\(\int \frac {(A+B x) (a+b x+c x^2)^{5/2}}{x^7} \, dx\) [140]

Optimal result

Integrand size = 23, antiderivative size = 332 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^7} \, dx=\frac {\left (2 a \left (4 a b B \left (3 b^2-28 a c\right )-5 A \left (b^2-4 a c\right )^2\right )-\left (5 A b \left (b^2-4 a c\right )^2-4 a B \left (3 b^4-28 a b^2 c-128 a^2 c^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{512 a^3 x^2}-\frac {\left (2 a \left (12 a b B-5 A \left (b^2-4 a c\right )\right )+\left (4 a B \left (3 b^2+16 a c\right )-5 A \left (b^3-4 a b c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 a^2 x^4}-\frac {(10 a A+(5 A b+12 a B) x) \left (a+b x+c x^2\right )^{5/2}}{60 a x^6}+\frac {\left (5 A \left (b^2-4 a c\right )^3-4 a b B \left (3 b^4-40 a b^2 c+240 a^2 c^2\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{1024 a^{7/2}}+B c^{5/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \] Output:

1/512*(2*a*(4*a*b*B*(-28*a*c+3*b^2)-5*A*(-4*a*c+b^2)^2)-(5*A*b*(-4*a*c+b^2 
)^2-4*a*B*(-128*a^2*c^2-28*a*b^2*c+3*b^4))*x)*(c*x^2+b*x+a)^(1/2)/a^3/x^2- 
1/192*(2*a*(12*a*b*B-5*A*(-4*a*c+b^2))+(4*a*B*(16*a*c+3*b^2)-5*A*(-4*a*b*c 
+b^3))*x)*(c*x^2+b*x+a)^(3/2)/a^2/x^4-1/60*(10*a*A+(5*A*b+12*B*a)*x)*(c*x^ 
2+b*x+a)^(5/2)/a/x^6+1/1024*(5*A*(-4*a*c+b^2)^3-4*a*b*B*(240*a^2*c^2-40*a* 
b^2*c+3*b^4))*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(7/2)+B 
*c^(5/2)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))
 

Mathematica [A] (verified)

Time = 4.68 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^7} \, dx=-\frac {\sqrt {a+x (b+c x)} \left (75 A b^5 x^5+256 a^5 (5 A+6 B x)-10 a b^3 x^4 (18 b B x+5 A (b+16 c x))+64 a^4 x (A (50 b+65 c x)+B x (63 b+88 c x))+40 a^2 b x^3 \left (3 b B x (b+18 c x)+A \left (b^2+12 b c x+66 c^2 x^2\right )\right )+16 a^3 x^2 \left (15 A \left (9 b^2+26 b c x+22 c^2 x^2\right )+2 B x \left (93 b^2+311 b c x+368 c^2 x^2\right )\right )\right )}{7680 a^3 x^6}-\frac {5 A b^6 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{512 a^{7/2}}-\frac {\left (3 b^5 B+15 A b^4 c-40 a b^3 B c-60 a A b^2 c^2+240 a^2 b B c^2+80 a^2 A c^3\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{128 a^{5/2}}-B c^{5/2} \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right ) \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^7,x]
 

Output:

-1/7680*(Sqrt[a + x*(b + c*x)]*(75*A*b^5*x^5 + 256*a^5*(5*A + 6*B*x) - 10* 
a*b^3*x^4*(18*b*B*x + 5*A*(b + 16*c*x)) + 64*a^4*x*(A*(50*b + 65*c*x) + B* 
x*(63*b + 88*c*x)) + 40*a^2*b*x^3*(3*b*B*x*(b + 18*c*x) + A*(b^2 + 12*b*c* 
x + 66*c^2*x^2)) + 16*a^3*x^2*(15*A*(9*b^2 + 26*b*c*x + 22*c^2*x^2) + 2*B* 
x*(93*b^2 + 311*b*c*x + 368*c^2*x^2))))/(a^3*x^6) - (5*A*b^6*ArcTanh[(Sqrt 
[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(512*a^(7/2)) - ((3*b^5*B + 15*A* 
b^4*c - 40*a*b^3*B*c - 60*a*A*b^2*c^2 + 240*a^2*b*B*c^2 + 80*a^2*A*c^3)*Ar 
cTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(128*a^(5/2)) - B*c 
^(5/2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.08, 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, 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.

