\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^{13/2}} \, dx\) [213]

Optimal result

Integrand size = 46, antiderivative size = 365 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\frac {(3 c e f-19 c d g+8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 e^2 (d+e x)^{7/2}}-\frac {c (3 c e f-115 c d g+56 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{96 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {c^2 (3 c e f+13 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 e^2 (2 c d-b e)^2 (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (d+e x)^{11/2}}-\frac {c^3 (3 c e f+13 c d g-8 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{64 e^2 (2 c d-b e)^{5/2}} \] Output:

1/24*(8*b*e*g-19*c*d*g+3*c*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)-1/96*c*(56*b*e*g-115*c*d*g+3*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)-1/64*c^2*(-8*b*e*g+13*c*d*g 
+3*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)-1/4*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(e*x+d)^ 
(11/2)-1/64*c^3*(-8*b*e*g+13*c*d*g+3*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 = 2.33 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.93 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\frac {c^3 ((d+e x) (-b e+c (d-e x)))^{3/2} \left (\frac {16 b^3 e^3 (3 e f+d g+4 e g x)-8 b^2 c e^2 \left (7 d^2 g-e^2 x (9 f+14 g x)+d e (27 f+29 g x)\right )+2 b c^2 e \left (25 d^3 g+3 e^3 x^2 (f+4 g x)+d^2 e (147 f+110 g x)-d e^2 x (138 f+191 g x)\right )+c^3 \left (5 d^4 g-9 e^4 f x^3+3 d^3 e (-39 f+g x)-39 d e^3 x^2 (f+g x)+d^2 e^2 x (237 f+343 g x)\right )}{c^3 (-2 c d+b e)^2 (d+e x)^4 (-b e+c (d-e x))}+\frac {3 (3 c e f+13 c d g-8 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{5/2} (-b e+c (d-e x))^{3/2}}\right )}{192 e^2 (d+e x)^{3/2}} \] Input:

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x 
)^(13/2),x]
 

Output:

(c^3*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*((16*b^3*e^3*(3*e*f + d*g + 
4*e*g*x) - 8*b^2*c*e^2*(7*d^2*g - e^2*x*(9*f + 14*g*x) + d*e*(27*f + 29*g* 
x)) + 2*b*c^2*e*(25*d^3*g + 3*e^3*x^2*(f + 4*g*x) + d^2*e*(147*f + 110*g*x 
) - d*e^2*x*(138*f + 191*g*x)) + c^3*(5*d^4*g - 9*e^4*f*x^3 + 3*d^3*e*(-39 
*f + g*x) - 39*d*e^3*x^2*(f + g*x) + d^2*e^2*x*(237*f + 343*g*x)))/(c^3*(- 
2*c*d + b*e)^2*(d + e*x)^4*(-(b*e) + c*(d - e*x))) + (3*(3*c*e*f + 13*c*d* 
g - 8*b*e*g)*ArcTan[Sqrt[c*d - b*e - c*e*x]/Sqrt[-2*c*d + b*e]])/((-2*c*d 
+ b*e)^(5/2)*(-(b*e) + c*(d - e*x))^(3/2))))/(192*e^2*(d + e*x)^(3/2))
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1220, 1130, 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.

\(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx\)

\(\Big \downarrow \) 1220

\(\displaystyle \frac {(-8 b e g+13 c d g+3 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^{11/2}}dx}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (d+e x)^{13/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {(-8 b e g+13 c d g+3 c e f) \left (-\frac {1}{2} c \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{(d+e x)^{7/2}}dx-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\right )}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (d+e x)^{13/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {(-8 b e g+13 c d g+3 c e f) \left (-\frac {1}{2} c \left (-\frac {1}{4} c \int \frac {1}{(d+e x)^{3/2} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\right )}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (d+e x)^{13/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {(-8 b e g+13 c d g+3 c e f) \left (-\frac {1}{2} c \left (-\frac {1}{4} c \left (\frac {c \int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2} (2 c d-b e)}\right )-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\right )}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (d+e x)^{13/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1136

