\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{17/2}} \, dx\) [225]

Optimal result

Integrand size = 46, antiderivative size = 430 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=-\frac {c (3 c e f-47 c d g+22 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{48 e^2 (d+e x)^{7/2}}+\frac {c^2 (3 c e f-239 c d g+118 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{192 e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac {c^3 (3 c e f+17 c d g-10 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{128 e^2 (2 c d-b e)^2 (d+e x)^{3/2}}+\frac {(c e f-5 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{15/2}}+\frac {c^4 (3 c e f+17 c d g-10 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{128 e^2 (2 c d-b e)^{5/2}} \] Output:

-1/48*c*(22*b*e*g-47*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)+1/192*c^2*(118*b*e*g-239*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)+1/128*c^3*(-10*b*e* 
g+17*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)+1/8*(2*b*e*g-5*c*d*g+c*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)-1/5*(-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)^(15/2)+1/128*c^4*(-10*b*e*g+17*c*d*g+3*c*e*f)*arctan 
h((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 = 3.33 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.03 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=\frac {c^4 ((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {-96 b^4 e^4 (4 e f+d g+5 e g x)+16 b^3 c e^3 \left (26 d^2 g-e^2 x (63 f+85 g x)+d e (129 f+133 g x)\right )-4 b^2 c^2 e^2 \left (157 d^3 g+e^3 x^2 (186 f+295 g x)-3 d e^2 x (380 f+447 g x)+d^2 e (978 f+825 g x)\right )+c^4 \left (-269 d^5 g+45 e^5 f x^4+15 d e^4 x^3 (16 f+17 g x)+2 d^3 e^2 x (1236 f+523 g x)-d^4 e (951 f+1352 g x)-2 d^2 e^3 x^2 (1263 f+1880 g x)\right )+2 b c^3 e \left (236 d^4 g-15 e^4 x^3 (f+5 g x)+3 d^3 e (523 f+419 g x)-3 d^2 e^2 x (1039 f+991 g x)+d e^3 x^2 (1443 f+2075 g x)\right )}{c^4 (-2 c d+b e)^2 (d+e x)^5 (-c d+b e+c e x)^2}-\frac {15 (3 c e f+17 c d g-10 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{5/2} (-b e+c (d-e x))^{5/2}}\right )}{1920 e^2 (d+e x)^{5/2}} \] 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 
)^(17/2),x]
 

Output:

(c^4*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((-96*b^4*e^4*(4*e*f + d*g + 
 5*e*g*x) + 16*b^3*c*e^3*(26*d^2*g - e^2*x*(63*f + 85*g*x) + d*e*(129*f + 
133*g*x)) - 4*b^2*c^2*e^2*(157*d^3*g + e^3*x^2*(186*f + 295*g*x) - 3*d*e^2 
*x*(380*f + 447*g*x) + d^2*e*(978*f + 825*g*x)) + c^4*(-269*d^5*g + 45*e^5 
*f*x^4 + 15*d*e^4*x^3*(16*f + 17*g*x) + 2*d^3*e^2*x*(1236*f + 523*g*x) - d 
^4*e*(951*f + 1352*g*x) - 2*d^2*e^3*x^2*(1263*f + 1880*g*x)) + 2*b*c^3*e*( 
236*d^4*g - 15*e^4*x^3*(f + 5*g*x) + 3*d^3*e*(523*f + 419*g*x) - 3*d^2*e^2 
*x*(1039*f + 991*g*x) + d*e^3*x^2*(1443*f + 2075*g*x)))/(c^4*(-2*c*d + b*e 
)^2*(d + e*x)^5*(-(c*d) + b*e + c*e*x)^2) - (15*(3*c*e*f + 17*c*d*g - 10*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))^(5/2))))/(1920*e^2*(d + e*x)^(5/2))
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.91, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1220, 1130, 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.

