\(\int \frac {(A+B x) (a+b x+c x^2)^{5/2}}{x^8} \, dx\) [141]

Optimal result

Integrand size = 23, antiderivative size = 219 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^8} \, dx=\frac {5 (A b-2 a B) \left (b^2-4 a c\right )^2 (2 a+b x) \sqrt {a+b x+c x^2}}{1024 a^4 x^2}-\frac {5 (A b-2 a B) \left (b^2-4 a c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{384 a^3 x^4}+\frac {(A b-2 a B) (2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{24 a^2 x^6}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{7 a x^7}-\frac {5 (A b-2 a B) \left (b^2-4 a c\right )^3 \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2048 a^{9/2}} \] Output:

5/1024*(A*b-2*B*a)*(-4*a*c+b^2)^2*(b*x+2*a)*(c*x^2+b*x+a)^(1/2)/a^4/x^2-5/ 
384*(A*b-2*B*a)*(-4*a*c+b^2)*(b*x+2*a)*(c*x^2+b*x+a)^(3/2)/a^3/x^4+1/24*(A 
*b-2*B*a)*(b*x+2*a)*(c*x^2+b*x+a)^(5/2)/a^2/x^6-1/7*A*(c*x^2+b*x+a)^(7/2)/ 
a/x^7-5/2048*(A*b-2*B*a)*(-4*a*c+b^2)^3*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 = 5.63 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.84 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^8} \, dx=\frac {-\sqrt {a} \sqrt {a+x (b+c x)} \left (-105 A b^6 x^6+512 a^6 (6 A+7 B x)+128 a^5 x \left (58 A b+70 b B x+72 A c x+91 B c x^2\right )+70 a b^4 x^5 (3 b B x+A (b+16 c x))-28 a^2 b^2 x^4 \left (5 b B x (b+16 c x)+2 A \left (b^2+12 b c x+66 c^2 x^2\right )\right )+32 a^4 x^2 \left (21 B x \left (9 b^2+26 b c x+22 c^2 x^2\right )+2 A \left (74 b^2+197 b c x+144 c^2 x^2\right )\right )+16 a^3 x^3 \left (7 b B x \left (b^2+12 b c x+66 c^2 x^2\right )+3 A \left (b^3+10 b^2 c x+38 b c^2 x^2+64 c^3 x^3\right )\right )\right )+105 \left (A b^7+128 a^4 B c^3\right ) x^7 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+210 a b \left (b^5 B+6 A b^4 c-12 a b^3 B c-24 a A b^2 c^2+48 a^2 b B c^2+32 a^2 A c^3\right ) x^7 \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{21504 a^{9/2} x^7} \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^8,x]
 

Output:

(-(Sqrt[a]*Sqrt[a + x*(b + c*x)]*(-105*A*b^6*x^6 + 512*a^6*(6*A + 7*B*x) + 
 128*a^5*x*(58*A*b + 70*b*B*x + 72*A*c*x + 91*B*c*x^2) + 70*a*b^4*x^5*(3*b 
*B*x + A*(b + 16*c*x)) - 28*a^2*b^2*x^4*(5*b*B*x*(b + 16*c*x) + 2*A*(b^2 + 
 12*b*c*x + 66*c^2*x^2)) + 32*a^4*x^2*(21*B*x*(9*b^2 + 26*b*c*x + 22*c^2*x 
^2) + 2*A*(74*b^2 + 197*b*c*x + 144*c^2*x^2)) + 16*a^3*x^3*(7*b*B*x*(b^2 + 
 12*b*c*x + 66*c^2*x^2) + 3*A*(b^3 + 10*b^2*c*x + 38*b*c^2*x^2 + 64*c^3*x^ 
3)))) + 105*(A*b^7 + 128*a^4*B*c^3)*x^7*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b 
 + c*x)])/Sqrt[a]] + 210*a*b*(b^5*B + 6*A*b^4*c - 12*a*b^3*B*c - 24*a*A*b^ 
2*c^2 + 48*a^2*b*B*c^2 + 32*a^2*A*c^3)*x^7*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a 
+ x*(b + c*x)])/Sqrt[a]])/(21504*a^(9/2)*x^7)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1228, 1152, 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.

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1152

\(\displaystyle -\frac {(A b-2 a B) \left (-\frac {5 \left (b^2-4 a c\right ) \int \frac {\left (c x^2+b x+a\right )^{3/2}}{x^5}dx}{24 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{12 a x^6}\right )}{2 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{7 a x^7}\)

\(\Big \downarrow \) 1152

\(\displaystyle -\frac {(A b-2 a B) \left (-\frac {5 \left (b^2-4 a c\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \int \frac {\sqrt {c x^2+b x+a}}{x^3}dx}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{24 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{12 a x^6}\right )}{2 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{7 a x^7}\)

\(\Big \downarrow \) 1152

\(\displaystyle -\frac {(A b-2 a B) \left (-\frac {5 \left (b^2-4 a c\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \left (-\frac {\left (b^2-4 a c\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{24 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{12 a x^6}\right )}{2 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{7 a x^7}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {(A b-2 a B) \left (-\frac {5 \left (b^2-4 a c\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{4 a}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{24 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{12 a x^6}\right )}{2 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{7 a x^7}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {(A b-2 a B) \left (-\frac {5 \left (b^2-4 a c\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2}}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{24 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{12 a x^6}\right )}{2 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{7 a x^7}\)

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^8,x]
 

Output:

-1/7*(A*(a + b*x + c*x^2)^(7/2))/(a*x^7) - ((A*b - 2*a*B)*(-1/12*((2*a + b 
*x)*(a + b*x + c*x^2)^(5/2))/(a*x^6) - (5*(b^2 - 4*a*c)*(-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)*Sq 
rt[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))))/(16*a)))/(24*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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(445\) vs. \(2(193)=386\).

Time = 1.82 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.04

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (3072 A \,a^{3} c^{3} x^{6}-3696 A \,a^{2} b^{2} c^{2} x^{6}+1120 A a \,b^{4} c \,x^{6}-105 A \,b^{6} x^{6}+7392 B \,a^{3} b \,c^{2} x^{6}-2240 B \,a^{2} b^{3} c \,x^{6}+210 B a \,b^{5} x^{6}+1824 A \,a^{3} b \,c^{2} x^{5}-672 A \,a^{2} b^{3} c \,x^{5}+70 A a \,b^{5} x^{5}+14784 B \,a^{4} c^{2} x^{5}+1344 B \,a^{3} b^{2} c \,x^{5}-140 B \,a^{2} b^{4} x^{5}+9216 A \,a^{4} c^{2} x^{4}+480 A \,a^{3} b^{2} c \,x^{4}-56 A \,a^{2} b^{4} x^{4}+17472 B \,a^{4} b c \,x^{4}+112 B \,a^{3} b^{3} x^{4}+12608 A \,a^{4} b c \,x^{3}+48 A \,a^{3} b^{3} x^{3}+11648 B \,a^{5} c \,x^{3}+6048 B \,a^{4} b^{2} x^{3}+9216 A \,a^{5} c \,x^{2}+4736 A \,a^{4} b^{2} x^{2}+8960 B \,a^{5} b \,x^{2}+7424 A \,a^{5} b x +3584 B \,a^{6} x +3072 A \,a^{6}\right )}{21504 x^{7} a^{4}}+\frac {5 \left (64 A \,a^{3} b \,c^{3}-48 A \,a^{2} b^{3} c^{2}+12 A a \,b^{5} c -A \,b^{7}-128 B \,a^{4} c^{3}+96 B \,a^{3} b^{2} c^{2}-24 B \,a^{2} b^{4} c +2 B a \,b^{6}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2048 a^{\frac {9}{2}}}\)	\(446\)
default	\(\text {Expression too large to display}\)	\(11035\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^8,x,method=_RETURNVERBOSE)
 

Output:

-1/21504*(c*x^2+b*x+a)^(1/2)*(3072*A*a^3*c^3*x^6-3696*A*a^2*b^2*c^2*x^6+11 
20*A*a*b^4*c*x^6-105*A*b^6*x^6+7392*B*a^3*b*c^2*x^6-2240*B*a^2*b^3*c*x^6+2 
10*B*a*b^5*x^6+1824*A*a^3*b*c^2*x^5-672*A*a^2*b^3*c*x^5+70*A*a*b^5*x^5+147 
84*B*a^4*c^2*x^5+1344*B*a^3*b^2*c*x^5-140*B*a^2*b^4*x^5+9216*A*a^4*c^2*x^4 
+480*A*a^3*b^2*c*x^4-56*A*a^2*b^4*x^4+17472*B*a^4*b*c*x^4+112*B*a^3*b^3*x^ 
4+12608*A*a^4*b*c*x^3+48*A*a^3*b^3*x^3+11648*B*a^5*c*x^3+6048*B*a^4*b^2*x^ 
3+9216*A*a^5*c*x^2+4736*A*a^4*b^2*x^2+8960*B*a^5*b*x^2+7424*A*a^5*b*x+3584 
*B*a^6*x+3072*A*a^6)/x^7/a^4+5/2048*(64*A*a^3*b*c^3-48*A*a^2*b^3*c^2+12*A* 
a*b^5*c-A*b^7-128*B*a^4*c^3+96*B*a^3*b^2*c^2-24*B*a^2*b^4*c+2*B*a*b^6)/a^( 
9/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (197) = 394\).

