\(\int \frac {\sqrt {d+e x} (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\) [246]

Optimal result

Integrand size = 46, antiderivative size = 291 \[ \int \frac {\sqrt {d+e x} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 (c e f+c d g-b e g) (d+e x)^{3/2}}{3 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {2 (2 c f-b g) \sqrt {d+e x}}{e (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e)^3 (d+e x)^{3/2}}-\frac {(5 c e f-c d g-2 b e g) \text {arctanh}\left (\frac {\sqrt {2 c d-b e} \sqrt {d+e x}}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e^2 (2 c d-b e)^{7/2}} \] Output:

2/3*(-b*e*g+c*d*g+c*e*f)*(e*x+d)^(3/2)/e^2/(-b*e+2*c*d)^2/(d*(-b*e+c*d)-b* 
e^2*x-c*e^2*x^2)^(3/2)+2*(-b*g+2*c*f)*(e*x+d)^(1/2)/e/(-b*e+2*c*d)^3/(d*(- 
b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)-(-d*g+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)^3/(e*x+d)^(3/2)-(-2*b*e*g-c*d*g+5*c*e*f)*arctan 
h((-b*e+2*c*d)^(1/2)*(e*x+d)^(1/2)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)) 
/e^2/(-b*e+2*c*d)^(7/2)
 

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {d+e x} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {(d+e x)^{5/2} \left (\frac {(-b e+c (d-e x)) \left (b^2 e^2 (-3 e f+11 d g+8 e g x)-2 b c e \left (4 d e f+9 d^2 g+e^2 x (10 f-3 g x)\right )+c^2 \left (7 d^3 g-15 e^3 f x^2+d^2 e (13 f-2 g x)+d e^2 x (10 f+3 g x)\right )\right )}{(2 c d-b e)^3 (d+e x)}+\frac {3 (-5 c e f+c d g+2 b e g) (-b e+c (d-e x))^{5/2} \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{7/2}}\right )}{3 e^2 ((d+e x) (-b e+c (d-e x)))^{5/2}} \] Input:

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

Output:

((d + e*x)^(5/2)*(((-(b*e) + c*(d - e*x))*(b^2*e^2*(-3*e*f + 11*d*g + 8*e* 
g*x) - 2*b*c*e*(4*d*e*f + 9*d^2*g + e^2*x*(10*f - 3*g*x)) + c^2*(7*d^3*g - 
 15*e^3*f*x^2 + d^2*e*(13*f - 2*g*x) + d*e^2*x*(10*f + 3*g*x))))/((2*c*d - 
 b*e)^3*(d + e*x)) + (3*(-5*c*e*f + c*d*g + 2*b*e*g)*(-(b*e) + c*(d - e*x) 
)^(5/2)*ArcTan[Sqrt[c*d - b*e - c*e*x]/Sqrt[-2*c*d + b*e]])/(-2*c*d + b*e) 
^(7/2)))/(3*e^2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2))
 

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1218, 1135, 1132, 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.

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

\(\Big \downarrow \) 1218

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

\(\Big \downarrow \) 1135

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

\(\Big \downarrow \) 1132

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

\(\Big \downarrow \) 1136

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(2*(c*e*f + c*d*g - b*e*g)*Sqrt[d + e*x])/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - 
 b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + ((5*c*e*f - c*d*g - 2*b*e*g)*(-(1/(e 
*(2*c*d - b*e)*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])) + 
 (3*c*((2*Sqrt[d + e*x])/(e*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c 
*e^2*x^2]) - (2*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*(2*c*d - b*e)^(3/2))))/(2*(2*c*d - b*e))))/ 
(3*c*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 1132

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(2*c*d - b*e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 
 1)*(b^2 - 4*a*c))), x] - Simp[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(b^2 - 
 4*a*c)))   Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; Free 
Q[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && LtQ 
[0, m, 1] && 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 1218

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + ( 
c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(( 
a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e))), x] - Simp[e*((m*(g*(c* 
d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e)))   I 
nt[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d 
, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(895\) vs. \(2(269)=538\).

