\(\int \frac {(A+B x) (a+b x+c x^2)^{5/2}}{x^3} \, dx\) [136]

Optimal result

Integrand size = 23, antiderivative size = 273 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^3} \, dx=\frac {5 \left (b^3 B+40 A b^2 c+44 a b B c+32 a A c^2+2 c \left (b^2 B+16 A b c+12 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c}-\frac {5 (6 (A b+a B)-(b B+4 A c) x) \left (a+b x+c x^2\right )^{3/2}}{24 x}-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}-\frac {5}{8} \sqrt {a} \left (3 A b^2+4 a b B+4 a A c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )-\frac {5 \left (b^4 B-8 A b^3 c-24 a b^2 B c-96 a A b c^2-48 a^2 B c^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{3/2}} \] Output:

5/64*(B*b^3+40*A*b^2*c+44*B*a*b*c+32*A*a*c^2+2*c*(16*A*b*c+12*B*a*c+B*b^2) 
*x)*(c*x^2+b*x+a)^(1/2)/c-5/24*(6*A*b+6*B*a-(4*A*c+B*b)*x)*(c*x^2+b*x+a)^( 
3/2)/x-1/4*(-B*x+2*A)*(c*x^2+b*x+a)^(5/2)/x^2-5/8*a^(1/2)*(4*A*a*c+3*A*b^2 
+4*B*a*b)*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))-5/128*(-96*A* 
a*b*c^2-8*A*b^3*c-48*B*a^2*c^2-24*B*a*b^2*c+B*b^4)*arctanh(1/2*(2*c*x+b)/c 
^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)
 

Mathematica [A] (verified)

Time = 2.45 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^3} \, dx=\frac {\sqrt {a+x (b+c x)} \left (-96 a^2 c (A+2 B x)+x^2 \left (15 b^3 B+16 c^3 x^2 (4 A+3 B x)+8 b c^2 x (26 A+17 B x)+2 b^2 c (132 A+59 B x)\right )+4 a c x (-4 A (27 b-28 c x)+B x (139 b+54 c x))\right )}{192 c x^2}+\frac {5 \left (-b^4 B+8 A b^3 c+24 a b^2 B c+96 a A b c^2+48 a^2 B c^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{128 c^{3/2}}-\frac {5}{4} \sqrt {a} \left (3 A b^2+4 a b B+4 a A c\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right ) \] Input:

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

Output:

(Sqrt[a + x*(b + c*x)]*(-96*a^2*c*(A + 2*B*x) + x^2*(15*b^3*B + 16*c^3*x^2 
*(4*A + 3*B*x) + 8*b*c^2*x*(26*A + 17*B*x) + 2*b^2*c*(132*A + 59*B*x)) + 4 
*a*c*x*(-4*A*(27*b - 28*c*x) + B*x*(139*b + 54*c*x))))/(192*c*x^2) + (5*(- 
(b^4*B) + 8*A*b^3*c + 24*a*b^2*B*c + 96*a*A*b*c^2 + 48*a^2*B*c^2)*ArcTanh[ 
(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(128*c^(3/2)) - (5*Sqrt[a] 
*(3*A*b^2 + 4*a*b*B + 4*a*A*c)*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x 
)])/Sqrt[a]])/4
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {1230, 27, 1230, 25, 1231, 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.

