\(\int \frac {(A+B x) (a+b x+c x^2)^{5/2}}{x^9} \, dx\) [142]

Optimal result

Integrand size = 23, antiderivative size = 288 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^9} \, dx=-\frac {5 \left (b^2-4 a c\right )^2 \left (9 A b^2-16 a b B-4 a A c\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{16384 a^5 x^2}+\frac {5 \left (b^2-4 a c\right ) \left (9 A b^2-16 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{6144 a^4 x^4}-\frac {\left (9 A b^2-16 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{384 a^3 x^6}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}+\frac {(9 A b-16 a B) \left (a+b x+c x^2\right )^{7/2}}{112 a^2 x^7}+\frac {5 \left (b^2-4 a c\right )^3 \left (9 A b^2-16 a b B-4 a A c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{32768 a^{11/2}} \] Output:

-5/16384*(-4*a*c+b^2)^2*(-4*A*a*c+9*A*b^2-16*B*a*b)*(b*x+2*a)*(c*x^2+b*x+a 
)^(1/2)/a^5/x^2+5/6144*(-4*a*c+b^2)*(-4*A*a*c+9*A*b^2-16*B*a*b)*(b*x+2*a)* 
(c*x^2+b*x+a)^(3/2)/a^4/x^4-1/384*(-4*A*a*c+9*A*b^2-16*B*a*b)*(b*x+2*a)*(c 
*x^2+b*x+a)^(5/2)/a^3/x^6-1/8*A*(c*x^2+b*x+a)^(7/2)/a/x^8+1/112*(9*A*b-16* 
B*a)*(c*x^2+b*x+a)^(7/2)/a^2/x^7+5/32768*(-4*a*c+b^2)^3*(-4*A*a*c+9*A*b^2- 
16*B*a*b)*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(11/2)
 

Mathematica [A] (verified)

Time = 7.67 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^9} \, dx=\frac {-\sqrt {a} \sqrt {a+x (b+c x)} \left (945 A b^7 x^7+6144 a^7 (7 A+8 B x)-210 a b^5 x^6 (3 A b+8 b B x+50 A c x)+1024 a^6 x (4 B x (29 b+36 c x)+A (99 b+119 c x))+256 a^5 x^2 \left (4 B x \left (74 b^2+197 b c x+144 c^2 x^2\right )+A \left (243 b^2+614 b c x+413 c^2 x^2\right )\right )+56 a^2 b^3 x^5 \left (20 b B x (b+16 c x)+A \left (9 b^2+113 b c x+674 c^2 x^2\right )\right )+384 a^4 x^3 \left (A \left (b^3+9 b^2 c x+29 b c^2 x^2+35 c^3 x^3\right )+2 B x \left (b^3+10 b^2 c x+38 b c^2 x^2+64 c^3 x^3\right )\right )-16 a^3 b x^4 \left (56 b B x \left (b^2+12 b c x+66 c^2 x^2\right )+A \left (27 b^3+284 b^2 c x+1194 b c^2 x^2+2652 c^3 x^3\right )\right )\right )-945 A b^8 x^8 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+1680 a \left (-b^7 B-7 A b^6 c+12 a b^5 B c+30 a A b^4 c^2-48 a^2 b^3 B c^2-48 a^2 A b^2 c^3+64 a^3 b B c^3+16 a^3 A c^4\right ) x^8 \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{344064 a^{11/2} x^8} \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^9,x]
 

Output:

(-(Sqrt[a]*Sqrt[a + x*(b + c*x)]*(945*A*b^7*x^7 + 6144*a^7*(7*A + 8*B*x) - 
 210*a*b^5*x^6*(3*A*b + 8*b*B*x + 50*A*c*x) + 1024*a^6*x*(4*B*x*(29*b + 36 
*c*x) + A*(99*b + 119*c*x)) + 256*a^5*x^2*(4*B*x*(74*b^2 + 197*b*c*x + 144 
*c^2*x^2) + A*(243*b^2 + 614*b*c*x + 413*c^2*x^2)) + 56*a^2*b^3*x^5*(20*b* 
B*x*(b + 16*c*x) + A*(9*b^2 + 113*b*c*x + 674*c^2*x^2)) + 384*a^4*x^3*(A*( 
b^3 + 9*b^2*c*x + 29*b*c^2*x^2 + 35*c^3*x^3) + 2*B*x*(b^3 + 10*b^2*c*x + 3 
8*b*c^2*x^2 + 64*c^3*x^3)) - 16*a^3*b*x^4*(56*b*B*x*(b^2 + 12*b*c*x + 66*c 
^2*x^2) + A*(27*b^3 + 284*b^2*c*x + 1194*b*c^2*x^2 + 2652*c^3*x^3)))) - 94 
5*A*b^8*x^8*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]] + 1680*a* 
(-(b^7*B) - 7*A*b^6*c + 12*a*b^5*B*c + 30*a*A*b^4*c^2 - 48*a^2*b^3*B*c^2 - 
 48*a^2*A*b^2*c^3 + 64*a^3*b*B*c^3 + 16*a^3*A*c^4)*x^8*ArcTanh[(-(Sqrt[c]* 
x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(344064*a^(11/2)*x^8)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.92, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {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.

