d \left (b^2 e^2-4 b c d e+8 c^2 d^2\right )-A e \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{7/2} (c d-b e)^{7/2}}-\frac{e \sqrt{b x+c x^2} \left (b^2 (-e) (B d-5 A e)-4 b c d (2 A e+B d)+8 A c^2 d^2\right )}{2 b^2 d^2 (d+e x)^2 (c d-b e)^2}-\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^2 (c d-b e)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.589581, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {822, 834, 806, 724, 206} \[ -\frac{e \sqrt{b x+c x^2} \left (-2 b^2 c d e (5 B d-19 A e)+3 b^3 e^2 (B d-5 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )}{4 b^2 d^3 (d+e x) (c d-b e)^3}-\frac{3 e \left (B d \left (b^2 e^2-4 b c d e+8 c^2 d^2\right )-A e \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{7/2} (c d-b e)^{7/2}}-\frac{e \sqrt{b x+c x^2} \left (b^2 (-e) (B d-5 A e)-4 b c d (2 A e+B d)+8 A c^2 d^2\right )}{2 b^2 d^2 (d+e x)^2 (c d-b e)^2}-\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^2 (c d-b e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*(d + e*x)^2*Sqrt[b*x + c*x^2]) - (e*
(8*A*c^2*d^2 - b^2*e*(B*d - 5*A*e) - 4*b*c*d*(B*d + 2*A*e))*Sqrt[b*x + c*x^2])/(2*b^2*d^2*(c*d - b*e)^2*(d + e
*x)^2) - (e*(16*A*c^3*d^3 - 2*b^2*c*d*e*(5*B*d - 19*A*e) + 3*b^3*e^2*(B*d - 5*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e)
)*Sqrt[b*x + c*x^2])/(4*b^2*d^3*(c*d - b*e)^3*(d + e*x)) - (3*e*(B*d*(8*c^2*d^2 - 4*b*c*d*e + b^2*e^2) - A*e*(
16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*
x^2])])/(8*d^(7/2)*(c*d - b*e)^(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 (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x)^2 \sqrt{b x+c x^2}}-\frac{2 \int \frac{\frac{1}{2} b e (b B d+4 A c d-5 A b e)-2 c e (b B d-2 A c d+A b e) x}{(d+e x)^3 \sqrt{b x+c x^2}} \, dx}{b^2 d (c d-b e)}\\ &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x)^2 \sqrt{b x+c x^2}}-\frac{e \left (8 A c^2 d^2-b^2 e (B d-5 A e)-4 b c d (B d+2 A e)\right ) \sqrt{b x+c x^2}}{2 b^2 d^2 (c d-b e)^2 (d+e x)^2}+\frac{\int \frac{-\frac{1}{4} b e \left (8 A c^2 d^2+4 b c d (2 B d-7 A e)-3 b^2 e (B d-5 A e)\right )-\frac{1}{2} c e \left (8 A c^2 d^2-b^2 e (B d-5 A e)-4 b c d (B d+2 A e)\right ) x}{(d+e x)^2 \sqrt{b x+c x^2}} \, dx}{b^2 d^2 (c d-b e)^2}\\ &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x)^2 \sqrt{b x+c x^2}}-\frac{e \left (8 A c^2 d^2-b^2 e (B d-5 A e)-4 b c d (B d+2 A e)\right ) \sqrt{b x+c x^2}}{2 b^2 d^2 (c d-b e)^2 (d+e x)^2}-\frac{e \left (16 A c^3 d^3-2 b^2 c d e (5 B d-19 A e)+3 b^3 e^2 (B d-5 A e)-8 b c^2 d^2 (B d+3 A e)\right ) \sqrt{b x+c x^2}}{4 b^2 d^3 (c d-b e)^3 (d+e x)}-\frac{\left (3 e \left (B d \left (8 c^2 d^2-4 b c d e+b^2 e^2\right )-A e \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right )\right )\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{8 d^3 (c d-b e)^3}\\ &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x)^2 \sqrt{b x+c x^2}}-\frac{e \left (8 A c^2 d^2-b^2 e (B d-5 A e)-4 b c d (B d+2 A e)\right ) \sqrt{b x+c x^2}}{2 b^2 d^2 (c d-b e)^2 (d+e x)^2}-\frac{e \left (16 A c^3 d^3-2 b^2 c d e (5 B d-19 A e)+3 b^3 e^2 (B d-5 A e)-8 b c^2 d^2 (B d+3 A e)\right ) \sqrt{b x+c x^2}}{4 b^2 d^3 (c d-b e)^3 (d+e