\(\int \frac {(A+B x) (a+b x+c x^2)^{3/2}}{x^7} \, dx\) [127]

Optimal result

Integrand size = 23, antiderivative size = 230 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx=\frac {\left (b^2-4 a c\right ) \left (7 A b^2-12 a b B-4 a A c\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{512 a^4 x^2}-\frac {\left (7 A b^2-12 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{192 a^3 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{6 a x^6}+\frac {(7 A b-12 a B) \left (a+b x+c x^2\right )^{5/2}}{60 a^2 x^5}-\frac {\left (b^2-4 a c\right )^2 \left (7 A b^2-12 a b B-4 a A c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{1024 a^{9/2}} \] Output:

1/512*(-4*a*c+b^2)*(-4*A*a*c+7*A*b^2-12*B*a*b)*(b*x+2*a)*(c*x^2+b*x+a)^(1/ 
2)/a^4/x^2-1/192*(-4*A*a*c+7*A*b^2-12*B*a*b)*(b*x+2*a)*(c*x^2+b*x+a)^(3/2) 
/a^3/x^4-1/6*A*(c*x^2+b*x+a)^(5/2)/a/x^6+1/60*(7*A*b-12*B*a)*(c*x^2+b*x+a) 
^(5/2)/a^2/x^5-1/1024*(-4*a*c+b^2)^2*(-4*A*a*c+7*A*b^2-12*B*a*b)*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 = 4.10 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx=\frac {-\sqrt {a} \sqrt {a+x (b+c x)} \left (-105 A b^5 x^5+256 a^5 (5 A+6 B x)+10 a b^3 x^4 (7 A b+18 b B x+76 A c x)+64 a^4 x \left (26 A b+33 b B x+35 A c x+48 B c x^2\right )+48 a^3 x^2 \left (A \left (b^2+6 b c x+10 c^2 x^2\right )+2 B x \left (b^2+7 b c x+16 c^2 x^2\right )\right )-8 a^2 b x^3 \left (15 b B x (b+10 c x)+A \left (7 b^2+54 b c x+162 c^2 x^2\right )\right )\right )+105 A b^6 x^6 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+60 a \left (3 b^5 B+15 A b^4 c-24 a b^3 B c-36 a A b^2 c^2+48 a^2 b B c^2+16 a^2 A c^3\right ) x^6 \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{7680 a^{9/2} x^6} \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^7,x]
 

Output:

(-(Sqrt[a]*Sqrt[a + x*(b + c*x)]*(-105*A*b^5*x^5 + 256*a^5*(5*A + 6*B*x) + 
 10*a*b^3*x^4*(7*A*b + 18*b*B*x + 76*A*c*x) + 64*a^4*x*(26*A*b + 33*b*B*x 
+ 35*A*c*x + 48*B*c*x^2) + 48*a^3*x^2*(A*(b^2 + 6*b*c*x + 10*c^2*x^2) + 2* 
B*x*(b^2 + 7*b*c*x + 16*c^2*x^2)) - 8*a^2*b*x^3*(15*b*B*x*(b + 10*c*x) + A 
*(7*b^2 + 54*b*c*x + 162*c^2*x^2)))) + 105*A*b^6*x^6*ArcTanh[(Sqrt[c]*x - 
Sqrt[a + x*(b + c*x)])/Sqrt[a]] + 60*a*(3*b^5*B + 15*A*b^4*c - 24*a*b^3*B* 
c - 36*a*A*b^2*c^2 + 48*a^2*b*B*c^2 + 16*a^2*A*c^3)*x^6*ArcTanh[(-(Sqrt[c] 
*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(7680*a^(9/2)*x^6)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {1237, 27, 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.

\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx\)

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1152

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

\(\Big \downarrow \) 1152

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^7,x]
 

Output:

-1/6*(A*(a + b*x + c*x^2)^(5/2))/(a*x^6) - (-1/5*((7*A*b - 12*a*B)*(a + b* 
x + c*x^2)^(5/2))/(a*x^5) - ((7*A*b^2 - 12*a*b*B - 4*a*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*S 
qrt[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(3/2))))/(16*a)))/(2*a))/(12*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 = 1.57 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.50