\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^9} \, dx\)

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

\(\displaystyle -\frac {-\frac {\left (-4 a A c-16 a b B+9 A b^2\right ) \int \frac {\left (c x^2+b x+a\right )^{5/2}}{x^7}dx}{2 a}-\frac {(9 A b-16 a B) \left (a+b x+c x^2\right )^{7/2}}{7 a x^7}}{16 a}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}\)

\(\Big \downarrow \) 1152

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

\(\Big \downarrow \) 1152

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

\(\Big \downarrow \) 1152

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^9,x]
 

Output:

-1/8*(A*(a + b*x + c*x^2)^(7/2))/(a*x^8) - (-1/7*((9*A*b - 16*a*B)*(a + b* 
x + c*x^2)^(7/2))/(a*x^7) - ((9*A*b^2 - 16*a*b*B - 4*a*A*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)*ArcTanh[(2*a + b*x)/(2*Sqr 
t[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(3/2))))/(16*a)))/(24*a)))/(2*a))/(16*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. \(562\) vs. \(2(258)=516\).

Time = 1.85 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.95

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-42432 A \,a^{3} b \,c^{3} x^{7}+37744 A \,a^{2} b^{3} c^{2} x^{7}-10500 A a \,b^{5} c \,x^{7}+945 A \,b^{7} x^{7}+49152 B \,a^{4} c^{3} x^{7}-59136 B \,a^{3} b^{2} c^{2} x^{7}+17920 B \,a^{2} b^{4} c \,x^{7}-1680 B a \,b^{6} x^{7}+13440 A \,a^{4} c^{3} x^{6}-19104 A \,a^{3} b^{2} c^{2} x^{6}+6328 A \,a^{2} b^{4} c \,x^{6}-630 A a \,b^{6} x^{6}+29184 B \,a^{4} b \,c^{2} x^{6}-10752 B \,a^{3} b^{3} c \,x^{6}+1120 B \,a^{2} b^{5} x^{6}+11136 A \,a^{4} b \,c^{2} x^{5}-4544 A \,a^{3} b^{3} c \,x^{5}+504 A \,a^{2} b^{5} x^{5}+147456 B \,a^{5} c^{2} x^{5}+7680 B \,a^{4} b^{2} c \,x^{5}-896 B \,a^{3} b^{4} x^{5}+105728 A \,a^{5} c^{2} x^{4}+3456 A \,a^{4} b^{2} c \,x^{4}-432 A \,a^{3} b^{4} x^{4}+201728 B \,a^{5} b c \,x^{4}+768 B \,a^{4} b^{3} x^{4}+157184 A \,a^{5} b c \,x^{3}+384 A \,a^{4} b^{3} x^{3}+147456 B \,a^{6} c \,x^{3}+75776 B \,a^{5} b^{2} x^{3}+121856 A \,a^{6} c \,x^{2}+62208 A \,a^{5} b^{2} x^{2}+118784 B \,a^{6} b \,x^{2}+101376 A \,a^{6} b x +49152 B \,a^{7} x +43008 A \,a^{7}\right )}{344064 x^{8} a^{5}}+\frac {5 \left (256 A \,a^{4} c^{4}-768 A \,a^{3} b^{2} c^{3}+480 A \,a^{2} b^{4} c^{2}-112 A a \,b^{6} c +9 A \,b^{8}+1024 B \,a^{4} b \,c^{3}-768 B \,a^{3} b^{3} c^{2}+192 B \,a^{2} b^{5} c -16 B a \,b^{7}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{32768 a^{\frac {11}{2}}}\)	\(563\)
default	\(\text {Expression too large to display}\)	\(16596\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^9,x,method=_RETURNVERBOSE)
 

