\(\int \frac {A+B x}{\sqrt {x} (a+b x+c x^2)^3} \, dx\) [102]

Optimal result

Integrand size = 23, antiderivative size = 468 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )^3} \, dx=\frac {\sqrt {x} \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {x} \left (a b B \left (b^2+8 a c\right )+A \left (3 b^4-25 a b^2 c+28 a^2 c^2\right )+c \left (a B \left (b^2+20 a c\right )+3 A \left (b^3-8 a b c\right )\right ) x\right )}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\sqrt {c} \left (a B \left (b^2+20 a c\right )+3 A \left (b^3-8 a b c\right )+\frac {a b B \left (b^2-52 a c\right )+3 A \left (b^4-10 a b^2 c+56 a^2 c^2\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a^2 \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (a B \left (b^2+20 a c\right )+3 A \left (b^3-8 a b c\right )-\frac {a b B \left (b^2-52 a c\right )+3 A \left (b^4-10 a b^2 c+56 a^2 c^2\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a^2 \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

1/2*x^(1/2)*(A*b^2-a*b*B-2*A*a*c+(A*b-2*B*a)*c*x)/a/(-4*a*c+b^2)/(c*x^2+b* 
x+a)^2+1/4*x^(1/2)*(a*b*B*(8*a*c+b^2)+A*(28*a^2*c^2-25*a*b^2*c+3*b^4)+c*(a 
*B*(20*a*c+b^2)+3*A*(-8*a*b*c+b^3))*x)/a^2/(-4*a*c+b^2)^2/(c*x^2+b*x+a)+1/ 
8*c^(1/2)*(a*B*(20*a*c+b^2)+3*A*(-8*a*b*c+b^3)+(a*b*B*(-52*a*c+b^2)+3*A*(5 
6*a^2*c^2-10*a*b^2*c+b^4))/(-4*a*c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x^(1 
/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a^2/(-4*a*c+b^2)^2/(b-(-4*a*c+b^ 
2)^(1/2))^(1/2)+1/8*c^(1/2)*(a*B*(20*a*c+b^2)+3*A*(-8*a*b*c+b^3)-(a*b*B*(- 
52*a*c+b^2)+3*A*(56*a^2*c^2-10*a*b^2*c+b^4))/(-4*a*c+b^2)^(1/2))*arctan(2^ 
(1/2)*c^(1/2)*x^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a^2/(-4*a*c+b^ 
2)^2/(b+(-4*a*c+b^2)^(1/2))^(1/2)
 

Mathematica [A] (verified)

Time = 4.90 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.12 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {2 \sqrt {x} \left (3 A b^3 x (b+c x)^2+4 a^3 c (4 b B+11 A c+9 B c x)+a^2 \left (-b^3 B-4 b c^2 x (A-7 B x)+4 c^3 x^2 (7 A+5 B x)+b^2 (-37 A c+5 B c x)\right )+a b (b+c x) \left (b B x (b+c x)+A \left (5 b^2-25 b c x-24 c^2 x^2\right )\right )\right )}{(a+x (b+c x))^2}+\frac {\sqrt {2} \sqrt {c} \left (a B \left (b^3-52 a b c+b^2 \sqrt {b^2-4 a c}+20 a c \sqrt {b^2-4 a c}\right )+3 A \left (b^4-10 a b^2 c+56 a^2 c^2+b^3 \sqrt {b^2-4 a c}-8 a b c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (a B \left (-b^3+52 a b c+b^2 \sqrt {b^2-4 a c}+20 a c \sqrt {b^2-4 a c}\right )+3 A \left (-b^4+10 a b^2 c-56 a^2 c^2+b^3 \sqrt {b^2-4 a c}-8 a b c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{8 a^2 \left (b^2-4 a c\right )^2} \] Input:

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

Output:

