3.1028 \(\int \frac{A+B x}{x^{3/2} (a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=664 \[ -\frac{-A \left (36 a^2 c^2-35 a b^2 c+5 b^4\right )+c x \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right )+a b B \left (b^2-16 a c\right )}{4 a^2 \sqrt{x} \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{3 \left (a b B \left (b^2-8 a c\right )-A \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )\right )}{4 a^3 \sqrt{x} \left (b^2-4 a c\right )^2}+\frac{3 \sqrt{c} \left (a B \left (56 a^2 c^2+b^3 \sqrt{b^2-4 a c}-10 a b^2 c-8 a b c \sqrt{b^2-4 a c}+b^4\right )-A \left (60 a^2 c^2 \sqrt{b^2-4 a c}+124 a^2 b c^2+5 b^4 \sqrt{b^2-4 a c}-47 a b^3 c-37 a b^2 c \sqrt{b^2-4 a c}+5 b^5\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left (a B \left (56 a^2 c^2-b^3 \sqrt{b^2-4 a c}-10 a b^2 c+8 a b c \sqrt{b^2-4 a c}+b^4\right )-A \left (-60 a^2 c^2 \sqrt{b^2-4 a c}+124 a^2 b c^2-5 b^4 \sqrt{b^2-4 a c}-47 a b^3 c+37 a b^2 c \sqrt{b^2-4 a c}+5 b^5\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{c x (A b-2 a B)-2 a A c-a b B+A b^2}{2 a \sqrt{x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

(3*(a*b*B*(b^2 - 8*a*c) - A*(5*b^4 - 37*a*b^2*c + 60*a^2*c^2)))/(4*a^3*(b^2 - 4*a*c)^2*Sqrt[x]) + (A*b^2 - a*b
*B - 2*a*A*c + (A*b - 2*a*B)*c*x)/(2*a*(b^2 - 4*a*c)*Sqrt[x]*(a + b*x + c*x^2)^2) - (a*b*B*(b^2 - 16*a*c) - A*
(5*b^4 - 35*a*b^2*c + 36*a^2*c^2) + c*(a*B*(b^2 - 28*a*c) - A*(5*b^3 - 32*a*b*c))*x)/(4*a^2*(b^2 - 4*a*c)^2*Sq
rt[x]*(a + b*x + c*x^2)) + (3*Sqrt[c]*(a*B*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sq
rt[b^2 - 4*a*c]) - A*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a
*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]
*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[c]*(a*B*(b^4 - 10*a*b^2*c + 56*a^2*c^2 - b^3*S
qrt[b^2 - 4*a*c] + 8*a*b*c*Sqrt[b^2 - 4*a*c]) - A*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 - 5*b^4*Sqrt[b^2 - 4*a*c
] + 37*a*b^2*c*Sqrt[b^2 - 4*a*c] - 60*a^2*c^2*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sq
rt[b^2 - 4*a*c]]])/(4*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

