\(\int \frac {A+B x}{x^4 (a+b x+c x^2)^{3/2}} \, dx\) [161]

Optimal result

Integrand size = 23, antiderivative size = 292 \[ \int \frac {A+B x}{x^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {A}{3 a x^3 \sqrt {a+b x+c x^2}}+\frac {7 A b-6 a B}{12 a^2 x^2 \sqrt {a+b x+c x^2}}-\frac {35 A b^2-30 a b B-32 a A c}{24 a^3 x \sqrt {a+b x+c x^2}}+\frac {6 a B \left (15 b^4-62 a b^2 c+24 a^2 c^2\right )-A \left (105 b^5-530 a b^3 c+488 a^2 b c^2\right )+c \left (6 a b B \left (15 b^2-52 a c\right )-A \left (105 b^4-460 a b^2 c+256 a^2 c^2\right )\right ) x}{24 a^4 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\left (35 A b^3-30 a b^2 B-60 a A b c+24 a^2 B c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{9/2}} \] Output:

-1/3*A/a/x^3/(c*x^2+b*x+a)^(1/2)+1/12*(7*A*b-6*B*a)/a^2/x^2/(c*x^2+b*x+a)^ 
(1/2)-1/24*(-32*A*a*c+35*A*b^2-30*B*a*b)/a^3/x/(c*x^2+b*x+a)^(1/2)+1/24*(6 
*a*B*(24*a^2*c^2-62*a*b^2*c+15*b^4)-A*(488*a^2*b*c^2-530*a*b^3*c+105*b^5)+ 
c*(6*a*b*B*(-52*a*c+15*b^2)-A*(256*a^2*c^2-460*a*b^2*c+105*b^4))*x)/a^4/(- 
4*a*c+b^2)/(c*x^2+b*x+a)^(1/2)+1/16*(-60*A*a*b*c+35*A*b^3+24*B*a^2*c-30*B* 
a*b^2)*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(9/2)
 

Mathematica [A] (verified)

Time = 2.44 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x}{x^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {a} \left (-16 a^4 c (2 A+3 B x)+105 A b^4 x^3 (b+c x)+4 a^3 \left (3 B x \left (b^2+10 b c x-12 c^2 x^2\right )+2 A \left (b^2+7 b c x+16 c^2 x^2\right )\right )-5 a b^2 x^2 \left (18 b B x (b+c x)+A \left (-7 b^2+106 b c x+92 c^2 x^2\right )\right )+2 a^2 x \left (3 b B x \left (-5 b^2+62 b c x+52 c^2 x^2\right )+A \left (-7 b^3-86 b^2 c x+244 b c^2 x^2+128 c^3 x^3\right )\right )\right )}{x^3 \sqrt {a+x (b+c x)}}+3 \left (b^2-4 a c\right ) \left (6 a B \left (-5 b^2+4 a c\right )+5 A \left (7 b^3-12 a b c\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{24 a^{9/2} \left (-b^2+4 a c\right )} \] Input:

Integrate[(A + B*x)/(x^4*(a + b*x + c*x^2)^(3/2)),x]
 

Output:

((Sqrt[a]*(-16*a^4*c*(2*A + 3*B*x) + 105*A*b^4*x^3*(b + c*x) + 4*a^3*(3*B* 
x*(b^2 + 10*b*c*x - 12*c^2*x^2) + 2*A*(b^2 + 7*b*c*x + 16*c^2*x^2)) - 5*a* 
b^2*x^2*(18*b*B*x*(b + c*x) + A*(-7*b^2 + 106*b*c*x + 92*c^2*x^2)) + 2*a^2 
*x*(3*b*B*x*(-5*b^2 + 62*b*c*x + 52*c^2*x^2) + A*(-7*b^3 - 86*b^2*c*x + 24 
4*b*c^2*x^2 + 128*c^3*x^3))))/(x^3*Sqrt[a + x*(b + c*x)]) + 3*(b^2 - 4*a*c 
)*(6*a*B*(-5*b^2 + 4*a*c) + 5*A*(7*b^3 - 12*a*b*c))*ArcTanh[(Sqrt[c]*x - S 
qrt[a + x*(b + c*x)])/Sqrt[a]])/(24*a^(9/2)*(-b^2 + 4*a*c))
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1235, 27, 1237, 27, 25, 1237, 27, 1228, 1154, 219}

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.