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a B c+16 A b c+b^2 B\right )+32 a A c^2+44 a b B c+40 A b^2 c+b^3 B\right )}{4 c}-\frac {\int -\frac {16 a c \left (3 A b^2+4 a B b+4 a A c\right )-\left (B b^4-8 A c b^3-24 a B c b^2-96 a A c^2 b-48 a^2 B c^2\right ) x}{2 x \sqrt {c x^2+b x+a}}dx}{4 c}\right )-\frac {\left (a+b x+c x^2\right )^{3/2} (6 (a B+A b)-x (4 A c+b B))}{3 x}\right )-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {\int \frac {16 a c \left (3 A b^2+4 a B b+4 a A c\right )-\left (B b^4-8 A c b^3-24 a B c b^2-96 a A c^2 b-48 a^2 B c^2\right ) x}{x \sqrt {c x^2+b x+a}}dx}{8 c}+\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a B c+16 A b c+b^2 B\right )+32 a A c^2+44 a b B c+40 A b^2 c+b^3 B\right )}{4 c}\right )-\frac {\left (a+b x+c x^2\right )^{3/2} (6 (a B+A b)-x (4 A c+b B))}{3 x}\right )-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {16 a c \left (4 a A c+4 a b B+3 A b^2\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx-\left (-48 a^2 B c^2-96 a A b c^2-24 a b^2 B c-8 A b^3 c+b^4 B\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}+\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a B c+16 A b c+b^2 B\right )+32 a A c^2+44 a b B c+40 A b^2 c+b^3 B\right )}{4 c}\right )-\frac {\left (a+b x+c x^2\right )^{3/2} (6 (a B+A b)-x (4 A c+b B))}{3 x}\right )-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {16 a c \left (4 a A c+4 a b B+3 A b^2\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx-2 \left (-48 a^2 B c^2-96 a A b c^2-24 a b^2 B c-8 A b^3 c+b^4 B\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{8 c}+\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a B c+16 A b c+b^2 B\right )+32 a A c^2+44 a b B c+40 A b^2 c+b^3 B\right )}{4 c}\right )-\frac {\left (a+b x+c x^2\right )^{3/2} (6 (a B+A b)-x (4 A c+b B))}{3 x}\right )-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {16 a c \left (4 a A c+4 a b B+3 A b^2\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx-\frac {\left (-48 a^2 B c^2-96 a A b c^2-24 a b^2 B c-8 A b^3 c+b^4 B\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}}{8 c}+\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a B c+16 A b c+b^2 B\right )+32 a A c^2+44 a b B c+40 A b^2 c+b^3 B\right )}{4 c}\right )-\frac {\left (a+b x+c x^2\right )^{3/2} (6 (a B+A b)-x (4 A c+b B))}{3 x}\right )-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {-32 a c \left (4 a A c+4 a b B+3 A b^2\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}-\frac {\left (-48 a^2 B c^2-96 a A b c^2-24 a b^2 B c-8 A b^3 c+b^4 B\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}}{8 c}+\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a B c+16 A b c+b^2 B\right )+32 a A c^2+44 a b B c+40 A b^2 c+b^3 B\right )}{4 c}\right )-\frac {\left (a+b x+c x^2\right )^{3/2} (6 (a B+A b)-x (4 A c+b B))}{3 x}\right )-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {-\frac {\left (-48 a^2 B c^2-96 a A b c^2-24 a b^2 B c-8 A b^3 c+b^4 B\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}-16 \sqrt {a} c \left (4 a A c+4 a b B+3 A b^2\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 c}+\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a B c+16 A b c+b^2 B\right )+32 a A c^2+44 a b B c+40 A b^2 c+b^3 B\right )}{4 c}\right )-\frac {\left (a+b x+c x^2\right )^{3/2} (6 (a B+A b)-x (4 A c+b B))}{3 x}\right )-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}\)

Input:

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

Output:

-1/4*((2*A - B*x)*(a + b*x + c*x^2)^(5/2))/x^2 + (5*(-1/3*((6*(A*b + a*B) 
- (b*B + 4*A*c)*x)*(a + b*x + c*x^2)^(3/2))/x + (((b^3*B + 40*A*b^2*c + 44 
*a*b*B*c + 32*a*A*c^2 + 2*c*(b^2*B + 16*A*b*c + 12*a*B*c)*x)*Sqrt[a + b*x 
+ c*x^2])/(4*c) + (-16*Sqrt[a]*c*(3*A*b^2 + 4*a*b*B + 4*a*A*c)*ArcTanh[(2* 
a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])] - ((b^4*B - 8*A*b^3*c - 24*a*b 
^2*B*c - 96*a*A*b*c^2 - 48*a^2*B*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[ 
a + b*x + c*x^2])])/Sqrt[c])/(8*c))/2))/8
 

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 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 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1231

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(504\) vs. \(2(240)=480\).

