\(\int \frac {(A+B x) (a+b x+c x^2)^{3/2}}{x^5} \, dx\) [125]

Optimal result

Integrand size = 23, antiderivative size = 217 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx=-\frac {\left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+\left (8 a B \left (b^2+8 a c\right )-3 A \left (b^3-4 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a^2 x^2}-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}+\frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{5/2}}+B c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \] Output:

-1/64*(2*a*(8*a*b*B-3*A*(-4*a*c+b^2))+(8*a*B*(8*a*c+b^2)-3*A*(-4*a*b*c+b^3 
))*x)*(c*x^2+b*x+a)^(1/2)/a^2/x^2-1/24*(6*a*A+(3*A*b+8*B*a)*x)*(c*x^2+b*x+ 
a)^(3/2)/a/x^4+1/128*(8*a*b*B*(-12*a*c+b^2)-3*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^(5/2)+B*c^(3/2)*arctanh(1/2*(2* 
c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))
 

Mathematica [A] (verified)

Time = 2.30 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.11 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx=-\frac {\sqrt {a+x (b+c x)} \left (-9 A b^3 x^3+16 a^3 (3 A+4 B x)+6 a b x^2 (4 b B x+A (b+10 c x))+8 a^2 x (3 A (3 b+5 c x)+2 B x (7 b+16 c x))\right )}{192 a^2 x^4}+\frac {3 A b^4 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{64 a^{5/2}}+\frac {\left (b^3 B+3 A b^2 c-12 a b B c-6 a A c^2\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{8 a^{3/2}}-B c^{3/2} \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right ) \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^5,x]
 

Output:

-1/192*(Sqrt[a + x*(b + c*x)]*(-9*A*b^3*x^3 + 16*a^3*(3*A + 4*B*x) + 6*a*b 
*x^2*(4*b*B*x + A*(b + 10*c*x)) + 8*a^2*x*(3*A*(3*b + 5*c*x) + 2*B*x*(7*b 
+ 16*c*x))))/(a^2*x^4) + (3*A*b^4*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x 
)])/Sqrt[a]])/(64*a^(5/2)) + ((b^3*B + 3*A*b^2*c - 12*a*b*B*c - 6*a*A*c^2) 
*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(8*a^(3/2)) - B* 
c^(3/2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1229, 27, 1229, 27, 1269, 1092, 219, 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.

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

\(\Big \downarrow \) 1229

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1229

\(\displaystyle \frac {-\frac {\int \frac {-128 a^2 B x c^2-3 A \left (b^2-4 a c\right )^2+8 a b B \left (b^2-12 a c\right )}{2 x \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+x \left (8 a B \left (8 a c+b^2\right )-3 A \left (b^3-4 a b c\right )\right )\right )}{4 a x^2}}{16 a}-\frac {\left (a+b x+c x^2\right )^{3/2} (x (8 a B+3 A b)+6 a A)}{24 a x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {-128 a^2 B x c^2-3 A \left (b^2-4 a c\right )^2+8 a b B \left (b^2-12 a c\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+x \left (8 a B \left (8 a c+b^2\right )-3 A \left (b^3-4 a b c\right )\right )\right )}{4 a x^2}}{16 a}-\frac {\left (a+b x+c x^2\right )^{3/2} (x (8 a B+3 A b)+6 a A)}{24 a x^4}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {-\frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx-128 a^2 B c^2 \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+x \left (8 a B \left (8 a c+b^2\right )-3 A \left (b^3-4 a b c\right )\right )\right )}{4 a x^2}}{16 a}-\frac {\left (a+b x+c x^2\right )^{3/2} (x (8 a B+3 A b)+6 a A)}{24 a x^4}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {-\frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx-256 a^2 B c^2 \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{8 a}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+x \left (8 a B \left (8 a c+b^2\right )-3 A \left (b^3-4 a b c\right )\right )\right )}{4 a x^2}}{16 a}-\frac {\left (a+b x+c x^2\right )^{3/2} (x (8 a B+3 A b)+6 a A)}{24 a x^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx-128 a^2 B c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 a}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+x \left (8 a B \left (8 a c+b^2\right )-3 A \left (b^3-4 a b c\right )\right )\right )}{4 a x^2}}{16 a}-\frac {\left (a+b x+c x^2\right )^{3/2} (x (8 a B+3 A b)+6 a A)}{24 a x^4}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {-\frac {-2 \left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}-128 a^2 B c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 a}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+x \left (8 a B \left (8 a c+b^2\right )-3 A \left (b^3-4 a b c\right )\right )\right )}{4 a x^2}}{16 a}-\frac {\left (a+b x+c x^2\right )^{3/2} (x (8 a B+3 A b)+6 a A)}{24 a x^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {-128 a^2 B c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )-\frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a}}}{8 a}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+x \left (8 a B \left (8 a c+b^2\right )-3 A \left (b^3-4 a b c\right )\right )\right )}{4 a x^2}}{16 a}-\frac {\left (a+b x+c x^2\right )^{3/2} (x (8 a B+3 A b)+6 a A)}{24 a x^4}\)

