\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{9/2}} \, dx\) [221]

Optimal result

Integrand size = 46, antiderivative size = 331 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {(2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)^{3/2}}-\frac {2 (2 c d-b e) (2 c e f-4 c d g+b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {2 (c e f-3 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}+\frac {(2 c d-b e)^{3/2} (5 c e f-9 c d g+2 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2} \] Output:

-(-b*e+2*c*d)^2*(-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)^(3/2)-2*(-b*e+2*c*d)*(b*e*g-4*c*d*g+2*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e 
^2*x^2)^(1/2)/e^2/(e*x+d)^(1/2)-2/3*(b*e*g-3*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)^(3/2)+2/5*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2 
*x^2)^(5/2)/e^2/(e*x+d)^(5/2)+(-b*e+2*c*d)^(3/2)*(2*b*e*g-9*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
 

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.77 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {b^2 e^2 (-15 e f+61 d g+46 e g x)+2 b c e \left (-146 d^2 g+5 d e (13 f-21 g x)+e^2 x (35 f+11 g x)\right )+2 c^2 \left (168 d^3 g+e^3 x^2 (5 f+3 g x)-6 d e^2 x (10 f+3 g x)+d^2 e (-95 f+117 g x)\right )}{(d+e x) (-c d+b e+c e x)^2}+\frac {15 (-2 c d+b e)^{3/2} (-5 c e f+9 c d g-2 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{(-b e+c (d-e x))^{5/2}}\right )}{15 e^2 (d+e x)^{5/2}} \] 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 
)^(9/2),x]
 

Output:

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((b^2*e^2*(-15*e*f + 61*d*g + 46 
*e*g*x) + 2*b*c*e*(-146*d^2*g + 5*d*e*(13*f - 21*g*x) + e^2*x*(35*f + 11*g 
*x)) + 2*c^2*(168*d^3*g + e^3*x^2*(5*f + 3*g*x) - 6*d*e^2*x*(10*f + 3*g*x) 
 + d^2*e*(-95*f + 117*g*x)))/((d + e*x)*(-(c*d) + b*e + c*e*x)^2) + (15*(- 
2*c*d + b*e)^(3/2)*(-5*c*e*f + 9*c*d*g - 2*b*e*g)*ArcTan[Sqrt[c*d - b*e - 
c*e*x]/Sqrt[-2*c*d + b*e]])/(-(b*e) + c*(d - e*x))^(5/2)))/(15*e^2*(d + e* 
x)^(5/2))
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1220, 1131, 1131, 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.

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

\(\Big \downarrow \) 1220

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

\(\Big \downarrow \) 1131

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

\(\Big \downarrow \) 1131

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

\(\Big \downarrow \) 1131

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

\(\Big \downarrow \) 1136

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

\(\Big \downarrow \) 221

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

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)^(9/2 
),x]
 

Output:

-(((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)^(9/2))) - ((5*c*e*f - 9*c*d*g + 2*b*e*g)*((2*(d*(c*d - b*e) 
 - b*e^2*x - c*e^2*x^2)^(5/2))/(5*e*(d + e*x)^(5/2)) + (2*c*d - b*e)*((2*( 
d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e*(d + e*x)^(3/2)) + (2*c*d 
 - b*e)*((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]/(S 
qrt[2*c*d - b*e]*Sqrt[d + e*x])])/e))))/(2*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 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. \(1127\) vs. \(2(303)=606\).

Time = 1.50 (sec) , antiderivative size = 1128, normalized size of antiderivative = 3.41

method	result	size
default	\(\text {Expression too large to display}\)	\(1128\)

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)^(9/2),x,method= 
_RETURNVERBOSE)
 

Output:

-1/15*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-22*b*c*e^3*g*x^2*(-c*e*x-b*e+c*d) 
^(1/2)*(b*e-2*c*d)^(1/2)+30*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2 
))*b^3*d*e^3*g+36*c^2*d*e^2*g*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2) 
-255*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*d*e^3*g*x+210* 
b*c*d*e^2*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+75*arctan((-c*e*x-b 
*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*d*e^3*f-300*arctan((-c*e*x-b*e+c*d) 
^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^2*e^2*f-336*c^2*d^3*g*(-c*e*x-b*e+c*d)^( 
1/2)*(b*e-2*c*d)^(1/2)+300*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2) 
)*c^3*d^3*e*f-300*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d 
*e^3*f*x+660*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+292*b*c*d^2*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-130*b*c*d*e 
^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+30*arctan((-c*e*x-b*e+c*d)^( 
1/2)/(b*e-2*c*d)^(1/2))*b^3*e^4*g*x+120*c^2*d*e^2*f*x*(-c*e*x-b*e+c*d)^(1/ 
2)*(b*e-2*c*d)^(1/2)-234*c^2*d^2*e*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^ 
(1/2)-70*b*c*e^3*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-540*arctan(( 
-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^4*g-255*arctan((-c*e*x-b*e+ 
c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*d^2*e^2*g+75*arctan((-c*e*x-b*e+c*d)^( 
1/2)/(b*e-2*c*d)^(1/2))*b^2*c*e^4*f*x+15*b^2*e^3*f*(-c*e*x-b*e+c*d)^(1/2)* 
(b*e-2*c*d)^(1/2)-6*c^2*e^3*g*x^3*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2) 
-540*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^3*e*g*x+300...
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 992, normalized size of antiderivative = 3.00 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx =\text {Too large to display} \] 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)^(9/2),x, 
algorithm="fricas")
 

