3.1253 \(\int \frac{A+B x}{(d+e x)^{3/2} (b x+c x^2)^3} \, dx\)

Optimal. Leaf size=506 \[ \frac{c x \left (b^2 c d e (2 A e+21 B d)+b^3 \left (-e^2\right ) (4 B d-5 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )+b (c d-b e) \left (b^2 e (4 B d-5 A e)-b c d (5 A e+6 B d)+12 A c^2 d^2\right )}{4 b^4 d^2 \left (b x+c x^2\right ) \sqrt{d+e x} (c d-b e)^2}+\frac{3 e \left (b^2 c^2 d^2 e (5 A e+9 B d)-b^3 c d e^2 (4 B d-3 A e)+b^4 e^3 (4 B d-5 A e)-4 b c^3 d^3 (4 A e+B d)+8 A c^4 d^4\right )}{4 b^4 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{3 c^{5/2} \left (3 b^2 c e (11 A e+8 B d)-4 b c^2 d (11 A e+2 B d)+16 A c^3 d^2-21 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (b^2 (-e) (4 B d-5 A e)-4 b c d (2 B d-3 A e)+16 A c^2 d^2\right )}{4 b^5 d^{7/2}}-\frac{c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt{d+e x} (c d-b e)} \]

[Out]

(3*e*(8*A*c^4*d^4 + b^4*e^3*(4*B*d - 5*A*e) - b^3*c*d*e^2*(4*B*d - 3*A*e) - 4*b*c^3*d^3*(B*d + 4*A*e) + b^2*c^
2*d^2*e*(9*B*d + 5*A*e)))/(4*b^4*d^3*(c*d - b*e)^3*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)*Sqrt[d + e*x]*(b*x + c*x^2)^2) + (b*(c*d - b*e)*(12*A*c^2*d^2 + b^2*e*(4*B*d - 5*
A*e) - b*c*d*(6*B*d + 5*A*e)) + c*(24*A*c^3*d^3 - b^3*e^2*(4*B*d - 5*A*e) + b^2*c*d*e*(21*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*Sqrt[d + e*x]*(b*x + c*x^2)) - (3*(16*A*c^2*d^2 - b^2*e*(4*
B*d - 5*A*e) - 4*b*c*d*(2*B*d - 3*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(7/2)) + (3*c^(5/2)*(16*A*c^3
*d^2 - 21*b^3*B*e^2 - 4*b*c^2*d*(2*B*d + 11*A*e) + 3*b^2*c*e*(8*B*d + 11*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])
/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 1.45107, antiderivative size = 506, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{c x \left (b^2 c d e (2 A e+21 B d)+b^3 \left (-e^2\right ) (4 B d-5 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )+b (c d-b e) \left (b^2 e (4 B d-5 A e)-b c d (5 A e+6 B d)+12 A c^2 d^2\right )}{4 b^4 d^2 \left (b x+c x^2\right ) \sqrt{d+e x} (c d-b e)^2}+\frac{3 e \left (b^2 c^2 d^2 e (5 A e+9 B d)-b^3 c d e^2 (4 B d-3 A e)+b^4 e^3 (4 B d-5 A e)-4 b c^3 d^3 (4 A e+B d)+8 A c^4 d^4\right )}{4 b^4 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{3 c^{5/2} \left (3 b^2 c e (11 A e+8 B d)-4 b c^2 d (11 A e+2 B d)+16 A c^3 d^2-21 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (b^2 (-e) (4 B d-5 A e)-4 b c d (2 B d-3 A e)+16 A c^2 d^2\right )}{4 b^5 d^{7/2}}-\frac{c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt{d+e x} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*e*(8*A*c^4*d^4 + b^4*e^3*(4*B*d - 5*A*e) - b^3*c*d*e^2*(4*B*d - 3*A*e) - 4*b*c^3*d^3*(B*d + 4*A*e) + b^2*c^
2*d^2*e*(9*B*d + 5*A*e)))/(4*b^4*d^3*(c*d - b*e)^3*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)*Sqrt[d + e*x]*(b*x + c*x^2)^2) + (b*(c*d - b*e)*(12*A*c^2*d^2 + b^2*e*(4*B*d - 5*
A*e) - b*c*d*(6*B*d + 5*A*e)) + c*(24*A*c^3*d^3 - b^3*e^2*(4*B*d - 5*A*e) + b^2*c*d*e*(21*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*Sqrt[d + e*x]*(b*x + c*x^2)) - (3*(16*A*c^2*d^2 - b^2*e*(4*
B*d - 5*A*e) - 4*b*c*d*(2*B*d - 3*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(7/2)) + (3*c^(5/2)*(16*A*c^3
*d^2 - 21*b^3*B*e^2 - 4*b*c^2*d*(2*B*d + 11*A*e) + 3*b^2*c*e*(8*B*d + 11*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])
/Sqrt[c*d - b*e]])/(4*b^5*(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 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)^{3/2} \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) \sqrt{d+e x} \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-5 A e)-b c d (6 B d+5 A e)\right )-\frac{7}{2} c e (b B d-2 A c d+A b e) x}{(d+e x)^{3/2} \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) \sqrt{d+e x} \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-5 A e)-b c d (6 B d+5 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-5 A e)+b^2 c d e (21 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 \sqrt{d+e x} \left (b x+c x^2\right )}+\frac{\int \frac{\frac{3}{4} (c d-b e)^2 \left (16 A c^2 d^2-b^2 e (4 B d-5 A e)-4 b c d (2 B d-3 A e)\right )+\frac{3}{4} c e \left (24 A c^3 d^3-b^3 e^2 (4 B d-5 A e)+b^2 c d e (21 B d+2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{2 b^4 d^2 (c d-b e)^2}\\ &=\frac{3 e \left (8 A c^4 d^4+b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)-4 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (9 B d+5 A e)\right )}{4 b^4 d^3 (c d-b e)^3 \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) \sqrt{d+e x} \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-5 A e)-b c d (6 B d+5 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-5 