\(\int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{11/2}} \, dx\) [203]

Optimal result

Integrand size = 46, antiderivative size = 375 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{11/2}} \, dx=-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (d+e x)^{9/2}}+\frac {(c e f-17 c d g+8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {c (5 c e f+11 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{96 e^2 (2 c d-b e)^2 (d+e x)^{5/2}}+\frac {c^2 (5 c e f+11 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 e^2 (2 c d-b e)^3 (d+e x)^{3/2}}+\frac {c^3 (5 c e f+11 c d g-8 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{64 e^2 (2 c d-b e)^{7/2}} \] Output:

-1/4*(-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)^(9/2)+1 
/24*(8*b*e*g-17*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)^(7/2)+1/96*c*(-8*b*e*g+11*c*d*g+5*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)^(5/2)+1/64*c^2*(-8*b*e 
*g+11*c*d*g+5*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)^3/(e*x+d)^(3/2)+1/64*c^3*(-8*b*e*g+11*c*d*g+5*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)^(7/2)
 

Mathematica [A] (verified)

Time = 1.76 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.88 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{11/2}} \, dx=\frac {c^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (\frac {16 b^3 e^3 (3 e f+d g+4 e g x)-8 b^2 c e^2 \left (11 d^2 g-e^2 x (f+2 g x)+5 d e (7 f+9 g x)\right )+2 b c^2 e \left (73 d^3 g-e^3 x^2 (5 f+12 g x)-d e^2 x (26 f+63 g x)+d^2 e (267 f+310 g x)\right )+c^3 \left (-83 d^4 g+15 e^4 f x^3+13 d^2 e^2 x (9 f+11 g x)+d e^3 x^2 (65 f+33 g x)-d^3 e (317 f+357 g x)\right )}{c^3 (2 c d-b e)^3 (d+e x)^4}+\frac {3 (5 c e f+11 c d g-8 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{7/2} \sqrt {-b e+c (d-e x)}}\right )}{192 e^2 \sqrt {d+e x}} \] Input:

Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^ 
(11/2),x]
 

Output:

(c^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((16*b^3*e^3*(3*e*f + d*g + 4* 
e*g*x) - 8*b^2*c*e^2*(11*d^2*g - e^2*x*(f + 2*g*x) + 5*d*e*(7*f + 9*g*x)) 
+ 2*b*c^2*e*(73*d^3*g - e^3*x^2*(5*f + 12*g*x) - d*e^2*x*(26*f + 63*g*x) + 
 d^2*e*(267*f + 310*g*x)) + c^3*(-83*d^4*g + 15*e^4*f*x^3 + 13*d^2*e^2*x*( 
9*f + 11*g*x) + d*e^3*x^2*(65*f + 33*g*x) - d^3*e*(317*f + 357*g*x)))/(c^3 
*(2*c*d - b*e)^3*(d + e*x)^4) + (3*(5*c*e*f + 11*c*d*g - 8*b*e*g)*ArcTan[S 
qrt[c*d - b*e - c*e*x]/Sqrt[-2*c*d + b*e]])/((-2*c*d + b*e)^(7/2)*Sqrt[-(b 
*e) + c*(d - e*x)])))/(192*e^2*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1220, 1130, 1135, 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.

\(\displaystyle \int \frac {(f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{(d+e x)^{11/2}} \, dx\)

\(\Big \downarrow \) 1220

\(\displaystyle \frac {(-8 b e g+11 c d g+5 c e f) \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{(d+e x)^{9/2}}dx}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (d+e x)^{11/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {(-8 b e g+11 c d g+5 c e f) \left (-\frac {1}{6} c \int \frac {1}{(d+e x)^{5/2} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e (d+e x)^{7/2}}\right )}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (d+e x)^{11/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {(-8 b e g+11 c d g+5 c e f) \left (-\frac {1}{6} c \left (\frac {3 c \int \frac {1}{(d+e x)^{3/2} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{4 (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2} (2 c d-b e)}\right )-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e (d+e x)^{7/2}}\right )}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (d+e x)^{11/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {(-8 b e g+11 c d g+5 c e f) \left (-\frac {1}{6} c \left (\frac {3 c \left (\frac {c \int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2} (2 c d-b e)}\right )}{4 (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2} (2 c d-b e)}\right )-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e (d+e x)^{7/2}}\right )}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (d+e x)^{11/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1136

