\(\int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx\) [611]

Optimal result

Integrand size = 28, antiderivative size = 430 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=-\frac {\left (b^2-4 a c\right ) e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}+\frac {(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac {\left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}+\frac {(2 c d-b e) \left (8 c^2 d^2+35 b^2 e^2-4 c e (2 b d+33 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}+\frac {\left (b^2-4 a c\right )^2 e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{256 \left (c d^2-b d e+a e^2\right )^{9/2}} \] Output:

-1/128*(-4*a*c+b^2)*e*(24*c^2*d^2+7*b^2*e^2-4*c*e*(a*e+6*b*d))*(b*d-2*a*e+ 
(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)^4/(e*x+d)^2+1/5*(- 
b*e+2*c*d)*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^5+1/40*(8*c^2*d 
^2+7*b^2*e^2-4*c*e*(5*a*e+2*b*d))*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d^2)^ 
2/(e*x+d)^4+1/240*(-b*e+2*c*d)*(8*c^2*d^2+35*b^2*e^2-4*c*e*(33*a*e+2*b*d)) 
*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)^3+1/256*(-4*a*c+b^2)^2* 
e*(24*c^2*d^2+7*b^2*e^2-4*c*e*(a*e+6*b*d))*arctanh(1/2*(b*d-2*a*e+(-b*e+2* 
c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2) 
^(9/2)
 

Mathematica [A] (verified)

Time = 13.03 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.93 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\frac {\frac {(2 c d-b e) (a+x (b+c x))^{3/2}}{(d+e x)^5}+\frac {\left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) (a+x (b+c x))^{3/2}}{8 \left (c d^2+e (-b d+a e)\right ) (d+e x)^4}+\frac {(2 c d-b e) \left (8 c^2 d^2+35 b^2 e^2-4 c e (2 b d+33 a e)\right ) (a+x (b+c x))^{3/2}}{48 \left (c d^2+e (-b d+a e)\right )^2 (d+e x)^3}-\frac {5 \left (b^2-4 a c\right ) e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \left (2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))+\left (b^2-4 a c\right ) (d+e x)^2 \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )\right )}{256 \left (c d^2+e (-b d+a e)\right )^{7/2} (d+e x)^2}}{5 \left (c d^2+e (-b d+a e)\right )} \] Input:

Integrate[((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^6,x]
 

Output:

(((2*c*d - b*e)*(a + x*(b + c*x))^(3/2))/(d + e*x)^5 + ((8*c^2*d^2 + 7*b^2 
*e^2 - 4*c*e*(2*b*d + 5*a*e))*(a + x*(b + c*x))^(3/2))/(8*(c*d^2 + e*(-(b* 
d) + a*e))*(d + e*x)^4) + ((2*c*d - b*e)*(8*c^2*d^2 + 35*b^2*e^2 - 4*c*e*( 
2*b*d + 33*a*e))*(a + x*(b + c*x))^(3/2))/(48*(c*d^2 + e*(-(b*d) + a*e))^2 
*(d + e*x)^3) - (5*(b^2 - 4*a*c)*e*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d 
+ a*e))*(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]*(-2*a*e + 
2*c*d*x + b*(d - e*x)) + (b^2 - 4*a*c)*(d + e*x)^2*ArcTanh[(-(b*d) + 2*a*e 
 - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x) 
])]))/(256*(c*d^2 + e*(-(b*d) + a*e))^(7/2)*(d + e*x)^2))/(5*(c*d^2 + e*(- 
(b*d) + a*e)))
 

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {1237, 27, 1237, 27, 25, 1228, 1152, 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.

\(\displaystyle \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {\left (-7 e b^2+4 c d b+20 a c e+4 c (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)^5}dx}{5 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (-7 e b^2+4 c d b+20 a c e+4 c (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^5}dx}{10 \left (a e^2-b d e+c d^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {\left (35 e^2 b^3-64 c d e b^2+8 c^2 d^2 b-132 a c e^2 b+224 a c^2 d e+2 c \left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)^4}dx}{4 \left (a e^2-b d e+c d^2\right )}}{10 \left (a e^2-b d e+c d^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int -\frac {\left (-35 e^2 b^3+64 c d e b^2-4 c \left (2 c d^2-33 a e^2\right ) b-224 a c^2 d e-2 c \left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^4}dx}{8 \left (a e^2-b d e+c d^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{10 \left (a e^2-b d e+c d^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {\left (-35 e^2 b^3+64 c d e b^2-4 c \left (2 c d^2-33 a e^2\right ) b-224 a c^2 d e-2 c \left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^4}dx}{8 \left (a e^2-b d e+c d^2\right )}}{10 \left (a e^2-b d e+c d^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {\frac {5 e \left (b^2-4 a c\right ) \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \int \frac {\sqrt {c x^2+b x+a}}{(d+e x)^3}dx}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (33 a e+2 b d)+35 b^2 e^2+8 c^2 d^2\right )}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}}{8 \left (a e^2-b d e+c d^2\right )}}{10 \left (a e^2-b d e+c d^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {\frac {5 e \left (b^2-4 a c\right ) \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (33 a e+2 b d)+35 b^2 e^2+8 c^2 d^2\right )}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}}{8 \left (a e^2-b d e+c d^2\right )}}{10 \left (a e^2-b d e+c d^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {\frac {5 e \left (b^2-4 a c\right ) \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (33 a e+2 b d)+35 b^2 e^2+8 c^2 d^2\right )}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}}{8 \left (a e^2-b d e+c d^2\right )}}{10 \left (a e^2-b d e+c d^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {\frac {5 e \left (b^2-4 a c\right ) \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{3/2}}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (33 a e+2 b d)+35 b^2 e^2+8 c^2 d^2\right )}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}}{8 \left (a e^2-b d e+c d^2\right )}}{10 \left (a e^2-b d e+c d^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\)