A e)+b^2 c d e (21 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 \sqrt{d+e x} \left (b x+c x^2\right )}+\frac{\int \frac{\frac{3}{4} (c d-b e)^3 \left (16 A c^2 d^2-b^2 e (4 B d-5 A e)-4 b c d (2 B d-3 A e)\right )+\frac{3}{4} c e \left (8 A c^4 d^4+b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)-4 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (9 B d+5 A e)\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d^3 (c d-b e)^3}\\ &=\frac{3 e \left (8 A c^4 d^4+b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)-4 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (9 B d+5 A e)\right )}{4 b^4 d^3 (c d-b e)^3 \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) \sqrt{d+e x} \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-5 A e)-b c d (6 B d+5 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-5 A e)+b^2 c d e (21 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 \sqrt{d+e x} \left (b x+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{3}{4} e (c d-b e)^3 \left (16 A c^2 d^2-b^2 e (4 B d-5 A e)-4 b c d (2 B d-3 A e)\right )-\frac{3}{4} c d e \left (8 A c^4 d^4+b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)-4 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (9 B d+5 A e)\right )+\frac{3}{4} c e \left (8 A c^4 d^4+b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)-4 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (9 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^3 (c d-b e)^3}\\ &=\frac{3 e \left (8 A c^4 d^4+b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)-4 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (9 B d+5 A e)\right )}{4 b^4 d^3 (c d-b e)^3 \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) \sqrt{d+e x} \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-5 A e)-b c d (6 B d+5 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-5 A e)+b^2 c d e (21 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 \sqrt{d+e x} \left (b x+c x^2\right )}+\frac{\left (3 c \left (16 A c^2 d^2-b^2 e (4 B d-5 A e)-4 b c d (2 B d-3 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^3}-\frac{\left (3 c^3 \left (16 A c^3 d^2-21 b^3 B e^2-4 b c^2 d (2 B d+11 A e)+3 b^2 c e (8 B d+11 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)^3}\\ &=\frac{3 e \left (8 A c^4 d^4+b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)-4 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (9 B d+5 A e)\right )}{4 b^4 d^3 (c d-b e)^3 \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) \sqrt{d+e x} \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-5 A e)-b c d (6 B d+5 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-5 A e)+b^2 c d e (21 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 \sqrt{d+e x} \left (b x+c x^2\right )}-\frac{3 \left (16 A c^2 d^2-b^2 e (4 B d-5 A e)-4 b c d (2 B d-3 A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{7/2}}+\frac{3 c^{5/2} \left (16 A c^3 d^2-21 b^3 B e^2-4 b c^2 d (2 B d+11 A e)+3 b^2 c e (8 B d+11 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)^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.555336, size = 387, normalized size = 0.76 \[ \frac{x^2 \left ((b+c x) \left (b c d (b e-c d) \left (-b^2 c d e (2 A e+21 B d)+b^3 e^2 (4 B d-5 A e)+12 b c^2 d^2 (3 A e+B d)-24 A c^3 d^3\right )-(b+c x) \left (3 c^2 d^3 \left (3 b^2 c e (11 A e+8 B d)-4 b c^2 d (11 A e+2 B d)+16 A c^3 d^2-21 b^3 B e^2\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{c (d+e x)}{c d-b e}\right )-3 (c d-b e)^3 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{e x}{d}+1\right ) \left (b^2 e (5 A e-4 B d)+4 b c d (3 A e-2 B d)+16 A c^2 d^2\right )\right )\right )+b^2 c d (c d-b e)^2 \left (b^2 e (4 B d-5 A e)-b c d (5 A e+6 B d)+12 A c^2 d^2\right )\right )+b^3 d x (b e-c d)^3 (-5 A b e-8 A c d+4 b B d)+2 A b^4 d^2 (b e-c d)^3}{4 b^5 d^3 x^2 (b+c x)^2 \sqrt{d+e x} (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^4*d^2*(-(c*d) + b*e)^3 + b^3*d*(-(c*d) + b*e)^3*(4*b*B*d - 8*A*c*d - 5*A*b*e)*x + x^2*(b^2*c*d*(c*d - b
*e)^2*(12*A*c^2*d^2 + b^2*e*(4*B*d - 5*A*e) - b*c*d*(6*B*d + 5*A*e)) + (b + c*x)*(b*c*d*(-(c*d) + b*e)*(-24*A*
c^3*d^3 + b^3*e^2*(4*B*d - 5*A*e) - b^2*c*d*e*(21*B*d + 2*A*e) + 12*b*c^2*d^2*(B*d + 3*A*e)) - (b + c*x)*(3*c^
2*d^3*(16*A*c^3*d^2 - 21*b^3*B*e^2 - 4*b*c^2*d*(2*B*d + 11*A*e) + 3*b^2*c*e*(8*B*d + 11*A*e))*Hypergeometric2F
1[-1/2, 1, 1/2, (c*(d + e*x))/(c*d - b*e)] - 3*(c*d - b*e)^3*(16*A*c^2*d^2 + 4*b*c*d*(-2*B*d + 3*A*e) + b^2*e*
(-4*B*d + 5*A*e))*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (e*x)/d]))))/(4*b^5*d^3*(c*d - b*e)^3*x^2*(b + c*x)^2*Sq
rt[d + e*x])

