\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^{9/2}} \, dx\) [211]

Optimal result

Integrand size = 46, antiderivative size = 257 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\frac {(3 c e f-11 c d g+4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (d+e x)^{3/2}}-\frac {2 c g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2}}-\frac {3 c (c e f-9 c d g+4 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2 \sqrt {2 c d-b e}} \] Output:

1/4*(4*b*e*g-11*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)^(3/2)-2*c*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(e*x+d)^(1/ 
2)-1/2*(-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)^(7/2) 
-3/4*c*(4*b*e*g-9*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 = 0.78 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.75 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\frac {c ((d+e x) (-b e+c (d-e x)))^{3/2} \left (\frac {c \left (-17 d^2 g+d e (f-29 g x)+e^2 x (5 f-8 g x)\right )+2 b e (d g+e (f+2 g x))}{c (d+e x)^2 (-b e+c (d-e x))}+\frac {3 (c e f-9 c d g+4 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{\sqrt {-2 c d+b e} (-b e+c (d-e x))^{3/2}}\right )}{4 e^2 (d+e x)^{3/2}} \] 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 
)^(9/2),x]
 

Output:

(c*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*((c*(-17*d^2*g + d*e*(f - 29*g 
*x) + e^2*x*(5*f - 8*g*x)) + 2*b*e*(d*g + e*(f + 2*g*x)))/(c*(d + e*x)^2*( 
-(b*e) + c*(d - e*x))) + (3*(c*e*f - 9*c*d*g + 4*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 
))^(3/2))))/(4*e^2*(d + e*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1220, 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.

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

\(\Big \downarrow \) 1220

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

\(\Big \downarrow \) 1130

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

\(\Big \downarrow \) 1131

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

\(\Big \downarrow \) 1136

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

\(\Big \downarrow \) 221

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

Input:

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^(9/2 
),x]
 

Output:

-1/2*((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)^(9/2)) - ((c*e*f - 9*c*d*g + 4*b*e*g)*(-((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]*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))/(4*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. \(656\) vs. \(2(231)=462\).

Time = 1.54 (sec) , antiderivative size = 657, normalized size of antiderivative = 2.56

method	result	size
default	\(\frac {\sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (12 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{3} g \,x^{2}-27 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} g \,x^{2}+3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} e^{3} f \,x^{2}+24 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} g x -54 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e g x +6 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} f x -8 c \,e^{2} g \,x^{2} \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}+12 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,d^{2} e g -27 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{3} g +3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e f +4 b \,e^{2} g x \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}-29 c d e g x \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}+5 c \,e^{2} f x \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}+2 b d e g \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}+2 b \,e^{2} f \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}-17 c \,d^{2} g \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}+c d e f \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\right )}{4 \left (e x +d \right )^{\frac {5}{2}} \sqrt {-c e x -b e +c d}\, e^{2} \sqrt {b e -2 c d}}\)	\(657\)

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

Output:

1/4*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(12*arctan((-c*e*x-b*e+c*d)^(1/2)/(b* 
e-2*c*d)^(1/2))*b*c*e^3*g*x^2-27*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d) 
^(1/2))*c^2*d*e^2*g*x^2+3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2)) 
*c^2*e^3*f*x^2+24*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*d*e 
^2*g*x-54*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*g*x+6 
*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d*e^2*f*x-8*c*e^2*g* 
x^2*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+12*arctan((-c*e*x-b*e+c*d)^(1 
/2)/(b*e-2*c*d)^(1/2))*b*c*d^2*e*g-27*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2 
*c*d)^(1/2))*c^2*d^3*g+3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))* 
c^2*d^2*e*f+4*b*e^2*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-29*c*d*e* 
g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+5*c*e^2*f*x*(-c*e*x-b*e+c*d)^ 
(1/2)*(b*e-2*c*d)^(1/2)+2*b*d*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2) 
+2*b*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-17*c*d^2*g*(-c*e*x-b*e 
+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+c*d*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^( 
1/2))/(e*x+d)^(5/2)/(-c*e*x-b*e+c*d)^(1/2)/e^2/(b*e-2*c*d)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 485 vs. \(2 (231) = 462\).

