\(\int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^5} \, dx\) [113]

Optimal result

Integrand size = 23, antiderivative size = 172 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^5} \, dx=-\frac {\left (5 A b^2-8 a b B-4 a A c\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{64 a^3 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}+\frac {(5 A b-8 a B) \left (a+b x+c x^2\right )^{3/2}}{24 a^2 x^3}+\frac {\left (b^2-4 a c\right ) \left (5 A b^2-8 a b B-4 a A c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{7/2}} \] Output:

-1/64*(-4*A*a*c+5*A*b^2-8*B*a*b)*(b*x+2*a)*(c*x^2+b*x+a)^(1/2)/a^3/x^2-1/4 
*A*(c*x^2+b*x+a)^(3/2)/a/x^4+1/24*(5*A*b-8*B*a)*(c*x^2+b*x+a)^(3/2)/a^2/x^ 
3+1/128*(-4*a*c+b^2)*(-4*A*a*c+5*A*b^2-8*B*a*b)*arctanh(1/2*(b*x+2*a)/a^(1 
/2)/(c*x^2+b*x+a)^(1/2))/a^(7/2)
 

Mathematica [A] (verified)

Time = 1.79 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.15 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^5} \, dx=\frac {-\frac {\sqrt {a} \sqrt {a+x (b+c x)} \left (15 A b^3 x^3+16 a^3 (3 A+4 B x)-2 a b x^2 (5 A b+12 b B x+26 A c x)+8 a^2 x (A (b+3 c x)+2 B x (b+4 c x))\right )}{x^4}-15 A b^4 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+24 a \left (-b^3 B-3 A b^2 c+4 a b B c+2 a A c^2\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{192 a^{7/2}} \] Input:

Integrate[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^5,x]
 

Output:

(-((Sqrt[a]*Sqrt[a + x*(b + c*x)]*(15*A*b^3*x^3 + 16*a^3*(3*A + 4*B*x) - 2 
*a*b*x^2*(5*A*b + 12*b*B*x + 26*A*c*x) + 8*a^2*x*(A*(b + 3*c*x) + 2*B*x*(b 
 + 4*c*x))))/x^4) - 15*A*b^4*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/S 
qrt[a]] + 24*a*(-(b^3*B) - 3*A*b^2*c + 4*a*b*B*c + 2*a*A*c^2)*ArcTanh[(-(S 
qrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(192*a^(7/2))
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1237, 27, 1228, 1152, 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.

\(\displaystyle \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^5} \, dx\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {\int \frac {(5 A b-8 a B+2 A c x) \sqrt {c x^2+b x+a}}{2 x^4}dx}{4 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(5 A b-8 a B+2 A c x) \sqrt {c x^2+b x+a}}{x^4}dx}{8 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}\)

\(\Big \downarrow \) 1228

\(\displaystyle -\frac {-\frac {\left (-4 a A c-8 a b B+5 A b^2\right ) \int \frac {\sqrt {c x^2+b x+a}}{x^3}dx}{2 a}-\frac {(5 A b-8 a B) \left (a+b x+c x^2\right )^{3/2}}{3 a x^3}}{8 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}\)

\(\Big \downarrow \) 1152

\(\displaystyle -\frac {-\frac {\left (-4 a A c-8 a b B+5 A b^2\right ) \left (-\frac {\left (b^2-4 a c\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{2 a}-\frac {(5 A b-8 a B) \left (a+b x+c x^2\right )^{3/2}}{3 a x^3}}{8 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {-\frac {\left (-4 a A c-8 a b B+5 A b^2\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{4 a}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{2 a}-\frac {(5 A b-8 a B) \left (a+b x+c x^2\right )^{3/2}}{3 a x^3}}{8 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {\left (-4 a A c-8 a b B+5 A b^2\right ) \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2}}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{2 a}-\frac {(5 A b-8 a B) \left (a+b x+c x^2\right )^{3/2}}{3 a x^3}}{8 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}\)

Input:

Int[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^5,x]
 

Output:

-1/4*(A*(a + b*x + c*x^2)^(3/2))/(a*x^4) - (-1/3*((5*A*b - 8*a*B)*(a + b*x 
 + c*x^2)^(3/2))/(a*x^3) - ((5*A*b^2 - 8*a*b*B - 4*a*A*c)*(-1/4*((2*a + b* 
x)*Sqrt[a + b*x + c*x^2])/(a*x^2) + ((b^2 - 4*a*c)*ArcTanh[(2*a + b*x)/(2* 
Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(3/2))))/(2*a))/(8*a)
 