\(\displaystyle \frac {(-8 b e g+13 c d g+3 c e f) \left (-\frac {1}{2} c \left (-\frac {1}{4} c \left (\frac {c e \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}}{2 c d-b e}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2} (2 c d-b e)}\right )-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\right )}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (d+e x)^{13/2} (2 c d-b e)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (-\frac {1}{2} c \left (-\frac {1}{4} c \left (-\frac {c \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e (2 c d-b e)^{3/2}}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2} (2 c d-b e)}\right )-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\right ) (-8 b e g+13 c d g+3 c e f)}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (d+e x)^{13/2} (2 c d-b e)}\)

Input:

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^(13/ 
2),x]
 

Output:

-1/4*((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(e^2*(2*c*d 
 - b*e)*(d + e*x)^(13/2)) + ((3*c*e*f + 13*c*d*g - 8*b*e*g)*(-1/3*(d*(c*d 
- b*e) - b*e^2*x - c*e^2*x^2)^(3/2)/(e*(d + e*x)^(9/2)) - (c*(-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*Arc 
Tanh[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))/(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. \(1508\) vs. \(2(331)=662\).

Time = 1.50 (sec) , antiderivative size = 1509, normalized size of antiderivative = 4.13

method	result	size
default	\(\text {Expression too large to display}\)	\(1509\)

Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(13/2),x,method 
=_RETURNVERBOSE)
 

Output:

-1/192*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-50*b*c^2*d^3*e*g*(-c*e*x-b*e+c*d 
)^(1/2)*(b*e-2*c*d)^(1/2)+382*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)+39*c^3*d*e^3*f*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2) 
-343*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)-220*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)-112*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-294*b*c^2*d^2*e^2*f*(-c*e*x-b*e+c*d)^( 
1/2)*(b*e-2*c*d)^(1/2)-39*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)+117*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)-5*c^3*d^4*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c* 
d)^(1/2)-237*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^2-3*c^3 
*d^3*e*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+216*b^2*c*d*e^3*f*(-c* 
e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-6*b*c^2*e^4*f*x^2*(-c*e*x-b*e+c*d)^(1 
/2)*(b*e-2*c*d)^(1/2)+39*c^3*d*e^3*g*x^3*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d 
)^(1/2)+56*b^2*c*d^2*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-72*b^2 
*c*e^4*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+276*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)+232*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. 1038 vs. \(2 (331) = 662\).

Time = 0.21 (sec) , antiderivative size = 2106, normalized size of antiderivative = 5.77 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(13/2),x, 
 algorithm="fricas")
 

Output:

[-1/384*(3*(3*c^4*d^5*e*f + (3*c^4*e^6*f + (13*c^4*d*e^5 - 8*b*c^3*e^6)*g) 
*x^5 + 5*(3*c^4*d*e^5*f + (13*c^4*d^2*e^4 - 8*b*c^3*d*e^5)*g)*x^4 + 10*(3* 
c^4*d^2*e^4*f + (13*c^4*d^3*e^3 - 8*b*c^3*d^2*e^4)*g)*x^3 + 10*(3*c^4*d^3* 
e^3*f + (13*c^4*d^4*e^2 - 8*b*c^3*d^3*e^3)*g)*x^2 + (13*c^4*d^6 - 8*b*c^3* 
d^5*e)*g + 5*(3*c^4*d^4*e^2*f + (13*c^4*d^5*e - 8*b*c^3*d^4*e^2)*g)*x)*sqr 
t(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*(3*(2*c^4*d*e^4 - b*c^3*e^5)*f + (26*c^4*d^2*e^3 - 29*b*c^3*d*e^4 
+ 8*b^2*c^2*e^5)*g)*x^3 + (3*(26*c^4*d^2*e^3 - 17*b*c^3*d*e^4 + 2*b^2*c^2* 
e^5)*f - (686*c^4*d^3*e^2 - 1107*b*c^3*d^2*e^3 + 606*b^2*c^2*d*e^4 - 112*b 
^3*c*e^5)*g)*x^2 + 3*(78*c^4*d^4*e - 235*b*c^3*d^3*e^2 + 242*b^2*c^2*d^2*e 
^3 - 104*b^3*c*d*e^4 + 16*b^4*e^5)*f - (10*c^4*d^5 + 95*b*c^3*d^4*e - 162* 
b^2*c^2*d^3*e^2 + 88*b^3*c*d^2*e^3 - 16*b^4*d*e^4)*g - (3*(158*c^4*d^3*e^2 
 - 263*b*c^3*d^2*e^3 + 140*b^2*c^2*d*e^4 - 24*b^3*c*e^5)*f + (6*c^4*d^4*e 
+ 437*b*c^3*d^3*e^2 - 684*b^2*c^2*d^2*e^3 + 360*b^3*c*d*e^4 - 64*b^4*e^5)* 
g)*x)*sqrt(e*x + d))/(8*c^3*d^8*e^2 - 12*b*c^2*d^7*e^3 + 6*b^2*c*d^6*e^4 - 
 b^3*d^5*e^5 + (8*c^3*d^3*e^7 - 12*b*c^2*d^2*e^8 + 6*b^2*c*d*e^9 - b^3*e^1 
0)*x^5 + 5*(8*c^3*d^4*e^6 - 12*b*c^2*d^3*e^7 + 6*b^2*c*d^2*e^8 - b^3*d*e^9 
)*x^4 + 10*(8*c^3*d^5*e^5 - 12*b*c^2*d^4*e^6 + 6*b^2*c*d^3*e^7 - b^3*d^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2)/(e*x+d)**(13 
/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {13}{2}}} \,d x } \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(13/2),x, 
 algorithm="maxima")
 

Output:

integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(g*x + f)/(e*x + d) 
^(13/2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 983 vs. \(2 (331) = 662\).

Time = 0.39 (sec) , antiderivative size = 983, normalized size of antiderivative = 2.69 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx =\text {Too large to display} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(13/2),x, 
 algorithm="giac")
 

Output:

1/192*(3*(3*c^5*e*f + 13*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))/((4*c^2*d^2 - 4*b*c*d*e + b^2*e^2)*sqrt( 
-2*c*d + b*e)) + (72*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^8*d^3*e*f - 108*sq 
rt(-(e*x + d)*c + 2*c*d - b*e)*b*c^7*d^2*e^2*f + 54*sqrt(-(e*x + d)*c + 2* 
c*d - b*e)*b^2*c^6*d*e^3*f - 9*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^3*c^5*e^ 
4*f + 312*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^8*d^4*g - 660*sqrt(-(e*x + d) 
*c + 2*c*d - b*e)*b*c^7*d^3*e*g + 522*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2 
*c^6*d^2*e^2*g - 183*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 - 132*(-(e*x + d)*c + 2*c* 
d - b*e)^(3/2)*c^7*d^2*e*f + 132*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^6* 
d*e^2*f - 33*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*c^5*e^3*f - 572*(-(e*x 
 + d)*c + 2*c*d - b*e)^(3/2)*c^7*d^3*g + 924*(-(e*x + d)*c + 2*c*d - b*e)^ 
(3/2)*b*c^6*d^2*e*g - 495*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*c^5*d*e^2 
*g + 88*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^3*c^4*e^3*g - 66*((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 + 33*((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 + 226*( 
(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 - 
193*((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 + 40*((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 - 9*((e*x + d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{13/2}} \,d x \] Input:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(d + e*x)^(13/ 
2),x)
 

Output:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(d + e*x)^(13/ 
2), x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 1836, normalized size of antiderivative = 5.03 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx =\text {Too large to display} \] Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(13/2),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 + 39*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 + 9*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 + 156*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 + 
36*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 + 234*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/s 
qrt(b*e - 2*c*d))*c**4*d**3*e**2*g*x**2 + 54*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 + 156*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 + 36*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 + 39*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 + 9*sqrt(b*e - 2*c*d)*ata 
n(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**4*e**5*f*x**4 + 16*sqrt 
( - b*e + c*d - c*e*x)*b**4*d*e**4*g + 48*sqrt( - b*e + c*d - c*e*x)*b*...