Output:

-1/344064*(c*x^2+b*x+a)^(1/2)*(-42432*A*a^3*b*c^3*x^7+37744*A*a^2*b^3*c^2* 
x^7-10500*A*a*b^5*c*x^7+945*A*b^7*x^7+49152*B*a^4*c^3*x^7-59136*B*a^3*b^2* 
c^2*x^7+17920*B*a^2*b^4*c*x^7-1680*B*a*b^6*x^7+13440*A*a^4*c^3*x^6-19104*A 
*a^3*b^2*c^2*x^6+6328*A*a^2*b^4*c*x^6-630*A*a*b^6*x^6+29184*B*a^4*b*c^2*x^ 
6-10752*B*a^3*b^3*c*x^6+1120*B*a^2*b^5*x^6+11136*A*a^4*b*c^2*x^5-4544*A*a^ 
3*b^3*c*x^5+504*A*a^2*b^5*x^5+147456*B*a^5*c^2*x^5+7680*B*a^4*b^2*c*x^5-89 
6*B*a^3*b^4*x^5+105728*A*a^5*c^2*x^4+3456*A*a^4*b^2*c*x^4-432*A*a^3*b^4*x^ 
4+201728*B*a^5*b*c*x^4+768*B*a^4*b^3*x^4+157184*A*a^5*b*c*x^3+384*A*a^4*b^ 
3*x^3+147456*B*a^6*c*x^3+75776*B*a^5*b^2*x^3+121856*A*a^6*c*x^2+62208*A*a^ 
5*b^2*x^2+118784*B*a^6*b*x^2+101376*A*a^6*b*x+49152*B*a^7*x+43008*A*a^7)/x 
^8/a^5+5/32768*(256*A*a^4*c^4-768*A*a^3*b^2*c^3+480*A*a^2*b^4*c^2-112*A*a* 
b^6*c+9*A*b^8+1024*B*a^4*b*c^3-768*B*a^3*b^3*c^2+192*B*a^2*b^5*c-16*B*a*b^ 
7)/a^(11/2)*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. 541 vs. \(2 (258) = 516\).

Time = 4.08 (sec) , antiderivative size = 1091, normalized size of antiderivative = 3.79 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^9} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^9,x, algorithm="fricas")
 

Output:

[-1/1376256*(105*(16*B*a*b^7 - 9*A*b^8 - 256*A*a^4*c^4 - 256*(4*B*a^4*b - 
3*A*a^3*b^2)*c^3 + 96*(8*B*a^3*b^3 - 5*A*a^2*b^4)*c^2 - 16*(12*B*a^2*b^5 - 
 7*A*a*b^6)*c)*sqrt(a)*x^8*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*(43008*A*a^8 - (1680*B* 
a^2*b^6 - 945*A*a*b^7 - 192*(256*B*a^5 - 221*A*a^4*b)*c^3 + 112*(528*B*a^4 
*b^2 - 337*A*a^3*b^3)*c^2 - 140*(128*B*a^3*b^4 - 75*A*a^2*b^5)*c)*x^7 + 2* 
(560*B*a^3*b^5 - 315*A*a^2*b^6 + 6720*A*a^5*c^3 + 48*(304*B*a^5*b - 199*A* 
a^4*b^2)*c^2 - 28*(192*B*a^4*b^3 - 113*A*a^3*b^4)*c)*x^6 - 8*(112*B*a^4*b^ 
4 - 63*A*a^3*b^5 - 48*(384*B*a^6 + 29*A*a^5*b)*c^2 - 8*(120*B*a^5*b^2 - 71 
*A*a^4*b^3)*c)*x^5 + 16*(48*B*a^5*b^3 - 27*A*a^4*b^4 + 6608*A*a^6*c^2 + 8* 
(1576*B*a^6*b + 27*A*a^5*b^2)*c)*x^4 + 128*(592*B*a^6*b^2 + 3*A*a^5*b^3 + 
4*(288*B*a^7 + 307*A*a^6*b)*c)*x^3 + 256*(464*B*a^7*b + 243*A*a^6*b^2 + 47 
6*A*a^7*c)*x^2 + 3072*(16*B*a^8 + 33*A*a^7*b)*x)*sqrt(c*x^2 + b*x + a))/(a 
^6*x^8), 1/688128*(105*(16*B*a*b^7 - 9*A*b^8 - 256*A*a^4*c^4 - 256*(4*B*a^ 
4*b - 3*A*a^3*b^2)*c^3 + 96*(8*B*a^3*b^3 - 5*A*a^2*b^4)*c^2 - 16*(12*B*a^2 
*b^5 - 7*A*a*b^6)*c)*sqrt(-a)*x^8*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*(43008*A*a^8 - (1680*B*a^2*b^6 
- 945*A*a*b^7 - 192*(256*B*a^5 - 221*A*a^4*b)*c^3 + 112*(528*B*a^4*b^2 - 3 
37*A*a^3*b^3)*c^2 - 140*(128*B*a^3*b^4 - 75*A*a^2*b^5)*c)*x^7 + 2*(560*B*a 
^3*b^5 - 315*A*a^2*b^6 + 6720*A*a^5*c^3 + 48*(304*B*a^5*b - 199*A*a^4*b...
 