\(\displaystyle \frac {(-8 b e g+11 c d g+5 c e f) \left (-\frac {1}{6} c \left (\frac {3 c \left (\frac {c e \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}}{2 c d-b e}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2} (2 c d-b e)}\right )}{4 (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2} (2 c d-b e)}\right )-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e (d+e x)^{7/2}}\right )}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (d+e x)^{11/2} (2 c d-b e)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (-\frac {1}{6} c \left (\frac {3 c \left (-\frac {c \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e (2 c d-b e)^{3/2}}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2} (2 c d-b e)}\right )}{4 (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2} (2 c d-b e)}\right )-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e (d+e x)^{7/2}}\right ) (-8 b e g+11 c d g+5 c e f)}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (d+e x)^{11/2} (2 c d-b e)}\)

Input:

Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^(11/2) 
,x]
 

Output:

-1/4*((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)^(11/2)) + ((5*c*e*f + 11*c*d*g - 8*b*e*g)*(-1/3*Sqrt[d*( 
c*d - b*e) - b*e^2*x - c*e^2*x^2]/(e*(d + e*x)^(7/2)) - (c*(-1/2*Sqrt[d*(c 
*d - b*e) - b*e^2*x - c*e^2*x^2]/(e*(2*c*d - b*e)*(d + e*x)^(5/2)) + (3*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*c*d - b*e))))/6))/( 
8*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. \(1532\) vs. \(2(341)=682\).

Time = 2.27 (sec) , antiderivative size = 1533, normalized size of antiderivative = 4.09

method	result	size
default	\(\text {Expression too large to display}\)	\(1533\)

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)^(11/2),x,method 
=_RETURNVERBOSE)
 

Output:

-1/192*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(146*b*c^2*d^3*e*g*(-c*e*x-b*e+c*d 
)^(1/2)*(b*e-2*c*d)^(1/2)-126*b*c^2*d*e^3*g*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b* 
e-2*c*d)^(1/2)+65*c^3*d*e^3*f*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2) 
+143*c^3*d^2*e^2*g*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+620*b*c^2* 
d^2*e^2*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+16*b^2*c*e^4*g*x^2*(- 
c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+96*arctan((-c*e*x-b*e+c*d)^(1/2)/(b 
*e-2*c*d)^(1/2))*b*c^3*d^3*e^2*g*x+534*b*c^2*d^2*e^2*f*(-c*e*x-b*e+c*d)^(1 
/2)*(b*e-2*c*d)^(1/2)-33*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))* 
c^4*d^5*g+16*b^3*d*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-317*c^3* 
d^3*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+48*b^3*e^4*f*(-c*e*x-b*e+ 
c*d)^(1/2)*(b*e-2*c*d)^(1/2)-83*c^3*d^4*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c* 
d)^(1/2)+117*c^3*d^2*e^2*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+144* 
arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^3*d^2*e^3*g*x^2-357*c 
^3*d^3*e*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-280*b^2*c*d*e^3*f*(- 
c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-10*b*c^2*e^4*f*x^2*(-c*e*x-b*e+c*d) 
^(1/2)*(b*e-2*c*d)^(1/2)+33*c^3*d*e^3*g*x^3*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2* 
c*d)^(1/2)-88*b^2*c*d^2*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+8*b 
^2*c*e^4*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-52*b*c^2*d*e^3*f*x*( 
-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-24*b*c^2*e^4*g*x^3*(-c*e*x-b*e+c*d 
)^(1/2)*(b*e-2*c*d)^(1/2)-360*b^2*c*d*e^3*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1114 vs. \(2 (341) = 682\).