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Maple [B]  time = 0.036, size = 1022, normalized size = 2. \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)^(3/2)/(c*x^2+b*x)^3,x)

[Out]

-9*e/d^(5/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c-1/e/d^2/b^3/x^2*B*(e*x+d)^(3/2)+1/e/d/b^3/x^2*(e*x+d)^(1/2
)*B-12/d^(3/2)/b^5*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c^2+6/d^(3/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*B*c+7/4/d
^3/b^3/x^2*A*(e*x+d)^(3/2)-9/4/d^2/b^3/x^2*(e*x+d)^(1/2)*A-2*e^4/d^2/(b*e-c*d)^3/(e*x+d)^(1/2)*B-15/4*e^2/d^(7
/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*A+3*e/d^(5/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*B+2*e^5/d^3/(b*e-c*d)^
3/(e*x+d)^(1/2)*A-33*e*c^5/(b*e-c*d)^3/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
+18*e*c^4/(b*e-c*d)^3/b^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d-3*e*c^6/(b*e-c*d
)^3/b^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*d+2*e*c^5/(b*e-c*d)^3/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*d-33/4*e^2*c^5/(
b*e-c*d)^3/b^3/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d+3*e*c^6/(b*e-c*d)^3/b^4/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d^2+25/4*
e^2*c^4/(b*e-c*d)^3/b^2/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*d-2*e*c^5/(b*e-c*d)^3/b^3/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*
d^2-17/4*e^3*c^3/(b*e-c*d)^3/b/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)+99/4*e^2*c^4/(b*e-c*d)^3/b^3/((b*e-c*d)*c)^(1/2)*
arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A-63/4*e^2*c^3/(b*e-c*d)^3/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/d^2/b^4/x^2*A*(e*x+d)^(3/2)*c-3/e/d/b^4/x^2*(e*x+d)^(1/2)*A*c+12*c^6/(b*e-c
*d)^3/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^5/(b*e-c*d)^3/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+19/4*e^2*c^5/(b*e-c*d)^3/b^3/(c*e*x+b*e)^2*(e*x+
d)^(3/2)*A-15/4*e^2*c^4/(b*e-c*d)^3/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B+21/4*e^3*c^4/(b*e-c*d)^3/b^2/(c*e*x+b*e)
^2*A*(e*x+d)^(1/2)

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

[Out]

Timed out

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Giac [B]  time = 1.5756, size = 1773, normalized size = 3.5 \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)^(3/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