((2*Sqrt[x]*(3*A*b^3*x*(b + c*x)^2 + 4*a^3*c*(4*b*B + 11*A*c + 9*B*c*x) + 
a^2*(-(b^3*B) - 4*b*c^2*x*(A - 7*B*x) + 4*c^3*x^2*(7*A + 5*B*x) + b^2*(-37 
*A*c + 5*B*c*x)) + a*b*(b + c*x)*(b*B*x*(b + c*x) + A*(5*b^2 - 25*b*c*x - 
24*c^2*x^2))))/(a + x*(b + c*x))^2 + (Sqrt[2]*Sqrt[c]*(a*B*(b^3 - 52*a*b*c 
 + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c]) + 3*A*(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]))*ArcT 
an[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a 
*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(a*B*(-b^3 + 52*a*b*c 
+ b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c]) + 3*A*(-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]))*ArcT 
an[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a 
*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(8*a^2*(b^2 - 4*a*c)^2)
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {1235, 27, 1235, 27, 1197, 1480, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {\sqrt {x} \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int -\frac {3 A b^2+a B b-14 a A c+5 (A b-2 a B) c x}{2 \sqrt {x} \left (c x^2+b x+a\right )^2}dx}{2 a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 A b^2+a B b-14 a A c+5 (A b-2 a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {\frac {\sqrt {x} \left (A \left (28 a^2 c^2-25 a b^2 c+3 b^4\right )+c x \left (3 A \left (b^3-8 a b c\right )+a B \left (20 a c+b^2\right )\right )+a b B \left (8 a c+b^2\right )\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int -\frac {a b B \left (b^2-16 a c\right )+3 A \left (b^4-9 a c b^2+28 a^2 c^2\right )+c \left (a B \left (b^2+20 a c\right )+3 A \left (b^3-8 a b c\right )\right ) x}{2 \sqrt {x} \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {a b B \left (b^2-16 a c\right )+3 A \left (b^4-9 a c b^2+28 a^2 c^2\right )+c \left (a B \left (b^2+20 a c\right )+3 A \left (b^3-8 a b c\right )\right ) x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (A \left (28 a^2 c^2-25 a b^2 c+3 b^4\right )+c x \left (3 A \left (b^3-8 a b c\right )+a B \left (20 a c+b^2\right )\right )+a b B \left (8 a c+b^2\right )\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1197

\(\displaystyle \frac {\frac {\int \frac {a b B \left (b^2-16 a c\right )+3 A \left (b^4-9 a c b^2+28 a^2 c^2\right )+c \left (a B \left (b^2+20 a c\right )+3 A \left (b^3-8 a b c\right )\right ) x}{c x^2+b x+a}d\sqrt {x}}{a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (A \left (28 a^2 c^2-25 a b^2 c+3 b^4\right )+c x \left (3 A \left (b^3-8 a b c\right )+a B \left (20 a c+b^2\right )\right )+a b B \left (8 a c+b^2\right )\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {\frac {\frac {1}{2} c \left (\frac {3 A \left (56 a^2 c^2-10 a b^2 c+b^4\right )+a b B \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}+3 A \left (b^3-8 a b c\right )+a B \left (20 a c+b^2\right )\right ) \int \frac {1}{\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}+\frac {1}{2} c \left (-\frac {3 A \left (56 a^2 c^2-10 a b^2 c+b^4\right )+a b B \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}+3 A \left (b^3-8 a b c\right )+a B \left (20 a c+b^2\right )\right ) \int \frac {1}{\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}}{a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (A \left (28 a^2 c^2-25 a b^2 c+3 b^4\right )+c x \left (3 A \left (b^3-8 a b c\right )+a B \left (20 a c+b^2\right )\right )+a b B \left (8 a c+b^2\right )\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {\sqrt {c} \left (\frac {3 A \left (56 a^2 c^2-10 a b^2 c+b^4\right )+a b B \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}+3 A \left (b^3-8 a b c\right )+a B \left (20 a c+b^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (-\frac {3 A \left (56 a^2 c^2-10 a b^2 c+b^4\right )+a b B \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}+3 A \left (b^3-8 a b c\right )+a B \left (20 a c+b^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (A \left (28 a^2 c^2-25 a b^2 c+3 b^4\right )+c x \left (3 A \left (b^3-8 a b c\right )+a B \left (20 a c+b^2\right )\right )+a b B \left (8 a c+b^2\right )\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

Input:

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

Output:

(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) 
*(a + b*x + c*x^2)^2) + ((Sqrt[x]*(a*b*B*(b^2 + 8*a*c) + A*(3*b^4 - 25*a*b 
^2*c + 28*a^2*c^2) + c*(a*B*(b^2 + 20*a*c) + 3*A*(b^3 - 8*a*b*c))*x))/(a*( 
b^2 - 4*a*c)*(a + b*x + c*x^2)) + ((Sqrt[c]*(a*B*(b^2 + 20*a*c) + 3*A*(b^3 
 - 8*a*b*c) + (a*b*B*(b^2 - 52*a*c) + 3*A*(b^4 - 10*a*b^2*c + 56*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[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(a*B*(b^2 + 20*a 
*c) + 3*A*(b^3 - 8*a*b*c) - (a*b*B*(b^2 - 52*a*c) + 3*A*(b^4 - 10*a*b^2*c 
+ 56*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[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(a*(b^2 - 4* 
a*c)))/(4*a*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

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

rule 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[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] /; Fr 
eeQ[{a, b, c, d, e, f, g}, x]
 

rule 1235

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] 
 + Simp[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] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1555\) vs. \(2(416)=832\).

Time = 1.54 (sec) , antiderivative size = 1556, normalized size of antiderivative = 3.32

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1556\)
default	\(\text {Expression too large to display}\)	\(1556\)

Input:

int((B*x+A)/x^(1/2)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

128*c^3*(-1/64/(-4*a*c+b^2)^(5/2)/c/(4*a*c-b^2)^2*((-1/32/c^2/a^2*(20*a*c* 
(-4*a*c+b^2)^(1/2)+b^2*(-4*a*c+b^2)^(1/2)+4*a*b*c-b^3)*(2880*A*(-4*a*c+b^2 
)^(1/2)*a^3*c^3-1968*A*(-4*a*c+b^2)^(1/2)*a^2*b^2*c^2+444*A*(-4*a*c+b^2)^( 
1/2)*a*b^4*c-33*A*(-4*a*c+b^2)^(1/2)*b^6-4416*A*a^3*b*c^3+2736*A*a^2*b^3*c 
^2-540*A*a*b^5*c+33*A*b^7+3200*B*a^4*c^3-1248*B*a^3*b^2*c^2+24*B*a^2*b^4*c 
+22*B*a*b^6)/(100*a*c+11*b^2)*x^(3/2)+1/16/c^2/a*(-6*b*(-4*a*c+b^2)^(1/2)+ 
28*a*c-7*b^2)*(4928*A*(-4*a*c+b^2)^(1/2)*a^3*c^3-3504*A*(-4*a*c+b^2)^(1/2) 
*a^2*b^2*c^2+828*A*(-4*a*c+b^2)^(1/2)*a*b^4*c-65*A*(-4*a*c+b^2)^(1/2)*b^6- 
6464*A*a^3*b*c^3+4272*A*a^2*b^3*c^2-924*A*a*b^5*c+65*A*b^7+6272*B*a^4*c^3- 
3552*B*a^3*b^2*c^2+600*B*a^2*b^4*c-26*B*a*b^6)/(196*a*c-13*b^2)*x^(1/2))/( 
x+1/2*b/c+1/2/c*(-4*a*c+b^2)^(1/2))^2-1/32*(20*a*c*(-4*a*c+b^2)^(1/2)+b^2* 
(-4*a*c+b^2)^(1/2)+52*a*b*c-b^3)*(26880*A*(-4*a*c+b^2)^(1/2)*a^4*c^4-26880 
*A*(-4*a*c+b^2)^(1/2)*a^3*b^2*c^3+10080*A*(-4*a*c+b^2)^(1/2)*a^2*b^4*c^2-1 
680*A*(-4*a*c+b^2)^(1/2)*a*b^6*c+105*A*(-4*a*c+b^2)^(1/2)*b^8-85248*A*a^4* 
b*c^4+61440*A*a^3*b^3*c^3-16416*A*a^2*b^5*c^2+2016*A*a*b^7*c-105*A*b^9+128 
00*B*a^5*c^4+13312*B*a^4*b^2*c^3-10176*B*a^3*b^4*c^2+1792*B*a^2*b^6*c-70*B 
*a*b^8)/a^2/(400*a^2*c^2+616*a*b^2*c-35*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)))+ 
1/64/(-4*a*c+b^2)^(5/2)/c/(4*a*c-b^2)^2*((-1/32/c^2/a^2*(-20*a*c*(-4*a*c+b 
^2)^(1/2)-b^2*(-4*a*c+b^2)^(1/2)+4*a*b*c-b^3)*(-2880*A*(-4*a*c+b^2)^(1/...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 9907 vs. \(2 (417) = 834\).