Output:

[-1/30*(15*((5*(2*c^2*d*e^3 - b*c*e^4)*f - (18*c^2*d^2*e^2 - 13*b*c*d*e^3 
+ 2*b^2*e^4)*g)*x^2 + 5*(2*c^2*d^3*e - b*c*d^2*e^2)*f - (18*c^2*d^4 - 13*b 
*c*d^3*e + 2*b^2*d^2*e^2)*g + 2*(5*(2*c^2*d^2*e^2 - b*c*d*e^3)*f - (18*c^2 
*d^3*e - 13*b*c*d^2*e^2 + 2*b^2*d*e^3)*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*(6*c^2*e^3*g*x^3 + 2*(5*c^2*e^3*f - (18*c^2*d*e^2 - 11*b*c*e^3)* 
g)*x^2 - 5*(38*c^2*d^2*e - 26*b*c*d*e^2 + 3*b^2*e^3)*f + (336*c^2*d^3 - 29 
2*b*c*d^2*e + 61*b^2*d*e^2)*g - 2*(5*(12*c^2*d*e^2 - 7*b*c*e^3)*f - (117*c 
^2*d^2*e - 105*b*c*d*e^2 + 23*b^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c 
*d^2 - b*d*e)*sqrt(e*x + d))/(e^4*x^2 + 2*d*e^3*x + d^2*e^2), 1/15*(15*((5 
*(2*c^2*d*e^3 - b*c*e^4)*f - (18*c^2*d^2*e^2 - 13*b*c*d*e^3 + 2*b^2*e^4)*g 
)*x^2 + 5*(2*c^2*d^3*e - b*c*d^2*e^2)*f - (18*c^2*d^4 - 13*b*c*d^3*e + 2*b 
^2*d^2*e^2)*g + 2*(5*(2*c^2*d^2*e^2 - b*c*d*e^3)*f - (18*c^2*d^3*e - 13*b* 
c*d^2*e^2 + 2*b^2*d*e^3)*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^2 - b*d 
*e + (2*c*d*e - b*e^2)*x)) + (6*c^2*e^3*g*x^3 + 2*(5*c^2*e^3*f - (18*c^2*d 
*e^2 - 11*b*c*e^3)*g)*x^2 - 5*(38*c^2*d^2*e - 26*b*c*d*e^2 + 3*b^2*e^3)*f 
+ (336*c^2*d^3 - 292*b*c*d^2*e + 61*b^2*d*e^2)*g - 2*(5*(12*c^2*d*e^2 - 7* 
b*c*e^3)*f - (117*c^2*d^2*e - 105*b*c*d*e^2 + 23*b^2*e^3)*g)*x)*sqrt(-c...
 

Sympy [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \] 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)**(9/ 
2),x)
 

Output:

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(f + g*x)/(d + e*x)**(9/2 
), x)
 

Maxima [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \] 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)^(9/2),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) 
^(9/2), x)
 

Giac [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.75 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {120 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d e f - 60 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} e^{2} f - 240 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g + 180 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} d e g - 30 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c e^{2} g + 10 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} e f - 30 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d g + 10 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c e g - 6 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c g + \frac {15 \, {\left (20 \, c^{4} d^{2} e f - 20 \, b c^{3} d e^{2} f + 5 \, b^{2} c^{2} e^{3} f - 36 \, c^{4} d^{3} g + 44 \, b c^{3} d^{2} e g - 17 \, b^{2} c^{2} d e^{2} g + 2 \, b^{3} c e^{3} g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {15 \, {\left (4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d^{2} e f - 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} d e^{2} f + \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{2} e^{3} f - 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d^{3} g + 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} d^{2} e g - \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{2} d e^{2} g\right )}}{{\left (e x + d\right )} c}}{15 \, c e^{2}} \] 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)^(9/2),x, 
algorithm="giac")
 

Output:

-1/15*(120*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d*e*f - 60*sqrt(-(e*x + d) 
*c + 2*c*d - b*e)*b*c^2*e^2*f - 240*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d 
^2*g + 180*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*d*e*g - 30*sqrt(-(e*x + 
d)*c + 2*c*d - b*e)*b^2*c*e^2*g + 10*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^ 
2*e*f - 30*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*d*g + 10*(-(e*x + d)*c + 
 2*c*d - b*e)^(3/2)*b*c*e*g - 6*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + 
 d)*c + 2*c*d - b*e)*c*g + 15*(20*c^4*d^2*e*f - 20*b*c^3*d*e^2*f + 5*b^2*c 
^2*e^3*f - 36*c^4*d^3*g + 44*b*c^3*d^2*e*g - 17*b^2*c^2*d*e^2*g + 2*b^3*c* 
e^3*g)*arctan(sqrt(-(e*x + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/sqrt(-2 
*c*d + b*e) + 15*(4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*d^2*e*f - 4*sqrt( 
-(e*x + d)*c + 2*c*d - b*e)*b*c^3*d*e^2*f + sqrt(-(e*x + d)*c + 2*c*d - b* 
e)*b^2*c^2*e^3*f - 4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*d^3*g + 4*sqrt(- 
(e*x + d)*c + 2*c*d - b*e)*b*c^3*d^2*e*g - sqrt(-(e*x + d)*c + 2*c*d - b*e 
)*b^2*c^2*d*e^2*g)/((e*x + d)*c))/(c*e^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \] 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)^(9/2 
),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)^(9/2 
), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 836, normalized size of antiderivative = 2.53 \[ \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)^{9/2}} \, dx=\frac {22 \sqrt {-c e x -b e +c d}\, b c \,e^{3} g \,x^{2}-36 \sqrt {-c e x -b e +c d}\, c^{2} d \,e^{2} g \,x^{2}+195 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} g x +150 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e f +70 \sqrt {-c e x -b e +c d}\, b c \,e^{3} f x -292 \sqrt {-c e x -b e +c d}\, b c \,d^{2} e g +130 \sqrt {-c e x -b e +c d}\, b c d \,e^{2} f +234 \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e g x -120 \sqrt {-c e x -b e +c d}\, c^{2} d \,e^{2} f x -30 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} d \,e^{2} g -30 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} e^{3} g x -75 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} f -75 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{3} f x -270 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{3} g +61 \sqrt {-c e x -b e +c d}\, b^{2} d \,e^{2} g +46 \sqrt {-c e x -b e +c d}\, b^{2} e^{3} g x -190 \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e f -15 \sqrt {-c e x -b e +c d}\, b^{2} e^{3} f +336 \sqrt {-c e x -b e +c d}\, c^{2} d^{3} g +10 \sqrt {-c e x -b e +c d}\, c^{2} e^{3} f \,x^{2}+6 \sqrt {-c e x -b e +c d}\, c^{2} e^{3} g \,x^{3}+195 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,d^{2} e g -270 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e g x +150 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} f x -210 \sqrt {-c e x -b e +c d}\, b c d \,e^{2} g x}{15 e^{2} \left (e x +d \right )} \] 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)^(9/2),x)
 

Output:

( - 30*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d) 
)*b**2*d*e**2*g - 30*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqr 
t(b*e - 2*c*d))*b**2*e**3*g*x + 195*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c 
*d - c*e*x)/sqrt(b*e - 2*c*d))*b*c*d**2*e*g - 75*sqrt(b*e - 2*c*d)*atan(sq 
rt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b*c*d*e**2*f + 195*sqrt(b*e - 
2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b*c*d*e**2*g*x - 
 75*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b 
*c*e**3*f*x - 270*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b 
*e - 2*c*d))*c**2*d**3*g + 150*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - 
c*e*x)/sqrt(b*e - 2*c*d))*c**2*d**2*e*f - 270*sqrt(b*e - 2*c*d)*atan(sqrt( 
 - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**2*d**2*e*g*x + 150*sqrt(b*e - 
2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**2*d*e**2*f*x 
+ 61*sqrt( - b*e + c*d - c*e*x)*b**2*d*e**2*g - 15*sqrt( - b*e + c*d - c*e 
*x)*b**2*e**3*f + 46*sqrt( - b*e + c*d - c*e*x)*b**2*e**3*g*x - 292*sqrt( 
- b*e + c*d - c*e*x)*b*c*d**2*e*g + 130*sqrt( - b*e + c*d - c*e*x)*b*c*d*e 
**2*f - 210*sqrt( - b*e + c*d - c*e*x)*b*c*d*e**2*g*x + 70*sqrt( - b*e + c 
*d - c*e*x)*b*c*e**3*f*x + 22*sqrt( - b*e + c*d - c*e*x)*b*c*e**3*g*x**2 + 
 336*sqrt( - b*e + c*d - c*e*x)*c**2*d**3*g - 190*sqrt( - b*e + c*d - c*e* 
x)*c**2*d**2*e*f + 234*sqrt( - b*e + c*d - c*e*x)*c**2*d**2*e*g*x - 120*sq 
rt( - b*e + c*d - c*e*x)*c**2*d*e**2*f*x - 36*sqrt( - b*e + c*d - c*e*x...