3/4*(8*B*b*c^5*d^2 - 16*A*c^6*d^2 - 24*B*b^2*c^4*d*e + 44*A*b*c^5*d*e + 21*B*b^3*c^3*e^2 - 33*A*b^2*c^4*e^2)*a
rctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^3*d^3 - 3*b^6*c^2*d^2*e + 3*b^7*c*d*e^2 - b^8*e^3)*sqrt(-c
^2*d + b*c*e)) + 2*(B*d*e^4 - A*e^5)/((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(x*e + d))
- 1/4*(12*(x*e + d)^(7/2)*B*b*c^5*d^4*e - 24*(x*e + d)^(7/2)*A*c^6*d^4*e - 36*(x*e + d)^(5/2)*B*b*c^5*d^5*e +
72*(x*e + d)^(5/2)*A*c^6*d^5*e + 36*(x*e + d)^(3/2)*B*b*c^5*d^6*e - 72*(x*e + d)^(3/2)*A*c^6*d^6*e - 12*sqrt(
x*e + d)*B*b*c^5*d^7*e + 24*sqrt(x*e + d)*A*c^6*d^7*e - 27*(x*e + d)^(7/2)*B*b^2*c^4*d^3*e^2 + 48*(x*e + d)^(7
/2)*A*b*c^5*d^3*e^2 + 99*(x*e + d)^(5/2)*B*b^2*c^4*d^4*e^2 - 180*(x*e + d)^(5/2)*A*b*c^5*d^4*e^2 - 117*(x*e +
d)^(3/2)*B*b^2*c^4*d^5*e^2 + 216*(x*e + d)^(3/2)*A*b*c^5*d^5*e^2 + 45*sqrt(x*e + d)*B*b^2*c^4*d^6*e^2 - 84*sqr
t(x*e + d)*A*b*c^5*d^6*e^2 + 12*(x*e + d)^(7/2)*B*b^3*c^3*d^2*e^3 - 15*(x*e + d)^(7/2)*A*b^2*c^4*d^2*e^3 - 77*
(x*e + d)^(5/2)*B*b^3*c^3*d^3*e^3 + 118*(x*e + d)^(5/2)*A*b^2*c^4*d^3*e^3 + 122*(x*e + d)^(3/2)*B*b^3*c^3*d^4*
e^3 - 199*(x*e + d)^(3/2)*A*b^2*c^4*d^4*e^3 - 57*sqrt(x*e + d)*B*b^3*c^3*d^5*e^3 + 96*sqrt(x*e + d)*A*b^2*c^4*
d^5*e^3 - 4*(x*e + d)^(7/2)*B*b^4*c^2*d*e^4 - 9*(x*e + d)^(7/2)*A*b^3*c^3*d*e^4 + 36*(x*e + d)^(5/2)*B*b^4*c^2
*d^2*e^4 + 3*(x*e + d)^(5/2)*A*b^3*c^3*d^2*e^4 - 72*(x*e + d)^(3/2)*B*b^4*c^2*d^3*e^4 + 38*(x*e + d)^(3/2)*A*b
^3*c^3*d^3*e^4 + 40*sqrt(x*e + d)*B*b^4*c^2*d^4*e^4 - 30*sqrt(x*e + d)*A*b^3*c^3*d^4*e^4 + 7*(x*e + d)^(7/2)*A
*b^4*c^2*e^5 - 8*(x*e + d)^(5/2)*B*b^5*c*d*e^5 - 41*(x*e + d)^(5/2)*A*b^4*c^2*d*e^5 + 28*(x*e + d)^(3/2)*B*b^5
*c*d^2*e^5 + 58*(x*e + d)^(3/2)*A*b^4*c^2*d^2*e^5 - 20*sqrt(x*e + d)*B*b^5*c*d^3*e^5 - 30*sqrt(x*e + d)*A*b^4*
c^2*d^3*e^5 + 14*(x*e + d)^(5/2)*A*b^5*c*e^6 - 4*(x*e + d)^(3/2)*B*b^6*d*e^6 - 41*(x*e + d)^(3/2)*A*b^5*c*d*e^
6 + 4*sqrt(x*e + d)*B*b^6*d^2*e^6 + 33*sqrt(x*e + d)*A*b^5*c*d^2*e^6 + 7*(x*e + d)^(3/2)*A*b^6*e^7 - 9*sqrt(x*
e + d)*A*b^6*d*e^7)/((b^4*c^3*d^6 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 - b^7*d^3*e^3)*((x*e + d)^2*c - 2*(x*e +
d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2) - 3/4*(8*B*b*c*d^2 - 16*A*c^2*d^2 + 4*B*b^2*d*e - 12*A*b*c*d*e - 5
*A*b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)*d^3)