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

\(\Big \downarrow \) 1229

\(\displaystyle -\frac {\int -\frac {\left (12 a b B+24 a c x B-5 A \left (b^2-4 a c\right )\right ) \left (c x^2+b x+a\right )^{3/2}}{2 x^5}dx}{12 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (x (12 a B+5 A b)+10 a A)}{60 a x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (12 a b B+24 a c x B-5 A \left (b^2-4 a c\right )\right ) \left (c x^2+b x+a\right )^{3/2}}{x^5}dx}{24 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (x (12 a B+5 A b)+10 a A)}{60 a x^6}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {-\frac {\int \frac {3 \left (-128 a^2 B x c^2-5 A \left (b^2-4 a c\right )^2+4 a b B \left (3 b^2-28 a c\right )\right ) \sqrt {c x^2+b x+a}}{2 x^3}dx}{8 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (2 a \left (12 a b B-5 A \left (b^2-4 a c\right )\right )+x \left (4 a B \left (16 a c+3 b^2\right )-5 A \left (b^3-4 a b c\right )\right )\right )}{8 a x^4}}{24 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (x (12 a B+5 A b)+10 a A)}{60 a x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \int \frac {\left (-128 a^2 B x c^2-5 A \left (b^2-4 a c\right )^2+4 a b B \left (3 b^2-28 a c\right )\right ) \sqrt {c x^2+b x+a}}{x^3}dx}{16 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (2 a \left (12 a b B-5 A \left (b^2-4 a c\right )\right )+x \left (4 a B \left (16 a c+3 b^2\right )-5 A \left (b^3-4 a b c\right )\right )\right )}{8 a x^4}}{24 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (x (12 a B+5 A b)+10 a A)}{60 a x^6}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {-\frac {3 \left (-\frac {\int -\frac {-1024 a^3 B x c^3+5 A \left (b^2-4 a c\right )^3-4 a b B \left (3 b^4-40 a c b^2+240 a^2 c^2\right )}{2 x \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (4 a b B \left (3 b^2-28 a c\right )-5 A \left (b^2-4 a c\right )^2\right )-x \left (5 A b \left (b^2-4 a c\right )^2-4 a B \left (-128 a^2 c^2-28 a b^2 c+3 b^4\right )\right )\right )}{4 a x^2}\right )}{16 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (2 a \left (12 a b B-5 A \left (b^2-4 a c\right )\right )+x \left (4 a B \left (16 a c+3 b^2\right )-5 A \left (b^3-4 a b c\right )\right )\right )}{8 a x^4}}{24 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (x (12 a B+5 A b)+10 a A)}{60 a x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \left (\frac {\int \frac {-1024 a^3 B x c^3+5 A \left (b^2-4 a c\right )^3-4 a b B \left (3 b^4-40 a c b^2+240 a^2 c^2\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (4 a b B \left (3 b^2-28 a c\right )-5 A \left (b^2-4 a c\right )^2\right )-x \left (5 A b \left (b^2-4 a c\right )^2-4 a B \left (-128 a^2 c^2-28 a b^2 c+3 b^4\right )\right )\right )}{4 a x^2}\right )}{16 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (2 a \left (12 a b B-5 A \left (b^2-4 a c\right )\right )+x \left (4 a B \left (16 a c+3 b^2\right )-5 A \left (b^3-4 a b c\right )\right )\right )}{8 a x^4}}{24 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (x (12 a B+5 A b)+10 a A)}{60 a x^6}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {-\frac {3 \left (\frac {\left (5 A \left (b^2-4 a c\right )^3-4 a b B \left (240 a^2 c^2-40 a b^2 c+3 b^4\right )\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx-1024 a^3 B c^3 \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (4 a b B \left (3 b^2-28 a c\right )-5 A \left (b^2-4 a c\right )^2\right )-x \left (5 A b \left (b^2-4 a c\right )^2-4 a B \left (-128 a^2 c^2-28 a b^2 c+3 b^4\right )\right )\right )}{4 a x^2}\right )}{16 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (2 a \left (12 a b B-5 A \left (b^2-4 a c\right )\right )+x \left (4 a B \left (16 a c+3 b^2\right )-5 A \left (b^3-4 a b c\right )\right )\right )}{8 a x^4}}{24 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (x (12 a B+5 A b)+10 a A)}{60 a x^6}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {-\frac {3 \left (\frac {\left (5 A \left (b^2-4 a c\right )^3-4 a b B \left (240 a^2 c^2-40 a b^2 c+3 b^4\right )\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx-2048 a^3 B c^3 \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{8 a}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (4 a b B \left (3 b^2-28 a c\right )-5 A \left (b^2-4 a c\right )^2\right )-x \left (5 A b \left (b^2-4 a c\right )^2-4 a B \left (-128 a^2 c^2-28 a b^2 c+3 b^4\right )\right )\right )}{4 a x^2}\right )}{16 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (2 a \left (12 a b B-5 A \left (b^2-4 a c\right )\right )+x \left (4 a B \left (16 a c+3 b^2\right )-5 A \left (b^3-4 a b c\right )\right )\right )}{8 a x^4}}{24 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (x (12 a B+5 A b)+10 a A)}{60 a x^6}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {3 \left (\frac {\left (5 A \left (b^2-4 a c\right )^3-4 a b B \left (240 a^2 c^2-40 a b^2 c+3 b^4\right )\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx-1024 a^3 B c^{5/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 a}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (4 a b B \left (3 b^2-28 a c\right )-5 A \left (b^2-4 a c\right )^2\right )-x \left (5 A b \left (b^2-4 a c\right )^2-4 a B \left (-128 a^2 c^2-28 a b^2 c+3 b^4\right )\right )\right )}{4 a x^2}\right )}{16 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (2 a \left (12 a b B-5 A \left (b^2-4 a c\right )\right )+x \left (4 a B \left (16 a c+3 b^2\right )-5 A \left (b^3-4 a b c\right )\right )\right )}{8 a x^4}}{24 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (x (12 a B+5 A b)+10 a A)}{60 a x^6}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {-\frac {3 \left (\frac {-2 \left (5 A \left (b^2-4 a c\right )^3-4 a b B \left (240 a^2 c^2-40 a b^2 c+3 b^4\right )\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}-1024 a^3 B c^{5/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 a}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (4 a b B \left (3 b^2-28 a c\right )-5 A \left (b^2-4 a c\right )^2\right )-x \left (5 A b \left (b^2-4 a c\right )^2-4 a B \left (-128 a^2 c^2-28 a b^2 c+3 b^4\right )\right )\right )}{4 a x^2}\right )}{16 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (2 a \left (12 a b B-5 A \left (b^2-4 a c\right )\right )+x \left (4 a B \left (16 a c+3 b^2\right )-5 A \left (b^3-4 a b c\right )\right )\right )}{8 a x^4}}{24 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (x (12 a B+5 A b)+10 a A)}{60 a x^6}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {3 \left (\frac {-1024 a^3 B c^{5/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )-\frac {\left (5 A \left (b^2-4 a c\right )^3-4 a b B \left (240 a^2 c^2-40 a b^2 c+3 b^4\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a}}}{8 a}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (4 a b B \left (3 b^2-28 a c\right )-5 A \left (b^2-4 a c\right )^2\right )-x \left (5 A b \left (b^2-4 a c\right )^2-4 a B \left (-128 a^2 c^2-28 a b^2 c+3 b^4\right )\right )\right )}{4 a x^2}\right )}{16 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (2 a \left (12 a b B-5 A \left (b^2-4 a c\right )\right )+x \left (4 a B \left (16 a c+3 b^2\right )-5 A \left (b^3-4 a b c\right )\right )\right )}{8 a x^4}}{24 a}-\frac {\left (a+b x+c x^2\right )^{5/2} (x (12 a B+5 A b)+10 a A)}{60 a x^6}\)

