\(\int \frac {f+g x}{\sqrt {d+e x} (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\) [247]

Optimal result

Integrand size = 46, antiderivative size = 377 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 c (c e f+c d g-b e g) (d+e x)^{3/2}}{3 e^2 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {2 c (3 c e f+c d g-2 b e g) \sqrt {d+e x}}{e^2 (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e^2 (2 c d-b e)^3 (d+e x)^{5/2}}-\frac {(11 c e f-3 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e)^4 (d+e x)^{3/2}}-\frac {5 c (7 c e f+c d g-4 b e g) \text {arctanh}\left (\frac {\sqrt {2 c d-b e} \sqrt {d+e x}}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{4 e^2 (2 c d-b e)^{9/2}} \] Output:

2/3*c*(-b*e*g+c*d*g+c*e*f)*(e*x+d)^(3/2)/e^2/(-b*e+2*c*d)^3/(d*(-b*e+c*d)- 
b*e^2*x-c*e^2*x^2)^(3/2)+2*c*(-2*b*e*g+c*d*g+3*c*e*f)*(e*x+d)^(1/2)/e^2/(- 
b*e+2*c*d)^4/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)-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)^(5/2)-1/4*(-4*b 
*e*g-3*c*d*g+11*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)^4/(e*x+d)^(3/2)-5/4*c*(-4*b*e*g+c*d*g+7*c*e*f)*arctanh((-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 
)^(9/2)
 

Mathematica [A] (verified)

Time = 1.83 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.87 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {c (d+e x)^{5/2} \left (\frac {(-b e+c (d-e x)) \left (6 b^3 e^3 (d g+e (f+2 g x))+c^3 \left (61 d^4 g-105 e^4 f x^3+d^2 e^2 x (161 f-5 g x)-5 d e^3 x^2 (7 f+3 g x)+d^3 e (43 f+23 g x)\right )-4 b c^2 e \left (33 d^3 g+49 d e^2 f x+5 e^3 x^2 (7 f-3 g x)+d^2 e (-4 f+30 g x)\right )+b^2 c e^2 \left (65 d^2 g+e^2 x (-21 f+80 g x)+d e (-57 f+109 g x)\right )\right )}{c (-2 c d+b e)^4 (d+e x)^2}+\frac {15 (7 c e f+c d g-4 b e g) (-b e+c (d-e x))^{5/2} \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{9/2}}\right )}{12 e^2 ((d+e x) (-b e+c (d-e x)))^{5/2}} \] Input:

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

Output:

(c*(d + e*x)^(5/2)*(((-(b*e) + c*(d - e*x))*(6*b^3*e^3*(d*g + e*(f + 2*g*x 
)) + c^3*(61*d^4*g - 105*e^4*f*x^3 + d^2*e^2*x*(161*f - 5*g*x) - 5*d*e^3*x 
^2*(7*f + 3*g*x) + d^3*e*(43*f + 23*g*x)) - 4*b*c^2*e*(33*d^3*g + 49*d*e^2 
*f*x + 5*e^3*x^2*(7*f - 3*g*x) + d^2*e*(-4*f + 30*g*x)) + b^2*c*e^2*(65*d^ 
2*g + e^2*x*(-21*f + 80*g*x) + d*e*(-57*f + 109*g*x))))/(c*(-2*c*d + b*e)^ 
4*(d + e*x)^2) + (15*(7*c*e*f + c*d*g - 4*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)^(9/ 
2)))/(12*e^2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2))
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1220, 1132, 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.

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

\(\Big \downarrow \) 1220

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

\(\Big \downarrow \) 1132

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

\(\Big \downarrow \) 1135

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

\(\Big \downarrow \) 1132

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

\(\Big \downarrow \) 1136

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/2*(e*f - d*g)/(e^2*(2*c*d - b*e)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x 
 - c*e^2*x^2)^(3/2)) + ((7*c*e*f + c*d*g - 4*b*e*g)*((2*Sqrt[d + e*x])/(3* 
e*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (5*(-(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*(2*c*d - b*e))))/(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 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 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. \(1519\) vs. \(2(345)=690\).

Time = 1.61 (sec) , antiderivative size = 1520, normalized size of antiderivative = 4.03

method	result	size
default	\(\text {Expression too large to display}\)	\(1520\)

Input:

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

Output:

1/12*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-196*(b*e-2*c*d)^(1/2)*b*c^2*d*e^3* 
f*x-90*(-c*e*x-b*e+c*d)^(1/2)*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-210*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*(-c*e*x-b*e+c*d)^(1/2)-105*(-c*e*x-b*e+c*d)^(1/2)*arctan 
((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*e^4*f*x^3+60*(b*e-2*c*d)^(1 
/2)*b*c^2*e^4*g*x^3-15*(b*e-2*c*d)^(1/2)*c^3*d*e^3*g*x^3+80*(b*e-2*c*d)^(1 
/2)*b^2*c*e^4*g*x^2-140*(b*e-2*c*d)^(1/2)*b*c^2*e^4*f*x^2-5*(b*e-2*c*d)^(1 
/2)*c^3*d^2*e^2*g*x^2+45*(-c*e*x-b*e+c*d)^(1/2)*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^2-105*(b*e-2*c*d)^(1/2)*c^3*e^4*f*x 
^3+15*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/ 
2))*c^3*d^4*g+12*(b*e-2*c*d)^(1/2)*b^3*e^4*g*x+6*(b*e-2*c*d)^(1/2)*b^3*d*e 
^3*g+43*(b*e-2*c*d)^(1/2)*c^3*d^3*e*f-75*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c 
*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^3*e*g-105*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*(-c*e*x-b*e+c*d)^(1/2)+60* 
arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*e^4*g*x^2*(-c*e*x-b 
*e+c*d)^(1/2)-105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*e 
^4*f*x^2*(-c*e*x-b*e+c*d)^(1/2)+120*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*(-c*e*x-b*e+c*d)^(1/2)+109*(b*e-2*c*d)^(1/2)*b^ 
2*c*d*e^3*g*x-120*(b*e-2*c*d)^(1/2)*b*c^2*d^2*e^2*g*x+60*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*(-c*e*x-b*e+c*d)^(1/2)+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1534 vs. \(2 (345) = 690\).

