\(\int \frac {(A+B x) (a+b x+c x^2)^{5/2}}{x^6} \, dx\) [139]

Optimal result

Integrand size = 23, antiderivative size = 346 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^6} \, dx=\frac {\left (10 a b B \left (b^2-20 a c\right )-A \left (3 b^4-28 a b^2 c+128 a^2 c^2\right )+2 c \left (10 a B \left (b^2+12 a c\right )-A \left (3 b^3-28 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 a^2 x}+\frac {\left (4 a \left (3 A b^2-10 a b B-16 a A c\right )-3 \left (10 a B \left (b^2+4 a c\right )-A \left (3 b^3-20 a b c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 a^2 x^3}-\frac {(8 a A+5 (A b+2 a B) x) \left (a+b x+c x^2\right )^{5/2}}{40 a x^5}+\frac {\left (10 a B \left (b^4-24 a b^2 c-48 a^2 c^2\right )-A \left (3 b^5-40 a b^3 c+240 a^2 b c^2\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{256 a^{5/2}}+\frac {1}{2} c^{3/2} (5 b B+2 A c) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \] Output:

1/128*(10*a*b*B*(-20*a*c+b^2)-A*(128*a^2*c^2-28*a*b^2*c+3*b^4)+2*c*(10*a*B 
*(12*a*c+b^2)-A*(-28*a*b*c+3*b^3))*x)*(c*x^2+b*x+a)^(1/2)/a^2/x+1/192*(4*a 
*(-16*A*a*c+3*A*b^2-10*B*a*b)-3*(10*a*B*(4*a*c+b^2)-A*(-20*a*b*c+3*b^3))*x 
)*(c*x^2+b*x+a)^(3/2)/a^2/x^3-1/40*(8*a*A+5*(A*b+2*B*a)*x)*(c*x^2+b*x+a)^( 
5/2)/a/x^5+1/256*(10*a*B*(-48*a^2*c^2-24*a*b^2*c+b^4)-A*(240*a^2*b*c^2-40* 
a*b^3*c+3*b^5))*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(5/2) 
+1/2*c^(3/2)*(2*A*c+5*B*b)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/ 
2))
 

Mathematica [A] (verified)

Time = 4.98 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.90 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^6} \, dx=\frac {-\frac {\sqrt {a+x (b+c x)} \left (-45 A b^4 x^4+96 a^4 (4 A+5 B x)+30 a b^2 x^3 (5 b B x+A (b+18 c x))+16 a^3 x (5 B x (17 b+27 c x)+A (63 b+88 c x))+4 a^2 x^2 \left (5 B x \left (59 b^2+278 b c x-96 c^2 x^2\right )+2 A \left (93 b^2+311 b c x+368 c^2 x^2\right )\right )\right )}{a^2 x^5}+\frac {45 \left (A b^5+160 a^3 B c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {150 b \left (b^3 B+4 A b^2 c-24 a b B c-24 a A c^2\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{a^{3/2}}-960 c^{3/2} (5 b B+2 A c) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{1920} \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^6,x]
 

Output:

(-((Sqrt[a + x*(b + c*x)]*(-45*A*b^4*x^4 + 96*a^4*(4*A + 5*B*x) + 30*a*b^2 
*x^3*(5*b*B*x + A*(b + 18*c*x)) + 16*a^3*x*(5*B*x*(17*b + 27*c*x) + A*(63* 
b + 88*c*x)) + 4*a^2*x^2*(5*B*x*(59*b^2 + 278*b*c*x - 96*c^2*x^2) + 2*A*(9 
3*b^2 + 311*b*c*x + 368*c^2*x^2))))/(a^2*x^5)) + (45*(A*b^5 + 160*a^3*B*c^ 
2)*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]])/a^(5/2) + (150*b* 
(b^3*B + 4*A*b^2*c - 24*a*b*B*c - 24*a*A*c^2)*ArcTanh[(-(Sqrt[c]*x) + Sqrt 
[a + x*(b + c*x)])/Sqrt[a]])/a^(3/2) - 960*c^(3/2)*(5*b*B + 2*A*c)*Log[b + 
 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/1920
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {1229, 27, 1229, 27, 1230, 25, 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.