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^7,x]
 

Output:

-1/60*((10*a*A + (5*A*b + 12*a*B)*x)*(a + b*x + c*x^2)^(5/2))/(a*x^6) + (- 
1/8*((2*a*(12*a*b*B - 5*A*(b^2 - 4*a*c)) + (4*a*B*(3*b^2 + 16*a*c) - 5*A*( 
b^3 - 4*a*b*c))*x)*(a + b*x + c*x^2)^(3/2))/(a*x^4) - (3*(-1/4*((2*a*(4*a* 
b*B*(3*b^2 - 28*a*c) - 5*A*(b^2 - 4*a*c)^2) - (5*A*b*(b^2 - 4*a*c)^2 - 4*a 
*B*(3*b^4 - 28*a*b^2*c - 128*a^2*c^2))*x)*Sqrt[a + b*x + c*x^2])/(a*x^2) + 
 (-(((5*A*(b^2 - 4*a*c)^3 - 4*a*b*B*(3*b^4 - 40*a*b^2*c + 240*a^2*c^2))*Ar 
cTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/Sqrt[a]) - 1024*a^3* 
B*c^(5/2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*a)))/ 
(16*a))/(24*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 = 1.55 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.16

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (2640 A \,a^{2} b \,c^{2} x^{5}-800 A a \,b^{3} c \,x^{5}+75 A \,b^{5} x^{5}+11776 B \,a^{3} c^{2} x^{5}+2160 B \,a^{2} b^{2} c \,x^{5}-180 B a \,b^{4} x^{5}+5280 A \,a^{3} c^{2} x^{4}+480 A \,a^{2} b^{2} c \,x^{4}-50 A a \,b^{4} x^{4}+9952 B \,a^{3} b c \,x^{4}+120 B \,a^{2} b^{3} x^{4}+6240 A \,a^{3} b c \,x^{3}+40 A \,a^{2} b^{3} x^{3}+5632 B \,a^{4} c \,x^{3}+2976 B \,a^{3} b^{2} x^{3}+4160 A \,a^{4} c \,x^{2}+2160 A \,a^{3} b^{2} x^{2}+4032 B \,a^{4} b \,x^{2}+3200 A \,a^{4} b x +1536 B \,a^{5} x +1280 A \,a^{5}\right )}{7680 x^{6} a^{3}}+\frac {-\frac {\left (320 a^{3} A \,c^{3}-240 A \,a^{2} b^{2} c^{2}+60 A a \,b^{4} c -5 A \,b^{6}+960 B \,a^{3} b \,c^{2}-160 B \,a^{2} b^{3} c +12 B a \,b^{5}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{\sqrt {a}}+1024 B \,a^{3} c^{\frac {5}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 a^{3}}\)	\(384\)
default	\(\text {Expression too large to display}\)	\(8893\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/7680*(c*x^2+b*x+a)^(1/2)*(2640*A*a^2*b*c^2*x^5-800*A*a*b^3*c*x^5+75*A*b 
^5*x^5+11776*B*a^3*c^2*x^5+2160*B*a^2*b^2*c*x^5-180*B*a*b^4*x^5+5280*A*a^3 
*c^2*x^4+480*A*a^2*b^2*c*x^4-50*A*a*b^4*x^4+9952*B*a^3*b*c*x^4+120*B*a^2*b 
^3*x^4+6240*A*a^3*b*c*x^3+40*A*a^2*b^3*x^3+5632*B*a^4*c*x^3+2976*B*a^3*b^2 
*x^3+4160*A*a^4*c*x^2+2160*A*a^3*b^2*x^2+4032*B*a^4*b*x^2+3200*A*a^4*b*x+1 
536*B*a^5*x+1280*A*a^5)/x^6/a^3+1/1024/a^3*(-(320*A*a^3*c^3-240*A*a^2*b^2* 
c^2+60*A*a*b^4*c-5*A*b^6+960*B*a^3*b*c^2-160*B*a^2*b^3*c+12*B*a*b^5)/a^(1/ 
2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+1024*B*a^3*c^(5/2)*ln((1/ 
2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 4.43 (sec) , antiderivative size = 1659, normalized size of antiderivative = 5.00 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^7} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^7,x, algorithm="fricas")
 

