\(\int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^6} \, dx\) [114]

Optimal result

Integrand size = 23, antiderivative size = 235 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^6} \, dx=\frac {\left (7 A b^3-10 a b^2 B-12 a A b c+8 a^2 B c\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{128 a^4 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}+\frac {(7 A b-10 a B) \left (a+b x+c x^2\right )^{3/2}}{40 a^2 x^4}-\frac {\left (35 A b^2-50 a b B-32 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{240 a^3 x^3}+\frac {\left (b^2-4 a c\right ) \left (2 a B \left (5 b^2-4 a c\right )-A \left (7 b^3-12 a b c\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{256 a^{9/2}} \] Output:

1/128*(-12*A*a*b*c+7*A*b^3+8*B*a^2*c-10*B*a*b^2)*(b*x+2*a)*(c*x^2+b*x+a)^( 
1/2)/a^4/x^2-1/5*A*(c*x^2+b*x+a)^(3/2)/a/x^5+1/40*(7*A*b-10*B*a)*(c*x^2+b* 
x+a)^(3/2)/a^2/x^4-1/240*(-32*A*a*c+35*A*b^2-50*B*a*b)*(c*x^2+b*x+a)^(3/2) 
/a^3/x^3+1/256*(-4*a*c+b^2)*(2*a*B*(-4*a*c+5*b^2)-A*(-12*a*b*c+7*b^3))*arc 
tanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(9/2)
 

Mathematica [A] (verified)

Time = 2.54 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.12 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^6} \, dx=\frac {\sqrt {a} \sqrt {a+x (b+c x)} \left (105 A b^4 x^4-96 a^4 (4 A+5 B x)-10 a b^2 x^3 (7 A b+15 b B x+46 A c x)-16 a^3 x (5 B x (b+3 c x)+A (3 b+8 c x))+4 a^2 x^2 \left (5 b B x (5 b+26 c x)+2 A \left (7 b^2+29 b c x+32 c^2 x^2\right )\right )\right )+15 \left (7 A b^5-32 a^3 B c^2\right ) x^5 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-30 a b \left (-5 b^3 B-20 A b^2 c+24 a b B c+24 a A c^2\right ) x^5 \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{1920 a^{9/2} x^5} \] Input:

Integrate[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^6,x]
 

Output:

(Sqrt[a]*Sqrt[a + x*(b + c*x)]*(105*A*b^4*x^4 - 96*a^4*(4*A + 5*B*x) - 10* 
a*b^2*x^3*(7*A*b + 15*b*B*x + 46*A*c*x) - 16*a^3*x*(5*B*x*(b + 3*c*x) + A* 
(3*b + 8*c*x)) + 4*a^2*x^2*(5*b*B*x*(5*b + 26*c*x) + 2*A*(7*b^2 + 29*b*c*x 
 + 32*c^2*x^2))) + 15*(7*A*b^5 - 32*a^3*B*c^2)*x^5*ArcTanh[(Sqrt[c]*x - Sq 
rt[a + x*(b + c*x)])/Sqrt[a]] - 30*a*b*(-5*b^3*B - 20*A*b^2*c + 24*a*b*B*c 
 + 24*a*A*c^2)*x^5*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]] 
)/(1920*a^(9/2)*x^5)
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {1237, 27, 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.

\(\displaystyle \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^6} \, dx\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {\int \frac {(7 A b-10 a B+4 A c x) \sqrt {c x^2+b x+a}}{2 x^5}dx}{5 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(7 A b-10 a B+4 A c x) \sqrt {c x^2+b x+a}}{x^5}dx}{10 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {-\frac {\int \frac {\left (35 A b^2-50 a B b-32 a A c+2 (7 A b-10 a B) c x\right ) \sqrt {c x^2+b x+a}}{2 x^4}dx}{4 a}-\frac {(7 A b-10 a B) \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}}{10 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {\left (35 A b^2-50 a B b-32 a A c+2 (7 A b-10 a B) c x\right ) \sqrt {c x^2+b x+a}}{x^4}dx}{8 a}-\frac {(7 A b-10 a B) \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}}{10 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}\)

