\(\int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{9/2}} \, dx\) [202]

Optimal result

Integrand size = 46, antiderivative size = 296 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{9/2}} \, dx=-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e^2 (d+e x)^{7/2}}+\frac {(c e f-13 c d g+6 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{12 e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac {c (c e f+3 c d g-2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e^2 (2 c d-b e)^2 (d+e x)^{3/2}}+\frac {c^2 (c e f+3 c d g-2 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{8 e^2 (2 c d-b e)^{5/2}} \] Output:

-1/3*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(e*x+d)^(7/2)+1 
/12*(6*b*e*g-13*c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(- 
b*e+2*c*d)/(e*x+d)^(5/2)+1/8*c*(-2*b*e*g+3*c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^ 
2*x-c*e^2*x^2)^(1/2)/e^2/(-b*e+2*c*d)^2/(e*x+d)^(3/2)+1/8*c^2*(-2*b*e*g+3* 
c*d*g+c*e*f)*arctanh((d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/(-b*e+2*c*d)^( 
1/2)/(e*x+d)^(1/2))/e^2/(-b*e+2*c*d)^(5/2)
 

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.83 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{9/2}} \, dx=\frac {c^2 \sqrt {(d+e x) (-b e+c (d-e x))} \left (\frac {-4 b^2 e^2 (2 e f+d g+3 e g x)+2 b c e \left (6 d^2 g-e^2 x (f+3 g x)+d e (15 f+19 g x)\right )+c^2 \left (-11 d^3 g+3 e^3 f x^2+d e^2 x (10 f+9 g x)-d^2 e (25 f+34 g x)\right )}{c^2 (-2 c d+b e)^2 (d+e x)^3}-\frac {3 (c e f+3 c d g-2 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{5/2} \sqrt {-b e+c (d-e x)}}\right )}{24 e^2 \sqrt {d+e x}} \] Input:

Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^ 
(9/2),x]
 

Output:

(c^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((-4*b^2*e^2*(2*e*f + d*g + 3* 
e*g*x) + 2*b*c*e*(6*d^2*g - e^2*x*(f + 3*g*x) + d*e*(15*f + 19*g*x)) + c^2 
*(-11*d^3*g + 3*e^3*f*x^2 + d*e^2*x*(10*f + 9*g*x) - d^2*e*(25*f + 34*g*x) 
))/(c^2*(-2*c*d + b*e)^2*(d + e*x)^3) - (3*(c*e*f + 3*c*d*g - 2*b*e*g)*Arc 
Tan[Sqrt[c*d - b*e - c*e*x]/Sqrt[-2*c*d + b*e]])/((-2*c*d + b*e)^(5/2)*Sqr 
t[-(b*e) + c*(d - e*x)])))/(24*e^2*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1220, 1130, 1135, 1136, 221}

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.

\(\displaystyle \int \frac {(f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{(d+e x)^{9/2}} \, dx\)

\(\Big \downarrow \) 1220

\(\displaystyle \frac {(-2 b e g+3 c d g+c e f) \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{(d+e x)^{7/2}}dx}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{9/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {(-2 b e g+3 c d g+c e f) \left (-\frac {1}{4} c \int \frac {1}{(d+e x)^{3/2} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2}}\right )}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{9/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {(-2 b e g+3 c d g+c e f) \left (-\frac {1}{4} c \left (\frac {c \int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2} (2 c d-b e)}\right )-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2}}\right )}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{9/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1136

\(\displaystyle \frac {(-2 b e g+3 c d g+c e f) \left (-\frac {1}{4} c \left (\frac {c e \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}}{2 c d-b e}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2} (2 c d-b e)}\right )-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2}}\right )}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{9/2} (2 c d-b e)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (-\frac {1}{4} c \left (-\frac {c \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e (2 c d-b e)^{3/2}}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2} (2 c d-b e)}\right )-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2}}\right ) (-2 b e g+3 c d g+c e f)}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{9/2} (2 c d-b e)}\)

Input:

Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^(9/2), 
x]
 

Output:

-1/3*((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(e^2*(2*c*d 
 - b*e)*(d + e*x)^(9/2)) + ((c*e*f + 3*c*d*g - 2*b*e*g)*(-1/2*Sqrt[d*(c*d 
- b*e) - b*e^2*x - c*e^2*x^2]/(e*(d + e*x)^(5/2)) - (c*(-(Sqrt[d*(c*d - b* 
e) - b*e^2*x - c*e^2*x^2]/(e*(2*c*d - b*e)*(d + e*x)^(3/2))) - (c*ArcTanh[ 
Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b*e]*Sqrt[d + e*x] 
)])/(e*(2*c*d - b*e)^(3/2))))/4))/(2*e*(2*c*d - b*e))
 

Defintions of rubi rules used

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 1130

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

rule 1135

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

rule 1136

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x 
_Symbol] :> Simp[2*e   Subst[Int[1/(2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + 
 b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 
- b*d*e + a*e^2, 0]
 

rule 1220

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1024\) vs. \(2(268)=536\).