Time = 1.35 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.85

method	result	size
risch	\(-\frac {a \sqrt {c \,x^{2}+b x +a}\, \left (9 A b x +4 B a x +2 A a \right )}{4 x^{2}}+\frac {c^{2} x^{2} \sqrt {c \,x^{2}+b x +a}\, A}{3}+\frac {59 b^{2} x \sqrt {c \,x^{2}+b x +a}\, B}{96}+\frac {5 b^{3} \sqrt {c \,x^{2}+b x +a}\, B}{64 c}+\frac {7 c a \sqrt {c \,x^{2}+b x +a}\, A}{3}+\frac {139 a \sqrt {c \,x^{2}+b x +a}\, B b}{48}-\frac {5 A \,a^{\frac {3}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) c}{2}-\frac {5 B \,a^{\frac {3}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b}{2}+\frac {17 c \,x^{2} \sqrt {c \,x^{2}+b x +a}\, B b}{24}+\frac {13 c b x \sqrt {c \,x^{2}+b x +a}\, A}{12}-\frac {5 b^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) B}{128 c^{\frac {3}{2}}}+\frac {15 B a \,b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 \sqrt {c}}+\frac {15 B \,a^{2} \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8}+\frac {15 A a b \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4}+\frac {B \,c^{2} x^{3} \sqrt {c \,x^{2}+b x +a}}{4}-\frac {15 A \sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b^{2}}{8}+\frac {9 c x \sqrt {c \,x^{2}+b x +a}\, a B}{8}+\frac {5 A \,b^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 \sqrt {c}}+\frac {11 b^{2} \sqrt {c \,x^{2}+b x +a}\, A}{8}\)	\(505\)
default	\(\text {Expression too large to display}\)	\(1244\)

Input:

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

Output:

-1/4*a*(c*x^2+b*x+a)^(1/2)*(9*A*b*x+4*B*a*x+2*A*a)/x^2+1/3*c^2*x^2*(c*x^2+ 
b*x+a)^(1/2)*A+59/96*b^2*x*(c*x^2+b*x+a)^(1/2)*B+5/64/c*b^3*(c*x^2+b*x+a)^ 
(1/2)*B+7/3*c*a*(c*x^2+b*x+a)^(1/2)*A+139/48*a*(c*x^2+b*x+a)^(1/2)*B*b-5/2 
*A*a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*c-5/2*B*a^(3/2)*l 
n((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*b+17/24*c*x^2*(c*x^2+b*x+a)^( 
1/2)*B*b+13/12*c*b*x*(c*x^2+b*x+a)^(1/2)*A-5/128/c^(3/2)*b^4*ln((1/2*b+c*x 
)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B+15/16*B*a*b^2*ln((1/2*b+c*x)/c^(1/2)+(c*x 
^2+b*x+a)^(1/2))/c^(1/2)+15/8*B*a^2*c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+ 
b*x+a)^(1/2))+15/4*A*a*b*c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2 
))+1/4*B*c^2*x^3*(c*x^2+b*x+a)^(1/2)-15/8*A*a^(1/2)*ln((2*a+b*x+2*a^(1/2)* 
(c*x^2+b*x+a)^(1/2))/x)*b^2+9/8*c*x*(c*x^2+b*x+a)^(1/2)*a*B+5/16*A*b^3*ln( 
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+11/8*b^2*(c*x^2+b*x+a)^(1 
/2)*A
 

Fricas [A] (verification not implemented)