\(\Big \downarrow \) 1228

\(\displaystyle -\frac {-\frac {-\frac {5 \left (8 a^2 B c-12 a A b c-10 a b^2 B+7 A b^3\right ) \int \frac {\sqrt {c x^2+b x+a}}{x^3}dx}{2 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-32 a A c-50 a b B+35 A b^2\right )}{3 a x^3}}{8 a}-\frac {(7 A b-10 a B) \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}}{10 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}\)

\(\Big \downarrow \) 1152

\(\displaystyle -\frac {-\frac {-\frac {5 \left (8 a^2 B c-12 a A b c-10 a b^2 B+7 A b^3\right ) \left (-\frac {\left (b^2-4 a c\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{2 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-32 a A c-50 a b B+35 A b^2\right )}{3 a x^3}}{8 a}-\frac {(7 A b-10 a B) \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}}{10 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {-\frac {-\frac {5 \left (8 a^2 B c-12 a A b c-10 a b^2 B+7 A b^3\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{4 a}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{2 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-32 a A c-50 a b B+35 A b^2\right )}{3 a x^3}}{8 a}-\frac {(7 A b-10 a B) \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}}{10 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {-\frac {5 \left (8 a^2 B c-12 a A b c-10 a b^2 B+7 A b^3\right ) \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2}}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{2 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-32 a A c-50 a b B+35 A b^2\right )}{3 a x^3}}{8 a}-\frac {(7 A b-10 a B) \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}}{10 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}\)

Input:

Int[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^6,x]
 

Output:

-1/5*(A*(a + b*x + c*x^2)^(3/2))/(a*x^5) - (-1/4*((7*A*b - 10*a*B)*(a + b* 
x + c*x^2)^(3/2))/(a*x^4) - (-1/3*((35*A*b^2 - 50*a*b*B - 32*a*A*c)*(a + b 
*x + c*x^2)^(3/2))/(a*x^3) - (5*(7*A*b^3 - 10*a*b^2*B - 12*a*A*b*c + 8*a^2 
*B*c)*(-1/4*((2*a + b*x)*Sqrt[a + b*x + c*x^2])/(a*x^2) + ((b^2 - 4*a*c)*A 
rcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(3/2))))/(2*a) 
)/(8*a))/(10*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 = 1.42 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.09

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-256 A \,a^{2} c^{2} x^{4}+460 A a \,b^{2} c \,x^{4}-105 A \,b^{4} x^{4}-520 B \,a^{2} b c \,x^{4}+150 B a \,b^{3} x^{4}-232 A \,a^{2} b c \,x^{3}+70 A a \,b^{3} x^{3}+240 B \,a^{3} c \,x^{3}-100 B \,a^{2} b^{2} x^{3}+128 A \,a^{3} c \,x^{2}-56 A \,a^{2} b^{2} x^{2}+80 B \,a^{3} b \,x^{2}+48 A \,a^{3} b x +480 a^{4} B x +384 a^{4} A \right )}{1920 x^{5} a^{4}}-\frac {\left (48 A \,a^{2} b \,c^{2}-40 A a \,b^{3} c +7 A \,b^{5}-32 B \,a^{3} c^{2}+48 B \,a^{2} b^{2} c -10 B a \,b^{4}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{256 a^{\frac {9}{2}}}\)	\(257\)
default	\(\text {Expression too large to display}\)	\(1616\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-1/1920*(c*x^2+b*x+a)^(1/2)*(-256*A*a^2*c^2*x^4+460*A*a*b^2*c*x^4-105*A*b^ 
4*x^4-520*B*a^2*b*c*x^4+150*B*a*b^3*x^4-232*A*a^2*b*c*x^3+70*A*a*b^3*x^3+2 
40*B*a^3*c*x^3-100*B*a^2*b^2*x^3+128*A*a^3*c*x^2-56*A*a^2*b^2*x^2+80*B*a^3 
*b*x^2+48*A*a^3*b*x+480*B*a^4*x+384*A*a^4)/x^5/a^4-1/256*(48*A*a^2*b*c^2-4 
0*A*a*b^3*c+7*A*b^5-32*B*a^3*c^2+48*B*a^2*b^2*c-10*B*a*b^4)/a^(9/2)*ln((2* 
a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)
 

Fricas [A] (verification not implemented)