\(\displaystyle \int \frac {A+B x}{x^4 \left (a+b x+c x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int -\frac {7 A b^2-6 a B b-16 a A c+6 (A b-2 a B) c x}{2 x^4 \sqrt {c x^2+b x+a}}dx}{a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {7 A b^2-6 a B b-16 a A c+6 (A b-2 a B) c x}{x^4 \sqrt {c x^2+b x+a}}dx}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {-\frac {\int -\frac {6 a B \left (5 b^2-12 a c\right )-2 A \left (\frac {35 b^3}{2}-58 a b c\right )-4 c \left (7 A b^2-6 a B b-16 a A c\right ) x}{2 x^3 \sqrt {c x^2+b x+a}}dx}{3 a}-\frac {\sqrt {a+b x+c x^2} \left (-16 a A c-6 a b B+7 A b^2\right )}{3 a x^3}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int -\frac {35 A b^3-30 a B b^2-116 a A c b+72 a^2 B c+4 c \left (7 A b^2-6 a B b-16 a A c\right ) x}{x^3 \sqrt {c x^2+b x+a}}dx}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (-16 a A c-6 a b B+7 A b^2\right )}{3 a x^3}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {\int \frac {35 A b^3-30 a B b^2-116 a A c b+72 a^2 B c+4 c \left (7 A b^2-6 a B b-16 a A c\right ) x}{x^3 \sqrt {c x^2+b x+a}}dx}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (-16 a A c-6 a b B+7 A b^2\right )}{3 a x^3}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {-\frac {-\frac {\int -\frac {6 a b B \left (15 b^2-52 a c\right )-2 A \left (\frac {105 b^4}{2}-230 a c b^2+128 a^2 c^2\right )-2 c \left (35 A b^3-30 a B b^2-116 a A c b+72 a^2 B c\right ) x}{2 x^2 \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {\sqrt {a+b x+c x^2} \left (72 a^2 B c-116 a A b c-30 a b^2 B+35 A b^3\right )}{2 a x^2}}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (-16 a A c-6 a b B+7 A b^2\right )}{3 a x^3}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\frac {\int \frac {6 a b B \left (15 b^2-52 a c\right )-A \left (105 b^4-460 a c b^2+256 a^2 c^2\right )-2 c \left (35 A b^3-30 a B b^2-116 a A c b+72 a^2 B c\right ) x}{x^2 \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (72 a^2 B c-116 a A b c-30 a b^2 B+35 A b^3\right )}{2 a x^2}}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (-16 a A c-6 a b B+7 A b^2\right )}{3 a x^3}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {-\frac {\frac {-\frac {3 \left (b^2-4 a c\right ) \left (6 a B \left (5 b^2-4 a c\right )-5 A \left (7 b^3-12 a b c\right )\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {\sqrt {a+b x+c x^2} \left (6 a b B \left (15 b^2-52 a c\right )-A \left (256 a^2 c^2-460 a b^2 c+105 b^4\right )\right )}{a x}}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (72 a^2 B c-116 a A b c-30 a b^2 B+35 A b^3\right )}{2 a x^2}}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (-16 a A c-6 a b B+7 A b^2\right )}{3 a x^3}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {-\frac {\frac {\frac {3 \left (b^2-4 a c\right ) \left (6 a B \left (5 b^2-4 a c\right )-5 A \left (7 b^3-12 a b c\right )\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{a}-\frac {\sqrt {a+b x+c x^2} \left (6 a b B \left (15 b^2-52 a c\right )-A \left (256 a^2 c^2-460 a b^2 c+105 b^4\right )\right )}{a x}}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (72 a^2 B c-116 a A b c-30 a b^2 B+35 A b^3\right )}{2 a x^2}}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (-16 a A c-6 a b B+7 A b^2\right )}{3 a x^3}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {\frac {\frac {3 \left (b^2-4 a c\right ) \left (6 a B \left (5 b^2-4 a c\right )-5 A \left (7 b^3-12 a b c\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2}}-\frac {\sqrt {a+b x+c x^2} \left (6 a b B \left (15 b^2-52 a c\right )-A \left (256 a^2 c^2-460 a b^2 c+105 b^4\right )\right )}{a x}}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (72 a^2 B c-116 a A b c-30 a b^2 B+35 A b^3\right )}{2 a x^2}}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (-16 a A c-6 a b B+7 A b^2\right )}{3 a x^3}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