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^5,x]
 

Output:

-1/24*((6*a*A + (3*A*b + 8*a*B)*x)*(a + b*x + c*x^2)^(3/2))/(a*x^4) + (-1/ 
4*((2*a*(8*a*b*B - 3*A*(b^2 - 4*a*c)) + (8*a*B*(b^2 + 8*a*c) - 3*A*(b^3 - 
4*a*b*c))*x)*Sqrt[a + b*x + c*x^2])/(a*x^2) - (-(((8*a*b*B*(b^2 - 12*a*c) 
- 3*A*(b^2 - 4*a*c)^2)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2 
])])/Sqrt[a]) - 128*a^2*B*c^(3/2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + 
b*x + c*x^2])])/(8*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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

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 1229

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 
)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.04

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (60 A a b c \,x^{3}-9 A \,b^{3} x^{3}+256 B \,a^{2} c \,x^{3}+24 B a \,b^{2} x^{3}+120 A \,a^{2} c \,x^{2}+6 A a \,b^{2} x^{2}+112 B \,a^{2} b \,x^{2}+72 A \,a^{2} b x +64 B \,a^{3} x +48 a^{3} A \right )}{192 x^{4} a^{2}}+\frac {-\frac {\left (48 a^{2} A \,c^{2}-24 A a \,b^{2} c +3 A \,b^{4}+96 a^{2} b B c -8 B a \,b^{3}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{\sqrt {a}}+128 B \,a^{2} c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 a^{2}}\)	\(225\)
default	\(\text {Expression too large to display}\)	\(2180\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^5,x,method=_RETURNVERBOSE)
 

Output:

-1/192*(c*x^2+b*x+a)^(1/2)*(60*A*a*b*c*x^3-9*A*b^3*x^3+256*B*a^2*c*x^3+24* 
B*a*b^2*x^3+120*A*a^2*c*x^2+6*A*a*b^2*x^2+112*B*a^2*b*x^2+72*A*a^2*b*x+64* 
B*a^3*x+48*A*a^3)/x^4/a^2+1/128/a^2*(-(48*A*a^2*c^2-24*A*a*b^2*c+3*A*b^4+9 
6*B*a^2*b*c-8*B*a*b^3)/a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/ 
x)+128*B*a^2*c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 1.32 (sec) , antiderivative size = 1083, normalized size of antiderivative = 4.99 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^5,x, algorithm="fricas")
 

Output:

[1/768*(384*B*a^3*c^(3/2)*x^4*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^ 
2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 3*(8*B*a*b^3 - 3*A*b^4 - 48*A* 
a^2*c^2 - 24*(4*B*a^2*b - A*a*b^2)*c)*sqrt(a)*x^4*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* 
(48*A*a^4 + (24*B*a^2*b^2 - 9*A*a*b^3 + 4*(64*B*a^3 + 15*A*a^2*b)*c)*x^3 + 
 2*(56*B*a^3*b + 3*A*a^2*b^2 + 60*A*a^3*c)*x^2 + 8*(8*B*a^4 + 9*A*a^3*b)*x 
)*sqrt(c*x^2 + b*x + a))/(a^3*x^4), -1/768*(768*B*a^3*sqrt(-c)*c*x^4*arcta 
n(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) 
+ 3*(8*B*a*b^3 - 3*A*b^4 - 48*A*a^2*c^2 - 24*(4*B*a^2*b - A*a*b^2)*c)*sqrt 
(a)*x^4*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*(48*A*a^4 + (24*B*a^2*b^2 - 9*A*a*b^3 + 4* 
(64*B*a^3 + 15*A*a^2*b)*c)*x^3 + 2*(56*B*a^3*b + 3*A*a^2*b^2 + 60*A*a^3*c) 
*x^2 + 8*(8*B*a^4 + 9*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^4), 1/384* 
(192*B*a^3*c^(3/2)*x^4*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x 
 + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 3*(8*B*a*b^3 - 3*A*b^4 - 48*A*a^2*c^2 
 - 24*(4*B*a^2*b - A*a*b^2)*c)*sqrt(-a)*x^4*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*(48*A*a^4 + (24*B*a^2 
*b^2 - 9*A*a*b^3 + 4*(64*B*a^3 + 15*A*a^2*b)*c)*x^3 + 2*(56*B*a^3*b + 3*A* 
a^2*b^2 + 60*A*a^3*c)*x^2 + 8*(8*B*a^4 + 9*A*a^3*b)*x)*sqrt(c*x^2 + b*x + 
a))/(a^3*x^4), -1/384*(384*B*a^3*sqrt(-c)*c*x^4*arctan(1/2*sqrt(c*x^2 +...
 