Time = 0.93 (sec) , antiderivative size = 553, normalized size of antiderivative = 2.35 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^6} \, dx=\left [-\frac {15 \, {\left (10 \, B a b^{4} - 7 \, A b^{5} + 16 \, {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} c^{2} - 8 \, {\left (6 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} c\right )} \sqrt {a} x^{5} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) + 4 \, {\left (384 \, A a^{5} + {\left (150 \, B a^{2} b^{3} - 105 \, A a b^{4} - 256 \, A a^{3} c^{2} - 20 \, {\left (26 \, B a^{3} b - 23 \, A a^{2} b^{2}\right )} c\right )} x^{4} - 2 \, {\left (50 \, B a^{3} b^{2} - 35 \, A a^{2} b^{3} - 4 \, {\left (30 \, B a^{4} - 29 \, A a^{3} b\right )} c\right )} x^{3} + 8 \, {\left (10 \, B a^{4} b - 7 \, A a^{3} b^{2} + 16 \, A a^{4} c\right )} x^{2} + 48 \, {\left (10 \, B a^{5} + A a^{4} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, a^{5} x^{5}}, -\frac {15 \, {\left (10 \, B a b^{4} - 7 \, A b^{5} + 16 \, {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} c^{2} - 8 \, {\left (6 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} c\right )} \sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) + 2 \, {\left (384 \, A a^{5} + {\left (150 \, B a^{2} b^{3} - 105 \, A a b^{4} - 256 \, A a^{3} c^{2} - 20 \, {\left (26 \, B a^{3} b - 23 \, A a^{2} b^{2}\right )} c\right )} x^{4} - 2 \, {\left (50 \, B a^{3} b^{2} - 35 \, A a^{2} b^{3} - 4 \, {\left (30 \, B a^{4} - 29 \, A a^{3} b\right )} c\right )} x^{3} + 8 \, {\left (10 \, B a^{4} b - 7 \, A a^{3} b^{2} + 16 \, A a^{4} c\right )} x^{2} + 48 \, {\left (10 \, B a^{5} + A a^{4} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, a^{5} x^{5}}\right ] \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^6,x, algorithm="fricas")
 

Output:

[-1/7680*(15*(10*B*a*b^4 - 7*A*b^5 + 16*(2*B*a^3 - 3*A*a^2*b)*c^2 - 8*(6*B 
*a^2*b^2 - 5*A*a*b^3)*c)*sqrt(a)*x^5*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*(384*A*a^5 + 
(150*B*a^2*b^3 - 105*A*a*b^4 - 256*A*a^3*c^2 - 20*(26*B*a^3*b - 23*A*a^2*b 
^2)*c)*x^4 - 2*(50*B*a^3*b^2 - 35*A*a^2*b^3 - 4*(30*B*a^4 - 29*A*a^3*b)*c) 
*x^3 + 8*(10*B*a^4*b - 7*A*a^3*b^2 + 16*A*a^4*c)*x^2 + 48*(10*B*a^5 + A*a^ 
4*b)*x)*sqrt(c*x^2 + b*x + a))/(a^5*x^5), -1/3840*(15*(10*B*a*b^4 - 7*A*b^ 
5 + 16*(2*B*a^3 - 3*A*a^2*b)*c^2 - 8*(6*B*a^2*b^2 - 5*A*a*b^3)*c)*sqrt(-a) 
*x^5*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*(384*A*a^5 + (150*B*a^2*b^3 - 105*A*a*b^4 - 256*A*a^3*c^2 - 
20*(26*B*a^3*b - 23*A*a^2*b^2)*c)*x^4 - 2*(50*B*a^3*b^2 - 35*A*a^2*b^3 - 4 
*(30*B*a^4 - 29*A*a^3*b)*c)*x^3 + 8*(10*B*a^4*b - 7*A*a^3*b^2 + 16*A*a^4*c 
)*x^2 + 48*(10*B*a^5 + A*a^4*b)*x)*sqrt(c*x^2 + b*x + a))/(a^5*x^5)]
 

Sympy [F]

\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^6} \, dx=\int \frac {\left (A + B x\right ) \sqrt {a + b x + c x^{2}}}{x^{6}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/x**6,x)
 

Output:

Integral((A + B*x)*sqrt(a + b*x + c*x**2)/x**6, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^6,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. 1407 vs. \(2 (209) = 418\).