Time = 1.61 (sec) , antiderivative size = 896, normalized size of antiderivative = 3.08

method	result	size
default	\(-\frac {\sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (6 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, b c \,e^{3} g \,x^{2}+3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d \,e^{2} g \,x^{2}-15 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} e^{3} f \,x^{2}+6 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} e^{3} g x \sqrt {-c e x -b e +c d}+3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, b c d \,e^{2} g x -15 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{3} f x \sqrt {-c e x -b e +c d}+6 \sqrt {b e -2 c d}\, b c \,e^{3} g \,x^{2}+3 \sqrt {b e -2 c d}\, c^{2} d \,e^{2} g \,x^{2}-15 \sqrt {b e -2 c d}\, c^{2} e^{3} f \,x^{2}+6 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} d \,e^{2} g \sqrt {-c e x -b e +c d}-3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, b c \,d^{2} e g -15 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} f \sqrt {-c e x -b e +c d}-3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d^{3} g +15 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e f +8 \sqrt {b e -2 c d}\, b^{2} e^{3} g x -20 \sqrt {b e -2 c d}\, b c \,e^{3} f x -2 \sqrt {b e -2 c d}\, c^{2} d^{2} e g x +10 \sqrt {b e -2 c d}\, c^{2} d \,e^{2} f x +11 \sqrt {b e -2 c d}\, b^{2} d \,e^{2} g -3 \sqrt {b e -2 c d}\, b^{2} e^{3} f -18 \sqrt {b e -2 c d}\, b c \,d^{2} e g -8 \sqrt {b e -2 c d}\, b c d \,e^{2} f +7 \sqrt {b e -2 c d}\, c^{2} d^{3} g +13 \sqrt {b e -2 c d}\, c^{2} d^{2} e f \right )}{3 \left (e x +d \right )^{\frac {3}{2}} \left (c e x +b e -c d \right )^{2} e^{2} \left (b e -2 c d \right )^{\frac {7}{2}}}\)	\(896\)

Input:

int((e*x+d)^(1/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x,method= 
_RETURNVERBOSE)
 

Output:

-1/3*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(6*arctan((-c*e*x-b*e+c*d)^(1/2)/(b* 
e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*b*c*e^3*g*x^2+3*arctan((-c*e*x-b*e+ 
c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c^2*d*e^2*g*x^2-15*ar 
ctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c^2* 
e^3*f*x^2+6*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*e^3*g*x*( 
-c*e*x-b*e+c*d)^(1/2)+3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*( 
-c*e*x-b*e+c*d)^(1/2)*b*c*d*e^2*g*x-15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e- 
2*c*d)^(1/2))*b*c*e^3*f*x*(-c*e*x-b*e+c*d)^(1/2)+6*(b*e-2*c*d)^(1/2)*b*c*e 
^3*g*x^2+3*(b*e-2*c*d)^(1/2)*c^2*d*e^2*g*x^2-15*(b*e-2*c*d)^(1/2)*c^2*e^3* 
f*x^2+6*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*d*e^2*g*(-c*e 
*x-b*e+c*d)^(1/2)-3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e 
*x-b*e+c*d)^(1/2)*b*c*d^2*e*g-15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d) 
^(1/2))*b*c*d*e^2*f*(-c*e*x-b*e+c*d)^(1/2)-3*arctan((-c*e*x-b*e+c*d)^(1/2) 
/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c^2*d^3*g+15*arctan((-c*e*x-b*e 
+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c^2*d^2*e*f+8*(b*e-2 
*c*d)^(1/2)*b^2*e^3*g*x-20*(b*e-2*c*d)^(1/2)*b*c*e^3*f*x-2*(b*e-2*c*d)^(1/ 
2)*c^2*d^2*e*g*x+10*(b*e-2*c*d)^(1/2)*c^2*d*e^2*f*x+11*(b*e-2*c*d)^(1/2)*b 
^2*d*e^2*g-3*(b*e-2*c*d)^(1/2)*b^2*e^3*f-18*(b*e-2*c*d)^(1/2)*b*c*d^2*e*g- 
8*(b*e-2*c*d)^(1/2)*b*c*d*e^2*f+7*(b*e-2*c*d)^(1/2)*c^2*d^3*g+13*(b*e-2*c* 
d)^(1/2)*c^2*d^2*e*f)/(e*x+d)^(3/2)/(c*e*x+b*e-c*d)^2/e^2/(b*e-2*c*d)^(...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1039 vs. \(2 (269) = 538\).