Time = 0.12 (sec) , antiderivative size = 1000, normalized size of antiderivative = 3.89 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(9/2),x, 
algorithm="fricas")
 

Output:

[1/8*(3*(c^2*d^3*e*f + (c^2*e^4*f - (9*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*( 
c^2*d*e^3*f - (9*c^2*d^2*e^2 - 4*b*c*d*e^3)*g)*x^2 - (9*c^2*d^4 - 4*b*c*d^ 
3*e)*g + 3*(c^2*d^2*e^2*f - (9*c^2*d^3*e - 4*b*c*d^2*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)*(8 
*(2*c^2*d*e^2 - b*c*e^3)*g*x^2 - (2*c^2*d^2*e + 3*b*c*d*e^2 - 2*b^2*e^3)*f 
 + (34*c^2*d^3 - 21*b*c*d^2*e + 2*b^2*d*e^2)*g - (5*(2*c^2*d*e^2 - b*c*e^3 
)*f - (58*c^2*d^2*e - 37*b*c*d*e^2 + 4*b^2*e^3)*g)*x)*sqrt(e*x + d))/(2*c* 
d^4*e^2 - b*d^3*e^3 + (2*c*d*e^5 - b*e^6)*x^3 + 3*(2*c*d^2*e^4 - b*d*e^5)* 
x^2 + 3*(2*c*d^3*e^3 - b*d^2*e^4)*x), -1/4*(3*(c^2*d^3*e*f + (c^2*e^4*f - 
(9*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*(c^2*d*e^3*f - (9*c^2*d^2*e^2 - 4*b*c 
*d*e^3)*g)*x^2 - (9*c^2*d^4 - 4*b*c*d^3*e)*g + 3*(c^2*d^2*e^2*f - (9*c^2*d 
^3*e - 4*b*c*d^2*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^2 - b*d*e 
+ (2*c*d*e - b*e^2)*x)) + sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(8*(2 
*c^2*d*e^2 - b*c*e^3)*g*x^2 - (2*c^2*d^2*e + 3*b*c*d*e^2 - 2*b^2*e^3)*f + 
(34*c^2*d^3 - 21*b*c*d^2*e + 2*b^2*d*e^2)*g - (5*(2*c^2*d*e^2 - b*c*e^3)*f 
 - (58*c^2*d^2*e - 37*b*c*d*e^2 + 4*b^2*e^3)*g)*x)*sqrt(e*x + d))/(2*c*d^4 
*e^2 - b*d^3*e^3 + (2*c*d*e^5 - b*e^6)*x^3 + 3*(2*c*d^2*e^4 - b*d*e^5)*...
 

Sympy [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \] Input:

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2)/(e*x+d)**(9/ 
2),x)
 

Output:

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(f + g*x)/(d + e*x)**(9/2 
), x)
 

Maxima [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(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)^(3/2)*(g*x + f)/(e*x + d) 
^(9/2), x)
 

Giac [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.30 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=-\frac {8 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{2} g - \frac {3 \, {\left (c^{3} e f - 9 \, c^{3} d g + 4 \, b c^{2} e g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} - \frac {6 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d e f - 3 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} e^{2} f - 22 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d^{2} g + 19 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} d e g - 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{2} e^{2} g - 5 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{3} e f + 13 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{3} d g - 4 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{2} e g}{{\left (e x + d\right )}^{2} c^{2}}}{4 \, c e^{2}} \] 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)^(9/2),x, 
algorithm="giac")
 

Output:

-1/4*(8*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^2*g - 3*(c^3*e*f - 9*c^3*d*g + 
4*b*c^2*e*g)*arctan(sqrt(-(e*x + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/s 
qrt(-2*c*d + b*e) - (6*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*d*e*f - 3*sqrt 
(-(e*x + d)*c + 2*c*d - b*e)*b*c^3*e^2*f - 22*sqrt(-(e*x + d)*c + 2*c*d - 
b*e)*c^4*d^2*g + 19*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^3*d*e*g - 4*sqrt( 
-(e*x + d)*c + 2*c*d - b*e)*b^2*c^2*e^2*g - 5*(-(e*x + d)*c + 2*c*d - b*e) 
^(3/2)*c^3*e*f + 13*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^3*d*g - 4*(-(e*x 
+ d)*c + 2*c*d - b*e)^(3/2)*b*c^2*e*g)/((e*x + d)^2*c^2))/(c*e^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \] Input:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(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)^(3/2))/(d + e*x)^(9/2 
), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 788, normalized size of antiderivative = 3.07 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\frac {12 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,d^{2} e g +24 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} g x +12 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{3} g \,x^{2}-27 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{3} g +3 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e f -54 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e g x +6 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} f x -27 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} g \,x^{2}+3 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} e^{3} f \,x^{2}+2 \sqrt {-c e x -b e +c d}\, b^{2} d \,e^{2} g +2 \sqrt {-c e x -b e +c d}\, b^{2} e^{3} f +4 \sqrt {-c e x -b e +c d}\, b^{2} e^{3} g x -21 \sqrt {-c e x -b e +c d}\, b c \,d^{2} e g -3 \sqrt {-c e x -b e +c d}\, b c d \,e^{2} f -37 \sqrt {-c e x -b e +c d}\, b c d \,e^{2} g x +5 \sqrt {-c e x -b e +c d}\, b c \,e^{3} f x -8 \sqrt {-c e x -b e +c d}\, b c \,e^{3} g \,x^{2}+34 \sqrt {-c e x -b e +c d}\, c^{2} d^{3} g -2 \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e f +58 \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e g x -10 \sqrt {-c e x -b e +c d}\, c^{2} d \,e^{2} f x +16 \sqrt {-c e x -b e +c d}\, c^{2} d \,e^{2} g \,x^{2}}{4 e^{2} \left (b \,e^{3} x^{2}-2 c d \,e^{2} x^{2}+2 b d \,e^{2} x -4 c \,d^{2} e x +b \,d^{2} e -2 c \,d^{3}\right )} \] 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)^(9/2),x)
 

Output:

(12*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 + 24*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 + 12*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*g*x**2 - 27*sqrt(b*e - 2*c*d)*atan(sqr 
t( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**2*d**3*g + 3*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 - 54*s 
qrt(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 + 6*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 - 27*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*g*x**2 + 3*sqrt(b*e - 2*c*d)*atan(sqr 
t( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**2*e**3*f*x**2 + 2*sqrt( - b* 
e + c*d - c*e*x)*b**2*d*e**2*g + 2*sqrt( - b*e + c*d - c*e*x)*b**2*e**3*f 
+ 4*sqrt( - b*e + c*d - c*e*x)*b**2*e**3*g*x - 21*sqrt( - b*e + c*d - c*e* 
x)*b*c*d**2*e*g - 3*sqrt( - b*e + c*d - c*e*x)*b*c*d*e**2*f - 37*sqrt( - b 
*e + c*d - c*e*x)*b*c*d*e**2*g*x + 5*sqrt( - b*e + c*d - c*e*x)*b*c*e**3*f 
*x - 8*sqrt( - b*e + c*d - c*e*x)*b*c*e**3*g*x**2 + 34*sqrt( - b*e + c*d - 
 c*e*x)*c**2*d**3*g - 2*sqrt( - b*e + c*d - c*e*x)*c**2*d**2*e*f + 58*sqrt 
( - b*e + c*d - c*e*x)*c**2*d**2*e*g*x - 10*sqrt( - b*e + c*d - c*e*x)*c** 
2*d*e**2*f*x + 16*sqrt( - b*e + c*d - c*e*x)*c**2*d*e**2*g*x**2)/(4*e**2*( 
b*d**2*e + 2*b*d*e**2*x + b*e**3*x**2 - 2*c*d**3 - 4*c*d**2*e*x - 2*c*d...