Time = 0.27 (sec) , antiderivative size = 1407, normalized size of antiderivative = 5.99 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^6} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^6,x, algorithm="giac")
 

Output:

-1/128*(10*B*a*b^4 - 7*A*b^5 - 48*B*a^2*b^2*c + 40*A*a*b^3*c + 32*B*a^3*c^ 
2 - 48*A*a^2*b*c^2)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/ 
(sqrt(-a)*a^4) + 1/1920*(150*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a*b^4 
 - 105*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*b^5 - 720*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^9*B*a^2*b^2*c + 600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) 
^9*A*a*b^3*c + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^3*c^2 - 720*( 
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^2*b*c^2 - 700*(sqrt(c)*x - sqrt(c 
*x^2 + b*x + a))^7*B*a^2*b^4 + 490*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A 
*a*b^5 + 3360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^3*b^2*c - 2800*(sq 
rt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^2*b^3*c + 2880*(sqrt(c)*x - sqrt(c* 
x^2 + b*x + a))^7*B*a^4*c^2 + 3360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A 
*a^3*b*c^2 + 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^4*b*c^(3/2) + 
 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^4*c^(5/2) + 1280*(sqrt(c)* 
x - sqrt(c*x^2 + b*x + a))^5*B*a^3*b^4 - 896*(sqrt(c)*x - sqrt(c*x^2 + b*x 
 + a))^5*A*a^2*b^5 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^4*b^2* 
c + 5120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^3*b^3*c + 15360*(sqrt(c 
)*x - sqrt(c*x^2 + b*x + a))^5*A*a^4*b*c^2 + 3840*(sqrt(c)*x - sqrt(c*x^2 
+ b*x + a))^4*B*a^4*b^3*sqrt(c) - 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) 
^4*B*a^5*b*c^(3/2) + 24320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^4*b^2 
*c^(3/2) + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^5*c^(5/2) - 5...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^6} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{x^6} \,d x \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^(1/2))/x^6,x)
 

Output:

int(((A + B*x)*(a + b*x + c*x^2)^(1/2))/x^6, x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.67 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^6} \, dx=\frac {-768 \sqrt {c \,x^{2}+b x +a}\, a^{5}-1056 \sqrt {c \,x^{2}+b x +a}\, a^{4} b x -256 \sqrt {c \,x^{2}+b x +a}\, a^{4} c \,x^{2}-48 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{2} x^{2}-16 \sqrt {c \,x^{2}+b x +a}\, a^{3} b c \,x^{3}+512 \sqrt {c \,x^{2}+b x +a}\, a^{3} c^{2} x^{4}+60 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{3} x^{3}+120 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} c \,x^{4}-90 \sqrt {c \,x^{2}+b x +a}\, a \,b^{4} x^{4}+240 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} b \,c^{2} x^{5}+120 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,b^{3} c \,x^{5}-45 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{5} x^{5}-240 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} b \,c^{2} x^{5}-120 \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{3} c \,x^{5}+45 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{5} x^{5}}{3840 a^{4} x^{5}} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^6,x)
 

Output:

( - 768*sqrt(a + b*x + c*x**2)*a**5 - 1056*sqrt(a + b*x + c*x**2)*a**4*b*x 
 - 256*sqrt(a + b*x + c*x**2)*a**4*c*x**2 - 48*sqrt(a + b*x + c*x**2)*a**3 
*b**2*x**2 - 16*sqrt(a + b*x + c*x**2)*a**3*b*c*x**3 + 512*sqrt(a + b*x + 
c*x**2)*a**3*c**2*x**4 + 60*sqrt(a + b*x + c*x**2)*a**2*b**3*x**3 + 120*sq 
rt(a + b*x + c*x**2)*a**2*b**2*c*x**4 - 90*sqrt(a + b*x + c*x**2)*a*b**4*x 
**4 + 240*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*b 
*c**2*x**5 + 120*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x) 
*a*b**3*c*x**5 - 45*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b 
*x)*b**5*x**5 - 240*sqrt(a)*log(x)*a**2*b*c**2*x**5 - 120*sqrt(a)*log(x)*a 
*b**3*c*x**5 + 45*sqrt(a)*log(x)*b**5*x**5)/(3840*a**4*x**5)