Time = 41.30 (sec) , antiderivative size = 9907, normalized size of antiderivative = 21.17 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)/x^(1/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)/x**(1/2)/(c*x**2+b*x+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )^3} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + b x + a\right )}^{3} \sqrt {x}} \,d x } \] Input:

integrate((B*x+A)/x^(1/2)/(c*x^2+b*x+a)^3,x, algorithm="maxima")
 

Output:

1/4*((3*(b^4*c^2 - 9*a*b^2*c^3 + 28*a^2*c^4)*A + (a*b^3*c^2 - 16*a^2*b*c^3 
)*B)*x^(9/2) + (3*(2*b^5*c - 17*a*b^3*c^2 + 48*a^2*b*c^3)*A + (2*a*b^4*c - 
 31*a^2*b^2*c^2 + 20*a^3*c^3)*B)*x^(7/2) + ((3*b^6 - 15*a*b^4*c - 19*a^2*b 
^2*c^2 + 196*a^3*c^3)*A + (a*b^5 - 12*a^2*b^3*c - 4*a^3*b*c^2)*B)*x^(5/2) 
+ 8*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*A*sqrt(x) + ((9*a*b^5 - 74*a^2*b^ 
3*c + 164*a^3*b*c^2)*A + 3*(a^2*b^4 - 9*a^3*b^2*c + 12*a^4*c^2)*B)*x^(3/2) 
)/(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16* 
a^5*c^4)*x^4 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^3 + (a^3*b^6 
 - 6*a^4*b^4*c + 32*a^6*c^3)*x^2 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2 
)*x) - integrate(1/8*((3*(b^4*c - 9*a*b^2*c^2 + 28*a^2*c^3)*A + (a*b^3*c - 
 16*a^2*b*c^2)*B)*x^(3/2) + (3*(b^5 - 10*a*b^3*c + 36*a^2*b*c^2)*A + (a*b^ 
4 - 17*a^2*b^2*c - 20*a^3*c^2)*B)*sqrt(x))/(a^4*b^4 - 8*a^5*b^2*c + 16*a^6 
*c^2 + (a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*x^2 + (a^3*b^5 - 8*a^4*b^3 
*c + 16*a^5*b*c^2)*x), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4621 vs. \(2 (417) = 834\).

Time = 1.36 (sec) , antiderivative size = 4621, normalized size of antiderivative = 9.87 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)/x^(1/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")
 

Output:

1/16*(3*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^8 - 17*sqrt(2)*sqrt(b*c 
 + sqrt(b^2 - 4*a*c)*c)*a*b^6*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c 
)*b^7*c - 2*b^8*c + 116*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^ 
2 + 26*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 + sqrt(2)*sqrt(b* 
c + sqrt(b^2 - 4*a*c)*c)*b^6*c^2 + 34*a*b^6*c^2 + 2*b^7*c^2 - 368*sqrt(2)* 
sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^3 - 128*sqrt(2)*sqrt(b*c + sqrt( 
b^2 - 4*a*c)*c)*a^2*b^3*c^3 - 13*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a 
*b^4*c^3 - 232*a^2*b^4*c^3 - 30*a*b^5*c^3 + 448*sqrt(2)*sqrt(b*c + sqrt(b^ 
2 - 4*a*c)*c)*a^4*c^4 + 224*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b* 
c^4 + 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^4 + 736*a^3*b^2 
*c^4 + 176*a^2*b^3*c^4 - 112*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c 
^5 - 896*a^4*c^5 - 352*a^3*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq 
rt(b^2 - 4*a*c)*c)*b^7 + 15*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 
- 4*a*c)*c)*a*b^5*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4* 
a*c)*c)*b^6*c - 88*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)* 
c)*a^2*b^3*c^2 - 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c) 
*c)*a*b^4*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)* 
b^5*c^2 + 176*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^ 
3*b*c^3 + 88*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2 
*b^2*c^3 + 11*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)...
 

