3.1254 \(\int \frac{A+B x}{(d+e x)^{5/2} (b x+c x^2)^3} \, dx\)

Optimal. Leaf size=644 \[ \frac{c x \left (b^2 c d e (23 B d-2 A e)+b^3 \left (-e^2\right ) (4 B d-7 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )+b (c d-b e) \left (b^2 e (4 B d-7 A e)-3 b c d (A e+2 B d)+12 A c^2 d^2\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)^2}+\frac{e \left (-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (4 A e+5 B d)+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-12 b c^4 d^4 (5 A e+B d)+24 A c^5 d^5\right )}{4 b^4 d^4 \sqrt{d+e x} (c d-b e)^4}+\frac{e \left (3 b^2 c^2 d^2 e (9 A e+29 B d)-9 b^3 c d e^2 (4 B d-5 A e)+5 b^4 e^3 (4 B d-7 A e)-36 b c^3 d^3 (4 A e+B d)+72 A c^4 d^4\right )}{12 b^4 d^3 (d+e x)^{3/2} (c d-b e)^3}+\frac{c^{7/2} \left (11 b^2 c e (13 A e+8 B d)-12 b c^2 d (13 A e+2 B d)+48 A c^3 d^2-99 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)+48 A c^2 d^2\right )}{4 b^5 d^{9/2}}-\frac{c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)} \]

[Out]

(e*(72*A*c^4*d^4 + 5*b^4*e^3*(4*B*d - 7*A*e) - 9*b^3*c*d*e^2*(4*B*d - 5*A*e) - 36*b*c^3*d^3*(B*d + 4*A*e) + 3*
b^2*c^2*d^2*e*(29*B*d + 9*A*e)))/(12*b^4*d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) + (e*(24*A*c^5*d^5 + 8*b^4*c*d*e^3
*(7*B*d - 10*A*e) - 5*b^5*e^4*(4*B*d - 7*A*e) - 6*b^3*c^2*d^2*e^2*(4*B*d - 3*A*e) + 7*b^2*c^3*d^3*e*(5*B*d + 4
*A*e) - 12*b*c^4*d^4*(B*d + 5*A*e)))/(4*b^4*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - (A*b*(c*d - b*e) + c*(2*A*c*d -
b*(B*d + A*e))*x)/(2*b^2*d*(c*d - b*e)*(d + e*x)^(3/2)*(b*x + c*x^2)^2) + (b*(c*d - b*e)*(12*A*c^2*d^2 + b^2*
e*(4*B*d - 7*A*e) - 3*b*c*d*(2*B*d + A*e)) + c*(24*A*c^3*d^3 - b^3*e^2*(4*B*d - 7*A*e) + b^2*c*d*e*(23*B*d - 2
*A*e) - 12*b*c^2*d^2*(B*d + 3*A*e))*x)/(4*b^4*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)*(b*x + c*x^2)) - ((48*A*c^2*d^
2 - 5*b^2*e*(4*B*d - 7*A*e) - 12*b*c*d*(2*B*d - 5*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(9/2)) + (c^(
7/2)*(48*A*c^3*d^2 - 99*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 13*A*e) + 11*b^2*c*e*(8*B*d + 13*A*e))*ArcTanh[(Sqrt[c
]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 1.79059, antiderivative size = 644, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {822, 828, 826, 1166, 208} \[ \frac{c x \left (b^2 c d e (23 B d-2 A e)+b^3 \left (-e^2\right ) (4 B d-7 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )+b (c d-b e) \left (b^2 e (4 B d-7 A e)-3 b c d (A e+2 B d)+12 A c^2 d^2\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)^2}+\frac{e \left (-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (4 A e+5 B d)+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-12 b c^4 d^4 (5 A e+B d)+24 A c^5 d^5\right )}{4 b^4 d^4 \sqrt{d+e x} (c d-b e)^4}+\frac{e \left (3 b^2 c^2 d^2 e (9 A e+29 B d)-9 b^3 c d e^2 (4 B d-5 A e)+5 b^4 e^3 (4 B d-7 A e)-36 b c^3 d^3 (4 A e+B d)+72 A c^4 d^4\right )}{12 b^4 d^3 (d+e x)^{3/2} (c d-b e)^3}+\frac{c^{7/2} \left (11 b^2 c e (13 A e+8 B d)-12 b c^2 d (13 A e+2 B d)+48 A c^3 d^2-99 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)+48 A c^2 d^2\right )}{4 b^5 d^{9/2}}-\frac{c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^(5/2)*(b*x + c*x^2)^3),x]