Time = 1.58 (sec) , antiderivative size = 3098, normalized size of antiderivative = 8.22 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.63 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {\frac {15 \, {\left (7 \, c^{2} e f + c^{2} d g - 4 \, b c e g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (16 \, c^{4} d^{4} e - 32 \, b c^{3} d^{3} e^{2} + 24 \, b^{2} c^{2} d^{2} e^{3} - 8 \, b^{3} c d e^{4} + b^{4} e^{5}\right )} \sqrt {-2 \, c d + b e}} - \frac {8 \, {\left (2 \, c^{3} d e f - b c^{2} e^{2} f + 2 \, c^{3} d^{2} g - 3 \, b c^{2} d e g + b^{2} c e^{2} g - 9 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} c^{2} e f - 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} c^{2} d g + 6 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} b c e g\right )}}{{\left (16 \, c^{4} d^{4} e - 32 \, b c^{3} d^{3} e^{2} + 24 \, b^{2} c^{2} d^{2} e^{3} - 8 \, b^{3} c d e^{4} + b^{4} e^{5}\right )} {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}} - \frac {3 \, {\left (26 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d e f - 13 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} e^{2} f - 10 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g - 3 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} d e g + 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c e^{2} g - 11 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} e f + 3 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d g + 4 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c e g\right )}}{{\left (16 \, c^{4} d^{4} e - 32 \, b c^{3} d^{3} e^{2} + 24 \, b^{2} c^{2} d^{2} e^{3} - 8 \, b^{3} c d e^{4} + b^{4} e^{5}\right )} {\left (e x + d\right )}^{2} c^{2}}}{12 \, e} \] Input:

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

Output:

1/12*(15*(7*c^2*e*f + c^2*d*g - 4*b*c*e*g)*arctan(sqrt(-(e*x + d)*c + 2*c* 
d - b*e)/sqrt(-2*c*d + b*e))/((16*c^4*d^4*e - 32*b*c^3*d^3*e^2 + 24*b^2*c^ 
2*d^2*e^3 - 8*b^3*c*d*e^4 + b^4*e^5)*sqrt(-2*c*d + b*e)) - 8*(2*c^3*d*e*f 
- b*c^2*e^2*f + 2*c^3*d^2*g - 3*b*c^2*d*e*g + b^2*c*e^2*g - 9*((e*x + d)*c 
 - 2*c*d + b*e)*c^2*e*f - 3*((e*x + d)*c - 2*c*d + b*e)*c^2*d*g + 6*((e*x 
+ d)*c - 2*c*d + b*e)*b*c*e*g)/((16*c^4*d^4*e - 32*b*c^3*d^3*e^2 + 24*b^2* 
c^2*d^2*e^3 - 8*b^3*c*d*e^4 + b^4*e^5)*((e*x + d)*c - 2*c*d + b*e)*sqrt(-( 
e*x + d)*c + 2*c*d - b*e)) - 3*(26*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d* 
e*f - 13*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*e^2*f - 10*sqrt(-(e*x + d) 
*c + 2*c*d - b*e)*c^3*d^2*g - 3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*d*e 
*g + 4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c*e^2*g - 11*(-(e*x + d)*c + 2 
*c*d - b*e)^(3/2)*c^2*e*f + 3*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*d*g + 
 4*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c*e*g)/((16*c^4*d^4*e - 32*b*c^3*d 
^3*e^2 + 24*b^2*c^2*d^2*e^3 - 8*b^3*c*d*e^4 + b^4*e^5)*(e*x + d)^2*c^2))/e
 

Mupad [F(-1)]

Timed out. \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {f+g\,x}{\sqrt {d+e\,x}\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}} \,d x \] 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 = 0.23 (sec) , antiderivative size = 1902, normalized size of antiderivative = 5.05 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 60*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*c*d**2*e**2*g - 120*sqrt(b*e - 2*c*d)*sq 
rt( - b*e + c*d - c*e*x)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d) 
)*b**2*c*d*e**3*g*x - 60*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*c*e**4*g*x**2 + 75*sqr 
t(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**2*d**3*e*g + 105*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**2*d**2 
*e**2*f + 90*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**2*d**2*e**2*g*x + 210*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**2*d*e**3*f*x - 45*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**2*d*e**3*g*x** 
2 + 105*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**2*e**4*f*x**2 - 60*sqrt(b*e - 2*c*d)*sq 
rt( - b*e + c*d - c*e*x)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d) 
)*b*c**2*e**4*g*x**3 - 15*sqrt(b*e - 2*c*d)*sqrt( - b*e + c*d - c*e*x)*ata 
n(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**3*d**4*g - 105*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**3*d**3*e*f - 15*sqrt(b*e - 2*c*d)*sqrt( - b*e + c*d - ...