Sympy [F]

\[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^9} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{x^{9}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**9,x)
 

Output:

Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**9, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^9} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^9,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. 3603 vs. \(2 (258) = 516\).

Time = 0.31 (sec) , antiderivative size = 3603, normalized size of antiderivative = 12.51 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^9} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^9,x, algorithm="giac")
 

Output:

5/16384*(16*B*a*b^7 - 9*A*b^8 - 192*B*a^2*b^5*c + 112*A*a*b^6*c + 768*B*a^ 
3*b^3*c^2 - 480*A*a^2*b^4*c^2 - 1024*B*a^4*b*c^3 + 768*A*a^3*b^2*c^3 - 256 
*A*a^4*c^4)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a 
)*a^5) - 1/344064*(1680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*B*a*b^7 - 9 
45*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*A*b^8 - 20160*(sqrt(c)*x - sqrt( 
c*x^2 + b*x + a))^15*B*a^2*b^5*c + 11760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a 
))^15*A*a*b^6*c + 80640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*B*a^3*b^3*c 
^2 - 50400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*A*a^2*b^4*c^2 - 107520*( 
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*B*a^4*b*c^3 + 80640*(sqrt(c)*x - sqr 
t(c*x^2 + b*x + a))^15*A*a^3*b^2*c^3 - 26880*(sqrt(c)*x - sqrt(c*x^2 + b*x 
 + a))^15*A*a^4*c^4 - 688128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*B*a^5* 
c^(7/2) - 12880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*B*a^2*b^7 + 7245*(s 
qrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*a*b^8 + 154560*(sqrt(c)*x - sqrt(c* 
x^2 + b*x + a))^13*B*a^3*b^5*c - 90160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) 
^13*A*a^2*b^6*c - 618240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*B*a^4*b^3* 
c^2 + 386400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*a^3*b^4*c^2 - 215756 
8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*B*a^5*b*c^3 - 618240*(sqrt(c)*x - 
 sqrt(c*x^2 + b*x + a))^13*A*a^4*b^2*c^3 - 711424*(sqrt(c)*x - sqrt(c*x^2 
+ b*x + a))^13*A*a^5*c^4 - 6193152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12* 
B*a^5*b^2*c^(5/2) + 688128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*B*a^6...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^9} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{x^9} \,d x \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^9,x)
 

Output:

int(((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^9, x)
 

Reduce [B] (verification not implemented)