[Out]

(e*(72*A*c^4*d^4 + 5*b^4*e^3*(4*B*d - 7*A*e) - 9*b^3*c*d*e^2*(4*B*d - 5*A*e) - 36*b*c^3*d^3*(B*d + 4*A*e) + 3*
b^2*c^2*d^2*e*(29*B*d + 9*A*e)))/(12*b^4*d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) + (e*(24*A*c^5*d^5 + 8*b^4*c*d*e^3
*(7*B*d - 10*A*e) - 5*b^5*e^4*(4*B*d - 7*A*e) - 6*b^3*c^2*d^2*e^2*(4*B*d - 3*A*e) + 7*b^2*c^3*d^3*e*(5*B*d + 4
*A*e) - 12*b*c^4*d^4*(B*d + 5*A*e)))/(4*b^4*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - (A*b*(c*d - b*e) + c*(2*A*c*d -
b*(B*d + A*e))*x)/(2*b^2*d*(c*d - b*e)*(d + e*x)^(3/2)*(b*x + c*x^2)^2) + (b*(c*d - b*e)*(12*A*c^2*d^2 + b^2*
e*(4*B*d - 7*A*e) - 3*b*c*d*(2*B*d + A*e)) + c*(24*A*c^3*d^3 - b^3*e^2*(4*B*d - 7*A*e) + b^2*c*d*e*(23*B*d - 2
*A*e) - 12*b*c^2*d^2*(B*d + 3*A*e))*x)/(4*b^4*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)*(b*x + c*x^2)) - ((48*A*c^2*d^
2 - 5*b^2*e*(4*B*d - 7*A*e) - 12*b*c*d*(2*B*d - 5*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(9/2)) + (c^(
7/2)*(48*A*c^3*d^2 - 99*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 13*A*e) + 11*b^2*c*e*(8*B*d + 13*A*e))*ArcTanh[(Sqrt[c
]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(9/2))

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 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), 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] && FractionQ[m] && LtQ[m, -1]

Rule 826

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

Rule 1166

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

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{A+B x}{(d+e x)^{5/2} \left (b x+c x^2\right )^3} \, dx &=-\frac{A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}-\frac{\int \frac{\frac{1}{2} \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )-\frac{9}{2} c e (b B d-2 A c d+A b e) x}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx}{2 b^2 d (c d-b e)}\\ &=-\frac{A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac{b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac{\int \frac{\frac{1}{4} (c d-b e)^2 \left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right )+\frac{5}{4} c e \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{2 b^4 d^2 (c d-b e)^2}\\ &=\frac{e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac{A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac{b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac{\int \frac{\frac{1}{4} (c d-b e)^3 \left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right )+\frac{1}{4} c e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{2 b^4 d^3 (c d-b e)^3}\\ &=\frac{e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac{e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac{b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac{\int \frac{\frac{1}{4} (c d-b e)^4 \left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right )+\frac{1}{4} c e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d^4 (c d-b e)^4}\\ &=\frac{e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac{e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac{b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{4} e (c d-b e)^4 \left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right )-\frac{1}{4} c d e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )+\frac{1}{4} c e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{b^4 d^4 (c d-b e)^4}\\ &=\frac{e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac{e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac{b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac{\left (c \left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 b^5 d^4}-\frac{\left (c^4 \left (48 A c^3 d^2-99 b^3 B e^2-12 b c^2 d (2 B d+13 A e)+11 b^2 c e (8 B d+13 A e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 b^5 (c d-b e)^4}\\ &=\frac{e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac{e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac{b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac{\left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{9/2}}+\frac{c^{7/2} \left (48 A c^3 d^2-99 b^3 B e^2-12 b c^2 d (2 B d+13 A e)+11 b^2 c e (8 B d+13 A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.588607, size = 388, normalized size = 0.6 \[ \frac{x^2 \left ((b+c x) \left (3 b c d (b e-c d) \left (b^2 c d e (2 A e-23 B d)+b^3 e^2 (4 B d-7 A e)+12 b c^2 d^2 (3 A e+B d)-24 A c^3 d^3\right )-(b+c x) \left (c^2 d^3 \left (11 b^2 c e (13 A e+8 B d)-12 b c^2 d (13 A e+2 B d)+48 A c^3 d^2-99 b^3 B e^2\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{c (d+e x)}{c d-b e}\right )-(c d-b e)^3 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{e x}{d}+1\right ) \left (5 b^2 e (7 A e-4 B d)+12 b c d (5 A e-2 B d)+48 A c^2 d^2\right )\right )\right )+3 b^2 c d (c d-b e)^2 \left (b^2 e (4 B d-7 A e)-3 b c d (A e+2 B d)+12 A c^2 d^2\right )\right )+3 b^3 d x (b e-c d)^3 (-7 A b e-8 A c d+4 b B d)+6 A b^4 d^2 (b e-c d)^3}{12 b^5 d^3 x^2 (b+c x)^2 (d+e x)^{3/2} (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^(5/2)*(b*x + c*x^2)^3),x]