Output:

[1/30720*(15360*B*a^4*c^(5/2)*x^6*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*(12*B*a*b^5 - 5*A*b^6 + 
 320*A*a^3*c^3 + 240*(4*B*a^3*b - A*a^2*b^2)*c^2 - 20*(8*B*a^2*b^3 - 3*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 
- 75*A*a*b^5 - 16*(736*B*a^4 + 165*A*a^3*b)*c^2 - 80*(27*B*a^3*b^2 - 10*A* 
a^2*b^3)*c)*x^5 + 2*(60*B*a^3*b^3 - 25*A*a^2*b^4 + 2640*A*a^4*c^2 + 16*(31 
1*B*a^4*b + 15*A*a^3*b^2)*c)*x^4 + 8*(372*B*a^4*b^2 + 5*A*a^3*b^3 + 4*(176 
*B*a^5 + 195*A*a^4*b)*c)*x^3 + 16*(252*B*a^5*b + 135*A*a^4*b^2 + 260*A*a^5 
*c)*x^2 + 128*(12*B*a^6 + 25*A*a^5*b)*x)*sqrt(c*x^2 + b*x + a))/(a^4*x^6), 
 -1/30720*(30720*B*a^4*sqrt(-c)*c^2*x^6*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*(12*B*a*b^5 - 5*A*b^6 + 
320*A*a^3*c^3 + 240*(4*B*a^3*b - A*a^2*b^2)*c^2 - 20*(8*B*a^2*b^3 - 3*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 - 
 75*A*a*b^5 - 16*(736*B*a^4 + 165*A*a^3*b)*c^2 - 80*(27*B*a^3*b^2 - 10*A*a 
^2*b^3)*c)*x^5 + 2*(60*B*a^3*b^3 - 25*A*a^2*b^4 + 2640*A*a^4*c^2 + 16*(311 
*B*a^4*b + 15*A*a^3*b^2)*c)*x^4 + 8*(372*B*a^4*b^2 + 5*A*a^3*b^3 + 4*(176* 
B*a^5 + 195*A*a^4*b)*c)*x^3 + 16*(252*B*a^5*b + 135*A*a^4*b^2 + 260*A*a^5* 
c)*x^2 + 128*(12*B*a^6 + 25*A*a^5*b)*x)*sqrt(c*x^2 + b*x + a))/(a^4*x^6...
 

Sympy [F]

\[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^7} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{x^{7}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**7,x)
 

Output:

Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**7, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^7} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/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. 2086 vs. \(2 (302) = 604\).

Time = 0.54 (sec) , antiderivative size = 2086, normalized size of antiderivative = 6.28 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^7} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^7,x, algorithm="giac")
 

Output:

-B*c^(5/2)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b)) + 1 
/512*(12*B*a*b^5 - 5*A*b^6 - 160*B*a^2*b^3*c + 60*A*a*b^4*c + 960*B*a^3*b* 
c^2 - 240*A*a^2*b^2*c^2 + 320*A*a^3*c^3)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + 
 b*x + a))/sqrt(-a))/(sqrt(-a)*a^3) - 1/7680*(180*(sqrt(c)*x - sqrt(c*x^2 
+ b*x + a))^11*B*a*b^5 - 75*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*b^6 - 
 2400*(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 - 31680*(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))^11*B*a^3*b*c^2 - 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^2 
*b^2*c^2 - 10560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^3*c^3 - 46080* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*B*a^3*b^2*c^(3/2) - 46080*(sqrt(c)* 
x - sqrt(c*x^2 + b*x + a))^10*B*a^4*c^(5/2) - 46080*(sqrt(c)*x - sqrt(c*x^ 
2 + b*x + a))^10*A*a^3*b*c^(5/2) - 1020*(sqrt(c)*x - sqrt(c*x^2 + b*x + a) 
)^9*B*a^2*b^5 + 425*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a*b^6 - 22240* 
(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 + 41280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a 
))^9*B*a^4*b*c^2 - 56400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^3*b^2*c 
^2 - 1600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^4*c^3 - 15360*(sqrt(c) 
*x - sqrt(c*x^2 + b*x + a))^8*B*a^3*b^4*sqrt(c) + 46080*(sqrt(c)*x - sqrt( 
c*x^2 + b*x + a))^8*B*a^4*b^2*c^(3/2) - 76800*(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))^8*A*a^3*b^3*c^(3/2) + 138240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^7} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{x^7} \,d x \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^7,x)
 