Time = 19.11 (sec) , antiderivative size = 710, normalized size of antiderivative = 2.47 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^9} \, dx=\frac {-12288 \sqrt {c \,x^{2}+b x +a}\, a^{8}-87040 \sqrt {c \,x^{2}+b x +a}\, a^{6} b c \,x^{3}-58624 \sqrt {c \,x^{2}+b x +a}\, a^{5} b^{2} c \,x^{4}-45312 \sqrt {c \,x^{2}+b x +a}\, a^{5} b \,c^{2} x^{5}-896 \sqrt {c \,x^{2}+b x +a}\, a^{4} b^{3} c \,x^{5}-2880 \sqrt {c \,x^{2}+b x +a}\, a^{4} b^{2} c^{2} x^{6}-1920 \sqrt {c \,x^{2}+b x +a}\, a^{4} b \,c^{3} x^{7}+1264 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{4} c \,x^{6}+6112 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{3} c^{2} x^{7}-2120 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{5} c \,x^{7}+3840 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{4} c^{4} x^{8}-3840 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{4} c^{4} x^{8}+3840 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{3} b^{2} c^{3} x^{8}-4320 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} b^{4} c^{2} x^{8}+1200 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,b^{6} c \,x^{8}-3840 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{3} b^{2} c^{3} x^{8}+4320 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} b^{4} c^{2} x^{8}-1200 \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{6} c \,x^{8}-43008 \sqrt {c \,x^{2}+b x +a}\, a^{7} b x -34816 \sqrt {c \,x^{2}+b x +a}\, a^{7} c \,x^{2}-51712 \sqrt {c \,x^{2}+b x +a}\, a^{6} b^{2} x^{2}-30208 \sqrt {c \,x^{2}+b x +a}\, a^{6} c^{2} x^{4}-21760 \sqrt {c \,x^{2}+b x +a}\, a^{5} b^{3} x^{3}-3840 \sqrt {c \,x^{2}+b x +a}\, a^{5} c^{3} x^{6}-96 \sqrt {c \,x^{2}+b x +a}\, a^{4} b^{4} x^{4}+112 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{5} x^{5}-140 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{6} x^{6}+210 \sqrt {c \,x^{2}+b x +a}\, a \,b^{7} x^{7}-105 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{8} x^{8}+105 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{8} x^{8}}{98304 a^{5} x^{8}} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^9,x)
 

Output:

( - 12288*sqrt(a + b*x + c*x**2)*a**8 - 43008*sqrt(a + b*x + c*x**2)*a**7* 
b*x - 34816*sqrt(a + b*x + c*x**2)*a**7*c*x**2 - 51712*sqrt(a + b*x + c*x* 
*2)*a**6*b**2*x**2 - 87040*sqrt(a + b*x + c*x**2)*a**6*b*c*x**3 - 30208*sq 
rt(a + b*x + c*x**2)*a**6*c**2*x**4 - 21760*sqrt(a + b*x + c*x**2)*a**5*b* 
*3*x**3 - 58624*sqrt(a + b*x + c*x**2)*a**5*b**2*c*x**4 - 45312*sqrt(a + b 
*x + c*x**2)*a**5*b*c**2*x**5 - 3840*sqrt(a + b*x + c*x**2)*a**5*c**3*x**6 
 - 96*sqrt(a + b*x + c*x**2)*a**4*b**4*x**4 - 896*sqrt(a + b*x + c*x**2)*a 
**4*b**3*c*x**5 - 2880*sqrt(a + b*x + c*x**2)*a**4*b**2*c**2*x**6 - 1920*s 
qrt(a + b*x + c*x**2)*a**4*b*c**3*x**7 + 112*sqrt(a + b*x + c*x**2)*a**3*b 
**5*x**5 + 1264*sqrt(a + b*x + c*x**2)*a**3*b**4*c*x**6 + 6112*sqrt(a + b* 
x + c*x**2)*a**3*b**3*c**2*x**7 - 140*sqrt(a + b*x + c*x**2)*a**2*b**6*x** 
6 - 2120*sqrt(a + b*x + c*x**2)*a**2*b**5*c*x**7 + 210*sqrt(a + b*x + c*x* 
*2)*a*b**7*x**7 + 3840*sqrt(a)*log( - 2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2 
*a - b*x)*a**4*c**4*x**8 + 3840*sqrt(a)*log( - 2*sqrt(a)*sqrt(a + b*x + c* 
x**2) - 2*a - b*x)*a**3*b**2*c**3*x**8 - 4320*sqrt(a)*log( - 2*sqrt(a)*sqr 
t(a + b*x + c*x**2) - 2*a - b*x)*a**2*b**4*c**2*x**8 + 1200*sqrt(a)*log( - 
 2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b**6*c*x**8 - 105*sqrt(a) 
*log( - 2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*b**8*x**8 - 3840*sqr 
t(a)*log(x)*a**4*c**4*x**8 - 3840*sqrt(a)*log(x)*a**3*b**2*c**3*x**8 + 432 
0*sqrt(a)*log(x)*a**2*b**4*c**2*x**8 - 1200*sqrt(a)*log(x)*a*b**6*c*x**...