Input:

Int[((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^6,x]
 

Output:

((2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(5*(c*d^2 - b*d*e + a*e^2)*(d + e* 
x)^5) + (((8*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(2*b*d + 5*a*e))*(a + b*x + c*x^2 
)^(3/2))/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) - (-1/3*((2*c*d - b*e)*(8 
*c^2*d^2 + 35*b^2*e^2 - 4*c*e*(2*b*d + 33*a*e))*(a + b*x + c*x^2)^(3/2))/( 
(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + (5*(b^2 - 4*a*c)*e*(24*c^2*d^2 + 7* 
b^2*e^2 - 4*c*e*(6*b*d + a*e))*(((b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + 
b*x + c*x^2])/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - ((b^2 - 4*a*c)*Arc 
Tanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a 
 + b*x + c*x^2])])/(8*(c*d^2 - b*d*e + a*e^2)^(3/2))))/(2*(c*d^2 - b*d*e + 
 a*e^2)))/(8*(c*d^2 - b*d*e + a*e^2)))/(10*(c*d^2 - b*d*e + a*e^2))
 

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 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 0]
 

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 1228

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

rule 1237

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(6446\) vs. \(2(404)=808\).

Time = 2.60 (sec) , antiderivative size = 6447, normalized size of antiderivative = 14.99

method	result	size
default	\(\text {Expression too large to display}\)	\(6447\)

Input:

int((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3408 vs. \(2 (404) = 808\).

Time = 46.41 (sec) , antiderivative size = 6858, normalized size of antiderivative = 15.95 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\text {Too large to display} \] Input:

integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\int \frac {\left (b + 2 c x\right ) \sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{6}}\, dx \] Input:

integrate((2*c*x+b)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**6,x)
 

Output:

Integral((b + 2*c*x)*sqrt(a + b*x + c*x**2)/(d + e*x)**6, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^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(a*e^2-b*d*e>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10398 vs. \(2 (404) = 808\).

Time = 8.87 (sec) , antiderivative size = 10398, normalized size of antiderivative = 24.18 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\text {Too large to display} \] Input:

integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x, algorithm="giac")
 

Output:

1/128*(24*b^4*c^2*d^2*e - 192*a*b^2*c^3*d^2*e + 384*a^2*c^4*d^2*e - 24*b^5 
*c*d*e^2 + 192*a*b^3*c^2*d*e^2 - 384*a^2*b*c^3*d*e^2 + 7*b^6*e^3 - 60*a*b^ 
4*c*e^3 + 144*a^2*b^2*c^2*e^3 - 64*a^3*c^3*e^3)*arctan(-((sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^4*d^8 
- 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e^3 - 
12*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 
- 4*a*b^3*d^3*e^5 - 12*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e 
^6 - 4*a^3*b*d*e^7 + a^4*e^8)*sqrt(-c*d^2 + b*d*e - a*e^2)) - 1/1920*(360* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^4*c^2*d^2*e^8 - 2880*(sqrt(c)*x - 
sqrt(c*x^2 + b*x + a))^9*a*b^2*c^3*d^2*e^8 + 5760*(sqrt(c)*x - sqrt(c*x^2 
+ b*x + a))^9*a^2*c^4*d^2*e^8 - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9* 
b^5*c*d*e^9 + 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^3*c^2*d*e^9 - 
 5760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b*c^3*d*e^9 + 105*(sqrt(c) 
*x - sqrt(c*x^2 + b*x + a))^9*b^6*e^10 - 900*(sqrt(c)*x - sqrt(c*x^2 + b*x 
 + a))^9*a*b^4*c*e^10 + 2160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b^2 
*c^2*e^10 - 960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^3*c^3*e^10 + 3240* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^4*c^(5/2)*d^3*e^7 - 25920*(sqrt(c) 
*x - sqrt(c*x^2 + b*x + a))^8*a*b^2*c^(7/2)*d^3*e^7 + 51840*(sqrt(c)*x - s 
qrt(c*x^2 + b*x + a))^8*a^2*c^(9/2)*d^3*e^7 - 3240*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))^8*b^5*c^(3/2)*d^2*e^8 + 25920*(sqrt(c)*x - sqrt(c*x^2 + b*...
 

Mupad [F(-1)]

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

int(((b + 2*c*x)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^6,x)
 

Output:

int(((b + 2*c*x)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^6, x)
 

Reduce [B] (verification not implemented)

Time = 67.36 (sec) , antiderivative size = 12069, normalized size of antiderivative = 28.07 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:

int((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x)
 

Output:

(960*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a* 
e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c**3*d**5*e** 
3 + 4800*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqr 
t(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c**3*d**4 
*e**4*x + 9600*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x** 
2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c** 
3*d**3*e**5*x**2 + 9600*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b* 
x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x) 
*a**3*c**3*d**2*e**6*x**3 + 4800*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sq 
rt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 
 2*c*d*x)*a**3*c**3*d*e**7*x**4 + 960*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 
 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b* 
e*x + 2*c*d*x)*a**3*c**3*e**8*x**5 - 2160*sqrt(a*e**2 - b*d*e + c*d**2)*lo 
g( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d 
- b*e*x + 2*c*d*x)*a**2*b**2*c**2*d**5*e**3 - 10800*sqrt(a*e**2 - b*d*e + 
c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2* 
a*e + b*d - b*e*x + 2*c*d*x)*a**2*b**2*c**2*d**4*e**4*x - 21600*sqrt(a*e** 
2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + 
c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b**2*c**2*d**3*e**5*x**2 - 2 
1600*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt...