\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{13/2}} \, dx\) [223]

Optimal result

Integrand size = 46, antiderivative size = 325 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=-\frac {c (5 c e f-33 c d g+14 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e^2 (d+e x)^{3/2}}+\frac {2 c^2 g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {(5 c e f-17 c d g+6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (d+e x)^{7/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (d+e x)^{11/2}}+\frac {5 c^2 (c e f-13 c d g+6 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{8 e^2 \sqrt {2 c d-b e}} \] Output:

-1/8*c*(14*b*e*g-33*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)^(3/2)+2*c^2*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(e*x+ 
d)^(1/2)+1/12*(6*b*e*g-17*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)^(7/2)-1/3*(-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)^(11/2)+5/8*c^2*(6*b*e*g-13*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)^(1/2)
 

Mathematica [A] (verified)

Time = 1.42 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.79 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\frac {c^2 ((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {-4 b^2 e^2 (2 e f+d g+3 e g x)-2 b c e \left (6 d^2 g+d e (-3 f+17 g x)+e^2 x (13 f+27 g x)\right )+c^2 \left (121 d^3 g+3 e^3 x^2 (-11 f+16 g x)+d e^2 x (-14 f+285 g x)+d^2 e (-13 f+326 g x)\right )}{c^2 (d+e x)^3 (-c d+b e+c e x)^2}-\frac {15 (c e f-13 c d g+6 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{\sqrt {-2 c d+b e} (-b e+c (d-e x))^{5/2}}\right )}{24 e^2 (d+e x)^{5/2}} \] 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 
)^(13/2),x]
 

Output:

(c^2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((-4*b^2*e^2*(2*e*f + d*g + 
3*e*g*x) - 2*b*c*e*(6*d^2*g + d*e*(-3*f + 17*g*x) + e^2*x*(13*f + 27*g*x)) 
 + c^2*(121*d^3*g + 3*e^3*x^2*(-11*f + 16*g*x) + d*e^2*x*(-14*f + 285*g*x) 
 + d^2*e*(-13*f + 326*g*x)))/(c^2*(d + e*x)^3*(-(c*d) + b*e + c*e*x)^2) - 
(15*(c*e*f - 13*c*d*g + 6*b*e*g)*ArcTan[Sqrt[c*d - b*e - c*e*x]/Sqrt[-2*c* 
d + b*e]])/(Sqrt[-2*c*d + b*e]*(-(b*e) + c*(d - e*x))^(5/2))))/(24*e^2*(d 
+ e*x)^(5/2))
 

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.99, 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, 1131, 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.

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

\(\Big \downarrow \) 1220

\(\displaystyle -\frac {(6 b e g-13 c d g+c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{11/2}}dx}{6 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^{13/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1130

\(\displaystyle -\frac {(6 b e g-13 c d g+c e f) \left (-\frac {5}{4} c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^{7/2}}dx-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}\right )}{6 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^{13/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1130

\(\displaystyle -\frac {(6 b e g-13 c d g+c e f) \left (-\frac {5}{4} c \left (-\frac {3}{2} c \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{(d+e x)^{3/2}}dx-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e (d+e x)^{5/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}\right )}{6 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^{13/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1131

\(\displaystyle -\frac {(6 b e g-13 c d g+c e f) \left (-\frac {5}{4} c \left (-\frac {3}{2} c \left ((2 c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx+\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e (d+e x)^{5/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}\right )}{6 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^{13/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1136

\(\displaystyle -\frac {(6 b e g-13 c d g+c e f) \left (-\frac {5}{4} c \left (-\frac {3}{2} c \left (2 e (2 c d-b e) \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}+\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e (d+e x)^{5/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}\right )}{6 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^{13/2} (2 c d-b e)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\left (-\frac {5}{4} c \left (-\frac {3}{2} c \left (\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}-\frac {2 \sqrt {2 c d-b e} \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e (d+e x)^{5/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}\right ) (6 b e g-13 c d g+c e f)}{6 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^{13/2} (2 c d-b e)}\)

Input:

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(13/ 
2),x]
 

Output:

-1/3*((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)^(13/2)) - ((c*e*f - 13*c*d*g + 6*b*e*g)*(-1/2*(d*(c*d - 
b*e) - b*e^2*x - c*e^2*x^2)^(5/2)/(e*(d + e*x)^(9/2)) - (5*c*(-((d*(c*d - 
b*e) - b*e^2*x - c*e^2*x^2)^(3/2)/(e*(d + e*x)^(5/2))) - (3*c*((2*Sqrt[d*( 
c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e*Sqrt[d + e*x]) - (2*Sqrt[2*c*d - b*e 
]*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))/4))/(6*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 1131

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 + 2*p + 1))), x 
] - Simp[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1)))   Int[(d + e*x)^(m + 1)*(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] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && Ne 
Q[m + 2*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. \(1061\) vs. \(2(293)=586\).

Time = 1.50 (sec) , antiderivative size = 1062, normalized size of antiderivative = 3.27

method	result	size
default	\(\text {Expression too large to display}\)	\(1062\)

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)^(13/2),x,method 
=_RETURNVERBOSE)
 

Output:

-1/24*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(54*b*c*e^3*g*x^2*(-c*e*x-b*e+c*d)^ 
(1/2)*(b*e-2*c*d)^(1/2)-285*c^2*d*e^2*g*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2* 
c*d)^(1/2)+34*b*c*d*e^2*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-121*c 
^2*d^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+15*arctan((-c*e*x-b*e+c* 
d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*e^4*f*x^3+15*arctan((-c*e*x-b*e+c*d)^(1/2) 
/(b*e-2*c*d)^(1/2))*c^3*d^3*e*f+270*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+270*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)-6*b*c*d*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+14*c^2*d*e^ 
2*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-326*c^2*d^2*e*g*x*(-c*e*x-b 
*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+26*b*c*e^3*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e 
-2*c*d)^(1/2)-195*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^4 
*g+90*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*e^4*g*x^3-195 
*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)+45*arctan((-c*e*x-b*e+c*d)^(1 
/2)/(b*e-2*c*d)^(1/2))*c^3*d*e^3*f*x^2-48*c^2*e^3*g*x^3*(-c*e*x-b*e+c*d)^( 
1/2)*(b*e-2*c*d)^(1/2)-585*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2) 
)*c^3*d^3*e*g*x+45*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)+13*c^2*d^2*e*f*(-c*e*x-b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 680 vs. \(2 (293) = 586\).

Time = 0.17 (sec) , antiderivative size = 1390, normalized size of antiderivative = 4.28 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(13/2),x, 
 algorithm="fricas")
 

Output:

[1/48*(15*(c^3*d^4*e*f + (c^3*e^5*f - (13*c^3*d*e^4 - 6*b*c^2*e^5)*g)*x^4 
+ 4*(c^3*d*e^4*f - (13*c^3*d^2*e^3 - 6*b*c^2*d*e^4)*g)*x^3 + 6*(c^3*d^2*e^ 
3*f - (13*c^3*d^3*e^2 - 6*b*c^2*d^2*e^3)*g)*x^2 - (13*c^3*d^5 - 6*b*c^2*d^ 
4*e)*g + 4*(c^3*d^3*e^2*f - (13*c^3*d^4*e - 6*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*s 
qrt(-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) 
*(48*(2*c^3*d*e^3 - b*c^2*e^4)*g*x^3 - 3*(11*(2*c^3*d*e^3 - b*c^2*e^4)*f - 
 (190*c^3*d^2*e^2 - 131*b*c^2*d*e^3 + 18*b^2*c*e^4)*g)*x^2 - (26*c^3*d^3*e 
 - 25*b*c^2*d^2*e^2 + 22*b^2*c*d*e^3 - 8*b^3*e^4)*f + (242*c^3*d^4 - 145*b 
*c^2*d^3*e + 4*b^2*c*d^2*e^2 + 4*b^3*d*e^3)*g - 2*((14*c^3*d^2*e^2 + 19*b* 
c^2*d*e^3 - 13*b^2*c*e^4)*f - (326*c^3*d^3*e - 197*b*c^2*d^2*e^2 + 5*b^2*c 
*d*e^3 + 6*b^3*e^4)*g)*x)*sqrt(e*x + d))/(2*c*d^5*e^2 - b*d^4*e^3 + (2*c*d 
*e^6 - b*e^7)*x^4 + 4*(2*c*d^2*e^5 - b*d*e^6)*x^3 + 6*(2*c*d^3*e^4 - b*d^2 
*e^5)*x^2 + 4*(2*c*d^4*e^3 - b*d^3*e^4)*x), 1/24*(15*(c^3*d^4*e*f + (c^3*e 
^5*f - (13*c^3*d*e^4 - 6*b*c^2*e^5)*g)*x^4 + 4*(c^3*d*e^4*f - (13*c^3*d^2* 
e^3 - 6*b*c^2*d*e^4)*g)*x^3 + 6*(c^3*d^2*e^3*f - (13*c^3*d^3*e^2 - 6*b*c^2 
*d^2*e^3)*g)*x^2 - (13*c^3*d^5 - 6*b*c^2*d^4*e)*g + 4*(c^3*d^3*e^2*f - (13 
*c^3*d^4*e - 6*b*c^2*d^3*e^2)*g)*x)*sqrt(-2*c*d + b*e)*arctan(-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)/(2*c*d...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(13 
/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {13}{2}}} \,d x } \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(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)^(5/2)*(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. 596 vs. \(2 (293) = 586\).