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

\(\Big \downarrow \) 1229

\(\displaystyle -\frac {\int \frac {\left (3 A b^2-10 a B b-16 a A c-2 (A b+10 a B) c x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 x^4}dx}{8 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\left (3 A b^2-10 a B b-16 a A c-2 (A b+10 a B) c x\right ) \left (c x^2+b x+a\right )^{3/2}}{x^4}dx}{16 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5}\)

\(\Big \downarrow \) 1229

\(\displaystyle -\frac {-\frac {\int -\frac {\left (10 a b B \left (b^2-20 a c\right )-2 A \left (\frac {3 b^4}{2}-14 a c b^2+64 a^2 c^2\right )-2 c \left (10 a B \left (b^2+12 a c\right )-A \left (3 b^3-28 a b c\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{2 x^2}dx}{4 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (4 a \left (-16 a A c-10 a b B+3 A b^2\right )-3 x \left (10 a B \left (4 a c+b^2\right )-A \left (3 b^3-20 a b c\right )\right )\right )}{12 a x^3}}{16 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\int \frac {\left (10 a b B \left (b^2-20 a c\right )-A \left (3 b^4-28 a c b^2+128 a^2 c^2\right )-2 c \left (10 a B \left (b^2+12 a c\right )-A \left (3 b^3-28 a b c\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{x^2}dx}{8 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (4 a \left (-16 a A c-10 a b B+3 A b^2\right )-3 x \left (10 a B \left (4 a c+b^2\right )-A \left (3 b^3-20 a b c\right )\right )\right )}{12 a x^3}}{16 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5}\)

\(\Big \downarrow \) 1230

\(\displaystyle -\frac {\frac {-\frac {1}{2} \int -\frac {-128 a^2 (5 b B+2 A c) x c^2+10 a B \left (b^4-24 a c b^2-48 a^2 c^2\right )-A \left (3 b^5-40 a c b^3+240 a^2 c^2 b\right )}{x \sqrt {c x^2+b x+a}}dx-\frac {\sqrt {a+b x+c x^2} \left (-A \left (128 a^2 c^2-28 a b^2 c+3 b^4\right )+2 c x \left (10 a B \left (12 a c+b^2\right )-A \left (3 b^3-28 a b c\right )\right )+10 a b B \left (b^2-20 a c\right )\right )}{x}}{8 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (4 a \left (-16 a A c-10 a b B+3 A b^2\right )-3 x \left (10 a B \left (4 a c+b^2\right )-A \left (3 b^3-20 a b c\right )\right )\right )}{12 a x^3}}{16 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\frac {1}{2} \int \frac {-128 a^2 (5 b B+2 A c) x c^2+10 a B \left (b^4-24 a c b^2-48 a^2 c^2\right )-A \left (3 b^5-40 a c b^3+240 a^2 c^2 b\right )}{x \sqrt {c x^2+b x+a}}dx-\frac {\sqrt {a+b x+c x^2} \left (-A \left (128 a^2 c^2-28 a b^2 c+3 b^4\right )+2 c x \left (10 a B \left (12 a c+b^2\right )-A \left (3 b^3-28 a b c\right )\right )+10 a b B \left (b^2-20 a c\right )\right )}{x}}{8 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (4 a \left (-16 a A c-10 a b B+3 A b^2\right )-3 x \left (10 a B \left (4 a c+b^2\right )-A \left (3 b^3-20 a b c\right )\right )\right )}{12 a x^3}}{16 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5}\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {\frac {\frac {1}{2} \left (\left (10 a B \left (-48 a^2 c^2-24 a b^2 c+b^4\right )-A \left (240 a^2 b c^2-40 a b^3 c+3 b^5\right )\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx-128 a^2 c^2 (2 A c+5 b B) \int \frac {1}{\sqrt {c x^2+b x+a}}dx\right )-\frac {\sqrt {a+b x+c x^2} \left (-A \left (128 a^2 c^2-28 a b^2 c+3 b^4\right )+2 c x \left (10 a B \left (12 a c+b^2\right )-A \left (3 b^3-28 a b c\right )\right )+10 a b B \left (b^2-20 a c\right )\right )}{x}}{8 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (4 a \left (-16 a A c-10 a b B+3 A b^2\right )-3 x \left (10 a B \left (4 a c+b^2\right )-A \left (3 b^3-20 a b c\right )\right )\right )}{12 a x^3}}{16 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5}\)