Time = 2.20 (sec) , antiderivative size = 887, normalized size of antiderivative = 4.05 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^8} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^8,x, algorithm="fricas")
 

Output:

[1/86016*(105*(2*B*a*b^6 - A*b^7 - 64*(2*B*a^4 - A*a^3*b)*c^3 + 48*(2*B*a^ 
3*b^2 - A*a^2*b^3)*c^2 - 12*(2*B*a^2*b^4 - A*a*b^5)*c)*sqrt(a)*x^7*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*(3072*A*a^7 + (210*B*a^2*b^5 - 105*A*a*b^6 + 3072*A*a^4*c 
^3 + 3696*(2*B*a^4*b - A*a^3*b^2)*c^2 - 1120*(2*B*a^3*b^3 - A*a^2*b^4)*c)* 
x^6 - 2*(70*B*a^3*b^4 - 35*A*a^2*b^5 - 48*(154*B*a^5 + 19*A*a^4*b)*c^2 - 3 
36*(2*B*a^4*b^2 - A*a^3*b^3)*c)*x^5 + 8*(14*B*a^4*b^3 - 7*A*a^3*b^4 + 1152 
*A*a^5*c^2 + 12*(182*B*a^5*b + 5*A*a^4*b^2)*c)*x^4 + 16*(378*B*a^5*b^2 + 3 
*A*a^4*b^3 + 4*(182*B*a^6 + 197*A*a^5*b)*c)*x^3 + 128*(70*B*a^6*b + 37*A*a 
^5*b^2 + 72*A*a^6*c)*x^2 + 256*(14*B*a^7 + 29*A*a^6*b)*x)*sqrt(c*x^2 + b*x 
 + a))/(a^5*x^7), -1/43008*(105*(2*B*a*b^6 - A*b^7 - 64*(2*B*a^4 - A*a^3*b 
)*c^3 + 48*(2*B*a^3*b^2 - A*a^2*b^3)*c^2 - 12*(2*B*a^2*b^4 - A*a*b^5)*c)*s 
qrt(-a)*x^7*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*(3072*A*a^7 + (210*B*a^2*b^5 - 105*A*a*b^6 + 3072*A*a 
^4*c^3 + 3696*(2*B*a^4*b - A*a^3*b^2)*c^2 - 1120*(2*B*a^3*b^3 - A*a^2*b^4) 
*c)*x^6 - 2*(70*B*a^3*b^4 - 35*A*a^2*b^5 - 48*(154*B*a^5 + 19*A*a^4*b)*c^2 
 - 336*(2*B*a^4*b^2 - A*a^3*b^3)*c)*x^5 + 8*(14*B*a^4*b^3 - 7*A*a^3*b^4 + 
1152*A*a^5*c^2 + 12*(182*B*a^5*b + 5*A*a^4*b^2)*c)*x^4 + 16*(378*B*a^5*b^2 
 + 3*A*a^4*b^3 + 4*(182*B*a^6 + 197*A*a^5*b)*c)*x^3 + 128*(70*B*a^6*b + 37 
*A*a^5*b^2 + 72*A*a^6*c)*x^2 + 256*(14*B*a^7 + 29*A*a^6*b)*x)*sqrt(c*x^...
 

Sympy [F]

\[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^8} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{x^{8}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**8,x)
 

Output:

Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**8, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^8} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^8,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. 2598 vs. \(2 (197) = 394\).

Time = 0.29 (sec) , antiderivative size = 2598, normalized size of antiderivative = 11.86 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^8} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^8,x, algorithm="giac")
 

Output:

-5/1024*(2*B*a*b^6 - A*b^7 - 24*B*a^2*b^4*c + 12*A*a*b^5*c + 96*B*a^3*b^2* 
c^2 - 48*A*a^2*b^3*c^2 - 128*B*a^4*c^3 + 64*A*a^3*b*c^3)*arctan(-(sqrt(c)* 
x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^4) + 1/21504*(210*(sqrt(c 
)*x - sqrt(c*x^2 + b*x + a))^13*B*a*b^6 - 105*(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))^13*A*b^7 - 2520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*B*a^2*b^4*c 
 + 1260*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*a*b^5*c + 10080*(sqrt(c)* 
x - sqrt(c*x^2 + b*x + a))^13*B*a^3*b^2*c^2 - 5040*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))^13*A*a^2*b^3*c^2 + 29568*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 
13*B*a^4*c^3 + 6720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*a^3*b*c^3 + 1 
29024*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*B*a^4*b*c^(5/2) + 43008*(sqrt 
(c)*x - sqrt(c*x^2 + b*x + a))^12*A*a^4*c^(7/2) - 1400*(sqrt(c)*x - sqrt(c 
*x^2 + b*x + a))^11*B*a^2*b^6 + 700*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11 
*A*a*b^7 + 16800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*B*a^3*b^4*c - 8400 
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^2*b^5*c + 147840*(sqrt(c)*x - 
sqrt(c*x^2 + b*x + a))^11*B*a^4*b^2*c^2 + 33600*(sqrt(c)*x - sqrt(c*x^2 + 
b*x + a))^11*A*a^3*b^3*c^2 - 25088*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11* 
B*a^5*c^3 + 141568*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^4*b*c^3 + 21 
5040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*B*a^4*b^3*c^(3/2) - 129024*(sq 
rt(c)*x - sqrt(c*x^2 + b*x + a))^10*B*a^5*b*c^(5/2) + 387072*(sqrt(c)*x - 
sqrt(c*x^2 + b*x + a))^10*A*a^4*b^2*c^(5/2) + 3962*(sqrt(c)*x - sqrt(c*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^8} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{x^8} \,d x \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^8,x)
 

