\(\int \frac {(A+B x) (a+b x+c x^2)^{3/2}}{x^6} \, dx\) [126]

Optimal result

Integrand size = 23, antiderivative size = 170 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx=-\frac {3 (A b-2 a B) \left (b^2-4 a c\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{128 a^3 x^2}+\frac {(A b-2 a B) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{16 a^2 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{5 a x^5}+\frac {3 (A b-2 a B) \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{256 a^{7/2}} \] Output:

-3/128*(A*b-2*B*a)*(-4*a*c+b^2)*(b*x+2*a)*(c*x^2+b*x+a)^(1/2)/a^3/x^2+1/16 
*(A*b-2*B*a)*(b*x+2*a)*(c*x^2+b*x+a)^(3/2)/a^2/x^4-1/5*A*(c*x^2+b*x+a)^(5/ 
2)/a/x^5+3/256*(A*b-2*B*a)*(-4*a*c+b^2)^2*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 = 2.78 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx=\frac {-\sqrt {a} \sqrt {a+x (b+c x)} \left (15 A b^4 x^4+32 a^4 (4 A+5 B x)-10 a b^2 x^3 (3 b B x+A (b+10 c x))+16 a^3 x (5 B x (3 b+5 c x)+A (11 b+16 c x))+4 a^2 x^2 \left (5 b B x (b+10 c x)+2 A \left (b^2+7 b c x+16 c^2 x^2\right )\right )\right )-15 \left (A b^5-32 a^3 B c^2\right ) x^5 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+30 a b \left (-b^3 B-4 A b^2 c+8 a b B c+8 a A c^2\right ) x^5 \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{640 a^{7/2} x^5} \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^6,x]
 

Output:

(-(Sqrt[a]*Sqrt[a + x*(b + c*x)]*(15*A*b^4*x^4 + 32*a^4*(4*A + 5*B*x) - 10 
*a*b^2*x^3*(3*b*B*x + A*(b + 10*c*x)) + 16*a^3*x*(5*B*x*(3*b + 5*c*x) + A* 
(11*b + 16*c*x)) + 4*a^2*x^2*(5*b*B*x*(b + 10*c*x) + 2*A*(b^2 + 7*b*c*x + 
16*c^2*x^2)))) - 15*(A*b^5 - 32*a^3*B*c^2)*x^5*ArcTanh[(Sqrt[c]*x - Sqrt[a 
 + x*(b + c*x)])/Sqrt[a]] + 30*a*b*(-(b^3*B) - 4*A*b^2*c + 8*a*b*B*c + 8*a 
*A*c^2)*x^5*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(640* 
a^(7/2)*x^5)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1228, 1152, 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.

\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx\)

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1152

\(\displaystyle -\frac {(A b-2 a B) \left (-\frac {3 \left (b^2-4 a c\right ) \int \frac {\sqrt {c x^2+b x+a}}{x^3}dx}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{2 a}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{5 a x^5}\)

\(\Big \downarrow \) 1152

\(\displaystyle -\frac {(A b-2 a B) \left (-\frac {3 \left (b^2-4 a c\right ) \left (-\frac {\left (b^2-4 a c\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{2 a}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{5 a x^5}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {(A b-2 a B) \left (-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{4 a}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{2 a}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{5 a x^5}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {(A b-2 a B) \left (-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2}}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{2 a}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{5 a x^5}\)

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^6,x]
 

Output:

-1/5*(A*(a + b*x + c*x^2)^(5/2))/(a*x^5) - ((A*b - 2*a*B)*(-1/8*((2*a + b* 
x)*(a + b*x + c*x^2)^(3/2))/(a*x^4) - (3*(b^2 - 4*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*Sqr 
t[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(3/2))))/(16*a)))/(2*a)
 

Defintions of rubi rules used

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]
 

Maple [A] (verified)

Time = 1.48 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.51

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (128 A \,a^{2} c^{2} x^{4}-100 A a \,b^{2} c \,x^{4}+15 A \,b^{4} x^{4}+200 B \,a^{2} b c \,x^{4}-30 B a \,b^{3} x^{4}+56 A \,a^{2} b c \,x^{3}-10 A a \,b^{3} x^{3}+400 B \,a^{3} c \,x^{3}+20 B \,a^{2} b^{2} x^{3}+256 A \,a^{3} c \,x^{2}+8 A \,a^{2} b^{2} x^{2}+240 B \,a^{3} b \,x^{2}+176 A \,a^{3} b x +160 a^{4} B x +128 a^{4} A \right )}{640 x^{5} a^{3}}+\frac {3 \left (16 A \,a^{2} b \,c^{2}-8 A a \,b^{3} c +A \,b^{5}-32 B \,a^{3} c^{2}+16 B \,a^{2} b^{2} c -2 B a \,b^{4}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{256 a^{\frac {7}{2}}}\)	\(256\)
default	\(\text {Expression too large to display}\)	\(2733\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-1/640*(c*x^2+b*x+a)^(1/2)*(128*A*a^2*c^2*x^4-100*A*a*b^2*c*x^4+15*A*b^4*x 
^4+200*B*a^2*b*c*x^4-30*B*a*b^3*x^4+56*A*a^2*b*c*x^3-10*A*a*b^3*x^3+400*B* 
a^3*c*x^3+20*B*a^2*b^2*x^3+256*A*a^3*c*x^2+8*A*a^2*b^2*x^2+240*B*a^3*b*x^2 
+176*A*a^3*b*x+160*B*a^4*x+128*A*a^4)/x^5/a^3+3/256*(16*A*a^2*b*c^2-8*A*a* 
b^3*c+A*b^5-32*B*a^3*c^2+16*B*a^2*b^2*c-2*B*a*b^4)/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 = 0.92 (sec) , antiderivative size = 555, normalized size of antiderivative = 3.26 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx=\left [-\frac {15 \, {\left (2 \, B a b^{4} - A b^{5} + 16 \, {\left (2 \, B a^{3} - A a^{2} b\right )} c^{2} - 8 \, {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} c\right )} \sqrt {a} x^{5} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) + 4 \, {\left (128 \, A a^{5} - {\left (30 \, B a^{2} b^{3} - 15 \, A a b^{4} - 128 \, A a^{3} c^{2} - 100 \, {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} c\right )} x^{4} + 2 \, {\left (10 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3} + 4 \, {\left (50 \, B a^{4} + 7 \, A a^{3} b\right )} c\right )} x^{3} + 8 \, {\left (30 \, B a^{4} b + A a^{3} b^{2} + 32 \, A a^{4} c\right )} x^{2} + 16 \, {\left (10 \, B a^{5} + 11 \, A a^{4} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2560 \, a^{4} x^{5}}, \frac {15 \, {\left (2 \, B a b^{4} - A b^{5} + 16 \, {\left (2 \, B a^{3} - A a^{2} b\right )} c^{2} - 8 \, {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} c\right )} \sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - 2 \, {\left (128 \, A a^{5} - {\left (30 \, B a^{2} b^{3} - 15 \, A a b^{4} - 128 \, A a^{3} c^{2} - 100 \, {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} c\right )} x^{4} + 2 \, {\left (10 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3} + 4 \, {\left (50 \, B a^{4} + 7 \, A a^{3} b\right )} c\right )} x^{3} + 8 \, {\left (30 \, B a^{4} b + A a^{3} b^{2} + 32 \, A a^{4} c\right )} x^{2} + 16 \, {\left (10 \, B a^{5} + 11 \, A a^{4} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1280 \, a^{4} x^{5}}\right ] \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^6,x, algorithm="fricas")
 

Output:

[-1/2560*(15*(2*B*a*b^4 - A*b^5 + 16*(2*B*a^3 - A*a^2*b)*c^2 - 8*(2*B*a^2* 
b^2 - 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*(128*A*a^5 - (30*B*a 
^2*b^3 - 15*A*a*b^4 - 128*A*a^3*c^2 - 100*(2*B*a^3*b - A*a^2*b^2)*c)*x^4 + 
 2*(10*B*a^3*b^2 - 5*A*a^2*b^3 + 4*(50*B*a^4 + 7*A*a^3*b)*c)*x^3 + 8*(30*B 
*a^4*b + A*a^3*b^2 + 32*A*a^4*c)*x^2 + 16*(10*B*a^5 + 11*A*a^4*b)*x)*sqrt( 
c*x^2 + b*x + a))/(a^4*x^5), 1/1280*(15*(2*B*a*b^4 - A*b^5 + 16*(2*B*a^3 - 
 A*a^2*b)*c^2 - 8*(2*B*a^2*b^2 - 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*(128*A* 
a^5 - (30*B*a^2*b^3 - 15*A*a*b^4 - 128*A*a^3*c^2 - 100*(2*B*a^3*b - A*a^2* 
b^2)*c)*x^4 + 2*(10*B*a^3*b^2 - 5*A*a^2*b^3 + 4*(50*B*a^4 + 7*A*a^3*b)*c)* 
x^3 + 8*(30*B*a^4*b + A*a^3*b^2 + 32*A*a^4*c)*x^2 + 16*(10*B*a^5 + 11*A*a^ 
4*b)*x)*sqrt(c*x^2 + b*x + a))/(a^4*x^5)]
 