Time = 0.41 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.83 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\frac {48 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} g - \frac {15 \, {\left (c^{4} e f - 13 \, c^{4} d g + 6 \, b c^{3} e g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} - \frac {60 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{6} d^{2} e f - 60 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{5} d e^{2} f + 15 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{4} e^{3} f - 396 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{6} d^{3} g + 564 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{5} d^{2} e g - 267 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{4} d e^{2} g + 42 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{3} c^{3} e^{3} g - 80 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{5} d e f + 40 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{4} e^{2} f + 464 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{5} d^{2} g - 424 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{4} d e g + 96 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} c^{3} e^{2} g + 33 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} e f - 141 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d g + 54 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} e g}{{\left (e x + d\right )}^{3} c^{3}}}{24 \, c e^{2}} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(13/2),x, 
 algorithm="giac")
 

Output:

1/24*(48*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*g - 15*(c^4*e*f - 13*c^4*d*g 
 + 6*b*c^3*e*g)*arctan(sqrt(-(e*x + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e) 
)/sqrt(-2*c*d + b*e) - (60*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^6*d^2*e*f - 
60*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^5*d*e^2*f + 15*sqrt(-(e*x + d)*c + 
 2*c*d - b*e)*b^2*c^4*e^3*f - 396*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^6*d^3 
*g + 564*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^5*d^2*e*g - 267*sqrt(-(e*x + 
 d)*c + 2*c*d - b*e)*b^2*c^4*d*e^2*g + 42*sqrt(-(e*x + d)*c + 2*c*d - b*e) 
*b^3*c^3*e^3*g - 80*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^5*d*e*f + 40*(-(e 
*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^4*e^2*f + 464*(-(e*x + d)*c + 2*c*d - b 
*e)^(3/2)*c^5*d^2*g - 424*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^4*d*e*g + 
 96*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*c^3*e^2*g + 33*((e*x + d)*c - 2 
*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*e*f - 141*((e*x + d)*c 
- 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*d*g + 54*((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)/((e*x + d)^ 
3*c^3))/(c*e^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{13/2}} \,d x \] Input:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(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)^(5/2))/(d + e*x)^(13/ 
2), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1230, normalized size of antiderivative = 3.78 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx =\text {Too large to display} \] Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(13/2),x)
 

Output:

( - 90*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 - 270*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 - 270*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 - 90*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 + 195*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 - 15*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 + 585*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 - 45*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 + 58 
5*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 - 45*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 + 195*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 - 15*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 - 4*sqrt( - b 
*e + c*d - c*e*x)*b**2*c*d**2*e**2*g + 22*sqrt( - b*e + c*d - c*e*x)*b**2* 
c*d*e**3*f - 10*sqrt( - b*e + c*d - c*e*x)*b**2*c*d*e**3*g*x - 26*sqrt( - 
b*e + c*d - c*e*x)*b**2*c*e**4*f*x - 54*sqrt( - b*e + c*d - c*e*x)*b**2...