\(\Big \downarrow \) 1092

\(\displaystyle -\frac {\frac {\frac {1}{2} \left (\left (10 a B \left (-48 a^2 c^2-24 a b^2 c+b^4\right )-A \left (240 a^2 b c^2-40 a b^3 c+3 b^5\right )\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx-256 a^2 c^2 (2 A c+5 b B) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}\right )-\frac {\sqrt {a+b x+c x^2} \left (-A \left (128 a^2 c^2-28 a b^2 c+3 b^4\right )+2 c x \left (10 a B \left (12 a c+b^2\right )-A \left (3 b^3-28 a b c\right )\right )+10 a b B \left (b^2-20 a c\right )\right )}{x}}{8 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (4 a \left (-16 a A c-10 a b B+3 A b^2\right )-3 x \left (10 a B \left (4 a c+b^2\right )-A \left (3 b^3-20 a b c\right )\right )\right )}{12 a x^3}}{16 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {\frac {1}{2} \left (\left (10 a B \left (-48 a^2 c^2-24 a b^2 c+b^4\right )-A \left (240 a^2 b c^2-40 a b^3 c+3 b^5\right )\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx-128 a^2 c^{3/2} (2 A c+5 b B) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )\right )-\frac {\sqrt {a+b x+c x^2} \left (-A \left (128 a^2 c^2-28 a b^2 c+3 b^4\right )+2 c x \left (10 a B \left (12 a c+b^2\right )-A \left (3 b^3-28 a b c\right )\right )+10 a b B \left (b^2-20 a c\right )\right )}{x}}{8 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (4 a \left (-16 a A c-10 a b B+3 A b^2\right )-3 x \left (10 a B \left (4 a c+b^2\right )-A \left (3 b^3-20 a b c\right )\right )\right )}{12 a x^3}}{16 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {\frac {\frac {1}{2} \left (-2 \left (10 a B \left (-48 a^2 c^2-24 a b^2 c+b^4\right )-A \left (240 a^2 b c^2-40 a b^3 c+3 b^5\right )\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}-128 a^2 c^{3/2} (2 A c+5 b B) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )\right )-\frac {\sqrt {a+b x+c x^2} \left (-A \left (128 a^2 c^2-28 a b^2 c+3 b^4\right )+2 c x \left (10 a B \left (12 a c+b^2\right )-A \left (3 b^3-28 a b c\right )\right )+10 a b B \left (b^2-20 a c\right )\right )}{x}}{8 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (4 a \left (-16 a A c-10 a b B+3 A b^2\right )-3 x \left (10 a B \left (4 a c+b^2\right )-A \left (3 b^3-20 a b c\right )\right )\right )}{12 a x^3}}{16 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {\frac {1}{2} \left (-\frac {\left (10 a B \left (-48 a^2 c^2-24 a b^2 c+b^4\right )-A \left (240 a^2 b c^2-40 a b^3 c+3 b^5\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a}}-128 a^2 c^{3/2} (2 A c+5 b B) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )\right )-\frac {\sqrt {a+b x+c x^2} \left (-A \left (128 a^2 c^2-28 a b^2 c+3 b^4\right )+2 c x \left (10 a B \left (12 a c+b^2\right )-A \left (3 b^3-28 a b c\right )\right )+10 a b B \left (b^2-20 a c\right )\right )}{x}}{8 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (4 a \left (-16 a A c-10 a b B+3 A b^2\right )-3 x \left (10 a B \left (4 a c+b^2\right )-A \left (3 b^3-20 a b c\right )\right )\right )}{12 a x^3}}{16 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5}\)

