\(\int \frac {(A+B x) (a+b x+c x^2)^{5/2}}{x^{10}} \, dx\) [143]

Optimal result

Integrand size = 23, antiderivative size = 375 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {5 \left (b^2-4 a c\right )^2 \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{32768 a^6 x^2}+\frac {5 \left (b^2-4 a c\right ) \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{12288 a^5 x^4}+\frac {\left (11 A b^3-18 a b^2 B-12 a A b c+8 a^2 B c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{768 a^4 x^6}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}+\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{144 a^2 x^8}-\frac {\left (99 A b^2-162 a b B-64 a A c\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 a^3 x^7}+\frac {5 \left (b^2-4 a c\right )^3 \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{65536 a^{13/2}} \] Output:

-5/32768*(-4*a*c+b^2)^2*(2*a*B*(-4*a*c+9*b^2)-A*(-12*a*b*c+11*b^3))*(b*x+2 
*a)*(c*x^2+b*x+a)^(1/2)/a^6/x^2+5/12288*(-4*a*c+b^2)*(2*a*B*(-4*a*c+9*b^2) 
-A*(-12*a*b*c+11*b^3))*(b*x+2*a)*(c*x^2+b*x+a)^(3/2)/a^5/x^4+1/768*(-12*A* 
a*b*c+11*A*b^3+8*B*a^2*c-18*B*a*b^2)*(b*x+2*a)*(c*x^2+b*x+a)^(5/2)/a^4/x^6 
-1/9*A*(c*x^2+b*x+a)^(7/2)/a/x^9+1/144*(11*A*b-18*B*a)*(c*x^2+b*x+a)^(7/2) 
/a^2/x^8-1/2016*(-64*A*a*c+99*A*b^2-162*B*a*b)*(c*x^2+b*x+a)^(7/2)/a^3/x^7 
+5/65536*(-4*a*c+b^2)^3*(2*a*B*(-4*a*c+9*b^2)-A*(-12*a*b*c+11*b^3))*arctan 
h(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(13/2)
 

Mathematica [A] (verified)

Time = 10.97 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.78 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^{10}} \, dx=\frac {-\frac {A (a+x (b+c x))^{7/2}}{x^9}+\frac {(11 A b-18 a B) (a+x (b+c x))^{7/2}}{16 a x^8}+\frac {\left (-99 A b^2+162 a b B+64 a A c\right ) (a+x (b+c x))^{7/2}}{224 a^2 x^7}+\frac {3 \left (2 a B \left (-9 b^2+4 a c\right )+A \left (11 b^3-12 a b c\right )\right ) \left (256 a^{5/2} (2 a+b x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) x^2 \left (16 a^{3/2} (2 a+b x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) x^2 \left (2 \sqrt {a} (2 a+b x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) x^2 \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )\right )\right )\right )}{65536 a^{11/2} x^6}}{9 a} \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^10,x]
 

Output:

(-((A*(a + x*(b + c*x))^(7/2))/x^9) + ((11*A*b - 18*a*B)*(a + x*(b + c*x)) 
^(7/2))/(16*a*x^8) + ((-99*A*b^2 + 162*a*b*B + 64*a*A*c)*(a + x*(b + c*x)) 
^(7/2))/(224*a^2*x^7) + (3*(2*a*B*(-9*b^2 + 4*a*c) + A*(11*b^3 - 12*a*b*c) 
)*(256*a^(5/2)*(2*a + b*x)*(a + x*(b + c*x))^(5/2) - 5*(b^2 - 4*a*c)*x^2*( 
16*a^(3/2)*(2*a + b*x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4*a*c)*x^2*(2*Sq 
rt[a]*(2*a + b*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*x^2*ArcTanh[(2*a + 
 b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])]))))/(65536*a^(11/2)*x^6))/(9*a)
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.86, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1237, 27, 1237, 27, 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.