\(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx\)

\(\Big \downarrow \) 1220

\(\displaystyle \frac {(-10 b e g+17 c d g+3 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{15/2}}dx}{10 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^{17/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {(-10 b e g+17 c d g+3 c e f) \left (-\frac {5}{8} c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^{11/2}}dx-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}\right )}{10 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^{17/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {(-10 b e g+17 c d g+3 c e f) \left (-\frac {5}{8} c \left (-\frac {1}{2} c \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{(d+e x)^{7/2}}dx-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}\right )}{10 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^{17/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {(-10 b e g+17 c d g+3 c e f) \left (-\frac {5}{8} c \left (-\frac {1}{2} c \left (-\frac {1}{4} c \int \frac {1}{(d+e x)^{3/2} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}\right )}{10 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^{17/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {(-10 b e g+17 c d g+3 c e f) \left (-\frac {5}{8} c \left (-\frac {1}{2} c \left (-\frac {1}{4} c \left (\frac {c \int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2} (2 c d-b e)}\right )-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}\right )}{10 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^{17/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1136

\(\displaystyle \frac {(-10 b e g+17 c d g+3 c e f) \left (-\frac {5}{8} c \left (-\frac {1}{2} c \left (-\frac {1}{4} c \left (\frac {c e \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}}{2 c d-b e}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2} (2 c d-b e)}\right )-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}\right )}{10 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^{17/2} (2 c d-b e)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (-\frac {5}{8} c \left (-\frac {1}{2} c \left (-\frac {1}{4} c \left (-\frac {c \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e (2 c d-b e)^{3/2}}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2} (2 c d-b e)}\right )-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e (d+e x)^{5/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}\right ) (-10 b e g+17 c d g+3 c e f)}{10 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^{17/2} (2 c d-b e)}\)

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)^(17/ 
2),x]
 

Output:

-1/5*((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)^(17/2)) + ((3*c*e*f + 17*c*d*g - 10*b*e*g)*(-1/4*(d*(c*d 
 - b*e) - b*e^2*x - c*e^2*x^2)^(5/2)/(e*(d + e*x)^(13/2)) - (5*c*(-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*Sq 
rt[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]*Sqr 
t[d + e*x])])/(e*(2*c*d - b*e)^(3/2))))/4))/2))/8))/(10*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. \(2078\) vs. \(2(390)=780\).

Time = 1.53 (sec) , antiderivative size = 2079, normalized size of antiderivative = 4.83

method	result	size
default	\(\text {Expression too large to display}\)	\(2079\)

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)^(17/2),x,method 
=_RETURNVERBOSE)
 

Output:

1/1920*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(4560*b^2*c^2*d*e^4*f*x*(b*e-2*c*d 
)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+2514*b*c^3*d^3*e^2*g*x*(b*e-2*c*d)^(1/2)*(- 
c*e*x-b*e+c*d)^(1/2)+4150*b*c^3*d*e^4*g*x^3*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+ 
c*d)^(1/2)-45*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^5*e^6*f*x 
^5-45*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^5*d^5*e*f-3760*c^ 
4*d^2*e^3*g*x^3*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+750*arctan((-c*e* 
x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^4*d*e^5*g*x^4+1500*arctan((-c*e*x- 
b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^4*d^2*e^4*g*x^3+1500*arctan((-c*e*x- 
b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^4*d^3*e^3*g*x^2+750*arctan((-c*e*x-b 
*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^4*d^4*e^2*g*x+1046*c^4*d^3*e^2*g*x^2* 
(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+416*b^3*c*d^2*e^3*g*(b*e-2*c*d)^( 
1/2)*(-c*e*x-b*e+c*d)^(1/2)+2064*b^3*c*d*e^4*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b 
*e+c*d)^(1/2)-3912*b^2*c^2*d^2*e^3*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1 
/2)+472*b*c^3*d^4*e*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-6234*b*c^3* 
d^2*e^3*f*x*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+3138*b*c^3*d^3*e^2*f* 
(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-5946*b*c^3*d^2*e^3*g*x^2*(b*e-2*c 
*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+2886*b*c^3*d*e^4*f*x^2*(b*e-2*c*d)^(1/2)* 
(-c*e*x-b*e+c*d)^(1/2)+240*c^4*d*e^4*f*x^3*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c 
*d)^(1/2)-1352*c^4*d^4*e*g*x*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+2472 
*c^4*d^3*e^2*f*x*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-255*arctan((-...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1311 vs. \(2 (390) = 780\).