Input:

Int[(A + B*x)/(x^4*(a + b*x + c*x^2)^(3/2)),x]
 

Output:

(2*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x))/(a*(b^2 - 4*a*c)*x^3*Sqr 
t[a + b*x + c*x^2]) + (-1/3*((7*A*b^2 - 6*a*b*B - 16*a*A*c)*Sqrt[a + b*x + 
 c*x^2])/(a*x^3) - (-1/2*((35*A*b^3 - 30*a*b^2*B - 116*a*A*b*c + 72*a^2*B* 
c)*Sqrt[a + b*x + c*x^2])/(a*x^2) + (-(((6*a*b*B*(15*b^2 - 52*a*c) - A*(10 
5*b^4 - 460*a*b^2*c + 256*a^2*c^2))*Sqrt[a + b*x + c*x^2])/(a*x)) + (3*(b^ 
2 - 4*a*c)*(6*a*B*(5*b^2 - 4*a*c) - 5*A*(7*b^3 - 12*a*b*c))*ArcTanh[(2*a + 
 b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*a^(3/2)))/(4*a))/(6*a))/(a*(b 
^2 - 4*a*c))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-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]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(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] - Simp[(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 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

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 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(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] + Simp[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] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.58

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-40 A a c \,x^{2}+57 x^{2} b^{2} A -42 B a \,x^{2} b -22 a b A x +12 a^{2} B x +8 a^{2} A \right )}{24 a^{4} x^{3}}+\frac {c \left (28 A a b c -19 A \,b^{3}-8 B \,a^{2} c +14 B a \,b^{2}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+a \left (60 A a b c -35 A \,b^{3}-24 B \,a^{2} c +30 B a \,b^{2}\right ) \left (\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}\right )-\frac {38 A \,b^{4} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {28 B a \,b^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {32 a^{2} A \,c^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {24 A a \,b^{2} c \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {16 a^{2} b B c \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}}{16 a^{4}}\)	\(460\)
default	\(A \left (-\frac {1}{3 a \,x^{3} \sqrt {c \,x^{2}+b x +a}}-\frac {7 b \left (-\frac {1}{2 a \,x^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {5 b \left (-\frac {1}{a x \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}-\frac {4 c \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{4 a}-\frac {3 c \left (\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )}{6 a}-\frac {4 c \left (-\frac {1}{a x \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}-\frac {4 c \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{3 a}\right )+B \left (-\frac {1}{2 a \,x^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {5 b \left (-\frac {1}{a x \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}-\frac {4 c \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{4 a}-\frac {3 c \left (\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )\)	\(729\)

Input:

int((B*x+A)/x^4/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/24*(c*x^2+b*x+a)^(1/2)*(-40*A*a*c*x^2+57*A*b^2*x^2-42*B*a*b*x^2-22*A*a* 
b*x+12*B*a^2*x+8*A*a^2)/a^4/x^3+1/16/a^4*(c*(28*A*a*b*c-19*A*b^3-8*B*a^2*c 
+14*B*a*b^2)*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b* 
x+a)^(1/2))+a*(60*A*a*b*c-35*A*b^3-24*B*a^2*c+30*B*a*b^2)*(1/a/(c*x^2+b*x+ 
a)^(1/2)-b/a*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-1/a^(3/2)*ln((2*a+b 
*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))-38*A*b^4*(2*c*x+b)/(4*a*c-b^2)/(c*x^ 
2+b*x+a)^(1/2)+28*B*a*b^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+32*a^2 
*A*c^2*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+24*A*a*b^2*c*(2*c*x+b)/(4 
*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+16*a^2*b*B*c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b* 
x+a)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (266) = 532\).