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 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 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 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 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 = 1.49 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.08

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-52 A a b c \,x^{3}+15 A \,b^{3} x^{3}+64 B \,a^{2} c \,x^{3}-24 B a \,b^{2} x^{3}+24 A \,a^{2} c \,x^{2}-10 A a \,b^{2} x^{2}+16 B \,a^{2} b \,x^{2}+8 A \,a^{2} b x +64 B \,a^{3} x +48 a^{3} A \right )}{192 x^{4} a^{3}}+\frac {\left (16 a^{2} A \,c^{2}-24 A a \,b^{2} c +5 A \,b^{4}+32 a^{2} b B c -8 B a \,b^{3}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{128 a^{\frac {7}{2}}}\)	\(185\)
default	\(A \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {5 b \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 a \,x^{3}}-\frac {b \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{2 a \,x^{2}}-\frac {b \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{a x}+\frac {b \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )}{2 a}+\frac {2 c \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{a}\right )}{4 a}+\frac {c \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )}{2 a}\right )}{2 a}\right )}{8 a}-\frac {c \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{2 a \,x^{2}}-\frac {b \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{a x}+\frac {b \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )}{2 a}+\frac {2 c \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{a}\right )}{4 a}+\frac {c \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )}{2 a}\right )}{4 a}\right )+B \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 a \,x^{3}}-\frac {b \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{2 a \,x^{2}}-\frac {b \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{a x}+\frac {b \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )}{2 a}+\frac {2 c \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{a}\right )}{4 a}+\frac {c \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )}{2 a}\right )}{2 a}\right )\)	\(951\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^5,x,method=_RETURNVERBOSE)
 

Output:

-1/192*(c*x^2+b*x+a)^(1/2)*(-52*A*a*b*c*x^3+15*A*b^3*x^3+64*B*a^2*c*x^3-24 
*B*a*b^2*x^3+24*A*a^2*c*x^2-10*A*a*b^2*x^2+16*B*a^2*b*x^2+8*A*a^2*b*x+64*B 
*a^3*x+48*A*a^3)/x^4/a^3+1/128*(16*A*a^2*c^2-24*A*a*b^2*c+5*A*b^4+32*B*a^2 
*b*c-8*B*a*b^3)/a^(7/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)
 

Fricas [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.47 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^5} \, dx=\left [-\frac {3 \, {\left (8 \, B a b^{3} - 5 \, A b^{4} - 16 \, A a^{2} c^{2} - 8 \, {\left (4 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} \sqrt {a} x^{4} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) + 4 \, {\left (48 \, A a^{4} - {\left (24 \, B a^{2} b^{2} - 15 \, A a b^{3} - 4 \, {\left (16 \, B a^{3} - 13 \, A a^{2} b\right )} c\right )} x^{3} + 2 \, {\left (8 \, B a^{3} b - 5 \, A a^{2} b^{2} + 12 \, A a^{3} c\right )} x^{2} + 8 \, {\left (8 \, B a^{4} + A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, a^{4} x^{4}}, \frac {3 \, {\left (8 \, B a b^{3} - 5 \, A b^{4} - 16 \, A a^{2} c^{2} - 8 \, {\left (4 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - 2 \, {\left (48 \, A a^{4} - {\left (24 \, B a^{2} b^{2} - 15 \, A a b^{3} - 4 \, {\left (16 \, B a^{3} - 13 \, A a^{2} b\right )} c\right )} x^{3} + 2 \, {\left (8 \, B a^{3} b - 5 \, A a^{2} b^{2} + 12 \, A a^{3} c\right )} x^{2} + 8 \, {\left (8 \, B a^{4} + A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, a^{4} x^{4}}\right ] \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^5,x, algorithm="fricas")
 

Output:

[-1/768*(3*(8*B*a*b^3 - 5*A*b^4 - 16*A*a^2*c^2 - 8*(4*B*a^2*b - 3*A*a*b^2) 
*c)*sqrt(a)*x^4*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*(48*A*a^4 - (24*B*a^2*b^2 - 15*A*a 
*b^3 - 4*(16*B*a^3 - 13*A*a^2*b)*c)*x^3 + 2*(8*B*a^3*b - 5*A*a^2*b^2 + 12* 
A*a^3*c)*x^2 + 8*(8*B*a^4 + A*a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^4*x^4), 
1/384*(3*(8*B*a*b^3 - 5*A*b^4 - 16*A*a^2*c^2 - 8*(4*B*a^2*b - 3*A*a*b^2)*c 
)*sqrt(-a)*x^4*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*(48*A*a^4 - (24*B*a^2*b^2 - 15*A*a*b^3 - 4*(16*B*a 
^3 - 13*A*a^2*b)*c)*x^3 + 2*(8*B*a^3*b - 5*A*a^2*b^2 + 12*A*a^3*c)*x^2 + 8 
*(8*B*a^4 + A*a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^4*x^4)]
 