________________________________________________________________________________________

Rubi [A]  time = 1.71029, antiderivative size = 664, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {822, 828, 826, 1166, 205} \[ -\frac{-A \left (36 a^2 c^2-35 a b^2 c+5 b^4\right )+c x \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right )+a b B \left (b^2-16 a c\right )}{4 a^2 \sqrt{x} \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{3 \left (a b B \left (b^2-8 a c\right )-A \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )\right )}{4 a^3 \sqrt{x} \left (b^2-4 a c\right )^2}+\frac{3 \sqrt{c} \left (a B \left (56 a^2 c^2+b^3 \sqrt{b^2-4 a c}-10 a b^2 c-8 a b c \sqrt{b^2-4 a c}+b^4\right )-A \left (60 a^2 c^2 \sqrt{b^2-4 a c}+124 a^2 b c^2+5 b^4 \sqrt{b^2-4 a c}-47 a b^3 c-37 a b^2 c \sqrt{b^2-4 a c}+5 b^5\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left (a B \left (56 a^2 c^2-b^3 \sqrt{b^2-4 a c}-10 a b^2 c+8 a b c \sqrt{b^2-4 a c}+b^4\right )-A \left (-60 a^2 c^2 \sqrt{b^2-4 a c}+124 a^2 b c^2-5 b^4 \sqrt{b^2-4 a c}-47 a b^3 c+37 a b^2 c \sqrt{b^2-4 a c}+5 b^5\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{c x (A b-2 a B)-2 a A c-a b B+A b^2}{2 a \sqrt{x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a*b*B*(b^2 - 8*a*c) - A*(5*b^4 - 37*a*b^2*c + 60*a^2*c^2)))/(4*a^3*(b^2 - 4*a*c)^2*Sqrt[x]) + (A*b^2 - a*b
*B - 2*a*A*c + (A*b - 2*a*B)*c*x)/(2*a*(b^2 - 4*a*c)*Sqrt[x]*(a + b*x + c*x^2)^2) - (a*b*B*(b^2 - 16*a*c) - A*
(5*b^4 - 35*a*b^2*c + 36*a^2*c^2) + c*(a*B*(b^2 - 28*a*c) - A*(5*b^3 - 32*a*b*c))*x)/(4*a^2*(b^2 - 4*a*c)^2*Sq
rt[x]*(a + b*x + c*x^2)) + (3*Sqrt[c]*(a*B*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sq
rt[b^2 - 4*a*c]) - A*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a
*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]
*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[c]*(a*B*(b^4 - 10*a*b^2*c + 56*a^2*c^2 - b^3*S
qrt[b^2 - 4*a*c] + 8*a*b*c*Sqrt[b^2 - 4*a*c]) - A*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 - 5*b^4*Sqrt[b^2 - 4*a*c
] + 37*a*b^2*c*Sqrt[b^2 - 4*a*c] - 60*a^2*c^2*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sq
rt[b^2 - 4*a*c]]])/(4*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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 205

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

Rubi steps

\begin{align*} \int \frac{A+B x}{x^{3/2} \left (a+b x+c x^2\right )^3} \, dx &=\frac{A b^2-a b B-2 a A c+(A b-2 a B) c x}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x+c x^2\right )^2}-\frac{\int \frac{\frac{1}{2} \left (-5 A b^2+a b B+18 a A c\right )-\frac{7}{2} (A b-2 a B) c x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac{A b^2-a b B-2 a A c+(A b-2 a B) c x}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x+c x^2\right )^2}-\frac{a b B \left (b^2-16 a c\right )-A \left (5 b^4-35 a b^2 c+36 a^2 c^2\right )+c \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt{x} \left (a+b x+c x^2\right )}+\frac{\int \frac{-\frac{3}{4} \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )-\frac{3}{4} c \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right ) x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx}{2 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac{3 \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt{x}}+\frac{A b^2-a b B-2 a A c+(A b-2 a B) c x}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x+c x^2\right )^2}-\frac{a b B \left (b^2-16 a c\right )-A \left (5 b^4-35 a b^2 c+36 a^2 c^2\right )+c \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt{x} \left (a+b x+c x^2\right )}+\frac{\int \frac{\frac{3}{4} \left (a B \left (b^4-9 a b^2 c+28 a^2 c^2\right )-A \left (5 b^5-42 a b^3 c+92 a^2 b c^2\right )\right )+\frac{3}{4} c \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right ) x}{\sqrt{x} \left (a+b x+c x^2\right )} \, dx}{2 a^3 \left (b^2-4 a c\right )^2}\\ &=\frac{3 \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt{x}}+\frac{A b^2-a b B-2 a A c+(A b-2 a B) c x}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x+c x^2\right )^2}-\frac{a b B \left (b^2-16 a c\right )-A \left (5 b^4-35 a b^2 c+36 a^2 c^2\right )+c \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt{x} \left (a+b x+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{3}{4} \left (a B \left (b^4-9 a b^2 c+28 a^2 c^2\right )-A \left (5 b^5-42 a b^3 c+92 a^2 b c^2\right )\right )+\frac{3}{4} c \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right ) x^2}{a+b x^2+c x^4} \, dx,x,\sqrt{x}\right )}{a^3 \left (b^2-4 a c\right )^2}\\ &=\frac{3 \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt{x}}+\frac{A b^2-a b B-2 a A c+(A b-2 a B) c x}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x+c x^2\right )^2}-\frac{a b B \left (b^2-16 a c\right )-A \left (5 b^4-35 a b^2 c+36 a^2 c^2\right )+c \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt{x} \left (a+b x+c x^2\right )}-\frac{\left (3 c \left (a B \left (b^4-10 a b^2 c+56 a^2 c^2-b^3 \sqrt{b^2-4 a c}+8 a b c \sqrt{b^2-4 a c}\right )-A \left (5 b^5-47 a b^3 c+124 a^2 b c^2-5 b^4 \sqrt{b^2-4 a c}+37 a b^2 c \sqrt{b^2-4 a c}-60 a^2 c^2 \sqrt{b^2-4 a c}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx,x,\sqrt{x}\right )}{8 a^3 \left (b^2-4 a c\right )^{5/2}}+\frac{\left (3 c \left (a B \left (b^4-10 a b^2 c+56 a^2 c^2+b^3 \sqrt{b^2-4 a c}-8 a b c \sqrt{b^2-4 a c}\right )-A \left (5 b^5-47 a b^3 c+124 a^2 b c^2+5 b^4 \sqrt{b^2-4 a c}-37 a b^2 c \sqrt{b^2-4 a c}+60 a^2 c^2 \sqrt{b^2-4 a c}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx,x,\sqrt{x}\right )}{8 a^3 \left (b^2-4 a c\right )^{5/2}}\\ &=\frac{3 \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt{x}}+\frac{A b^2-a b B-2 a A c+(A b-2 a B) c x}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x+c x^2\right )^2}-\frac{a b B \left (b^2-16 a c\right )-A \left (5 b^4-35 a b^2 c+36 a^2 c^2\right )+c \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt{x} \left (a+b x+c x^2\right )}+\frac{3 \sqrt{c} \left (a B \left (b^4-10 a b^2 c+56 a^2 c^2+b^3 \sqrt{b^2-4 a c}-8 a b c \sqrt{b^2-4 a c}\right )-A \left (5 b^5-47 a b^3 c+124 a^2 b c^2+5 b^4 \sqrt{b^2-4 a c}-37 a b^2 c \sqrt{b^2-4 a c}+60 a^2 c^2 \sqrt{b^2-4 a c}\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left (a B \left (b^4-10 a b^2 c+56 a^2 c^2-b^3 \sqrt{b^2-4 a c}+8 a b c \sqrt{b^2-4 a c}\right )-A \left (5 b^5-47 a b^3 c+124 a^2 b c^2-5 b^4 \sqrt{b^2-4 a c}+37 a b^2 c \sqrt{b^2-4 a c}-60 a^2 c^2 \sqrt{b^2-4 a c}\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}