Time = 0.97 (sec) , antiderivative size = 1093, normalized size of antiderivative = 3.74 \[ \int \frac {A+B x}{x^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)/x^4/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
 

Output:

[1/96*(3*((48*(2*B*a^3 - 5*A*a^2*b)*c^3 - 8*(18*B*a^2*b^2 - 25*A*a*b^3)*c^ 
2 + 5*(6*B*a*b^4 - 7*A*b^5)*c)*x^5 + (30*B*a*b^5 - 35*A*b^6 + 48*(2*B*a^3* 
b - 5*A*a^2*b^2)*c^2 - 8*(18*B*a^2*b^3 - 25*A*a*b^4)*c)*x^4 + (30*B*a^2*b^ 
4 - 35*A*a*b^5 + 48*(2*B*a^4 - 5*A*a^3*b)*c^2 - 8*(18*B*a^3*b^2 - 25*A*a^2 
*b^3)*c)*x^3)*sqrt(a)*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b 
*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) - 4*(8*A*a^4*b^2 - 32*A*a^5*c + 
(256*A*a^3*c^3 + 4*(78*B*a^3*b - 115*A*a^2*b^2)*c^2 - 15*(6*B*a^2*b^3 - 7* 
A*a*b^4)*c)*x^4 - (90*B*a^2*b^4 - 105*A*a*b^5 + 8*(18*B*a^4 - 61*A*a^3*b)* 
c^2 - 2*(186*B*a^3*b^2 - 265*A*a^2*b^3)*c)*x^3 - (30*B*a^3*b^3 - 35*A*a^2* 
b^4 - 128*A*a^4*c^2 - 4*(30*B*a^4*b - 43*A*a^3*b^2)*c)*x^2 + 2*(6*B*a^4*b^ 
2 - 7*A*a^3*b^3 - 4*(6*B*a^5 - 7*A*a^4*b)*c)*x)*sqrt(c*x^2 + b*x + a))/((a 
^5*b^2*c - 4*a^6*c^2)*x^5 + (a^5*b^3 - 4*a^6*b*c)*x^4 + (a^6*b^2 - 4*a^7*c 
)*x^3), 1/48*(3*((48*(2*B*a^3 - 5*A*a^2*b)*c^3 - 8*(18*B*a^2*b^2 - 25*A*a* 
b^3)*c^2 + 5*(6*B*a*b^4 - 7*A*b^5)*c)*x^5 + (30*B*a*b^5 - 35*A*b^6 + 48*(2 
*B*a^3*b - 5*A*a^2*b^2)*c^2 - 8*(18*B*a^2*b^3 - 25*A*a*b^4)*c)*x^4 + (30*B 
*a^2*b^4 - 35*A*a*b^5 + 48*(2*B*a^4 - 5*A*a^3*b)*c^2 - 8*(18*B*a^3*b^2 - 2 
5*A*a^2*b^3)*c)*x^3)*sqrt(-a)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(b*x + 2*a) 
*sqrt(-a)/(a*c*x^2 + a*b*x + a^2)) - 2*(8*A*a^4*b^2 - 32*A*a^5*c + (256*A* 
a^3*c^3 + 4*(78*B*a^3*b - 115*A*a^2*b^2)*c^2 - 15*(6*B*a^2*b^3 - 7*A*a*b^4 
)*c)*x^4 - (90*B*a^2*b^4 - 105*A*a*b^5 + 8*(18*B*a^4 - 61*A*a^3*b)*c^2 ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{x^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)/x**4/(c*x**2+b*x+a)**(3/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x}{x^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)/x^4/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 798 vs. \(2 (266) = 532\).

Time = 0.25 (sec) , antiderivative size = 798, normalized size of antiderivative = 2.73 \[ \int \frac {A+B x}{x^4 \left (a+b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)/x^4/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
 

Output:

2*((B*a^5*b^3*c - A*a^4*b^4*c - 3*B*a^6*b*c^2 + 4*A*a^5*b^2*c^2 - 2*A*a^6* 
c^3)*x/(a^8*b^2 - 4*a^9*c) + (B*a^5*b^4 - A*a^4*b^5 - 4*B*a^6*b^2*c + 5*A* 
a^5*b^3*c + 2*B*a^7*c^2 - 5*A*a^6*b*c^2)/(a^8*b^2 - 4*a^9*c))/sqrt(c*x^2 + 
 b*x + a) + 1/8*(30*B*a*b^2 - 35*A*b^3 - 24*B*a^2*c + 60*A*a*b*c)*arctan(- 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^4) - 1/24*(42*(s 
qrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a*b^2 - 57*(sqrt(c)*x - sqrt(c*x^2 + 
 b*x + a))^5*A*b^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^2*c + 84 
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*b*c + 48*(sqrt(c)*x - sqrt(c*x^ 
2 + b*x + a))^4*B*a^2*b*sqrt(c) - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4 
*A*a*b^2*sqrt(c) + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^2*c^(3/2) 
- 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^2*b^2 + 136*(sqrt(c)*x - sq 
rt(c*x^2 + b*x + a))^3*A*a*b^3 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3 
*A*a^2*b*c - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^3*b*sqrt(c) + 1 
44*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^2*b^2*sqrt(c) - 192*(sqrt(c)* 
x - sqrt(c*x^2 + b*x + a))^2*A*a^3*c^(3/2) + 54*(sqrt(c)*x - sqrt(c*x^2 + 
b*x + a))*B*a^3*b^2 - 87*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*b^3 + 2 
4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^4*c - 36*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))*A*a^3*b*c + 96*B*a^4*b*sqrt(c) - 144*A*a^3*b^2*sqrt(c) + 80*A 
*a^4*c^(3/2))/(((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^3*a^4)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{x^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{x^4\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \] Input:

int((A + B*x)/(x^4*(a + b*x + c*x^2)^(3/2)),x)
                                                                                    
                                                                                    
 

Output:

int((A + B*x)/(x^4*(a + b*x + c*x^2)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 796, normalized size of antiderivative = 2.73 \[ \int \frac {A+B x}{x^4 \left (a+b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int((B*x+A)/x^4/(c*x^2+b*x+a)^(3/2),x)
 

Output:

( - 64*sqrt(a + b*x + c*x**2)*a**5*c + 16*sqrt(a + b*x + c*x**2)*a**4*b**2 
 + 16*sqrt(a + b*x + c*x**2)*a**4*b*c*x + 256*sqrt(a + b*x + c*x**2)*a**4* 
c**2*x**2 - 4*sqrt(a + b*x + c*x**2)*a**3*b**3*x - 104*sqrt(a + b*x + c*x* 
*2)*a**3*b**2*c*x**2 + 688*sqrt(a + b*x + c*x**2)*a**3*b*c**2*x**3 + 512*s 
qrt(a + b*x + c*x**2)*a**3*c**3*x**4 + 10*sqrt(a + b*x + c*x**2)*a**2*b**4 
*x**2 - 316*sqrt(a + b*x + c*x**2)*a**2*b**3*c*x**3 - 296*sqrt(a + b*x + c 
*x**2)*a**2*b**2*c**2*x**4 + 30*sqrt(a + b*x + c*x**2)*a*b**5*x**3 + 30*sq 
rt(a + b*x + c*x**2)*a*b**4*c*x**4 + 432*sqrt(a)*log(2*sqrt(a)*sqrt(a + b* 
x + c*x**2) - 2*a - b*x)*a**3*b*c**2*x**3 - 168*sqrt(a)*log(2*sqrt(a)*sqrt 
(a + b*x + c*x**2) - 2*a - b*x)*a**2*b**3*c*x**3 + 432*sqrt(a)*log(2*sqrt( 
a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*b**2*c**2*x**4 + 432*sqrt(a)*l 
og(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*b*c**3*x**5 + 15*sqr 
t(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b**5*x**3 - 168*s 
qrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b**4*c*x**4 - 1 
68*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b**3*c**2*x 
**5 + 15*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*b**6*x* 
*4 + 15*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*b**5*c*x 
**5 - 432*sqrt(a)*log(x)*a**3*b*c**2*x**3 + 168*sqrt(a)*log(x)*a**2*b**3*c 
*x**3 - 432*sqrt(a)*log(x)*a**2*b**2*c**2*x**4 - 432*sqrt(a)*log(x)*a**2*b 
*c**3*x**5 - 15*sqrt(a)*log(x)*a*b**5*x**3 + 168*sqrt(a)*log(x)*a*b**4*...