Time = 2.20 (sec) , antiderivative size = 1025, normalized size of antiderivative = 3.46

method	result	size
default	\(\text {Expression too large to display}\)	\(1025\)

Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(9/2),x,method= 
_RETURNVERBOSE)
 

Output:

1/24*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-6*b*c*e^3*g*x^2*(-c*e*x-b*e+c*d)^( 
1/2)*(b*e-2*c*d)^(1/2)+9*c^2*d*e^2*g*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d 
)^(1/2)+38*b*c*d*e^2*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-11*c^2*d 
^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-3*arctan((-c*e*x-b*e+c*d)^(1 
/2)/(b*e-2*c*d)^(1/2))*c^3*e^4*f*x^3-3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e- 
2*c*d)^(1/2))*c^3*d^3*e*f+18*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/ 
2))*b*c^2*d*e^3*g*x^2+18*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))* 
b*c^2*d^2*e^2*g*x+12*b*c*d^2*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+ 
30*b*c*d*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+10*c^2*d*e^2*f*x*( 
-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-34*c^2*d^2*e*g*x*(-c*e*x-b*e+c*d)^ 
(1/2)*(b*e-2*c*d)^(1/2)-2*b*c*e^3*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^( 
1/2)-9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^4*g+6*arctan 
((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*e^4*g*x^3-9*arctan((-c*e* 
x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d*e^3*g*x^3-8*b^2*e^3*f*(-c*e*x-b* 
e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d) 
^(1/2))*c^3*d*e^3*f*x^2-27*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2) 
)*c^3*d^3*e*g*x-9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^2 
*e^2*f*x-12*b^2*e^3*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-4*b^2*d*e 
^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-25*c^2*d^2*e*f*(-c*e*x-b*e+c 
*d)^(1/2)*(b*e-2*c*d)^(1/2)+3*c^2*e^3*f*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 788 vs. \(2 (268) = 536\).

Time = 0.16 (sec) , antiderivative size = 1606, normalized size of antiderivative = 5.43 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{9/2}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(9/2),x, 
algorithm="fricas")
 

Output:

[-1/48*(3*(c^3*d^4*e*f + (c^3*e^5*f + (3*c^3*d*e^4 - 2*b*c^2*e^5)*g)*x^4 + 
 4*(c^3*d*e^4*f + (3*c^3*d^2*e^3 - 2*b*c^2*d*e^4)*g)*x^3 + 6*(c^3*d^2*e^3* 
f + (3*c^3*d^3*e^2 - 2*b*c^2*d^2*e^3)*g)*x^2 + (3*c^3*d^5 - 2*b*c^2*d^4*e) 
*g + 4*(c^3*d^3*e^2*f + (3*c^3*d^4*e - 2*b*c^2*d^3*e^2)*g)*x)*sqrt(2*c*d - 
 b*e)*log(-(c*e^2*x^2 - 3*c*d^2 + 2*b*d*e - 2*(c*d*e - b*e^2)*x + 2*sqrt(- 
c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*sqrt(e*x + d))/(e^2 
*x^2 + 2*d*e*x + d^2)) - 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(3*( 
(2*c^3*d*e^3 - b*c^2*e^4)*f + (6*c^3*d^2*e^2 - 7*b*c^2*d*e^3 + 2*b^2*c*e^4 
)*g)*x^2 - (50*c^3*d^3*e - 85*b*c^2*d^2*e^2 + 46*b^2*c*d*e^3 - 8*b^3*e^4)* 
f - (22*c^3*d^4 - 35*b*c^2*d^3*e + 20*b^2*c*d^2*e^2 - 4*b^3*d*e^3)*g + 2*( 
(10*c^3*d^2*e^2 - 7*b*c^2*d*e^3 + b^2*c*e^4)*f - (34*c^3*d^3*e - 55*b*c^2* 
d^2*e^2 + 31*b^2*c*d*e^3 - 6*b^3*e^4)*g)*x)*sqrt(e*x + d))/(8*c^3*d^7*e^2 
- 12*b*c^2*d^6*e^3 + 6*b^2*c*d^5*e^4 - b^3*d^4*e^5 + (8*c^3*d^3*e^6 - 12*b 
*c^2*d^2*e^7 + 6*b^2*c*d*e^8 - b^3*e^9)*x^4 + 4*(8*c^3*d^4*e^5 - 12*b*c^2* 
d^3*e^6 + 6*b^2*c*d^2*e^7 - b^3*d*e^8)*x^3 + 6*(8*c^3*d^5*e^4 - 12*b*c^2*d 
^4*e^5 + 6*b^2*c*d^3*e^6 - b^3*d^2*e^7)*x^2 + 4*(8*c^3*d^6*e^3 - 12*b*c^2* 
d^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x), 1/24*(3*(c^3*d^4*e*f + (c^3*e 
^5*f + (3*c^3*d*e^4 - 2*b*c^2*e^5)*g)*x^4 + 4*(c^3*d*e^4*f + (3*c^3*d^2*e^ 
3 - 2*b*c^2*d*e^4)*g)*x^3 + 6*(c^3*d^2*e^3*f + (3*c^3*d^3*e^2 - 2*b*c^2*d^ 
2*e^3)*g)*x^2 + (3*c^3*d^5 - 2*b*c^2*d^4*e)*g + 4*(c^3*d^3*e^2*f + (3*c...
 