\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^{10}} \, dx\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {\int \frac {(11 A b-18 a B+4 A c x) \left (c x^2+b x+a\right )^{5/2}}{2 x^9}dx}{9 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(11 A b-18 a B+4 A c x) \left (c x^2+b x+a\right )^{5/2}}{x^9}dx}{18 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {-\frac {\int \frac {\left (99 A b^2-162 a B b-64 a A c+2 (11 A b-18 a B) c x\right ) \left (c x^2+b x+a\right )^{5/2}}{2 x^8}dx}{8 a}-\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}}{18 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {\left (99 A b^2-162 a B b-64 a A c+2 (11 A b-18 a B) c x\right ) \left (c x^2+b x+a\right )^{5/2}}{x^8}dx}{16 a}-\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}}{18 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}\)

\(\Big \downarrow \) 1228

\(\displaystyle -\frac {-\frac {-\frac {9 \left (8 a^2 B c-12 a A b c-18 a b^2 B+11 A b^3\right ) \int \frac {\left (c x^2+b x+a\right )^{5/2}}{x^7}dx}{2 a}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a A c-162 a b B+99 A b^2\right )}{7 a x^7}}{16 a}-\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}}{18 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}\)

\(\Big \downarrow \) 1152

\(\displaystyle -\frac {-\frac {-\frac {9 \left (8 a^2 B c-12 a A b c-18 a b^2 B+11 A b^3\right ) \left (-\frac {5 \left (b^2-4 a c\right ) \int \frac {\left (c x^2+b x+a\right )^{3/2}}{x^5}dx}{24 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{12 a x^6}\right )}{2 a}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a A c-162 a b B+99 A b^2\right )}{7 a x^7}}{16 a}-\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}}{18 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}\)

\(\Big \downarrow \) 1152

\(\displaystyle -\frac {-\frac {-\frac {9 \left (8 a^2 B c-12 a A b c-18 a b^2 B+11 A b^3\right ) \left (-\frac {5 \left (b^2-4 a c\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \int \frac {\sqrt {c x^2+b x+a}}{x^3}dx}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{24 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{12 a x^6}\right )}{2 a}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a A c-162 a b B+99 A b^2\right )}{7 a x^7}}{16 a}-\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}}{18 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}\)

\(\Big \downarrow \) 1152

\(\displaystyle -\frac {-\frac {-\frac {9 \left (8 a^2 B c-12 a A b c-18 a b^2 B+11 A b^3\right ) \left (-\frac {5 \left (b^2-4 a c\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \left (-\frac {\left (b^2-4 a c\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{24 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{12 a x^6}\right )}{2 a}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a A c-162 a b B+99 A b^2\right )}{7 a x^7}}{16 a}-\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}}{18 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {-\frac {-\frac {9 \left (8 a^2 B c-12 a A b c-18 a b^2 B+11 A b^3\right ) \left (-\frac {5 \left (b^2-4 a c\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{4 a}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{24 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{12 a x^6}\right )}{2 a}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a A c-162 a b B+99 A b^2\right )}{7 a x^7}}{16 a}-\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}}{18 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {-\frac {9 \left (8 a^2 B c-12 a A b c-18 a b^2 B+11 A b^3\right ) \left (-\frac {5 \left (b^2-4 a c\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2}}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{24 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{12 a x^6}\right )}{2 a}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a A c-162 a b B+99 A b^2\right )}{7 a x^7}}{16 a}-\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}}{18 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}\)

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^10,x]
 

Output:

-1/9*(A*(a + b*x + c*x^2)^(7/2))/(a*x^9) - (-1/8*((11*A*b - 18*a*B)*(a + b 
*x + c*x^2)^(7/2))/(a*x^8) - (-1/7*((99*A*b^2 - 162*a*b*B - 64*a*A*c)*(a + 
 b*x + c*x^2)^(7/2))/(a*x^7) - (9*(11*A*b^3 - 18*a*b^2*B - 12*a*A*b*c + 8* 
a^2*B*c)*(-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)*Sqrt[a + b*x + c*x^2])/(a*x^2) + ((b^2 - 4*a*c)*Arc 
Tanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(3/2))))/(16*a)) 
)/(24*a)))/(2*a))/(16*a))/(18*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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(694\) vs. \(2(341)=682\).