Sympy [F]

\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^5} \, dx=\int \frac {\left (A + B x\right ) \sqrt {a + b x + c x^{2}}}{x^{5}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/x**5,x)
 

Output:

Integral((A + B*x)*sqrt(a + b*x + c*x**2)/x**5, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^5} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^5,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. 991 vs. \(2 (150) = 300\).

Time = 0.26 (sec) , antiderivative size = 991, normalized size of antiderivative = 5.76 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^5} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^5,x, algorithm="giac")
 

Output:

1/64*(8*B*a*b^3 - 5*A*b^4 - 32*B*a^2*b*c + 24*A*a*b^2*c - 16*A*a^2*c^2)*ar 
ctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^3) - 1/192 
*(24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a*b^3 - 15*(sqrt(c)*x - sqrt( 
c*x^2 + b*x + a))^7*A*b^4 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^2 
*b*c + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a*b^2*c - 48*(sqrt(c)*x 
- sqrt(c*x^2 + b*x + a))^7*A*a^2*c^2 - 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
 a))^6*B*a^3*c^(3/2) - 88*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^2*b^3 
+ 55*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*b^4 - 288*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^5*B*a^3*b*c - 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5 
*A*a^2*b^2*c - 336*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^3*c^2 - 384*( 
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^3*b^2*sqrt(c) + 384*(sqrt(c)*x - 
sqrt(c*x^2 + b*x + a))^4*B*a^4*c^(3/2) - 1152*(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))^4*A*a^3*b*c^(3/2) + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^3 
*b^3 - 73*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^2*b^4 + 96*(sqrt(c)*x 
- sqrt(c*x^2 + b*x + a))^3*B*a^4*b*c - 648*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
 a))^3*A*a^3*b^2*c - 336*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^4*c^2 + 
 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^4*b^2*sqrt(c) - 384*(sqrt(c 
)*x - sqrt(c*x^2 + b*x + a))^2*A*a^3*b^3*sqrt(c) - 128*(sqrt(c)*x - sqrt(c 
*x^2 + b*x + a))^2*B*a^5*c^(3/2) - 256*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) 
^2*A*a^4*b*c^(3/2) + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^4*b^3 -...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^5} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{x^5} \,d x \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^(1/2))/x^5,x)
 

Output:

int(((A + B*x)*(a + b*x + c*x^2)^(1/2))/x^5, x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.59 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^5} \, dx=\frac {-32 \sqrt {c \,x^{2}+b x +a}\, a^{4}-48 \sqrt {c \,x^{2}+b x +a}\, a^{3} b x -16 \sqrt {c \,x^{2}+b x +a}\, a^{3} c \,x^{2}-4 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} x^{2}-8 \sqrt {c \,x^{2}+b x +a}\, a^{2} b c \,x^{3}+6 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} x^{3}+16 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} c^{2} x^{4}+8 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,b^{2} c \,x^{4}-3 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{4} x^{4}-16 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} c^{2} x^{4}-8 \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{2} c \,x^{4}+3 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{4} x^{4}}{128 a^{3} x^{4}} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^5,x)
 

Output:

( - 32*sqrt(a + b*x + c*x**2)*a**4 - 48*sqrt(a + b*x + c*x**2)*a**3*b*x - 
16*sqrt(a + b*x + c*x**2)*a**3*c*x**2 - 4*sqrt(a + b*x + c*x**2)*a**2*b**2 
*x**2 - 8*sqrt(a + b*x + c*x**2)*a**2*b*c*x**3 + 6*sqrt(a + b*x + c*x**2)* 
a*b**3*x**3 + 16*sqrt(a)*log( - 2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b 
*x)*a**2*c**2*x**4 + 8*sqrt(a)*log( - 2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2 
*a - b*x)*a*b**2*c*x**4 - 3*sqrt(a)*log( - 2*sqrt(a)*sqrt(a + b*x + c*x**2 
) - 2*a - b*x)*b**4*x**4 - 16*sqrt(a)*log(x)*a**2*c**2*x**4 - 8*sqrt(a)*lo 
g(x)*a*b**2*c*x**4 + 3*sqrt(a)*log(x)*b**4*x**4)/(128*a**3*x**4)