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^6,x]
 

Output:

-1/40*((8*a*A + 5*(A*b + 2*a*B)*x)*(a + b*x + c*x^2)^(5/2))/(a*x^5) - (-1/ 
12*((4*a*(3*A*b^2 - 10*a*b*B - 16*a*A*c) - 3*(10*a*B*(b^2 + 4*a*c) - A*(3* 
b^3 - 20*a*b*c))*x)*(a + b*x + c*x^2)^(3/2))/(a*x^3) + (-(((10*a*b*B*(b^2 
- 20*a*c) - A*(3*b^4 - 28*a*b^2*c + 128*a^2*c^2) + 2*c*(10*a*B*(b^2 + 12*a 
*c) - A*(3*b^3 - 28*a*b*c))*x)*Sqrt[a + b*x + c*x^2])/x) + (-(((10*a*B*(b^ 
4 - 24*a*b^2*c - 48*a^2*c^2) - A*(3*b^5 - 40*a*b^3*c + 240*a^2*b*c^2))*Arc 
Tanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/Sqrt[a]) - 128*a^2*c^ 
(3/2)*(5*b*B + 2*A*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2] 
)])/2)/(8*a))/(16*a)
 

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 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 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 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 = 1.50 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.12

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (2944 A \,a^{2} c^{2} x^{4}+540 A a \,b^{2} c \,x^{4}-45 A \,b^{4} x^{4}+5560 B \,a^{2} b c \,x^{4}+150 B a \,b^{3} x^{4}+2488 A \,a^{2} b c \,x^{3}+30 A a \,b^{3} x^{3}+2160 B \,a^{3} c \,x^{3}+1180 B \,a^{2} b^{2} x^{3}+1408 A \,a^{3} c \,x^{2}+744 A \,a^{2} b^{2} x^{2}+1360 B \,a^{3} b \,x^{2}+1008 A \,a^{3} b x +480 a^{4} B x +384 a^{4} A \right )}{1920 x^{5} a^{2}}+\frac {-\frac {\left (240 A \,a^{2} b \,c^{2}-40 A a \,b^{3} c +3 A \,b^{5}+480 B \,a^{3} c^{2}+240 B \,a^{2} b^{2} c -10 B a \,b^{4}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{\sqrt {a}}+256 a^{2} A \,c^{\frac {5}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )+768 B \,a^{2} b \,c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )+256 B \,a^{2} c^{3} \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{256 a^{2}}\)	\(389\)
default	\(\text {Expression too large to display}\)	\(5474\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-1/1920*(c*x^2+b*x+a)^(1/2)*(2944*A*a^2*c^2*x^4+540*A*a*b^2*c*x^4-45*A*b^4 
*x^4+5560*B*a^2*b*c*x^4+150*B*a*b^3*x^4+2488*A*a^2*b*c*x^3+30*A*a*b^3*x^3+ 
2160*B*a^3*c*x^3+1180*B*a^2*b^2*x^3+1408*A*a^3*c*x^2+744*A*a^2*b^2*x^2+136 
0*B*a^3*b*x^2+1008*A*a^3*b*x+480*B*a^4*x+384*A*a^4)/x^5/a^2+1/256/a^2*(-(2 
40*A*a^2*b*c^2-40*A*a*b^3*c+3*A*b^5+480*B*a^3*c^2+240*B*a^2*b^2*c-10*B*a*b 
^4)/a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+256*a^2*A*c^(5/2 
)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+768*B*a^2*b*c^(3/2)*ln((1/2* 
b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+256*B*a^2*c^3*(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 = 3.51 (sec) , antiderivative size = 1445, normalized size of antiderivative = 4.18 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^6} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^6,x, algorithm="fricas")
 