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-1296 A \,a^{2} b \,c^{2} x^{5}+760 A a \,b^{3} c \,x^{5}-105 A \,b^{5} x^{5}+1536 B \,a^{3} c^{2} x^{5}-1200 B \,a^{2} b^{2} c \,x^{5}+180 B a \,b^{4} x^{5}+480 A \,a^{3} c^{2} x^{4}-432 A \,a^{2} b^{2} c \,x^{4}+70 A a \,b^{4} x^{4}+672 B \,a^{3} b c \,x^{4}-120 B \,a^{2} b^{3} x^{4}+288 A \,a^{3} b c \,x^{3}-56 A \,a^{2} b^{3} x^{3}+3072 B \,a^{4} c \,x^{3}+96 B \,a^{3} b^{2} x^{3}+2240 A \,a^{4} c \,x^{2}+48 A \,a^{3} b^{2} x^{2}+2112 B \,a^{4} b \,x^{2}+1664 A \,a^{4} b x +1536 B \,a^{5} x +1280 A \,a^{5}\right )}{7680 x^{6} a^{4}}+\frac {\left (64 a^{3} A \,c^{3}-144 A \,a^{2} b^{2} c^{2}+60 A a \,b^{4} c -7 A \,b^{6}+192 B \,a^{3} b \,c^{2}-96 B \,a^{2} b^{3} c +12 B a \,b^{5}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{1024 a^{\frac {9}{2}}}\)	\(344\)
default	\(\text {Expression too large to display}\)	\(4143\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/7680*(c*x^2+b*x+a)^(1/2)*(-1296*A*a^2*b*c^2*x^5+760*A*a*b^3*c*x^5-105*A 
*b^5*x^5+1536*B*a^3*c^2*x^5-1200*B*a^2*b^2*c*x^5+180*B*a*b^4*x^5+480*A*a^3 
*c^2*x^4-432*A*a^2*b^2*c*x^4+70*A*a*b^4*x^4+672*B*a^3*b*c*x^4-120*B*a^2*b^ 
3*x^4+288*A*a^3*b*c*x^3-56*A*a^2*b^3*x^3+3072*B*a^4*c*x^3+96*B*a^3*b^2*x^3 
+2240*A*a^4*c*x^2+48*A*a^3*b^2*x^2+2112*B*a^4*b*x^2+1664*A*a^4*b*x+1536*B* 
a^5*x+1280*A*a^5)/x^6/a^4+1/1024*(64*A*a^3*c^3-144*A*a^2*b^2*c^2+60*A*a*b^ 
4*c-7*A*b^6+192*B*a^3*b*c^2-96*B*a^2*b^3*c+12*B*a*b^5)/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 = 1.83 (sec) , antiderivative size = 709, normalized size of antiderivative = 3.08 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^7,x, algorithm="fricas")
 

Output:

[1/30720*(15*(12*B*a*b^5 - 7*A*b^6 + 64*A*a^3*c^3 + 48*(4*B*a^3*b - 3*A*a^ 
2*b^2)*c^2 - 12*(8*B*a^2*b^3 - 5*A*a*b^4)*c)*sqrt(a)*x^6*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*(1280*A*a^6 + (180*B*a^2*b^4 - 105*A*a*b^5 + 48*(32*B*a^4 - 27*A*a^ 
3*b)*c^2 - 40*(30*B*a^3*b^2 - 19*A*a^2*b^3)*c)*x^5 - 2*(60*B*a^3*b^3 - 35* 
A*a^2*b^4 - 240*A*a^4*c^2 - 24*(14*B*a^4*b - 9*A*a^3*b^2)*c)*x^4 + 8*(12*B 
*a^4*b^2 - 7*A*a^3*b^3 + 12*(32*B*a^5 + 3*A*a^4*b)*c)*x^3 + 16*(132*B*a^5* 
b + 3*A*a^4*b^2 + 140*A*a^5*c)*x^2 + 128*(12*B*a^6 + 13*A*a^5*b)*x)*sqrt(c 
*x^2 + b*x + a))/(a^5*x^6), -1/15360*(15*(12*B*a*b^5 - 7*A*b^6 + 64*A*a^3* 
c^3 + 48*(4*B*a^3*b - 3*A*a^2*b^2)*c^2 - 12*(8*B*a^2*b^3 - 5*A*a*b^4)*c)*s 
qrt(-a)*x^6*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*(1280*A*a^6 + (180*B*a^2*b^4 - 105*A*a*b^5 + 48*(32*B 
*a^4 - 27*A*a^3*b)*c^2 - 40*(30*B*a^3*b^2 - 19*A*a^2*b^3)*c)*x^5 - 2*(60*B 
*a^3*b^3 - 35*A*a^2*b^4 - 240*A*a^4*c^2 - 24*(14*B*a^4*b - 9*A*a^3*b^2)*c) 
*x^4 + 8*(12*B*a^4*b^2 - 7*A*a^3*b^3 + 12*(32*B*a^5 + 3*A*a^4*b)*c)*x^3 + 
16*(132*B*a^5*b + 3*A*a^4*b^2 + 140*A*a^5*c)*x^2 + 128*(12*B*a^6 + 13*A*a^ 
5*b)*x)*sqrt(c*x^2 + b*x + a))/(a^5*x^6)]
 

Sympy [F]

\[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x^{7}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/x**7,x)
 

Output:

Integral((A + B*x)*(a + b*x + c*x**2)**(3/2)/x**7, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^7,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. 2059 vs. \(2 (204) = 408\).

Time = 0.28 (sec) , antiderivative size = 2059, normalized size of antiderivative = 8.95 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^7,x, algorithm="giac")
 

Output:

-1/512*(12*B*a*b^5 - 7*A*b^6 - 96*B*a^2*b^3*c + 60*A*a*b^4*c + 192*B*a^3*b 
*c^2 - 144*A*a^2*b^2*c^2 + 64*A*a^3*c^3)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + 
 b*x + a))/sqrt(-a))/(sqrt(-a)*a^4) + 1/7680*(180*(sqrt(c)*x - sqrt(c*x^2 
+ b*x + a))^11*B*a*b^5 - 105*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*b^6 
- 1440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*B*a^2*b^3*c + 900*(sqrt(c)*x 
 - sqrt(c*x^2 + b*x + a))^11*A*a*b^4*c + 2880*(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))^11*B*a^3*b*c^2 - 2160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^2 
*b^2*c^2 + 960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^3*c^3 + 15360*(s 
qrt(c)*x - sqrt(c*x^2 + b*x + a))^10*B*a^4*c^(5/2) - 1020*(sqrt(c)*x - sqr 
t(c*x^2 + b*x + a))^9*B*a^2*b^5 + 595*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 
9*A*a*b^6 + 8160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^3*b^3*c - 5100* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^2*b^4*c + 29760*(sqrt(c)*x - sqr 
t(c*x^2 + b*x + a))^9*B*a^4*b*c^2 + 12240*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
a))^9*A*a^3*b^2*c^2 + 15040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^4*c^ 
3 + 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^4*b^2*c^(3/2) - 15360* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^5*c^(5/2) + 76800*(sqrt(c)*x - s 
qrt(c*x^2 + b*x + a))^8*A*a^4*b*c^(5/2) + 2376*(sqrt(c)*x - sqrt(c*x^2 + b 
*x + a))^7*B*a^3*b^5 - 1386*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^2*b^ 
6 + 24000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^4*b^3*c + 11880*(sqrt( 
c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^3*b^4*c + 13440*(sqrt(c)*x - sqrt(c...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^7} \,d x \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^7,x)
 

Output:

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^7, x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 1.69 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.03 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx=\frac {-512 \sqrt {c \,x^{2}+b x +a}\, a^{6}-1280 \sqrt {c \,x^{2}+b x +a}\, a^{5} b x -896 \sqrt {c \,x^{2}+b x +a}\, a^{5} c \,x^{2}-864 \sqrt {c \,x^{2}+b x +a}\, a^{4} b^{2} x^{2}-1344 \sqrt {c \,x^{2}+b x +a}\, a^{4} b c \,x^{3}-192 \sqrt {c \,x^{2}+b x +a}\, a^{4} c^{2} x^{4}-16 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{3} x^{3}-96 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{2} c \,x^{4}-96 \sqrt {c \,x^{2}+b x +a}\, a^{3} b \,c^{2} x^{5}+20 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{4} x^{4}+176 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{3} c \,x^{5}-30 \sqrt {c \,x^{2}+b x +a}\, a \,b^{5} x^{5}+192 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{3} c^{3} x^{6}+144 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} b^{2} c^{2} x^{6}-108 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,b^{4} c \,x^{6}+15 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{6} x^{6}-192 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{3} c^{3} x^{6}-144 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} b^{2} c^{2} x^{6}+108 \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{4} c \,x^{6}-15 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{6} x^{6}}{3072 a^{4} x^{6}} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^7,x)
 

Output:

( - 512*sqrt(a + b*x + c*x**2)*a**6 - 1280*sqrt(a + b*x + c*x**2)*a**5*b*x 
 - 896*sqrt(a + b*x + c*x**2)*a**5*c*x**2 - 864*sqrt(a + b*x + c*x**2)*a** 
4*b**2*x**2 - 1344*sqrt(a + b*x + c*x**2)*a**4*b*c*x**3 - 192*sqrt(a + b*x 
 + c*x**2)*a**4*c**2*x**4 - 16*sqrt(a + b*x + c*x**2)*a**3*b**3*x**3 - 96* 
sqrt(a + b*x + c*x**2)*a**3*b**2*c*x**4 - 96*sqrt(a + b*x + c*x**2)*a**3*b 
*c**2*x**5 + 20*sqrt(a + b*x + c*x**2)*a**2*b**4*x**4 + 176*sqrt(a + b*x + 
 c*x**2)*a**2*b**3*c*x**5 - 30*sqrt(a + b*x + c*x**2)*a*b**5*x**5 + 192*sq 
rt(a)*log( - 2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**3*c**3*x**6 
+ 144*sqrt(a)*log( - 2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*b* 
*2*c**2*x**6 - 108*sqrt(a)*log( - 2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - 
 b*x)*a*b**4*c*x**6 + 15*sqrt(a)*log( - 2*sqrt(a)*sqrt(a + b*x + c*x**2) - 
 2*a - b*x)*b**6*x**6 - 192*sqrt(a)*log(x)*a**3*c**3*x**6 - 144*sqrt(a)*lo 
g(x)*a**2*b**2*c**2*x**6 + 108*sqrt(a)*log(x)*a*b**4*c*x**6 - 15*sqrt(a)*l 
og(x)*b**6*x**6)/(3072*a**4*x**6)