Time = 0.44 (sec) , antiderivative size = 2108, normalized size of antiderivative = 7.24 \[ \int \frac {\sqrt {d+e x} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(1/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, 
algorithm="fricas")
 

Output:

[1/6*(3*((5*c^3*e^5*f - (c^3*d*e^4 + 2*b*c^2*e^5)*g)*x^4 + 2*(5*b*c^2*e^5* 
f - (b*c^2*d*e^4 + 2*b^2*c*e^5)*g)*x^3 - (5*(2*c^3*d^2*e^3 - 2*b*c^2*d*e^4 
 - b^2*c*e^5)*f - (2*c^3*d^3*e^2 + 2*b*c^2*d^2*e^3 - 5*b^2*c*d*e^4 - 2*b^3 
*e^5)*g)*x^2 + 5*(c^3*d^4*e - 2*b*c^2*d^3*e^2 + b^2*c*d^2*e^3)*f - (c^3*d^ 
5 - 3*b^2*c*d^3*e^2 + 2*b^3*d^2*e^3)*g - 2*(5*(b*c^2*d^2*e^3 - b^2*c*d*e^4 
)*f - (b*c^2*d^3*e^2 + b^2*c*d^2*e^3 - 2*b^3*d*e^4)*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^3*d*e^3 - b*c^2*e^4)*f - (2*c^3*d^2*e^2 + 3*b*c^2*d*e^3 - 2*b^2*c*e^4)* 
g)*x^2 - (26*c^3*d^3*e - 29*b*c^2*d^2*e^2 + 2*b^2*c*d*e^3 + 3*b^3*e^4)*f - 
 (14*c^3*d^4 - 43*b*c^2*d^3*e + 40*b^2*c*d^2*e^2 - 11*b^3*d*e^3)*g - 2*(5* 
(2*c^3*d^2*e^2 - 5*b*c^2*d*e^3 + 2*b^2*c*e^4)*f - (2*c^3*d^3*e - b*c^2*d^2 
*e^2 - 8*b^2*c*d*e^3 + 4*b^3*e^4)*g)*x)*sqrt(e*x + d))/(16*c^6*d^8*e^2 - 6 
4*b*c^5*d^7*e^3 + 104*b^2*c^4*d^6*e^4 - 88*b^3*c^3*d^5*e^5 + 41*b^4*c^2*d^ 
4*e^6 - 10*b^5*c*d^3*e^7 + b^6*d^2*e^8 + (16*c^6*d^4*e^6 - 32*b*c^5*d^3*e^ 
7 + 24*b^2*c^4*d^2*e^8 - 8*b^3*c^3*d*e^9 + b^4*c^2*e^10)*x^4 + 2*(16*b*c^5 
*d^4*e^6 - 32*b^2*c^4*d^3*e^7 + 24*b^3*c^3*d^2*e^8 - 8*b^4*c^2*d*e^9 + b^5 
*c*e^10)*x^3 - (32*c^6*d^6*e^4 - 96*b*c^5*d^5*e^5 + 96*b^2*c^4*d^4*e^6 - 3 
2*b^3*c^3*d^3*e^7 - 6*b^4*c^2*d^2*e^8 + 6*b^5*c*d*e^9 - b^6*e^10)*x^2 -...
 

Sympy [F]

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

integrate((e*x+d)**(1/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/ 
2),x)
 

Output:

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

Maxima [F]

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

integrate((e*x+d)^(1/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, 
algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {d+e x} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {\frac {3 \, {\left (5 \, c e f - c d g - 2 \, b e g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (8 \, c^{3} d^{3} e - 12 \, b c^{2} d^{2} e^{2} + 6 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} \sqrt {-2 \, c d + b e}} - \frac {2 \, {\left (2 \, c^{2} d e f - b c e^{2} f + 2 \, c^{2} d^{2} g - 3 \, b c d e g + b^{2} e^{2} g - 6 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} c e f + 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} b e g\right )}}{{\left (8 \, c^{3} d^{3} e - 12 \, b c^{2} d^{2} e^{2} + 6 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}} - \frac {3 \, {\left (\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c e f - \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c d g\right )}}{{\left (8 \, c^{3} d^{3} e - 12 \, b c^{2} d^{2} e^{2} + 6 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} {\left (e x + d\right )} c}}{3 \, e} \] Input:

integrate((e*x+d)^(1/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, 
algorithm="giac")
 