Mathematica [A]  time = 2.38976, size = 628, normalized size = 0.95 \[ \frac{\frac{A \left (-36 a^2 c^2+35 a b^2 c+32 a b c^2 x-5 b^3 c x-5 b^4\right )+a B \left (-16 a b c-28 a c^2 x+b^2 c x+b^3\right )}{a \sqrt{x} \left (4 a c-b^2\right ) (a+x (b+c x))}+\frac{\frac{3 \left (A \left (-60 a^2 c^2+37 a b^2 c-5 b^4\right )+a b B \left (b^2-8 a c\right )\right )}{\sqrt{x}}+\frac{3 \sqrt{c} \left (-\frac{\left (A \left (60 a^2 c^2 \sqrt{b^2-4 a c}+124 a^2 b c^2+5 b^4 \sqrt{b^2-4 a c}-47 a b^3 c-37 a b^2 c \sqrt{b^2-4 a c}+5 b^5\right )-a B \left (56 a^2 c^2+b^3 \sqrt{b^2-4 a c}-10 a b^2 c-8 a b c \sqrt{b^2-4 a c}+b^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (A \left (60 a^2 c^2 \sqrt{b^2-4 a c}-124 a^2 b c^2+5 b^4 \sqrt{b^2-4 a c}+47 a b^3 c-37 a b^2 c \sqrt{b^2-4 a c}-5 b^5\right )+a B \left (56 a^2 c^2-b^3 \sqrt{b^2-4 a c}-10 a b^2 c+8 a b c \sqrt{b^2-4 a c}+b^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \sqrt{b^2-4 a c}}}{a^2 \left (b^2-4 a c\right )}+\frac{2 \left (A \left (-2 a c+b^2+b c x\right )-a B (b+2 c x)\right )}{\sqrt{x} (a+x (b+c x))^2}}{4 a \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*(-(a*B*(b + 2*c*x)) + A*(b^2 - 2*a*c + b*c*x)))/(Sqrt[x]*(a + x*(b + c*x))^2) + (a*B*(b^3 - 16*a*b*c + b^2
*c*x - 28*a*c^2*x) + A*(-5*b^4 + 35*a*b^2*c - 36*a^2*c^2 - 5*b^3*c*x + 32*a*b*c^2*x))/(a*(-b^2 + 4*a*c)*Sqrt[x
]*(a + x*(b + c*x))) + ((3*(a*b*B*(b^2 - 8*a*c) + A*(-5*b^4 + 37*a*b^2*c - 60*a^2*c^2)))/Sqrt[x] + (3*Sqrt[c]*
(-(((-(a*B*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sqrt[b^2 - 4*a*c])) + A*(5*b^5 - 4
7*a*b^3*c + 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a
*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((a*B*(b^4
- 10*a*b^2*c + 56*a^2*c^2 - b^3*Sqrt[b^2 - 4*a*c] + 8*a*b*c*Sqrt[b^2 - 4*a*c]) + A*(-5*b^5 + 47*a*b^3*c - 124
*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c]))*ArcTan[(S
qrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(Sqrt[2]*Sqrt[b^2 - 4*a*c]
))/(a^2*(b^2 - 4*a*c)))/(4*a*(b^2 - 4*a*c))