Time = 0.24 (sec) , antiderivative size = 2258, normalized size of antiderivative = 6.02 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{11/2}} \, dx=\text {Too large to display} \] 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)^(11/2),x, 
 algorithm="fricas")
 

Output:

[1/384*(3*(5*c^4*d^5*e*f + (5*c^4*e^6*f + (11*c^4*d*e^5 - 8*b*c^3*e^6)*g)* 
x^5 + 5*(5*c^4*d*e^5*f + (11*c^4*d^2*e^4 - 8*b*c^3*d*e^5)*g)*x^4 + 10*(5*c 
^4*d^2*e^4*f + (11*c^4*d^3*e^3 - 8*b*c^3*d^2*e^4)*g)*x^3 + 10*(5*c^4*d^3*e 
^3*f + (11*c^4*d^4*e^2 - 8*b*c^3*d^3*e^3)*g)*x^2 + (11*c^4*d^6 - 8*b*c^3*d 
^5*e)*g + 5*(5*c^4*d^4*e^2*f + (11*c^4*d^5*e - 8*b*c^3*d^4*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*(5*(2*c^4*d*e^4 - b*c^3*e^5)*f + (22*c^4*d^2*e^3 - 27*b*c^3*d*e^4 + 
 8*b^2*c^2*e^5)*g)*x^3 + (5*(26*c^4*d^2*e^3 - 17*b*c^3*d*e^4 + 2*b^2*c^2*e 
^5)*f + (286*c^4*d^3*e^2 - 395*b*c^3*d^2*e^3 + 158*b^2*c^2*d*e^4 - 16*b^3* 
c*e^5)*g)*x^2 - (634*c^4*d^4*e - 1385*b*c^3*d^3*e^2 + 1094*b^2*c^2*d^2*e^3 
 - 376*b^3*c*d*e^4 + 48*b^4*e^5)*f - (166*c^4*d^5 - 375*b*c^3*d^4*e + 322* 
b^2*c^2*d^3*e^2 - 120*b^3*c*d^2*e^3 + 16*b^4*d*e^4)*g + ((234*c^4*d^3*e^2 
- 221*b*c^3*d^2*e^3 + 68*b^2*c^2*d*e^4 - 8*b^3*c*e^5)*f - (714*c^4*d^4*e - 
 1597*b*c^3*d^3*e^2 + 1340*b^2*c^2*d^2*e^3 - 488*b^3*c*d*e^4 + 64*b^4*e^5) 
*g)*x)*sqrt(e*x + d))/(16*c^4*d^9*e^2 - 32*b*c^3*d^8*e^3 + 24*b^2*c^2*d^7* 
e^4 - 8*b^3*c*d^6*e^5 + b^4*d^5*e^6 + (16*c^4*d^4*e^7 - 32*b*c^3*d^3*e^8 + 
 24*b^2*c^2*d^2*e^9 - 8*b^3*c*d*e^10 + b^4*e^11)*x^5 + 5*(16*c^4*d^5*e^6 - 
 32*b*c^3*d^4*e^7 + 24*b^2*c^2*d^3*e^8 - 8*b^3*c*d^2*e^9 + b^4*d*e^10)*...
 

Sympy [F]

\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{11/2}} \, dx=\int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{\frac {11}{2}}}\, dx \] 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)**(11 
/2),x)
 

Output:

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**(11/2), 
 x)
 

Maxima [F]

\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{11/2}} \, dx=\int { \frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \] 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)^(11/2),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)^( 
11/2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1013 vs. \(2 (341) = 682\).

