\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{15/2}} \, dx\) [224]

Optimal result

Integrand size = 46, antiderivative size = 353 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=-\frac {c (5 c e f-53 c d g+24 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{32 e^2 (d+e x)^{5/2}}+\frac {c^2 (5 c e f-181 c d g+88 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac {(5 c e f-21 c d g+8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 e^2 (d+e x)^{9/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (d+e x)^{13/2}}+\frac {5 c^3 (c e f+15 c d g-8 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{64 e^2 (2 c d-b e)^{3/2}} \] Output:

-1/32*c*(24*b*e*g-53*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/(e*x+d)^(5/2)+1/64*c^2*(88*b*e*g-181*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)/(e*x+d)^(3/2)+1/24*(8*b*e*g-21*c*d* 
g+5*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(e*x+d)^(9/2)-1/4*(- 
d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/e^2/(e*x+d)^(13/2)+5/64*c^ 
3*(-8*b*e*g+15*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)^(3/2)
 

Mathematica [A] (verified)

Time = 2.24 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.97 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=\frac {c^3 ((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {16 b^3 e^3 (3 e f+d g+4 e g x)-8 b^2 c e^2 \left (3 d^2 g+d e (19 f+13 g x)-e^2 x (17 f+26 g x)\right )+2 b c^2 e \left (25 d^3 g+3 d^2 e (25 f+34 g x)-d e^2 x (154 f+79 g x)+e^3 x^2 (59 f+132 g x)\right )-c^3 \left (147 d^4 g-15 e^4 f x^3+61 d^3 e (f+9 g x)+3 d^2 e^2 x (-39 f+187 g x)+d e^3 x^2 (191 f+543 g x)\right )}{c^3 (2 c d-b e) (d+e x)^4 (-c d+b e+c e x)^2}+\frac {15 (c e f+15 c d g-8 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{3/2} (-b e+c (d-e x))^{5/2}}\right )}{192 e^2 (d+e x)^{5/2}} \] Input:

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x 
)^(15/2),x]
 

Output:

(c^3*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((16*b^3*e^3*(3*e*f + d*g + 
4*e*g*x) - 8*b^2*c*e^2*(3*d^2*g + d*e*(19*f + 13*g*x) - e^2*x*(17*f + 26*g 
*x)) + 2*b*c^2*e*(25*d^3*g + 3*d^2*e*(25*f + 34*g*x) - d*e^2*x*(154*f + 79 
*g*x) + e^3*x^2*(59*f + 132*g*x)) - c^3*(147*d^4*g - 15*e^4*f*x^3 + 61*d^3 
*e*(f + 9*g*x) + 3*d^2*e^2*x*(-39*f + 187*g*x) + d*e^3*x^2*(191*f + 543*g* 
x)))/(c^3*(2*c*d - b*e)*(d + e*x)^4*(-(c*d) + b*e + c*e*x)^2) + (15*(c*e*f 
 + 15*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)^(3/2)*(-(b*e) + c*(d - e*x))^(5/2))))/(192*e^2*(d + e*x)^ 
(5/2))
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.92, 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, 1130, 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.

\(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx\)

\(\Big \downarrow \) 1220

\(\displaystyle \frac {(-8 b e g+15 c d g+c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{13/2}}dx}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (d+e x)^{15/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {(-8 b e g+15 c d g+c e f) \left (-\frac {5}{6} c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^{9/2}}dx-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e (d+e x)^{11/2}}\right )}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (d+e x)^{15/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {(-8 b e g+15 c d g+c e f) \left (-\frac {5}{6} c \left (-\frac {3}{4} c \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{(d+e x)^{5/2}}dx-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e (d+e x)^{11/2}}\right )}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (d+e x)^{15/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {(-8 b e g+15 c d g+c e f) \left (-\frac {5}{6} c \left (-\frac {3}{4} c \left (-\frac {1}{2} c \int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e (d+e x)^{11/2}}\right )}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (d+e x)^{15/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1136