Output:

1/3*(3*(5*c*e*f - c*d*g - 2*b*e*g)*arctan(sqrt(-(e*x + d)*c + 2*c*d - b*e) 
/sqrt(-2*c*d + b*e))/((8*c^3*d^3*e - 12*b*c^2*d^2*e^2 + 6*b^2*c*d*e^3 - b^ 
3*e^4)*sqrt(-2*c*d + b*e)) - 2*(2*c^2*d*e*f - b*c*e^2*f + 2*c^2*d^2*g - 3* 
b*c*d*e*g + b^2*e^2*g - 6*((e*x + d)*c - 2*c*d + b*e)*c*e*f + 3*((e*x + d) 
*c - 2*c*d + b*e)*b*e*g)/((8*c^3*d^3*e - 12*b*c^2*d^2*e^2 + 6*b^2*c*d*e^3 
- b^3*e^4)*((e*x + d)*c - 2*c*d + b*e)*sqrt(-(e*x + d)*c + 2*c*d - b*e)) - 
 3*(sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*e*f - sqrt(-(e*x + d)*c + 2*c*d - b 
*e)*c*d*g)/((8*c^3*d^3*e - 12*b*c^2*d^2*e^2 + 6*b^2*c*d*e^3 - b^3*e^4)*(e* 
x + d)*c))/e
 

Mupad [F(-1)]

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

int(((f + g*x)*(d + e*x)^(1/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2 
),x)
 

Output:

int(((f + g*x)*(d + e*x)^(1/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2 
), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 1124, normalized size of antiderivative = 3.86 \[ \int \frac {\sqrt {d+e x} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^(1/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
 

Output:

(6*sqrt(b*e - 2*c*d)*sqrt( - b*e + c*d - c*e*x)*atan(sqrt( - b*e + c*d - c 
*e*x)/sqrt(b*e - 2*c*d))*b**2*d*e**2*g + 6*sqrt(b*e - 2*c*d)*sqrt( - b*e + 
 c*d - c*e*x)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b**2*e**3 
*g*x - 3*sqrt(b*e - 2*c*d)*sqrt( - b*e + c*d - c*e*x)*atan(sqrt( - b*e + c 
*d - c*e*x)/sqrt(b*e - 2*c*d))*b*c*d**2*e*g - 15*sqrt(b*e - 2*c*d)*sqrt( - 
 b*e + c*d - c*e*x)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b*c 
*d*e**2*f + 3*sqrt(b*e - 2*c*d)*sqrt( - b*e + c*d - c*e*x)*atan(sqrt( - b* 
e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b*c*d*e**2*g*x - 15*sqrt(b*e - 2*c*d)* 
sqrt( - b*e + c*d - c*e*x)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c* 
d))*b*c*e**3*f*x + 6*sqrt(b*e - 2*c*d)*sqrt( - b*e + c*d - c*e*x)*atan(sqr 
t( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b*c*e**3*g*x**2 - 3*sqrt(b*e - 
2*c*d)*sqrt( - b*e + c*d - c*e*x)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e 
 - 2*c*d))*c**2*d**3*g + 15*sqrt(b*e - 2*c*d)*sqrt( - b*e + c*d - c*e*x)*a 
tan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**2*d**2*e*f + 3*sqrt(b 
*e - 2*c*d)*sqrt( - b*e + c*d - c*e*x)*atan(sqrt( - b*e + c*d - c*e*x)/sqr 
t(b*e - 2*c*d))*c**2*d*e**2*g*x**2 - 15*sqrt(b*e - 2*c*d)*sqrt( - b*e + c* 
d - c*e*x)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**2*e**3*f* 
x**2 + 11*b**3*d*e**3*g - 3*b**3*e**4*f + 8*b**3*e**4*g*x - 40*b**2*c*d**2 
*e**2*g - 2*b**2*c*d*e**3*f - 16*b**2*c*d*e**3*g*x - 20*b**2*c*e**4*f*x + 
6*b**2*c*e**4*g*x**2 + 43*b*c**2*d**3*e*g + 29*b*c**2*d**2*e**2*f - 2*b...