Time = 0.37 (sec) , antiderivative size = 1013, normalized size of antiderivative = 2.70 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{11/2}} \, dx =\text {Too large to display} \] 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)^(11/2),x, 
 algorithm="giac")
 

Output:

-1/192*(3*(5*c^5*e*f + 11*c^5*d*g - 8*b*c^4*e*g)*arctan(sqrt(-(e*x + d)*c 
+ 2*c*d - b*e)/sqrt(-2*c*d + b*e))/((8*c^3*d^3 - 12*b*c^2*d^2*e + 6*b^2*c* 
d*e^2 - b^3*e^3)*sqrt(-2*c*d + b*e)) + (120*sqrt(-(e*x + d)*c + 2*c*d - b* 
e)*c^8*d^3*e*f - 180*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^7*d^2*e^2*f + 90 
*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^6*d*e^3*f - 15*sqrt(-(e*x + d)*c + 
 2*c*d - b*e)*b^3*c^5*e^4*f + 264*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^8*d^4 
*g - 588*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^7*d^3*e*g + 486*sqrt(-(e*x + 
 d)*c + 2*c*d - b*e)*b^2*c^6*d^2*e^2*g - 177*sqrt(-(e*x + d)*c + 2*c*d - b 
*e)*b^3*c^5*d*e^3*g + 24*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^4*c^4*e^4*g + 
292*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^7*d^2*e*f - 292*(-(e*x + d)*c + 2 
*c*d - b*e)^(3/2)*b*c^6*d*e^2*f + 73*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^ 
2*c^5*e^3*f + 28*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^7*d^3*g - 188*(-(e*x 
 + d)*c + 2*c*d - b*e)^(3/2)*b*c^6*d^2*e*g + 167*(-(e*x + d)*c + 2*c*d - b 
*e)^(3/2)*b^2*c^5*d*e^2*g - 40*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^3*c^4* 
e^3*g - 110*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e) 
*c^6*d*e*f + 55*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - 
b*e)*b*c^5*e^2*f - 242*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2 
*c*d - b*e)*c^6*d^2*g + 297*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)* 
c + 2*c*d - b*e)*b*c^5*d*e*g - 88*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x 
 + d)*c + 2*c*d - b*e)*b^2*c^4*e^2*g - 15*((e*x + d)*c - 2*c*d + b*e)^3...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{11/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^{11/2}} \,d x \] 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)^(11/ 
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)^(11/ 
2), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1916, normalized size of antiderivative = 5.11 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{11/2}} \, dx =\text {Too large to display} \] 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)^(11/2),x)
 

Output:

( - 24*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d) 
)*b*c**3*d**4*e*g - 96*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/s 
qrt(b*e - 2*c*d))*b*c**3*d**3*e**2*g*x - 144*sqrt(b*e - 2*c*d)*atan(sqrt( 
- b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b*c**3*d**2*e**3*g*x**2 - 96*sqrt( 
b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b*c**3*d*e 
**4*g*x**3 - 24*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e 
 - 2*c*d))*b*c**3*e**5*g*x**4 + 33*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c* 
d - c*e*x)/sqrt(b*e - 2*c*d))*c**4*d**5*g + 15*sqrt(b*e - 2*c*d)*atan(sqrt 
( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**4*d**4*e*f + 132*sqrt(b*e - 2 
*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**4*d**4*e*g*x + 
 60*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c 
**4*d**3*e**2*f*x + 198*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/ 
sqrt(b*e - 2*c*d))*c**4*d**3*e**2*g*x**2 + 90*sqrt(b*e - 2*c*d)*atan(sqrt( 
 - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**4*d**2*e**3*f*x**2 + 132*sqrt( 
b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**4*d**2* 
e**3*g*x**3 + 60*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b* 
e - 2*c*d))*c**4*d*e**4*f*x**3 + 33*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c 
*d - c*e*x)/sqrt(b*e - 2*c*d))*c**4*d*e**4*g*x**4 + 15*sqrt(b*e - 2*c*d)*a 
tan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**4*e**5*f*x**4 - 16*sq 
rt( - b*e + c*d - c*e*x)*b**4*d*e**4*g - 48*sqrt( - b*e + c*d - c*e*x)*...