x)}+\frac{\left (3 e \left (B d \left (8 c^2 d^2-4 b c d e+b^2 e^2\right )-A e \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac{-b d-(2 c d-b e) x}{\sqrt{b x+c x^2}}\right )}{4 d^3 (c d-b e)^3}\\ &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x)^2 \sqrt{b x+c x^2}}-\frac{e \left (8 A c^2 d^2-b^2 e (B d-5 A e)-4 b c d (B d+2 A e)\right ) \sqrt{b x+c x^2}}{2 b^2 d^2 (c d-b e)^2 (d+e x)^2}-\frac{e \left (16 A c^3 d^3-2 b^2 c d e (5 B d-19 A e)+3 b^3 e^2 (B d-5 A e)-8 b c^2 d^2 (B d+3 A e)\right ) \sqrt{b x+c x^2}}{4 b^2 d^3 (c d-b e)^3 (d+e x)}-\frac{3 e \left (B d \left (8 c^2 d^2-4 b c d e+b^2 e^2\right )-A e \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right )\right ) \tanh ^{-1}\left (\frac{b d+(2 c d-b e) x}{2 \sqrt{d} \sqrt{c d-b e} \sqrt{b x+c x^2}}\right )}{8 d^{7/2} (c d-b e)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 1.5434, size = 383, normalized size = 1.02 \[ \frac{x \left (\frac{\frac{\sqrt{x} \left (c \sqrt{x} (b+c x) \left (2 b^2 c d e (5 B d-19 A e)+3 b^3 e^2 (5 A e-B d)+8 b c^2 d^2 (3 A e+B d)-16 A c^3 d^3\right )+\frac{3 b^2 e (b+c x)^{3/2} \left (A e \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (b^2 e^2-4 b c d e+8 c^2 d^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{d} \sqrt{b e-c d}}\right )}{b (c d-b e)}-(b+c x) \left (3 b^2 e (5 A e-B d)+4 b c d (2 B d-7 A e)+8 A c^2 d^2\right )}{2 b d^2 (b e-c d)}-\frac{(b+c x) (5 A e (b e-2 c d)+B d (6 c d-b e))}{2 d (d+e x) (c d-b e)}+\frac{(b+c x) (A e-B d)}{(d+e x)^2}\right )}{2 d (x (b+c x))^{3/2} (b e-c d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(((-(B*d) + A*e)*(b + c*x))/(d + e*x)^2 - ((B*d*(6*c*d - b*e) + 5*A*e*(-2*c*d + b*e))*(b + c*x))/(2*d*(c*d
- b*e)*(d + e*x)) + (-((8*A*c^2*d^2 + 4*b*c*d*(2*B*d - 7*A*e) + 3*b^2*e*(-(B*d) + 5*A*e))*(b + c*x)) + (Sqrt[x
]*(c*(-16*A*c^3*d^3 + 2*b^2*c*d*e*(5*B*d - 19*A*e) + 8*b*c^2*d^2*(B*d + 3*A*e) + 3*b^3*e^2*(-(B*d) + 5*A*e))*S
qrt[x]*(b + c*x) + (3*b^2*e*(-(B*d*(8*c^2*d^2 - 4*b*c*d*e + b^2*e^2)) + A*e*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e
^2))*(b + c*x)^(3/2)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*Sqrt[-(c*d) + b*e]
)))/(b*(c*d - b*e)))/(2*b*d^2*(-(c*d) + b*e))))/(2*d*(-(c*d) + b*e)*(x*(b + c*x))^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 3735, normalized size = 10. \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)^(3/2),x)

[Out]

B/e/d/(b*e-c*d)/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)-3*B*e/d^2/(b*e-c*d)^2/((x+d/
e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b-5/4/d/(b*e-c*d)^2/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+
d/e)-d*(b*e-c*d)/e^2)^(1/2)*b*B-5/2/d/(b*e-c*d)^2/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^
(1/2)*c*A+30/(b*e-c*d)^3/b^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*c^4*A-15/4*e^3/d^3/(b
*e-c*d)^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b^2*A+15/4*e^2/d^2/(b*e-c*d)^3/((x+d/e)^2*
c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b^2*B-30*e/d/(b*e-c*d)^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(
b*e-c*d)/e^2)^(1/2)*c^2*A-8*e/d^2/(b*e-c*d)^2*c/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*A+1/
2/e/d/(b*e-c*d)/(x+d/e)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*A-19/e/(b*e-c*d)^2*c^2/b/(
(x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*B+5/2/e/(b*e-c*d)^2/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/
e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c*B+45/(b*e-c*d)^3/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2
)*x*c^3*B+13/d/(b*e-c*d)^2*c^2/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*A-9/2*c/d/(b*e-c*d)
^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2
*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*B+30/(b*e-c*d)^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-
d*(b*e-c*d)/e^2)^(1/2)*c^2*B-15/2/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x
+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*c^2*B-1/2
/e^2/(b*e-c*d)/(x+d/e)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*B+17/d/(b*e-c*d)^2*c/((x+d/
e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*B+15/(b*e-c*d)^3/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b
*e-c*d)/e^2)^(1/2)*c^3*A-8*B/e*c^2/d/(b*e-c*d)/b^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x
-30/e/(b*e-c*d)^3/b^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*c^4*B*d-15/2*e^2/d^2/(b*e-c*
d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)
^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b*c*A-15/4*e^3/d^3/(b*e-c*d)^3/((x+d/e)^2*c+(b*e-2
*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*c*b*A+15/4*e^2/d^2/(b*e-c*d)^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(
b*e-c*d)/e^2)^(1/2)*x*c*b*B-3*B*e/d^2/(b*e-c*d)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*
c+5/4*e/d^2/(b*e-c*d)^2/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b*A+75/4*e^2/d^2/(b*
e-c*d)^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b*c*A+45/2*e^2/d^2/(b*e-c*d)^3/((x+d/e)^2*c
+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*A-45/2*e/d/(b*e-c*d)^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-
d*(b*e-c*d)/e^2)^(1/2)*x*c^2*B-15/e/(b*e-c*d)^3/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c^
3*B*d+15/8*e^3/d^3/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*
e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b^2*A-15/8*e^2/d^2/(b*e-
c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/
e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b^2*B+3/2*B*e/d^2/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)
^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d
/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b-4*B/e*c/d/(b*e-c*d)/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^
2)^(1/2)-75/4*e/d/(b*e-c*d)^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b*c*B+15/2*e/d/(b*e-c*
d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)
^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*c^2*A+3/2*e*c/d^2/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(
1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e
)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*A-38/e/(b*e-c*d)^2*c^3/b^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e
^2)^(1/2)*x*B+25/d/(b*e-c*d)^2*c^2/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*B+26/d/(b*e-c
*d)^2*c^3/b^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*A-45*e/d/(b*e-c*d)^3/b/((x+d/e)^2*c+
(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*c^3*A+15/2*e/d/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b
*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^
2)^(1/2))/(x+d/e))*b*c*B-13*e/d^2/(b*e-c*d)^2*c^2/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*
x*A

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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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.42566, size = 4636, normalized size = 12.4 \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)^(3/2),x, algorithm="fricas")

[Out]

[-1/8*(3*((8*B*b^2*c^3*d^3*e^3 - 5*A*b^4*c*e^6 - 4*(B*b^3*c^2 + 4*A*b^2*c^3)*d^2*e^4 + (B*b^4*c + 16*A*b^3*c^2
)*d*e^5)*x^4 + (16*B*b^2*c^3*d^4*e^2 - 32*A*b^2*c^3*d^3*e^3 - 5*A*b^5*e^6 - 2*(B*b^4*c - 8*A*b^3*c^2)*d^2*e^4
+ (B*b^5 + 6*A*b^4*c)*d*e^5)*x^3 + (8*B*b^2*c^3*d^5*e - 10*A*b^5*d*e^5 + 4*(3*B*b^3*c^2 - 4*A*b^2*c^3)*d^4*e^2
- (7*B*b^4*c + 16*A*b^3*c^2)*d^3*e^3 + (2*B*b^5 + 27*A*b^4*c)*d^2*e^4)*x^2 + (8*B*b^3*c^2*d^5*e - 5*A*b^5*d^2
*e^4 - 4*(B*b^4*c + 4*A*b^3*c^2)*d^4*e^2 + (B*b^5 + 16*A*b^4*c)*d^3*e^3)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*
c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + 2*(8*A*b*c^4*d^7 - 32*A*b^2*c^3*d^6*e + 4
8*A*b^3*c^2*d^5*e^2 - 32*A*b^4*c*d^4*e^3 + 8*A*b^5*d^3*e^4 + (15*A*b^4*c*d*e^6 - 8*(B*b*c^4 - 2*A*c^5)*d^5*e^2
- 2*(B*b^2*c^3 + 20*A*b*c^4)*d^4*e^3 + (13*B*b^3*c^2 + 62*A*b^2*c^3)*d^3*e^4 - (3*B*b^4*c + 53*A*b^3*c^2)*d^2
*e^5)*x^3 + (15*A*b^5*d*e^6 - 16*(B*b*c^4 - 2*A*c^5)*d^6*e + 4*(B*b^2*c^3 - 18*A*b*c^4)*d^5*e^2 + (7*B*b^3*c^2
+ 80*A*b^2*c^3)*d^4*e^3 + (8*B*b^4*c - 27*A*b^3*c^2)*d^3*e^4 - (3*B*b^5 + 28*A*b^4*c)*d^2*e^5)*x^2 + (25*A*b^
5*d^2*e^5 - 8*(B*b*c^4 - 2*A*c^5)*d^7 + 8*(B*b^2*c^3 - 3*A*b*c^4)*d^6*e - 4*(3*B*b^3*c^2 + 4*A*b^2*c^3)*d^5*e^
2 + (17*B*b^4*c + 80*A*b^3*c^2)*d^4*e^3 - (5*B*b^5 + 81*A*b^4*c)*d^3*e^4)*x)*sqrt(c*x^2 + b*x))/((b^2*c^5*d^8*
e^2 - 4*b^3*c^4*d^7*e^3 + 6*b^4*c^3*d^6*e^4 - 4*b^5*c^2*d^5*e^5 + b^6*c*d^4*e^6)*x^4 + (2*b^2*c^5*d^9*e - 7*b^
3*c^4*d^8*e^2 + 8*b^4*c^3*d^7*e^3 - 2*b^5*c^2*d^6*e^4 - 2*b^6*c*d^5*e^5 + b^7*d^4*e^6)*x^3 + (b^2*c^5*d^10 - 2
*b^3*c^4*d^9*e - 2*b^4*c^3*d^8*e^2 + 8*b^5*c^2*d^7*e^3 - 7*b^6*c*d^6*e^4 + 2*b^7*d^5*e^5)*x^2 + (b^3*c^4*d^10
- 4*b^4*c^3*d^9*e + 6*b^5*c^2*d^8*e^2 - 4*b^6*c*d^7*e^3 + b^7*d^6*e^4)*x), -1/4*(3*((8*B*b^2*c^3*d^3*e^3 - 5*A
*b^4*c*e^6 - 4*(B*b^3*c^2 + 4*A*b^2*c^3)*d^2*e^4 + (B*b^4*c + 16*A*b^3*c^2)*d*e^5)*x^4 + (16*B*b^2*c^3*d^4*e^2
- 32*A*b^2*c^3*d^3*e^3 - 5*A*b^5*e^6 - 2*(B*b^4*c - 8*A*b^3*c^2)*d^2*e^4 + (B*b^5 + 6*A*b^4*c)*d*e^5)*x^3 + (
8*B*b^2*c^3*d^5*e - 10*A*b^5*d*e^5 + 4*(3*B*b^3*c^2 - 4*A*b^2*c^3)*d^4*e^2 - (7*B*b^4*c + 16*A*b^3*c^2)*d^3*e^
3 + (2*B*b^5 + 27*A*b^4*c)*d^2*e^4)*x^2 + (8*B*b^3*c^2*d^5*e - 5*A*b^5*d^2*e^4 - 4*(B*b^4*c + 4*A*b^3*c^2)*d^4
*e^2 + (B*b^5 + 16*A*b^4*c)*d^3*e^3)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((
c*d - b*e)*x)) + (8*A*b*c^4*d^7 - 32*A*b^2*c^3*d^6*e + 48*A*b^3*c^2*d^5*e^2 - 32*A*b^4*c*d^4*e^3 + 8*A*b^5*d^3