Time = 0.31 (sec) , antiderivative size = 2652, normalized size of antiderivative = 6.17 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=\text {Too large to display} \] 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)^(17/2),x, 
 algorithm="fricas")
 

Output:

[-1/3840*(15*(3*c^5*d^6*e*f + (3*c^5*e^7*f + (17*c^5*d*e^6 - 10*b*c^4*e^7) 
*g)*x^6 + 6*(3*c^5*d*e^6*f + (17*c^5*d^2*e^5 - 10*b*c^4*d*e^6)*g)*x^5 + 15 
*(3*c^5*d^2*e^5*f + (17*c^5*d^3*e^4 - 10*b*c^4*d^2*e^5)*g)*x^4 + 20*(3*c^5 
*d^3*e^4*f + (17*c^5*d^4*e^3 - 10*b*c^4*d^3*e^4)*g)*x^3 + 15*(3*c^5*d^4*e^ 
3*f + (17*c^5*d^5*e^2 - 10*b*c^4*d^4*e^3)*g)*x^2 + (17*c^5*d^7 - 10*b*c^4* 
d^6*e)*g + 6*(3*c^5*d^5*e^2*f + (17*c^5*d^6*e - 10*b*c^4*d^5*e^2)*g)*x)*sq 
rt(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)*(15*(3*(2*c^5*d*e^5 - b*c^4*e^6)*f + (34*c^5*d^2*e^4 - 37*b*c^4*d*e^ 
5 + 10*b^2*c^3*e^6)*g)*x^4 + 10*(3*(16*c^5*d^2*e^4 - 10*b*c^4*d*e^5 + b^2* 
c^3*e^6)*f - (752*c^5*d^3*e^3 - 1206*b*c^4*d^2*e^4 + 651*b^2*c^3*d*e^5 - 1 
18*b^3*c^2*e^6)*g)*x^3 - 2*(3*(842*c^5*d^3*e^3 - 1383*b*c^4*d^2*e^4 + 729* 
b^2*c^3*d*e^5 - 124*b^3*c^2*e^6)*f - (1046*c^5*d^4*e^2 - 6469*b*c^4*d^3*e^ 
3 + 8337*b^2*c^3*d^2*e^4 - 4042*b^3*c^2*d*e^5 + 680*b^4*c*e^6)*g)*x^2 - 3* 
(634*c^5*d^5*e - 2409*b*c^4*d^4*e^2 + 3654*b^2*c^3*d^3*e^3 - 2680*b^3*c^2* 
d^2*e^4 + 944*b^4*c*d*e^5 - 128*b^5*e^6)*f - (538*c^5*d^6 - 1213*b*c^4*d^5 
*e + 1728*b^2*c^3*d^4*e^2 - 1460*b^3*c^2*d^3*e^3 + 608*b^4*c*d^2*e^4 - 96* 
b^5*d*e^5)*g + 2*(3*(824*c^5*d^4*e^2 - 2490*b*c^4*d^3*e^3 + 2559*b^2*c^3*d 
^2*e^4 - 1096*b^3*c^2*d*e^5 + 168*b^4*c*e^6)*f - (1352*c^5*d^5*e - 3190...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=\text {Timed out} \] 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)**(17 
/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {17}{2}}} \,d x } \] 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)^(17/2),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) 
^(17/2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1449 vs. \(2 (390) = 780\).