Time = 2.04 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.85

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-65536 A \,a^{4} c^{4} x^{8}+234432 A \,a^{3} b^{2} c^{3} x^{8}-162288 A \,a^{2} b^{4} c^{2} x^{8}+40740 A a \,b^{6} c \,x^{8}-3465 A \,b^{8} x^{8}-254592 B \,a^{4} b \,c^{3} x^{8}+226464 B \,a^{3} b^{3} c^{2} x^{8}-63000 B \,a^{2} b^{5} c \,x^{8}+5670 B a \,b^{7} x^{8}-88192 A \,a^{4} b \,c^{3} x^{7}+84384 A \,a^{3} b^{3} c^{2} x^{7}-24696 A \,a^{2} b^{5} c \,x^{7}+2310 A a \,b^{7} x^{7}+80640 B \,a^{5} c^{3} x^{7}-114624 B \,a^{4} b^{2} c^{2} x^{7}+37968 B \,a^{3} b^{4} c \,x^{7}-3780 B \,a^{2} b^{6} x^{7}+32768 A \,a^{5} c^{3} x^{6}-51072 A \,a^{4} b^{2} c^{2} x^{6}+17856 A \,a^{3} b^{4} c \,x^{6}-1848 A \,a^{2} b^{6} x^{6}+66816 B \,a^{5} b \,c^{2} x^{6}-27264 B \,a^{4} b^{3} c \,x^{6}+3024 B \,a^{3} b^{5} x^{6}+31488 A \,a^{5} b \,c^{2} x^{5}-13696 A \,a^{4} b^{3} c \,x^{5}+1584 A \,a^{3} b^{5} x^{5}+634368 B \,a^{6} c^{2} x^{5}+20736 B \,a^{5} b^{2} c \,x^{5}-2592 B \,a^{4} b^{4} x^{5}+491520 A \,a^{6} c^{2} x^{4}+10752 A \,a^{5} b^{2} c \,x^{4}-1408 A \,a^{4} b^{4} x^{4}+943104 B \,a^{6} b c \,x^{4}+2304 B \,a^{5} b^{3} x^{4}+771072 A \,a^{6} b c \,x^{3}+1280 A \,a^{5} b^{3} x^{3}+731136 B \,a^{7} c \,x^{3}+373248 B \,a^{6} b^{2} x^{3}+622592 A \,a^{7} c \,x^{2}+316416 A \,a^{6} b^{2} x^{2}+608256 B \,a^{7} b \,x^{2}+530432 A \,a^{7} b x +258048 B \,a^{8} x +229376 A \,a^{8}\right )}{2064384 x^{9} a^{6}}-\frac {5 \left (768 A \,a^{4} b \,c^{4}-1280 A \,a^{3} b^{3} c^{3}+672 A \,a^{2} b^{5} c^{2}-144 A a \,b^{7} c +11 A \,b^{9}-512 B \,a^{5} c^{4}+1536 B \,a^{4} b^{2} c^{3}-960 B \,a^{3} b^{4} c^{2}+224 B \,a^{2} b^{6} c -18 B a \,b^{8}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{65536 a^{\frac {13}{2}}}\)	\(695\)
default	\(\text {Expression too large to display}\)	\(27691\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^10,x,method=_RETURNVERBOSE)
 

Output:

-1/2064384*(c*x^2+b*x+a)^(1/2)*(-65536*A*a^4*c^4*x^8+234432*A*a^3*b^2*c^3* 
x^8-162288*A*a^2*b^4*c^2*x^8+40740*A*a*b^6*c*x^8-3465*A*b^8*x^8-254592*B*a 
^4*b*c^3*x^8+226464*B*a^3*b^3*c^2*x^8-63000*B*a^2*b^5*c*x^8+5670*B*a*b^7*x 
^8-88192*A*a^4*b*c^3*x^7+84384*A*a^3*b^3*c^2*x^7-24696*A*a^2*b^5*c*x^7+231 
0*A*a*b^7*x^7+80640*B*a^5*c^3*x^7-114624*B*a^4*b^2*c^2*x^7+37968*B*a^3*b^4 
*c*x^7-3780*B*a^2*b^6*x^7+32768*A*a^5*c^3*x^6-51072*A*a^4*b^2*c^2*x^6+1785 
6*A*a^3*b^4*c*x^6-1848*A*a^2*b^6*x^6+66816*B*a^5*b*c^2*x^6-27264*B*a^4*b^3 
*c*x^6+3024*B*a^3*b^5*x^6+31488*A*a^5*b*c^2*x^5-13696*A*a^4*b^3*c*x^5+1584 
*A*a^3*b^5*x^5+634368*B*a^6*c^2*x^5+20736*B*a^5*b^2*c*x^5-2592*B*a^4*b^4*x 
^5+491520*A*a^6*c^2*x^4+10752*A*a^5*b^2*c*x^4-1408*A*a^4*b^4*x^4+943104*B* 
a^6*b*c*x^4+2304*B*a^5*b^3*x^4+771072*A*a^6*b*c*x^3+1280*A*a^5*b^3*x^3+731 
136*B*a^7*c*x^3+373248*B*a^6*b^2*x^3+622592*A*a^7*c*x^2+316416*A*a^6*b^2*x 
^2+608256*B*a^7*b*x^2+530432*A*a^7*b*x+258048*B*a^8*x+229376*A*a^8)/x^9/a^ 
6-5/65536*(768*A*a^4*b*c^4-1280*A*a^3*b^3*c^3+672*A*a^2*b^5*c^2-144*A*a*b^ 
7*c+11*A*b^9-512*B*a^5*c^4+1536*B*a^4*b^2*c^3-960*B*a^3*b^4*c^2+224*B*a^2* 
b^6*c-18*B*a*b^8)/a^(13/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)
 