Sympy [F]

\[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x^{5}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/x**5,x)
 

Output:

Integral((A + B*x)*(a + b*x + c*x**2)**(3/2)/x**5, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^5,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. 1016 vs. \(2 (191) = 382\).

Time = 0.35 (sec) , antiderivative size = 1016, normalized size of antiderivative = 4.68 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^5,x, algorithm="giac")
 

Output:

-B*c^(3/2)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b)) - 1 
/64*(8*B*a*b^3 - 3*A*b^4 - 96*B*a^2*b*c + 24*A*a*b^2*c - 48*A*a^2*c^2)*arc 
tan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^2) + 1/192* 
(24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a*b^3 - 9*(sqrt(c)*x - sqrt(c* 
x^2 + b*x + a))^7*A*b^4 + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^2* 
b*c + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a*b^2*c + 240*(sqrt(c)*x 
- sqrt(c*x^2 + b*x + a))^7*A*a^2*c^2 + 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
 a))^6*B*a^2*b^2*sqrt(c) + 768*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^3 
*c^(3/2) + 768*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^2*b*c^(3/2) + 40* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^2*b^3 + 33*(sqrt(c)*x - sqrt(c*x 
^2 + b*x + a))^5*A*a*b^4 - 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^3 
*b*c + 504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^2*b^2*c + 144*(sqrt(c 
)*x - sqrt(c*x^2 + b*x + a))^5*A*a^3*c^2 - 384*(sqrt(c)*x - sqrt(c*x^2 + b 
*x + a))^4*B*a^3*b^2*sqrt(c) + 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A 
*a^2*b^3*sqrt(c) - 1536*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^4*c^(3/2 
) - 88*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^3*b^3 + 33*(sqrt(c)*x - s 
qrt(c*x^2 + b*x + a))^3*A*a^2*b^4 + 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a) 
)^3*B*a^4*b*c + 504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^3*b^2*c + 14 
4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^4*c^2 + 1280*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^2*B*a^5*c^(3/2) + 768*(sqrt(c)*x - sqrt(c*x^2 + b*x ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^5} \,d x \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^5,x)
 

Output:

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^5, x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.43 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx=\frac {-96 \sqrt {c \,x^{2}+b x +a}\, a^{4}-272 \sqrt {c \,x^{2}+b x +a}\, a^{3} b x -240 \sqrt {c \,x^{2}+b x +a}\, a^{3} c \,x^{2}-236 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} x^{2}-632 \sqrt {c \,x^{2}+b x +a}\, a^{2} b c \,x^{3}-30 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} x^{3}+144 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} c^{2} x^{4}+216 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,b^{2} c \,x^{4}-15 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{4} x^{4}-144 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} c^{2} x^{4}-216 \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{2} c \,x^{4}+15 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{4} x^{4}+384 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) a^{2} b c \,x^{4}}{384 a^{2} x^{4}} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^5,x)
 

Output:

( - 96*sqrt(a + b*x + c*x**2)*a**4 - 272*sqrt(a + b*x + c*x**2)*a**3*b*x - 
 240*sqrt(a + b*x + c*x**2)*a**3*c*x**2 - 236*sqrt(a + b*x + c*x**2)*a**2* 
b**2*x**2 - 632*sqrt(a + b*x + c*x**2)*a**2*b*c*x**3 - 30*sqrt(a + b*x + c 
*x**2)*a*b**3*x**3 + 144*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2* 
a - b*x)*a**2*c**2*x**4 + 216*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) 
 - 2*a - b*x)*a*b**2*c*x**4 - 15*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x* 
*2) - 2*a - b*x)*b**4*x**4 - 144*sqrt(a)*log(x)*a**2*c**2*x**4 - 216*sqrt( 
a)*log(x)*a*b**2*c*x**4 + 15*sqrt(a)*log(x)*b**4*x**4 + 384*sqrt(c)*log( - 
 2*sqrt(c)*sqrt(a + b*x + c*x**2) - b - 2*c*x)*a**2*b*c*x**4)/(384*a**2*x* 
*4)