Output:

int(((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^7, x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 1.73 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.52 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^7} \, dx=\frac {-2560 \sqrt {c \,x^{2}+b x +a}\, a^{6}-9472 \sqrt {c \,x^{2}+b x +a}\, a^{5} b x -8320 \sqrt {c \,x^{2}+b x +a}\, a^{5} c \,x^{2}-12384 \sqrt {c \,x^{2}+b x +a}\, a^{4} b^{2} x^{2}-23744 \sqrt {c \,x^{2}+b x +a}\, a^{4} b c \,x^{3}-10560 \sqrt {c \,x^{2}+b x +a}\, a^{4} c^{2} x^{4}-6032 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{3} x^{3}-20864 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{2} c \,x^{4}-28832 \sqrt {c \,x^{2}+b x +a}\, a^{3} b \,c^{2} x^{5}-140 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{4} x^{4}-2720 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{3} c \,x^{5}+210 \sqrt {c \,x^{2}+b x +a}\, a \,b^{5} x^{5}+4800 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{3} c^{3} x^{6}+10800 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} b^{2} c^{2} x^{6}-1500 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,b^{4} c \,x^{6}+105 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{6} x^{6}-4800 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{3} c^{3} x^{6}-10800 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} b^{2} c^{2} x^{6}+1500 \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{4} c \,x^{6}-105 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{6} x^{6}+15360 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) a^{3} b \,c^{2} x^{6}}{15360 a^{3} x^{6}} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^7,x)
 

Output:

( - 2560*sqrt(a + b*x + c*x**2)*a**6 - 9472*sqrt(a + b*x + c*x**2)*a**5*b* 
x - 8320*sqrt(a + b*x + c*x**2)*a**5*c*x**2 - 12384*sqrt(a + b*x + c*x**2) 
*a**4*b**2*x**2 - 23744*sqrt(a + b*x + c*x**2)*a**4*b*c*x**3 - 10560*sqrt( 
a + b*x + c*x**2)*a**4*c**2*x**4 - 6032*sqrt(a + b*x + c*x**2)*a**3*b**3*x 
**3 - 20864*sqrt(a + b*x + c*x**2)*a**3*b**2*c*x**4 - 28832*sqrt(a + b*x + 
 c*x**2)*a**3*b*c**2*x**5 - 140*sqrt(a + b*x + c*x**2)*a**2*b**4*x**4 - 27 
20*sqrt(a + b*x + c*x**2)*a**2*b**3*c*x**5 + 210*sqrt(a + b*x + c*x**2)*a* 
b**5*x**5 + 4800*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x) 
*a**3*c**3*x**6 + 10800*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 - 1500*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c* 
x**2) - 2*a - b*x)*a*b**4*c*x**6 + 105*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x 
+ c*x**2) - 2*a - b*x)*b**6*x**6 - 4800*sqrt(a)*log(x)*a**3*c**3*x**6 - 10 
800*sqrt(a)*log(x)*a**2*b**2*c**2*x**6 + 1500*sqrt(a)*log(x)*a*b**4*c*x**6 
 - 105*sqrt(a)*log(x)*b**6*x**6 + 15360*sqrt(c)*log( - 2*sqrt(c)*sqrt(a + 
b*x + c*x**2) - b - 2*c*x)*a**3*b*c**2*x**6)/(15360*a**3*x**6)