Fricas [A] (verification not implemented)

Time = 5.00 (sec) , antiderivative size = 1315, normalized size of antiderivative = 3.51 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^{10}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^10,x, algorithm="fricas")
 

Output:

[-1/8257536*(315*(18*B*a*b^8 - 11*A*b^9 + 256*(2*B*a^5 - 3*A*a^4*b)*c^4 - 
256*(6*B*a^4*b^2 - 5*A*a^3*b^3)*c^3 + 96*(10*B*a^3*b^4 - 7*A*a^2*b^5)*c^2 
- 16*(14*B*a^2*b^6 - 9*A*a*b^7)*c)*sqrt(a)*x^9*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*(22 
9376*A*a^9 + (5670*B*a^2*b^7 - 3465*A*a*b^8 - 65536*A*a^5*c^4 - 576*(442*B 
*a^5*b - 407*A*a^4*b^2)*c^3 + 336*(674*B*a^4*b^3 - 483*A*a^3*b^4)*c^2 - 42 
0*(150*B*a^3*b^5 - 97*A*a^2*b^6)*c)*x^8 - 2*(1890*B*a^3*b^6 - 1155*A*a^2*b 
^7 - 64*(630*B*a^6 - 689*A*a^5*b)*c^3 + 144*(398*B*a^5*b^2 - 293*A*a^4*b^3 
)*c^2 - 84*(226*B*a^4*b^4 - 147*A*a^3*b^5)*c)*x^7 + 8*(378*B*a^4*b^5 - 231 
*A*a^3*b^6 + 4096*A*a^6*c^3 + 48*(174*B*a^6*b - 133*A*a^5*b^2)*c^2 - 24*(1 
42*B*a^5*b^3 - 93*A*a^4*b^4)*c)*x^6 - 16*(162*B*a^5*b^4 - 99*A*a^4*b^5 - 4 
8*(826*B*a^7 + 41*A*a^6*b)*c^2 - 8*(162*B*a^6*b^2 - 107*A*a^5*b^3)*c)*x^5 
+ 128*(18*B*a^6*b^3 - 11*A*a^5*b^4 + 3840*A*a^7*c^2 + 12*(614*B*a^7*b + 7* 
A*a^6*b^2)*c)*x^4 + 256*(1458*B*a^7*b^2 + 5*A*a^6*b^3 + 12*(238*B*a^8 + 25 
1*A*a^7*b)*c)*x^3 + 1024*(594*B*a^8*b + 309*A*a^7*b^2 + 608*A*a^8*c)*x^2 + 
 14336*(18*B*a^9 + 37*A*a^8*b)*x)*sqrt(c*x^2 + b*x + a))/(a^7*x^9), -1/412 
8768*(315*(18*B*a*b^8 - 11*A*b^9 + 256*(2*B*a^5 - 3*A*a^4*b)*c^4 - 256*(6* 
B*a^4*b^2 - 5*A*a^3*b^3)*c^3 + 96*(10*B*a^3*b^4 - 7*A*a^2*b^5)*c^2 - 16*(1 
4*B*a^2*b^6 - 9*A*a*b^7)*c)*sqrt(-a)*x^9*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*(229376*A*a^9 + (5670...
 

Sympy [F]

\[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^{10}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{x^{10}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**10,x)
 

Output:

Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**10, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^{10}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^10,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. 4427 vs. \(2 (341) = 682\).