[Out]

(6*A*b^4*d^2*(-(c*d) + b*e)^3 + 3*b^3*d*(-(c*d) + b*e)^3*(4*b*B*d - 8*A*c*d - 7*A*b*e)*x + x^2*(3*b^2*c*d*(c*d
- b*e)^2*(12*A*c^2*d^2 + b^2*e*(4*B*d - 7*A*e) - 3*b*c*d*(2*B*d + A*e)) + (b + c*x)*(3*b*c*d*(-(c*d) + b*e)*(
-24*A*c^3*d^3 + b^3*e^2*(4*B*d - 7*A*e) + b^2*c*d*e*(-23*B*d + 2*A*e) + 12*b*c^2*d^2*(B*d + 3*A*e)) - (b + c*x
)*(c^2*d^3*(48*A*c^3*d^2 - 99*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 13*A*e) + 11*b^2*c*e*(8*B*d + 13*A*e))*Hypergeom
etric2F1[-3/2, 1, -1/2, (c*(d + e*x))/(c*d - b*e)] - (c*d - b*e)^3*(48*A*c^2*d^2 + 12*b*c*d*(-2*B*d + 5*A*e) +
5*b^2*e*(-4*B*d + 7*A*e))*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (e*x)/d]))))/(12*b^5*d^3*(c*d - b*e)^3*x^2*(b
+ c*x)^2*(d + e*x)^(3/2))

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Maple [A]  time = 0.046, size = 1130, normalized size = 1.8 \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)^(5/2)/(c*x^2+b*x)^3,x)

[Out]