Time = 2.88 (sec) , antiderivative size = 1269, normalized size of antiderivative = 4.65 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/768*(15*(B*b^4 - 48*(B*a^2 + 2*A*a*b)*c^2 - 8*(3*B*a*b^2 + A*b^3)*c)*s 
qrt(c)*x^2*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) - 240*(4*A*a*c^3 + (4*B*a*b + 3*A*b^2)*c^2)*sqrt(a) 
*x^2*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*B*c^4*x^5 - 96*A*a^2*c^2 + 8*(17*B*b*c^3 
+ 8*A*c^4)*x^4 - 48*(4*B*a^2 + 9*A*a*b)*c^2*x + 2*(59*B*b^2*c^2 + 4*(27*B* 
a + 26*A*b)*c^3)*x^3 + (15*B*b^3*c + 448*A*a*c^3 + 4*(139*B*a*b + 66*A*b^2 
)*c^2)*x^2)*sqrt(c*x^2 + b*x + a))/(c^2*x^2), 1/384*(15*(B*b^4 - 48*(B*a^2 
 + 2*A*a*b)*c^2 - 8*(3*B*a*b^2 + A*b^3)*c)*sqrt(-c)*x^2*arctan(1/2*sqrt(c* 
x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 120*(4*A*a* 
c^3 + (4*B*a*b + 3*A*b^2)*c^2)*sqrt(a)*x^2*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) + 2*(48*B*c 
^4*x^5 - 96*A*a^2*c^2 + 8*(17*B*b*c^3 + 8*A*c^4)*x^4 - 48*(4*B*a^2 + 9*A*a 
*b)*c^2*x + 2*(59*B*b^2*c^2 + 4*(27*B*a + 26*A*b)*c^3)*x^3 + (15*B*b^3*c + 
 448*A*a*c^3 + 4*(139*B*a*b + 66*A*b^2)*c^2)*x^2)*sqrt(c*x^2 + b*x + a))/( 
c^2*x^2), 1/768*(480*(4*A*a*c^3 + (4*B*a*b + 3*A*b^2)*c^2)*sqrt(-a)*x^2*ar 
ctan(1/2*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(-a)/(a*c*x^2 + a*b*x + a^2 
)) - 15*(B*b^4 - 48*(B*a^2 + 2*A*a*b)*c^2 - 8*(3*B*a*b^2 + A*b^3)*c)*sqrt( 
c)*x^2*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) + 4*(48*B*c^4*x^5 - 96*A*a^2*c^2 + 8*(17*B*b*c^3 + 8...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^3,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. 527 vs. \(2 (239) = 478\).

Time = 0.29 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.93 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^3} \, dx=\frac {1}{192} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, B c^{2} x + \frac {17 \, B b c^{4} + 8 \, A c^{5}}{c^{3}}\right )} x + \frac {59 \, B b^{2} c^{3} + 108 \, B a c^{4} + 104 \, A b c^{4}}{c^{3}}\right )} x + \frac {15 \, B b^{3} c^{2} + 556 \, B a b c^{3} + 264 \, A b^{2} c^{3} + 448 \, A a c^{4}}{c^{3}}\right )} + \frac {5 \, {\left (4 \, B a^{2} b + 3 \, A a b^{2} + 4 \, A a^{2} c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a}} + \frac {5 \, {\left (B b^{4} - 24 \, B a b^{2} c - 8 \, A b^{3} c - 48 \, B a^{2} c^{2} - 96 \, A a b c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {3}{2}}} + \frac {4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{2} b + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{2} c + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{3} \sqrt {c} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{2} b \sqrt {c} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{3} b - 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{3} c - 8 \, B a^{4} \sqrt {c} - 16 \, A a^{3} b \sqrt {c}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{2}} \] Input:

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

Output:

1/192*sqrt(c*x^2 + b*x + a)*(2*(4*(6*B*c^2*x + (17*B*b*c^4 + 8*A*c^5)/c^3) 
*x + (59*B*b^2*c^3 + 108*B*a*c^4 + 104*A*b*c^4)/c^3)*x + (15*B*b^3*c^2 + 5 
56*B*a*b*c^3 + 264*A*b^2*c^3 + 448*A*a*c^4)/c^3) + 5/4*(4*B*a^2*b + 3*A*a* 
b^2 + 4*A*a^2*c)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/sqr 
t(-a) + 5/128*(B*b^4 - 24*B*a*b^2*c - 8*A*b^3*c - 48*B*a^2*c^2 - 96*A*a*b* 
c^2)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(3/2) + 
 1/4*(4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^2*b + 9*(sqrt(c)*x - sqr 
t(c*x^2 + b*x + a))^3*A*a*b^2 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A* 
a^2*c + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^3*sqrt(c) + 24*(sqrt(c 
)*x - sqrt(c*x^2 + b*x + a))^2*A*a^2*b*sqrt(c) - 4*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))*B*a^3*b - 7*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*b^2 + 4 
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^3*c - 8*B*a^4*sqrt(c) - 16*A*a^3* 
b*sqrt(c))/((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.59 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^3} \, dx=\frac {-192 \sqrt {c \,x^{2}+b x +a}\, a^{3} c^{2}-1248 \sqrt {c \,x^{2}+b x +a}\, a^{2} b \,c^{2} x +896 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{3} x^{2}+1640 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{2} x^{2}+848 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{3} x^{3}+128 \sqrt {c \,x^{2}+b x +a}\, a \,c^{4} x^{4}+30 \sqrt {c \,x^{2}+b x +a}\, b^{4} c \,x^{2}+236 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{2} x^{3}+272 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{3} x^{4}+96 \sqrt {c \,x^{2}+b x +a}\, b \,c^{4} x^{5}+960 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} c^{3} x^{2}+1680 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,b^{2} c^{2} x^{2}-960 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} c^{3} x^{2}-1680 \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{2} c^{2} x^{2}+2160 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) a^{2} b \,c^{2} x^{2}+480 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) a \,b^{3} c \,x^{2}-15 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) b^{5} x^{2}}{384 c^{2} x^{2}} \] Input:

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

Output:

( - 192*sqrt(a + b*x + c*x**2)*a**3*c**2 - 1248*sqrt(a + b*x + c*x**2)*a** 
2*b*c**2*x + 896*sqrt(a + b*x + c*x**2)*a**2*c**3*x**2 + 1640*sqrt(a + b*x 
 + c*x**2)*a*b**2*c**2*x**2 + 848*sqrt(a + b*x + c*x**2)*a*b*c**3*x**3 + 1 
28*sqrt(a + b*x + c*x**2)*a*c**4*x**4 + 30*sqrt(a + b*x + c*x**2)*b**4*c*x 
**2 + 236*sqrt(a + b*x + c*x**2)*b**3*c**2*x**3 + 272*sqrt(a + b*x + c*x** 
2)*b**2*c**3*x**4 + 96*sqrt(a + b*x + c*x**2)*b*c**4*x**5 + 960*sqrt(a)*lo 
g(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*c**3*x**2 + 1680*sqrt 
(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b**2*c**2*x**2 - 9 
60*sqrt(a)*log(x)*a**2*c**3*x**2 - 1680*sqrt(a)*log(x)*a*b**2*c**2*x**2 + 
2160*sqrt(c)*log( - 2*sqrt(c)*sqrt(a + b*x + c*x**2) - b - 2*c*x)*a**2*b*c 
**2*x**2 + 480*sqrt(c)*log( - 2*sqrt(c)*sqrt(a + b*x + c*x**2) - b - 2*c*x 
)*a*b**3*c*x**2 - 15*sqrt(c)*log( - 2*sqrt(c)*sqrt(a + b*x + c*x**2) - b - 
 2*c*x)*b**5*x**2)/(384*c**2*x**2)