Time = 0.31 (sec) , antiderivative size = 4427, normalized size of antiderivative = 11.81 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^{10}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^10,x, algorithm="giac")
 

Output:

-5/32768*(18*B*a*b^8 - 11*A*b^9 - 224*B*a^2*b^6*c + 144*A*a*b^7*c + 960*B* 
a^3*b^4*c^2 - 672*A*a^2*b^5*c^2 - 1536*B*a^4*b^2*c^3 + 1280*A*a^3*b^3*c^3 
+ 512*B*a^5*c^4 - 768*A*a^4*b*c^4)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + 
 a))/sqrt(-a))/(sqrt(-a)*a^6) + 1/2064384*(5670*(sqrt(c)*x - sqrt(c*x^2 + 
b*x + a))^17*B*a*b^8 - 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^17*A*b^9 - 
 70560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^17*B*a^2*b^6*c + 45360*(sqrt(c) 
*x - sqrt(c*x^2 + b*x + a))^17*A*a*b^7*c + 302400*(sqrt(c)*x - sqrt(c*x^2 
+ b*x + a))^17*B*a^3*b^4*c^2 - 211680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 
17*A*a^2*b^5*c^2 - 483840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^17*B*a^4*b^2 
*c^3 + 403200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^17*A*a^3*b^3*c^3 + 16128 
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^17*B*a^5*c^4 - 241920*(sqrt(c)*x - s 
qrt(c*x^2 + b*x + a))^17*A*a^4*b*c^4 - 49140*(sqrt(c)*x - sqrt(c*x^2 + b*x 
 + a))^15*B*a^2*b^8 + 30030*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*A*a*b^9 
 + 611520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*B*a^3*b^6*c - 393120*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))^15*A*a^2*b^7*c - 2620800*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^15*B*a^4*b^4*c^2 + 1834560*(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))^15*A*a^3*b^5*c^2 + 4193280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15* 
B*a^5*b^2*c^3 - 3494400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*A*a^4*b^3*c 
^3 + 4107264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*B*a^6*c^4 + 2096640*(s 
qrt(c)*x - sqrt(c*x^2 + b*x + a))^15*A*a^5*b*c^4 + 28901376*(sqrt(c)*x ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^{10}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{x^{10}} \,d x \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^10,x)
 

Output:

int(((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^10, x)
 

Reduce [B] (verification not implemented)

Time = 96.89 (sec) , antiderivative size = 829, normalized size of antiderivative = 2.21 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^{10}} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^10,x)
 

Output:

( - 458752*sqrt(a + b*x + c*x**2)*a**9 - 1576960*sqrt(a + b*x + c*x**2)*a* 
*8*b*x - 1245184*sqrt(a + b*x + c*x**2)*a**8*c*x**2 - 1849344*sqrt(a + b*x 
 + c*x**2)*a**7*b**2*x**2 - 3004416*sqrt(a + b*x + c*x**2)*a**7*b*c*x**3 - 
 983040*sqrt(a + b*x + c*x**2)*a**7*c**2*x**4 - 749056*sqrt(a + b*x + c*x* 
*2)*a**6*b**3*x**3 - 1907712*sqrt(a + b*x + c*x**2)*a**6*b**2*c*x**4 - 133 
1712*sqrt(a + b*x + c*x**2)*a**6*b*c**2*x**5 - 65536*sqrt(a + b*x + c*x**2 
)*a**6*c**3*x**6 - 1792*sqrt(a + b*x + c*x**2)*a**5*b**4*x**4 - 14080*sqrt 
(a + b*x + c*x**2)*a**5*b**3*c*x**5 - 31488*sqrt(a + b*x + c*x**2)*a**5*b* 
*2*c**2*x**6 + 15104*sqrt(a + b*x + c*x**2)*a**5*b*c**3*x**7 + 131072*sqrt 
(a + b*x + c*x**2)*a**5*c**4*x**8 + 2016*sqrt(a + b*x + c*x**2)*a**4*b**5* 
x**5 + 18816*sqrt(a + b*x + c*x**2)*a**4*b**4*c*x**6 + 60480*sqrt(a + b*x 
+ c*x**2)*a**4*b**3*c**2*x**7 + 40320*sqrt(a + b*x + c*x**2)*a**4*b**2*c** 
3*x**8 - 2352*sqrt(a + b*x + c*x**2)*a**3*b**6*x**6 - 26544*sqrt(a + b*x + 
 c*x**2)*a**3*b**5*c*x**7 - 128352*sqrt(a + b*x + c*x**2)*a**3*b**4*c**2*x 
**8 + 2940*sqrt(a + b*x + c*x**2)*a**2*b**7*x**7 + 44520*sqrt(a + b*x + c* 
x**2)*a**2*b**6*c*x**8 - 4410*sqrt(a + b*x + c*x**2)*a*b**8*x**8 + 80640*s 
qrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**4*b*c**4*x**9 
+ 80640*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**3*b** 
3*c**3*x**9 - 90720*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b 
*x)*a**2*b**5*c**2*x**9 + 25200*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*...