-1/e/b^3/d^3/x^2*B*(e*x+d)^(3/2)+1/e/b^3/d^2/x^2*(e*x+d)^(1/2)*B-12*e^5/d^3/(b*e-c*d)^4/(e*x+d)^(1/2)*A*c-4*e^
5/d^3/(b*e-c*d)^4/(e*x+d)^(1/2)*B*b+10*e^4/d^2/(b*e-c*d)^4/(e*x+d)^(1/2)*B*c+6*e^6/d^4/(b*e-c*d)^4/(e*x+d)^(1/
2)*A*b-15*e/b^4/d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c+6/b^4/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B*c+11
/4/b^3/d^4/x^2*A*(e*x+d)^(3/2)-13/4/b^3/d^3/x^2*(e*x+d)^(1/2)*A-2/3*e^4/d^2/(b*e-c*d)^3/(e*x+d)^(3/2)*B-35/4*e
^2/b^3/d^(9/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A+5*e/b^3/d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B+2/3*e^5/d^3/(
b*e-c*d)^3/(e*x+d)^(3/2)*A-12/b^5/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c^2+39*e*c^6/(b*e-c*d)^4/b^4/((b*e-
c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d-22*e*c^5/(b*e-c*d)^4/b^3/((b*e-c*d)*c)^(1/2)*arc
tan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d-2*e*c^6/(b*e-c*d)^4/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*d+37/4*e^2*
c^6/(b*e-c*d)^4/b^3/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d-3*e*c^7/(b*e-c*d)^4/b^4/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d^2+
3*e*c^7/(b*e-c*d)^4/b^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*d-29/4*e^2*c^5/(b*e-c*d)^4/b^2/(c*e*x+b*e)^2*B*(e*x+d)^(
1/2)*d+2*e*c^6/(b*e-c*d)^4/b^3/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*d^2-3/e/b^4/d^2/x^2*(e*x+d)^(1/2)*A*c-12*c^7/(b*e
-c*d)^4/b^5/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d^2+6*c^6/(b*e-c*d)^4/b^4/((b*e-
c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d^2-23/4*e^2*c^6/(b*e-c*d)^4/b^3/(c*e*x+b*e)^2*(e*
x+d)^(3/2)*A+19/4*e^2*c^5/(b*e-c*d)^4/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B-25/4*e^3*c^5/(b*e-c*d)^4/b^2/(c*e*x+b*
e)^2*A*(e*x+d)^(1/2)+21/4*e^3*c^4/(b*e-c*d)^4/b/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)-143/4*e^2*c^5/(b*e-c*d)^4/b^3/((
b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A+99/4*e^2*c^4/(b*e-c*d)^4/b^2/((b*e-c*d)*c)^(1/
2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B+3/e/b^4/d^3/x^2*A*(e*x+d)^(3/2)*c

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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)^(5/2)/(c*x^2+b*x)^3,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)^(5/2)/(c*x^2+b*x)^3,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)**(5/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Giac [B]  time = 2.61185, size = 2160, normalized size = 3.35 \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)^(5/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