Sympy [F]

\[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x^{6}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/x**6,x)
 

Output:

Integral((A + B*x)*(a + b*x + c*x**2)**(3/2)/x**6, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/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. 1357 vs. \(2 (151) = 302\).

Time = 0.25 (sec) , antiderivative size = 1357, normalized size of antiderivative = 7.98 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^6,x, algorithm="giac")
 

Output:

3/128*(2*B*a*b^4 - A*b^5 - 16*B*a^2*b^2*c + 8*A*a*b^3*c + 32*B*a^3*c^2 - 1 
6*A*a^2*b*c^2)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt 
(-a)*a^3) - 1/640*(30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a*b^4 - 15*( 
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*b^5 - 240*(sqrt(c)*x - sqrt(c*x^2 + 
 b*x + a))^9*B*a^2*b^2*c + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a*b 
^3*c - 800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^3*c^2 - 240*(sqrt(c)* 
x - sqrt(c*x^2 + b*x + a))^9*A*a^2*b*c^2 - 2560*(sqrt(c)*x - sqrt(c*x^2 + 
b*x + a))^8*B*a^3*b*c^(3/2) - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*A 
*a^3*c^(5/2) - 140*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^2*b^4 + 70*(s 
qrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a*b^5 - 1440*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))^7*B*a^3*b^2*c - 560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a 
^2*b^3*c + 320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^4*c^2 - 2720*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^3*b*c^2 - 1280*(sqrt(c)*x - sqrt(c*x 
^2 + b*x + a))^6*B*a^3*b^3*sqrt(c) + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
a))^6*B*a^4*b*c^(3/2) - 5120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^3*b 
^2*c^(3/2) - 128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^2*b^5 - 2560*(s 
qrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^3*b^3*c - 3840*(sqrt(c)*x - sqrt(c 
*x^2 + b*x + a))^5*A*a^4*b*c^2 + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 
4*B*a^4*b^3*sqrt(c) - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^3*b^4 
*sqrt(c) - 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^5*b*c^(3/2) -...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^6} \,d x \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^6,x)
 

Output:

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^6, x)
 

Reduce [B] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.02 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx=\frac {-256 \sqrt {c \,x^{2}+b x +a}\, a^{5}-672 \sqrt {c \,x^{2}+b x +a}\, a^{4} b x -512 \sqrt {c \,x^{2}+b x +a}\, a^{4} c \,x^{2}-496 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{2} x^{2}-912 \sqrt {c \,x^{2}+b x +a}\, a^{3} b c \,x^{3}-256 \sqrt {c \,x^{2}+b x +a}\, a^{3} c^{2} x^{4}-20 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{3} x^{3}-200 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} c \,x^{4}+30 \sqrt {c \,x^{2}+b x +a}\, a \,b^{4} x^{4}+240 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} b \,c^{2} x^{5}-120 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,b^{3} c \,x^{5}+15 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{5} x^{5}-240 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} b \,c^{2} x^{5}+120 \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{3} c \,x^{5}-15 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{5} x^{5}}{1280 a^{3} x^{5}} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^6,x)
 

Output:

( - 256*sqrt(a + b*x + c*x**2)*a**5 - 672*sqrt(a + b*x + c*x**2)*a**4*b*x 
- 512*sqrt(a + b*x + c*x**2)*a**4*c*x**2 - 496*sqrt(a + b*x + c*x**2)*a**3 
*b**2*x**2 - 912*sqrt(a + b*x + c*x**2)*a**3*b*c*x**3 - 256*sqrt(a + b*x + 
 c*x**2)*a**3*c**2*x**4 - 20*sqrt(a + b*x + c*x**2)*a**2*b**3*x**3 - 200*s 
qrt(a + b*x + c*x**2)*a**2*b**2*c*x**4 + 30*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 + 15*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 - 15*sqrt(a)*log(x)*b**5*x**5)/(1280*a**3*x**5)