Output:

int(((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^8, x)
 

Reduce [B] (verification not implemented)

Time = 6.77 (sec) , antiderivative size = 561, normalized size of antiderivative = 2.56 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^8} \, dx=\frac {-6144 \sqrt {c \,x^{2}+b x +a}\, a^{7}-22016 \sqrt {c \,x^{2}+b x +a}\, a^{6} b x -18432 \sqrt {c \,x^{2}+b x +a}\, a^{6} c \,x^{2}-27392 \sqrt {c \,x^{2}+b x +a}\, a^{5} b^{2} x^{2}-48512 \sqrt {c \,x^{2}+b x +a}\, a^{5} b c \,x^{3}-18432 \sqrt {c \,x^{2}+b x +a}\, a^{5} c^{2} x^{4}-12192 \sqrt {c \,x^{2}+b x +a}\, a^{4} b^{3} x^{3}-35904 \sqrt {c \,x^{2}+b x +a}\, a^{4} b^{2} c \,x^{4}-33216 \sqrt {c \,x^{2}+b x +a}\, a^{4} b \,c^{2} x^{5}-6144 \sqrt {c \,x^{2}+b x +a}\, a^{4} c^{3} x^{6}-112 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{4} x^{4}-1344 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{3} c \,x^{5}-7392 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{2} c^{2} x^{6}+140 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{5} x^{5}+2240 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{4} c \,x^{6}-210 \sqrt {c \,x^{2}+b x +a}\, a \,b^{6} x^{6}+6720 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{3} b \,c^{3} x^{7}-5040 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} b^{3} c^{2} x^{7}+1260 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,b^{5} c \,x^{7}-105 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{7} x^{7}-6720 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{3} b \,c^{3} x^{7}+5040 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} b^{3} c^{2} x^{7}-1260 \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{5} c \,x^{7}+105 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{7} x^{7}}{43008 a^{4} x^{7}} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^8,x)
 

Output:

( - 6144*sqrt(a + b*x + c*x**2)*a**7 - 22016*sqrt(a + b*x + c*x**2)*a**6*b 
*x - 18432*sqrt(a + b*x + c*x**2)*a**6*c*x**2 - 27392*sqrt(a + b*x + c*x** 
2)*a**5*b**2*x**2 - 48512*sqrt(a + b*x + c*x**2)*a**5*b*c*x**3 - 18432*sqr 
t(a + b*x + c*x**2)*a**5*c**2*x**4 - 12192*sqrt(a + b*x + c*x**2)*a**4*b** 
3*x**3 - 35904*sqrt(a + b*x + c*x**2)*a**4*b**2*c*x**4 - 33216*sqrt(a + b* 
x + c*x**2)*a**4*b*c**2*x**5 - 6144*sqrt(a + b*x + c*x**2)*a**4*c**3*x**6 
- 112*sqrt(a + b*x + c*x**2)*a**3*b**4*x**4 - 1344*sqrt(a + b*x + c*x**2)* 
a**3*b**3*c*x**5 - 7392*sqrt(a + b*x + c*x**2)*a**3*b**2*c**2*x**6 + 140*s 
qrt(a + b*x + c*x**2)*a**2*b**5*x**5 + 2240*sqrt(a + b*x + c*x**2)*a**2*b* 
*4*c*x**6 - 210*sqrt(a + b*x + c*x**2)*a*b**6*x**6 + 6720*sqrt(a)*log(2*sq 
rt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**3*b*c**3*x**7 - 5040*sqrt(a)* 
log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*b**3*c**2*x**7 + 12 
60*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b**5*c*x**7 
 - 105*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*b**7*x**7 
 - 6720*sqrt(a)*log(x)*a**3*b*c**3*x**7 + 5040*sqrt(a)*log(x)*a**2*b**3*c* 
*2*x**7 - 1260*sqrt(a)*log(x)*a*b**5*c*x**7 + 105*sqrt(a)*log(x)*b**7*x**7 
)/(43008*a**4*x**7)