Sympy [F]

\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{9/2}} \, dx=\int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \] Input:

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**(9/ 
2),x)
 

Output:

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**(9/2), 
x)
 

Maxima [F]

\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{9/2}} \, dx=\int { \frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(9/2),x, 
algorithm="maxima")
 

Output:

integrate(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^( 
9/2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 587 vs. \(2 (268) = 536\).

Time = 0.38 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.98 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{9/2}} \, dx=-\frac {\frac {3 \, {\left (c^{4} e f + 3 \, c^{4} d g - 2 \, b c^{3} e g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \sqrt {-2 \, c d + b e}} + \frac {12 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{6} d^{2} e f - 12 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{5} d e^{2} f + 3 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{4} e^{3} f + 36 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{6} d^{3} g - 60 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{5} d^{2} e g + 33 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{4} d e^{2} g - 6 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{3} c^{3} e^{3} g + 16 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{5} d e f - 8 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{4} e^{2} f - 16 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{5} d^{2} g + 8 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{4} d e g - 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} e f - 9 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d g + 6 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} e g}{{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} {\left (e x + d\right )}^{3} c^{3}}}{24 \, c e^{2}} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(9/2),x, 
algorithm="giac")
 

Output:

-1/24*(3*(c^4*e*f + 3*c^4*d*g - 2*b*c^3*e*g)*arctan(sqrt(-(e*x + d)*c + 2* 
c*d - b*e)/sqrt(-2*c*d + b*e))/((4*c^2*d^2 - 4*b*c*d*e + b^2*e^2)*sqrt(-2* 
c*d + b*e)) + (12*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^6*d^2*e*f - 12*sqrt(- 
(e*x + d)*c + 2*c*d - b*e)*b*c^5*d*e^2*f + 3*sqrt(-(e*x + d)*c + 2*c*d - b 
*e)*b^2*c^4*e^3*f + 36*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^6*d^3*g - 60*sqr 
t(-(e*x + d)*c + 2*c*d - b*e)*b*c^5*d^2*e*g + 33*sqrt(-(e*x + d)*c + 2*c*d 
 - b*e)*b^2*c^4*d*e^2*g - 6*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^3*c^3*e^3*g 
 + 16*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^5*d*e*f - 8*(-(e*x + d)*c + 2*c 
*d - b*e)^(3/2)*b*c^4*e^2*f - 16*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^5*d^ 
2*g + 8*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^4*d*e*g - 3*((e*x + d)*c - 
2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*e*f - 9*((e*x + d)*c - 
 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*d*g + 6*((e*x + d)*c 
- 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^3*e*g)/((4*c^2*d^2 - 
 4*b*c*d*e + b^2*e^2)*(e*x + d)^3*c^3))/(c*e^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^{9/2}} \,d x \] Input:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^(9/2 
),x)
 

Output:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^(9/2 
), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1302, normalized size of antiderivative = 4.40 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{9/2}} \, dx =\text {Too large to display} \] Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(9/2),x)
 

Output:

(6*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b* 
c**2*d**3*e*g + 18*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt( 
b*e - 2*c*d))*b*c**2*d**2*e**2*g*x + 18*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e 
 + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b*c**2*d*e**3*g*x**2 + 6*sqrt(b*e - 2*c 
*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b*c**2*e**4*g*x**3 
- 9*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c 
**3*d**4*g - 3*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e 
- 2*c*d))*c**3*d**3*e*f - 27*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c* 
e*x)/sqrt(b*e - 2*c*d))*c**3*d**3*e*g*x - 9*sqrt(b*e - 2*c*d)*atan(sqrt( - 
 b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**3*d**2*e**2*f*x - 27*sqrt(b*e - 
2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**3*d**2*e**2*g 
*x**2 - 9*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c 
*d))*c**3*d*e**3*f*x**2 - 9*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e 
*x)/sqrt(b*e - 2*c*d))*c**3*d*e**3*g*x**3 - 3*sqrt(b*e - 2*c*d)*atan(sqrt( 
 - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**3*e**4*f*x**3 - 4*sqrt( - b*e 
+ c*d - c*e*x)*b**3*d*e**3*g - 8*sqrt( - b*e + c*d - c*e*x)*b**3*e**4*f - 
12*sqrt( - b*e + c*d - c*e*x)*b**3*e**4*g*x + 20*sqrt( - b*e + c*d - c*e*x 
)*b**2*c*d**2*e**2*g + 46*sqrt( - b*e + c*d - c*e*x)*b**2*c*d*e**3*f + 62* 
sqrt( - b*e + c*d - c*e*x)*b**2*c*d*e**3*g*x - 2*sqrt( - b*e + c*d - c*e*x 
)*b**2*c*e**4*f*x - 6*sqrt( - b*e + c*d - c*e*x)*b**2*c*e**4*g*x**2 - 3...