1/4*(24*B*b*c^6*d^2 - 48*A*c^7*d^2 - 88*B*b^2*c^5*d*e + 156*A*b*c^6*d*e + 99*B*b^3*c^4*e^2 - 143*A*b^2*c^5*e^2
)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^4*d^4 - 4*b^6*c^3*d^3*e + 6*b^7*c^2*d^2*e^2 - 4*b^8*c*d
*e^3 + b^9*e^4)*sqrt(-c^2*d + b*c*e)) + 2/3*(15*(x*e + d)*B*c*d^2*e^4 + B*c*d^3*e^4 - 6*(x*e + d)*B*b*d*e^5 -
18*(x*e + d)*A*c*d*e^5 - B*b*d^2*e^5 - A*c*d^2*e^5 + 9*(x*e + d)*A*b*e^6 + A*b*d*e^6)/((c^4*d^8 - 4*b*c^3*d^7*
e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4)*(x*e + d)^(3/2)) - 1/4*(12*(x*e + d)^(7/2)*B*b*c^6*d^5*
e - 24*(x*e + d)^(7/2)*A*c^7*d^5*e - 36*(x*e + d)^(5/2)*B*b*c^6*d^6*e + 72*(x*e + d)^(5/2)*A*c^7*d^6*e + 36*(x
*e + d)^(3/2)*B*b*c^6*d^7*e - 72*(x*e + d)^(3/2)*A*c^7*d^7*e - 12*sqrt(x*e + d)*B*b*c^6*d^8*e + 24*sqrt(x*e +
d)*A*c^7*d^8*e - 35*(x*e + d)^(7/2)*B*b^2*c^5*d^4*e^2 + 60*(x*e + d)^(7/2)*A*b*c^6*d^4*e^2 + 123*(x*e + d)^(5/
2)*B*b^2*c^5*d^5*e^2 - 216*(x*e + d)^(5/2)*A*b*c^6*d^5*e^2 - 141*(x*e + d)^(3/2)*B*b^2*c^5*d^6*e^2 + 252*(x*e
+ d)^(3/2)*A*b*c^6*d^6*e^2 + 53*sqrt(x*e + d)*B*b^2*c^5*d^7*e^2 - 96*sqrt(x*e + d)*A*b*c^6*d^7*e^2 + 24*(x*e +
d)^(7/2)*B*b^3*c^4*d^3*e^3 - 28*(x*e + d)^(7/2)*A*b^2*c^5*d^3*e^3 - 125*(x*e + d)^(5/2)*B*b^3*c^4*d^4*e^3 + 1
75*(x*e + d)^(5/2)*A*b^2*c^5*d^4*e^3 + 182*(x*e + d)^(3/2)*B*b^3*c^4*d^5*e^3 - 274*(x*e + d)^(3/2)*A*b^2*c^5*d
^5*e^3 - 81*sqrt(x*e + d)*B*b^3*c^4*d^6*e^3 + 127*sqrt(x*e + d)*A*b^2*c^5*d^6*e^3 - 16*(x*e + d)^(7/2)*B*b^4*c
^3*d^2*e^4 - 18*(x*e + d)^(7/2)*A*b^3*c^4*d^2*e^4 + 96*(x*e + d)^(5/2)*B*b^4*c^3*d^3*e^4 + 10*(x*e + d)^(5/2)*
A*b^3*c^4*d^3*e^4 - 160*(x*e + d)^(3/2)*B*b^4*c^3*d^4*e^4 + 55*(x*e + d)^(3/2)*A*b^3*c^4*d^4*e^4 + 80*sqrt(x*e
+ d)*B*b^4*c^3*d^5*e^4 - 45*sqrt(x*e + d)*A*b^3*c^4*d^5*e^4 + 4*(x*e + d)^(7/2)*B*b^5*c^2*d*e^5 + 32*(x*e + d
)^(7/2)*A*b^4*c^3*d*e^5 - 44*(x*e + d)^(5/2)*B*b^5*c^2*d^2*e^5 - 140*(x*e + d)^(5/2)*A*b^4*c^3*d^2*e^5 + 100*(
x*e + d)^(3/2)*B*b^5*c^2*d^3*e^5 + 180*(x*e + d)^(3/2)*A*b^4*c^3*d^3*e^5 - 60*sqrt(x*e + d)*B*b^5*c^2*d^4*e^5
- 80*sqrt(x*e + d)*A*b^4*c^3*d^4*e^5 - 11*(x*e + d)^(7/2)*A*b^5*c^2*e^6 + 8*(x*e + d)^(5/2)*B*b^6*c*d*e^6 + 99
*(x*e + d)^(5/2)*A*b^5*c^2*d*e^6 - 32*(x*e + d)^(3/2)*B*b^6*c*d^2*e^6 - 199*(x*e + d)^(3/2)*A*b^5*c^2*d^2*e^6
+ 24*sqrt(x*e + d)*B*b^6*c*d^3*e^6 + 123*sqrt(x*e + d)*A*b^5*c^2*d^3*e^6 - 22*(x*e + d)^(5/2)*A*b^6*c*e^7 + 4*
(x*e + d)^(3/2)*B*b^7*d*e^7 + 80*(x*e + d)^(3/2)*A*b^6*c*d*e^7 - 4*sqrt(x*e + d)*B*b^7*d^2*e^7 - 66*sqrt(x*e +
d)*A*b^6*c*d^2*e^7 - 11*(x*e + d)^(3/2)*A*b^7*e^8 + 13*sqrt(x*e + d)*A*b^7*d*e^8)/((b^4*c^4*d^8 - 4*b^5*c^3*d
^7*e + 6*b^6*c^2*d^6*e^2 - 4*b^7*c*d^5*e^3 + b^8*d^4*e^4)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)
*b*e - b*d*e)^2) - 1/4*(24*B*b*c*d^2 - 48*A*c^2*d^2 + 20*B*b^2*d*e - 60*A*b*c*d*e - 35*A*b^2*e^2)*arctan(sqrt(
x*e + d)/sqrt(-d))/(b^5*sqrt(-d)*d^4)