________________________________________________________________________________________

Maple [B]  time = 0.06, size = 2918, normalized size = 4.4 \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)/x^(3/2)/(c*x^2+b*x+a)^3,x)

[Out]

-2*A/a^3/x^(1/2)+15/8/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^
(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^5-3/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c/
(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c
)^(1/2))*B*b^4-141/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)
^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^3+15/4/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^
2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/
2))*c)^(1/2))*B*b^2-3/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c
)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*B*b^4+93/2/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c
^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2)
)*c)^(1/2))*A*b+93/2/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(
1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b+15/8/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(-
4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^
(1/2))*A*b^5-141/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1
/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^3+15/4/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-4
*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(
1/2))*B*b^2+3/4/a^2/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*B*b^5-9/4/a^2/(c*x^2+b*x+a)^2/(16*a^2*c
^2-8*a*b^2*c+b^4)*x^(1/2)*A*b^5+11*a/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)*B*c^2+5/4/a/(c*x^2+b*x
+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)*B*b^4-7/4/a^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*A*b^
6-13/a/(c*x^2+b*x+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)*A-1/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x
^(3/2)*B*b*c^2-27/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)*A*b*c^2-37/4/(c*x^2+b*x+a)^2/(16*a^2*c^2-
8*a*b^2*c+b^4)*x^(1/2)*B*b^2*c+7/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)*B-17/(c*x^2+b*x+a)^2/(
16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*A*c^3-111/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/
2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^2+3/a/(16*a^2*c^2-8*a*b^2*c+b^4)
*c^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b*
B-3/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-
b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*B*b^3+111/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))
*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^2-3/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*
2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b*B+3/8/a^
2/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b
^2)^(1/2))*c)^(1/2))*B*b^3+15/8/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arc
tanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^4-15/8/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/
((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^4-21/(16*a^2*c
^2-8*a*b^2*c+b^4)*c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/(
(-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*B-21/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b
^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*B+45/2/a/(16*a^2*c^2-8*a*b^2*c+
b^4)*c^3*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)
)*A-45/2/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((
b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A+47/4/a^2/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)*A*b^2-6/a/(c
*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)*b*B+3/4/a^2/(c*x^2+b*x+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4
)*x^(7/2)*B*b^3-34/a/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)*A*b+99/4/a^2/(c*x^2+b*x+a)^2*c^2/(
16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)*A*b^3-49/4/a/(c*x^2+b*x+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)*B*b^2-7/
4/a^3/(c*x^2+b*x+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)*A*b^4-7/2/a^3/(c*x^2+b*x+a)^2*c/(16*a^2*c^2-8*a*b
^2*c+b^4)*x^(5/2)*A*b^5+3/2/a^2/(c*x^2+b*x+a)^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)*B*b^4-25/4/a/(c*x^2+b*x+a
)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*A*b^2*c^2+43/4/a^2/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*A
*b^4*c-5/a/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*B*b^3*c+33/2/a/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b
^2*c+b^4)*x^(1/2)*A*b^3*c

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out