Output:

[1/7680*(1920*(5*B*a^3*b*c + 2*A*a^3*c^2)*sqrt(c)*x^5*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) - 15*(10 
*B*a*b^4 - 3*A*b^5 - 240*(2*B*a^3 + A*a^2*b)*c^2 - 40*(6*B*a^2*b^2 - A*a*b 
^3)*c)*sqrt(a)*x^5*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*(1920*B*a^3*c^2*x^5 - 384*A*a^5 
 - (150*B*a^2*b^3 - 45*A*a*b^4 + 2944*A*a^3*c^2 + 20*(278*B*a^3*b + 27*A*a 
^2*b^2)*c)*x^4 - 2*(590*B*a^3*b^2 + 15*A*a^2*b^3 + 4*(270*B*a^4 + 311*A*a^ 
3*b)*c)*x^3 - 8*(170*B*a^4*b + 93*A*a^3*b^2 + 176*A*a^4*c)*x^2 - 48*(10*B* 
a^5 + 21*A*a^4*b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^5), -1/7680*(3840*(5*B* 
a^3*b*c + 2*A*a^3*c^2)*sqrt(-c)*x^5*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)) + 15*(10*B*a*b^4 - 3*A*b^5 - 240* 
(2*B*a^3 + A*a^2*b)*c^2 - 40*(6*B*a^2*b^2 - A*a*b^3)*c)*sqrt(a)*x^5*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*(1920*B*a^3*c^2*x^5 - 384*A*a^5 - (150*B*a^2*b^3 - 45*A* 
a*b^4 + 2944*A*a^3*c^2 + 20*(278*B*a^3*b + 27*A*a^2*b^2)*c)*x^4 - 2*(590*B 
*a^3*b^2 + 15*A*a^2*b^3 + 4*(270*B*a^4 + 311*A*a^3*b)*c)*x^3 - 8*(170*B*a^ 
4*b + 93*A*a^3*b^2 + 176*A*a^4*c)*x^2 - 48*(10*B*a^5 + 21*A*a^4*b)*x)*sqrt 
(c*x^2 + b*x + a))/(a^3*x^5), -1/3840*(15*(10*B*a*b^4 - 3*A*b^5 - 240*(2*B 
*a^3 + A*a^2*b)*c^2 - 40*(6*B*a^2*b^2 - A*a*b^3)*c)*sqrt(-a)*x^5*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)) -...
 

Sympy [F]

\[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^6} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{x^{6}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**6,x)
 

Output:

Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**6, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^6,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. 1525 vs. \(2 (314) = 628\).

Time = 0.41 (sec) , antiderivative size = 1525, normalized size of antiderivative = 4.41 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^6} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^6,x, algorithm="giac")
 

Output:

sqrt(c*x^2 + b*x + a)*B*c^2 - 1/2*(5*B*b*c^2 + 2*A*c^3)*log(abs(-2*(sqrt(c 
)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/sqrt(c) - 1/128*(10*B*a*b^4 - 3 
*A*b^5 - 240*B*a^2*b^2*c + 40*A*a*b^3*c - 480*B*a^3*c^2 - 240*A*a^2*b*c^2) 
*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^2) + 1/ 
1920*(150*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a*b^4 - 45*(sqrt(c)*x - 
sqrt(c*x^2 + b*x + a))^9*A*b^5 + 7920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 
9*B*a^2*b^2*c + 600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a*b^3*c + 4320 
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^3*c^2 + 7920*(sqrt(c)*x - sqrt( 
c*x^2 + b*x + a))^9*A*a^2*b*c^2 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) 
^8*B*a^2*b^3*sqrt(c) + 23040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^3*b 
*c^(3/2) + 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*A*a^2*b^2*c^(3/2) + 
 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*A*a^3*c^(5/2) + 580*(sqrt(c)* 
x - sqrt(c*x^2 + b*x + a))^7*B*a^2*b^4 + 210*(sqrt(c)*x - sqrt(c*x^2 + b*x 
 + a))^7*A*a*b^5 - 13920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^3*b^2*c 
 + 6160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^2*b^3*c - 4800*(sqrt(c)* 
x - sqrt(c*x^2 + b*x + a))^7*B*a^4*c^2 - 2400*(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))^7*A*a^3*b*c^2 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^3*b 
^3*sqrt(c) + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^2*b^4*sqrt(c) 
- 57600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^4*b*c^(3/2) - 23040*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^4*c^(5/2) - 1280*(sqrt(c)*x - sqr...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^6} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{x^6} \,d x \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^6,x)
 

Output:

int(((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^6, x)
 

Reduce [B] (verification not implemented)

Time = 0.79 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.27 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^6} \, dx=\frac {-768 \sqrt {c \,x^{2}+b x +a}\, a^{5}-2976 \sqrt {c \,x^{2}+b x +a}\, a^{4} b x -2816 \sqrt {c \,x^{2}+b x +a}\, a^{4} c \,x^{2}-4208 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{2} x^{2}-9296 \sqrt {c \,x^{2}+b x +a}\, a^{3} b c \,x^{3}-5888 \sqrt {c \,x^{2}+b x +a}\, a^{3} c^{2} x^{4}-2420 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{3} x^{3}-12200 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} c \,x^{4}+3840 \sqrt {c \,x^{2}+b x +a}\, a^{2} b \,c^{2} x^{5}-210 \sqrt {c \,x^{2}+b x +a}\, a \,b^{4} x^{4}+10800 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} b \,c^{2} x^{5}+3000 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,b^{3} c \,x^{5}-105 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{5} x^{5}-10800 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} b \,c^{2} x^{5}-3000 \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{3} c \,x^{5}+105 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{5} x^{5}+3840 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) a^{3} c^{2} x^{5}+9600 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) a^{2} b^{2} c \,x^{5}}{3840 a^{2} x^{5}} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^6,x)
 

Output:

( - 768*sqrt(a + b*x + c*x**2)*a**5 - 2976*sqrt(a + b*x + c*x**2)*a**4*b*x 
 - 2816*sqrt(a + b*x + c*x**2)*a**4*c*x**2 - 4208*sqrt(a + b*x + c*x**2)*a 
**3*b**2*x**2 - 9296*sqrt(a + b*x + c*x**2)*a**3*b*c*x**3 - 5888*sqrt(a + 
b*x + c*x**2)*a**3*c**2*x**4 - 2420*sqrt(a + b*x + c*x**2)*a**2*b**3*x**3 
- 12200*sqrt(a + b*x + c*x**2)*a**2*b**2*c*x**4 + 3840*sqrt(a + b*x + c*x* 
*2)*a**2*b*c**2*x**5 - 210*sqrt(a + b*x + c*x**2)*a*b**4*x**4 + 10800*sqrt 
(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*b*c**2*x**5 + 3 
000*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b**3*c*x** 
5 - 105*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*b**5*x** 
5 - 10800*sqrt(a)*log(x)*a**2*b*c**2*x**5 - 3000*sqrt(a)*log(x)*a*b**3*c*x 
**5 + 105*sqrt(a)*log(x)*b**5*x**5 + 3840*sqrt(c)*log( - 2*sqrt(c)*sqrt(a 
+ b*x + c*x**2) - b - 2*c*x)*a**3*c**2*x**5 + 9600*sqrt(c)*log( - 2*sqrt(c 
)*sqrt(a + b*x + c*x**2) - b - 2*c*x)*a**2*b**2*c*x**5)/(3840*a**2*x**5)