\(\int \frac {(A+B x) (a+b x+c x^2)^{3/2}}{x^4} \, dx\) [124]

Optimal result

Integrand size = 23, antiderivative size = 206 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx=\frac {\left (A b^2-6 a b B-8 a A c+2 (A b+6 a B) c x\right ) \sqrt {a+b x+c x^2}}{8 a x}-\frac {(4 a A+3 (A b+2 a B) x) \left (a+b x+c x^2\right )^{3/2}}{12 a x^3}-\frac {\left (6 a B \left (b^2+4 a c\right )-A \left (b^3-12 a b c\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{3/2}}+\frac {1}{2} \sqrt {c} (3 b B+2 A c) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \] Output:

1/8*(A*b^2-6*a*b*B-8*A*a*c+2*(A*b+6*B*a)*c*x)*(c*x^2+b*x+a)^(1/2)/a/x-1/12 
*(4*a*A+3*(A*b+2*B*a)*x)*(c*x^2+b*x+a)^(3/2)/a/x^3-1/16*(6*a*B*(4*a*c+b^2) 
-A*(-12*a*b*c+b^3))*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^( 
3/2)+1/2*c^(1/2)*(2*A*c+3*B*b)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a) 
^(1/2))
 

Mathematica [A] (verified)

Time = 1.76 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.88 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx=-\frac {\sqrt {a+x (b+c x)} \left (3 A b^2 x^2+4 a^2 (2 A+3 B x)+2 a x (3 B x (5 b-4 c x)+A (7 b+16 c x))\right )}{24 a x^3}+\frac {\left (-6 a B \left (b^2+4 a c\right )+A \left (b^3-12 a b c\right )\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{8 a^{3/2}}-\frac {1}{2} \sqrt {c} (3 b B+2 A c) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right ) \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^4,x]
 

Output:

-1/24*(Sqrt[a + x*(b + c*x)]*(3*A*b^2*x^2 + 4*a^2*(2*A + 3*B*x) + 2*a*x*(3 
*B*x*(5*b - 4*c*x) + A*(7*b + 16*c*x))))/(a*x^3) + ((-6*a*B*(b^2 + 4*a*c) 
+ A*(b^3 - 12*a*b*c))*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[ 
a]])/(8*a^(3/2)) - (Sqrt[c]*(3*b*B + 2*A*c)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt 
[a + x*(b + c*x)]])/2
 

Rubi [F]

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.

\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx\)

\(\Big \downarrow \) 1229

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^4,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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]
 

Maple [A] (verified)

Time = 1.33 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.23

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (32 A a c \,x^{2}+3 x^{2} b^{2} A +30 B a \,x^{2} b +14 a b A x +12 a^{2} B x +8 a^{2} A \right )}{24 x^{3} a}+\frac {-\frac {\left (12 A a b c -A \,b^{3}+24 B \,a^{2} c +6 B a \,b^{2}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{\sqrt {a}}+16 A a \,c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )+32 B a b \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )+16 B a \,c^{2} \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{16 a}\)	\(254\)
default	\(\text {Expression too large to display}\)	\(1323\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/24*(c*x^2+b*x+a)^(1/2)*(32*A*a*c*x^2+3*A*b^2*x^2+30*B*a*b*x^2+14*A*a*b* 
x+12*B*a^2*x+8*A*a^2)/x^3/a+1/16/a*(-(12*A*a*b*c-A*b^3+24*B*a^2*c+6*B*a*b^ 
2)/a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+16*A*a*c^(3/2)*ln 
((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+32*B*a*b*c^(1/2)*ln((1/2*b+c*x)/ 
c^(1/2)+(c*x^2+b*x+a)^(1/2))+16*B*a*c^2*(1/c*(c*x^2+b*x+a)^(1/2)-1/2*b/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 = 0.99 (sec) , antiderivative size = 953, normalized size of antiderivative = 4.63 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^4,x, algorithm="fricas")
 

Output:

[1/96*(24*(3*B*a^2*b + 2*A*a^2*c)*sqrt(c)*x^3*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*(6*B*a*b^2 - 
 A*b^3 + 12*(2*B*a^2 + A*a*b)*c)*sqrt(a)*x^3*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*(24*B 
*a^2*c*x^3 - 8*A*a^3 - (30*B*a^2*b + 3*A*a*b^2 + 32*A*a^2*c)*x^2 - 2*(6*B* 
a^3 + 7*A*a^2*b)*x)*sqrt(c*x^2 + b*x + a))/(a^2*x^3), -1/96*(48*(3*B*a^2*b 
 + 2*A*a^2*c)*sqrt(-c)*x^3*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sq 
rt(-c)/(c^2*x^2 + b*c*x + a*c)) - 3*(6*B*a*b^2 - A*b^3 + 12*(2*B*a^2 + A*a 
*b)*c)*sqrt(a)*x^3*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*(24*B*a^2*c*x^3 - 8*A*a^3 - (30 
*B*a^2*b + 3*A*a*b^2 + 32*A*a^2*c)*x^2 - 2*(6*B*a^3 + 7*A*a^2*b)*x)*sqrt(c 
*x^2 + b*x + a))/(a^2*x^3), 1/48*(3*(6*B*a*b^2 - A*b^3 + 12*(2*B*a^2 + A*a 
*b)*c)*sqrt(-a)*x^3*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)) + 12*(3*B*a^2*b + 2*A*a^2*c)*sqrt(c)*x^3*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) + 2*(24*B*a^2*c*x^3 - 8*A*a^3 - (30*B*a^2*b + 3*A*a*b^2 + 32*A*a^2*c)* 
x^2 - 2*(6*B*a^3 + 7*A*a^2*b)*x)*sqrt(c*x^2 + b*x + a))/(a^2*x^3), 1/48*(3 
*(6*B*a*b^2 - A*b^3 + 12*(2*B*a^2 + A*a*b)*c)*sqrt(-a)*x^3*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)) - 24*(3*B* 
a^2*b + 2*A*a^2*c)*sqrt(-c)*x^3*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x...
 