Mupad [B] (verification not implemented)

Time = 15.25 (sec) , antiderivative size = 22946, normalized size of antiderivative = 49.03 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int((A + B*x)/(x^(1/2)*(a + b*x + c*x^2)^3),x)
 

Output:

((x^(3/2)*(3*A*b^5 + 36*B*a^3*c^2 + B*a*b^4 - 20*A*a*b^3*c - 4*A*a^2*b*c^2 
 + 5*B*a^2*b^2*c))/(4*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^(1/2)*(5*A* 
b^4 + 44*A*a^2*c^2 - B*a*b^3 - 37*A*a*b^2*c + 16*B*a^2*b*c))/(4*a*(b^4 + 1 
6*a^2*c^2 - 8*a*b^2*c)) + (x^(5/2)*(28*A*a^2*c^3 + 6*A*b^4*c + 2*B*a*b^3*c 
 - 49*A*a*b^2*c^2 + 28*B*a^2*b*c^2))/(4*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c) 
) + (c*x^(7/2)*(20*B*a^2*c^2 + 3*A*b^3*c - 24*A*a*b*c^2 + B*a*b^2*c))/(4*a 
^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^2*(2*a*c + b^2) + a^2 + c^2*x^4 + 2 
*a*b*x + 2*b*c*x^3) + atan(((((1048576*B*a^9*b*c^8 - 5505024*A*a^9*c^9 + 1 
92*A*a^2*b^14*c^2 - 5568*A*a^3*b^12*c^3 + 70656*A*a^4*b^10*c^4 - 506880*A* 
a^5*b^8*c^5 + 2211840*A*a^6*b^6*c^6 - 5849088*A*a^7*b^4*c^7 + 8650752*A*a^ 
8*b^2*c^8 + 64*B*a^3*b^13*c^2 - 2304*B*a^4*b^11*c^3 + 30720*B*a^5*b^9*c^4 
- 204800*B*a^6*b^7*c^5 + 737280*B*a^7*b^5*c^6 - 1376256*B*a^8*b^3*c^7)/(64 
*(a^4*b^12 + 4096*a^10*c^6 - 24*a^5*b^10*c + 240*a^6*b^8*c^2 - 1280*a^7*b^ 
6*c^3 + 3840*a^8*b^4*c^4 - 6144*a^9*b^2*c^5)) - (x^(1/2)*(-(9*A^2*b^19 + B 
^2*a^2*b^17 + 9*A^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 6*A*B*a*b^18 + 6921*A^ 
2*a^2*b^15*c^2 - 77580*A^2*a^3*b^13*c^3 + 570960*A^2*a^4*b^11*c^4 - 285177 
6*A^2*a^5*b^9*c^5 + 9628416*A^2*a^6*b^7*c^6 - 21095424*A^2*a^7*b^5*c^7 + 2 
7095040*A^2*a^8*b^3*c^8 + 441*A^2*a^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + B^2* 
a^2*b^2*(-(4*a*c - b^2)^15)^(1/2) + 1140*B^2*a^4*b^13*c^2 - 10160*B^2*a^5* 
b^11*c^3 + 34880*B^2*a^6*b^9*c^4 + 43776*B^2*a^7*b^7*c^5 - 680960*B^2*a...
 

Reduce [B] (verification not implemented)

Time = 4.98 (sec) , antiderivative size = 6756, normalized size of antiderivative = 14.44 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((B*x+A)/x^(1/2)/(c*x^2+b*x+a)^3,x)
 

Output:

(184*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) 
 - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a**4*b*c**2 - 102*sqrt( 
a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt( 
x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b**3*c + 368*sqrt(a)*sqrt(2* 
sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c) 
)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b**2*c**2*x + 368*sqrt(a)*sqrt(2*sqrt( 
c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqr 
t(2*sqrt(c)*sqrt(a) + b))*a**3*b*c**3*x**2 + 8*sqrt(a)*sqrt(2*sqrt(c)*sqrt 
(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqr 
t(c)*sqrt(a) + b))*a**2*b**5 - 204*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*ata 
n((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) 
 + b))*a**2*b**4*c*x - 20*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2 
*sqrt(c)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a* 
*2*b**3*c**2*x**2 + 368*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*s 
qrt(c)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2 
*b**2*c**3*x**3 + 184*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqr 
t(c)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b 
*c**4*x**4 + 16*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*s 
qrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b**6*x - 8 
6*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b)...