*e^4 + (15*A*b^4*c*d*e^6 - 8*(B*b*c^4 - 2*A*c^5)*d^5*e^2 - 2*(B*b^2*c^3 + 20*A*b*c^4)*d^4*e^3 + (13*B*b^3*c^2
+ 62*A*b^2*c^3)*d^3*e^4 - (3*B*b^4*c + 53*A*b^3*c^2)*d^2*e^5)*x^3 + (15*A*b^5*d*e^6 - 16*(B*b*c^4 - 2*A*c^5)*d
^6*e + 4*(B*b^2*c^3 - 18*A*b*c^4)*d^5*e^2 + (7*B*b^3*c^2 + 80*A*b^2*c^3)*d^4*e^3 + (8*B*b^4*c - 27*A*b^3*c^2)*
d^3*e^4 - (3*B*b^5 + 28*A*b^4*c)*d^2*e^5)*x^2 + (25*A*b^5*d^2*e^5 - 8*(B*b*c^4 - 2*A*c^5)*d^7 + 8*(B*b^2*c^3 -
3*A*b*c^4)*d^6*e - 4*(3*B*b^3*c^2 + 4*A*b^2*c^3)*d^5*e^2 + (17*B*b^4*c + 80*A*b^3*c^2)*d^4*e^3 - (5*B*b^5 + 8
1*A*b^4*c)*d^3*e^4)*x)*sqrt(c*x^2 + b*x))/((b^2*c^5*d^8*e^2 - 4*b^3*c^4*d^7*e^3 + 6*b^4*c^3*d^6*e^4 - 4*b^5*c^
2*d^5*e^5 + b^6*c*d^4*e^6)*x^4 + (2*b^2*c^5*d^9*e - 7*b^3*c^4*d^8*e^2 + 8*b^4*c^3*d^7*e^3 - 2*b^5*c^2*d^6*e^4
- 2*b^6*c*d^5*e^5 + b^7*d^4*e^6)*x^3 + (b^2*c^5*d^10 - 2*b^3*c^4*d^9*e - 2*b^4*c^3*d^8*e^2 + 8*b^5*c^2*d^7*e^3
- 7*b^6*c*d^6*e^4 + 2*b^7*d^5*e^5)*x^2 + (b^3*c^4*d^10 - 4*b^4*c^3*d^9*e + 6*b^5*c^2*d^8*e^2 - 4*b^6*c*d^7*e^
3 + b^7*d^6*e^4)*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)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.70358, size = 1478, normalized size = 3.95 \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)^(3/2),x, algorithm="giac")

[Out]

2*((B*b*c^3*d^6 - 2*A*c^4*d^6 + 3*A*b*c^3*d^5*e - 3*A*b^2*c^2*d^4*e^2 + A*b^3*c*d^3*e^3)*x/(b^2*c^3*d^9 - 3*b^
3*c^2*d^8*e + 3*b^4*c*d^7*e^2 - b^5*d^6*e^3) - (A*b*c^3*d^6 - 3*A*b^2*c^2*d^5*e + 3*A*b^3*c*d^4*e^2 - A*b^4*d^
3*e^3)/(b^2*c^3*d^9 - 3*b^3*c^2*d^8*e + 3*b^4*c*d^7*e^2 - b^5*d^6*e^3))/sqrt(c*x^2 + b*x) - 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 + 16*A*b*c*d*e^3 - 5*A*b^2*e^4)*arctan(-((sqrt(c)*x - sqrt
(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)
*sqrt(-c*d^2 + b*d*e)) + 1/4*(40*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*c^(5/2)*d^4*e + 16*(sqrt(c)*x - sqrt(c*x^
2 + b*x))^3*B*c^2*d^3*e^2 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b*c^2*d^4*e - 28*(sqrt(c)*x - sqrt(c*x^2 + b*
x))^2*B*b*c^(3/2)*d^3*e^2 - 56*(sqrt(c)*x - sqrt(c*x^2 + b*x))^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))^3*B*b*c*d^2*e^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*c^2*d^2*e^3 - 24*(
sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^2*c*d^3*e^2 - 56*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b*c^2*d^3*e^2 + 9*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^2*B*b^2*sqrt(c)*d^2*e^3 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b*c^(3/2)*d^2*e^3 -
3*B*b^3*sqrt(c)*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))^3*B*b^2*d*e^4 + 24*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^3*A*b*c*d*e^4 + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^3*d^2*e^3 + 44*(sqrt(c)*x -
sqrt(c*x^2 + b*x))*A*b^2*c*d^2*e^3 - 13*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^2*sqrt(c)*d*e^4 + 7*A*b^3*sqrt(
c)*d^2*e^3 - 7*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^2*e^5 - 9*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^3*d*e^4)/((
c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*((sqrt(c)*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x -
sqrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^2)