Time = 0.49 (sec) , antiderivative size = 1449, normalized size of antiderivative = 3.37 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=\text {Too large to display} \] 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)^(17/2),x, 
 algorithm="giac")
 

Output:

-1/1920*(15*(3*c^6*e*f + 17*c^6*d*g - 10*b*c^5*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)*s 
qrt(-2*c*d + b*e)) + (720*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^10*d^4*e*f - 
1440*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^9*d^3*e^2*f + 1080*sqrt(-(e*x + 
d)*c + 2*c*d - b*e)*b^2*c^8*d^2*e^3*f - 360*sqrt(-(e*x + d)*c + 2*c*d - b* 
e)*b^3*c^7*d*e^4*f + 45*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^4*c^6*e^5*f + 4 
080*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^10*d^5*g - 10560*sqrt(-(e*x + d)*c 
+ 2*c*d - b*e)*b*c^9*d^4*e*g + 10920*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2* 
c^8*d^3*e^2*g - 5640*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^3*c^7*d^2*e^3*g + 
1455*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^4*c^6*d*e^4*g - 150*sqrt(-(e*x + d 
)*c + 2*c*d - b*e)*b^5*c^5*e^5*g - 1680*(-(e*x + d)*c + 2*c*d - b*e)^(3/2) 
*c^9*d^3*e*f + 2520*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^8*d^2*e^2*f - 1 
260*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*c^7*d*e^3*f + 210*(-(e*x + d)*c 
 + 2*c*d - b*e)^(3/2)*b^3*c^6*e^4*f - 9520*(-(e*x + d)*c + 2*c*d - b*e)^(3 
/2)*c^9*d^4*g + 19880*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^8*d^3*e*g - 1 
5540*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*c^7*d^2*e^2*g + 5390*(-(e*x + 
d)*c + 2*c*d - b*e)^(3/2)*b^3*c^6*d*e^3*g - 700*(-(e*x + d)*c + 2*c*d - b* 
e)^(3/2)*b^4*c^5*e^4*g + 1536*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d 
)*c + 2*c*d - b*e)*c^8*d^2*e*f - 1536*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(- 
(e*x + d)*c + 2*c*d - b*e)*b*c^7*d*e^2*f + 384*((e*x + d)*c - 2*c*d + b...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{17/2}} \,d x \] 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)^(17/ 
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)^(17/ 
2), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 2434, normalized size of antiderivative = 5.66 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx =\text {Too large to display} \] 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)^(17/2),x)
 

Output:

(150*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))* 
b*c**4*d**5*e*g + 750*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sq 
rt(b*e - 2*c*d))*b*c**4*d**4*e**2*g*x + 1500*sqrt(b*e - 2*c*d)*atan(sqrt( 
- b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b*c**4*d**3*e**3*g*x**2 + 1500*sqr 
t(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b*c**4*d 
**2*e**4*g*x**3 + 750*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sq 
rt(b*e - 2*c*d))*b*c**4*d*e**5*g*x**4 + 150*sqrt(b*e - 2*c*d)*atan(sqrt( - 
 b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b*c**4*e**6*g*x**5 - 255*sqrt(b*e - 
 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**5*d**6*g - 4 
5*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c** 
5*d**5*e*f - 1275*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b 
*e - 2*c*d))*c**5*d**5*e*g*x - 225*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c* 
d - c*e*x)/sqrt(b*e - 2*c*d))*c**5*d**4*e**2*f*x - 2550*sqrt(b*e - 2*c*d)* 
atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**5*d**4*e**2*g*x**2 - 
 450*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))* 
c**5*d**3*e**3*f*x**2 - 2550*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c* 
e*x)/sqrt(b*e - 2*c*d))*c**5*d**3*e**3*g*x**3 - 450*sqrt(b*e - 2*c*d)*atan 
(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**5*d**2*e**4*f*x**3 - 127 
5*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c** 
5*d**2*e**4*g*x**4 - 225*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e...