\(\displaystyle \frac {(-8 b e g+15 c d g+c e f) \left (-\frac {5}{6} c \left (-\frac {3}{4} c \left (-c e \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e (d+e x)^{11/2}}\right )}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (d+e x)^{15/2} (2 c d-b e)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (-\frac {5}{6} c \left (-\frac {3}{4} c \left (\frac {c \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e \sqrt {2 c d-b e}}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e (d+e x)^{11/2}}\right ) (-8 b e g+15 c d g+c e f)}{8 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (d+e x)^{15/2} (2 c d-b e)}\)

Input:

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(15/ 
2),x]
 

Output:

-1/4*((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(e^2*(2*c*d 
 - b*e)*(d + e*x)^(15/2)) + ((c*e*f + 15*c*d*g - 8*b*e*g)*(-1/3*(d*(c*d - 
b*e) - b*e^2*x - c*e^2*x^2)^(5/2)/(e*(d + e*x)^(11/2)) - (5*c*(-1/2*(d*(c* 
d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)/(e*(d + e*x)^(7/2)) - (3*c*(-(Sqrt[d 
*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(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*Sqrt[2*c*d - b*e])))/4))/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 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. \(2(319)=638\).

Time = 1.49 (sec) , antiderivative size = 1509, normalized size of antiderivative = 4.27

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)^(5/2)/(e*x+d)^(15/2),x,method 
=_RETURNVERBOSE)
 

Output:

-1/192*(50*b*c^2*d^3*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-158*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)-191*c^3*d*e^3*f*x^2 
*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-561*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)+204*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)+208*b^2*c*e^4*g*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d) 
^(1/2)+480*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+150*b*c^2*d^2*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-225*arcta 
n((-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)-61*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)-14 
7*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)+720*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^2-549*c^3*d^3*e*g*x*(-c*e*x-b*e+c*d)^ 
(1/2)*(b*e-2*c*d)^(1/2)-152*b^2*c*d*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c* 
d)^(1/2)+118*b*c^2*e^4*f*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-543* 
c^3*d*e^3*g*x^3*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-24*b^2*c*d^2*e^2* 
g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+136*b^2*c*e^4*f*x*(-c*e*x-b*e+c 
*d)^(1/2)*(b*e-2*c*d)^(1/2)-308*b*c^2*d*e^3*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b* 
e-2*c*d)^(1/2)+264*b*c^2*e^4*g*x^3*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2 
)-104*b^2*c*d*e^3*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+480*arct...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 941 vs. \(2 (319) = 638\).

Time = 0.18 (sec) , antiderivative size = 1912, normalized size of antiderivative = 5.42 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(15/2),x, 
 algorithm="fricas")
 

Output:

[1/384*(15*(c^4*d^5*e*f + (c^4*e^6*f + (15*c^4*d*e^5 - 8*b*c^3*e^6)*g)*x^5 
 + 5*(c^4*d*e^5*f + (15*c^4*d^2*e^4 - 8*b*c^3*d*e^5)*g)*x^4 + 10*(c^4*d^2* 
e^4*f + (15*c^4*d^3*e^3 - 8*b*c^3*d^2*e^4)*g)*x^3 + 10*(c^4*d^3*e^3*f + (1 
5*c^4*d^4*e^2 - 8*b*c^3*d^3*e^3)*g)*x^2 + (15*c^4*d^6 - 8*b*c^3*d^5*e)*g + 
 5*(c^4*d^4*e^2*f + (15*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 - (362*c^4*d^2*e^3 - 357*b*c^3*d*e^4 + 88*b^2*c 
^2*e^5)*g)*x^3 - ((382*c^4*d^2*e^3 - 427*b*c^3*d*e^4 + 118*b^2*c^2*e^5)*f 
+ (1122*c^4*d^3*e^2 - 245*b*c^3*d^2*e^3 - 574*b^2*c^2*d*e^4 + 208*b^3*c*e^ 
5)*g)*x^2 - (122*c^4*d^4*e - 361*b*c^3*d^3*e^2 + 454*b^2*c^2*d^2*e^3 - 248 
*b^3*c*d*e^4 + 48*b^4*e^5)*f - (294*c^4*d^5 - 247*b*c^3*d^4*e + 98*b^2*c^2 
*d^3*e^2 - 56*b^3*c*d^2*e^3 + 16*b^4*d*e^4)*g + ((234*c^4*d^3*e^2 - 733*b* 
c^3*d^2*e^3 + 580*b^2*c^2*d*e^4 - 136*b^3*c*e^5)*f - (1098*c^4*d^4*e - 957 
*b*c^3*d^3*e^2 + 412*b^2*c^2*d^2*e^3 - 232*b^3*c*d*e^4 + 64*b^4*e^5)*g)*x) 
*sqrt(e*x + d))/(4*c^2*d^7*e^2 - 4*b*c*d^6*e^3 + b^2*d^5*e^4 + (4*c^2*d^2* 
e^7 - 4*b*c*d*e^8 + b^2*e^9)*x^5 + 5*(4*c^2*d^3*e^6 - 4*b*c*d^2*e^7 + b^2* 
d*e^8)*x^4 + 10*(4*c^2*d^4*e^5 - 4*b*c*d^3*e^6 + b^2*d^2*e^7)*x^3 + 10*(4* 
c^2*d^5*e^4 - 4*b*c*d^4*e^5 + b^2*d^3*e^6)*x^2 + 5*(4*c^2*d^6*e^3 - 4*b...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(15 
/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {15}{2}}} \,d x } \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(15/2),x, 
 algorithm="maxima")
 

Output:

integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d) 
^(15/2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 956 vs. \(2 (319) = 638\).

Time = 0.42 (sec) , antiderivative size = 956, normalized size of antiderivative = 2.71 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx =\text {Too large to display} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(15/2),x, 
 algorithm="giac")
 

Output:

-1/192*(15*(c^5*e*f + 15*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))/((2*c*d - b*e)*sqrt(-2*c*d + b*e)) + (12 
0*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 + 1800*sqrt(-( 
e*x + d)*c + 2*c*d - b*e)*c^8*d^4*g - 3660*sqrt(-(e*x + d)*c + 2*c*d - b*e 
)*b*c^7*d^3*e*g + 2790*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^6*d^2*e^2*g 
- 945*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^3*c^5*d*e^3*g + 120*sqrt(-(e*x + 
d)*c + 2*c*d - b*e)*b^4*c^4*e^4*g - 220*(-(e*x + d)*c + 2*c*d - b*e)^(3/2) 
*c^7*d^2*e*f + 220*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^6*d*e^2*f - 55*( 
-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*c^5*e^3*f - 3300*(-(e*x + d)*c + 2*c 
*d - b*e)^(3/2)*c^7*d^3*g + 5060*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^6* 
d^2*e*g - 2585*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*c^5*d*e^2*g + 440*(- 
(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^3*c^4*e^3*g + 146*((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 - 73*((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 + 2190*((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 - 2263*((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 + 
584*((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*sqrt(-(e*x + d)*c + 2*c*d - b...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{15/2}} \,d x \] Input:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^(15/ 
2),x)
 

Output:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^(15/ 
2), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1756, normalized size of antiderivative = 4.97 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx =\text {Too large to display} \] Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(15/2),x)
 

Output:

( - 120*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 - 480*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**3*e**2*g*x - 720*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 - 480*sq 
rt(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 - 120*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 + 225*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 + 900*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 + 1350*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 + 900 
*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)/s 
qrt(b*e - 2*c*d))*c**4*d*e**4*f*x**3 + 225*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)*atan(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 - ...