Sympy [F]

\[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/x**4,x)
 

Output:

Integral((A + B*x)*(a + b*x + c*x**2)**(3/2)/x**4, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^4,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. 627 vs. \(2 (179) = 358\).

Time = 0.29 (sec) , antiderivative size = 627, normalized size of antiderivative = 3.04 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx=\sqrt {c x^{2} + b x + a} B c - \frac {{\left (3 \, B b c + 2 \, A c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2 \, \sqrt {c}} + \frac {{\left (6 \, B a b^{2} - A b^{3} + 24 \, B a^{2} c + 12 \, A a b c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a} + \frac {30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a b^{2} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A b^{3} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{2} c + 60 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a b c + 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{2} b \sqrt {c} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a b^{2} \sqrt {c} + 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a^{2} c^{\frac {3}{2}} - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{2} b^{2} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a b^{3} - 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{3} b \sqrt {c} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{3} c^{\frac {3}{2}} + 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{3} b^{2} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} b^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{4} c + 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{3} b c + 48 \, B a^{4} b \sqrt {c} + 64 \, A a^{4} c^{\frac {3}{2}}}{24 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{3} a} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^4,x, algorithm="giac")
 

Output:

sqrt(c*x^2 + b*x + a)*B*c - 1/2*(3*B*b*c + 2*A*c^2)*log(abs(-2*(sqrt(c)*x 
- sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/sqrt(c) + 1/8*(6*B*a*b^2 - A*b^3 + 
24*B*a^2*c + 12*A*a*b*c)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt( 
-a))/(sqrt(-a)*a) + 1/24*(30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a*b^2 
 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*b^3 + 24*(sqrt(c)*x - sqrt(c* 
x^2 + b*x + a))^5*B*a^2*c + 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*b 
*c + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^2*b*sqrt(c) + 48*(sqrt(c 
)*x - sqrt(c*x^2 + b*x + a))^4*A*a*b^2*sqrt(c) + 96*(sqrt(c)*x - sqrt(c*x^ 
2 + b*x + a))^4*A*a^2*c^(3/2) - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B 
*a^2*b^2 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a*b^3 - 144*(sqrt(c)* 
x - sqrt(c*x^2 + b*x + a))^2*B*a^3*b*sqrt(c) - 96*(sqrt(c)*x - sqrt(c*x^2 
+ b*x + a))^2*A*a^3*c^(3/2) + 18*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^3 
*b^2 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*b^3 - 24*(sqrt(c)*x - s 
qrt(c*x^2 + b*x + a))*B*a^4*c + 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a 
^3*b*c + 48*B*a^4*b*sqrt(c) + 64*A*a^4*c^(3/2))/(((sqrt(c)*x - sqrt(c*x^2 
+ b*x + a))^2 - a)^3*a)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^4} \,d x \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^4,x)
 

Output:

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^4, x)
 

Reduce [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.28 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx=\frac {-16 \sqrt {c \,x^{2}+b x +a}\, a^{3}-52 \sqrt {c \,x^{2}+b x +a}\, a^{2} b x -64 \sqrt {c \,x^{2}+b x +a}\, a^{2} c \,x^{2}-66 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} x^{2}+48 \sqrt {c \,x^{2}+b x +a}\, a b c \,x^{3}+108 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a b c \,x^{3}+15 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{3} x^{3}-108 \sqrt {a}\, \mathrm {log}\left (x \right ) a b c \,x^{3}-15 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{3} x^{3}+48 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) a^{2} c \,x^{3}+72 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) a \,b^{2} x^{3}}{48 a \,x^{3}} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^4,x)
 

Output:

( - 16*sqrt(a + b*x + c*x**2)*a**3 - 52*sqrt(a + b*x + c*x**2)*a**2*b*x - 
64*sqrt(a + b*x + c*x**2)*a**2*c*x**2 - 66*sqrt(a + b*x + c*x**2)*a*b**2*x 
**2 + 48*sqrt(a + b*x + c*x**2)*a*b*c*x**3 + 108*sqrt(a)*log(2*sqrt(a)*sqr 
t(a + b*x + c*x**2) - 2*a - b*x)*a*b*c*x**3 + 15*sqrt(a)*log(2*sqrt(a)*sqr 
t(a + b*x + c*x**2) - 2*a - b*x)*b**3*x**3 - 108*sqrt(a)*log(x)*a*b*c*x**3 
 - 15*sqrt(a)*log(x)*b**3*x**3 + 48*sqrt(c)*log( - 2*sqrt(c)*sqrt(a + b*x 
+ c*x**2) - b - 2*c*x)*a**2*c*x**3 + 72*sqrt(c)*log( - 2*sqrt(c)*sqrt(a + 
b*x + c*x**2) - b - 2*c*x)*a*b**2*x**3)/(48*a*x**3)