\(\int \frac {f+g x}{(d+e x)^{3/2} (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\) [238]

Optimal result

Integrand size = 46, antiderivative size = 303 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 c (c e f+c d g-b e g) \sqrt {d+e x}}{e^2 (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e^2 (2 c d-b e)^2 (d+e x)^{5/2}}-\frac {(7 c e f+c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e)^3 (d+e x)^{3/2}}-\frac {3 c (5 c e f+3 c d g-4 b e g) \text {arctanh}\left (\frac {\sqrt {2 c d-b e} \sqrt {d+e x}}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{4 e^2 (2 c d-b e)^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.34 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.84 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {c (d+e x)^{3/2} \left (\frac {(-b e+c (d-e x)) \left (b c e \left (-9 d^2 g+e^2 x (5 f-12 g x)+13 d e (f-g x)\right )-2 b^2 e^2 (d g+e (f+2 g x))+c^2 \left (11 d^3 g+15 e^3 f x^2-3 d^2 e (f-4 g x)+d e^2 x (20 f+9 g x)\right )\right )}{c (2 c d-b e)^3 (d+e x)^2}-\frac {3 (5 c e f+3 c d g-4 b e g) (-b e+c (d-e x))^{3/2} \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{7/2}}\right )}{4 e^2 ((d+e x) (-b e+c (d-e x)))^{3/2}} \] Input:

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

Output:

(c*(d + e*x)^(3/2)*(((-(b*e) + c*(d - e*x))*(b*c*e*(-9*d^2*g + e^2*x*(5*f 
- 12*g*x) + 13*d*e*(f - g*x)) - 2*b^2*e^2*(d*g + e*(f + 2*g*x)) + c^2*(11* 
d^3*g + 15*e^3*f*x^2 - 3*d^2*e*(f - 4*g*x) + d*e^2*x*(20*f + 9*g*x))))/(c* 
(2*c*d - b*e)^3*(d + e*x)^2) - (3*(5*c*e*f + 3*c*d*g - 4*b*e*g)*(-(b*e) + 
c*(d - e*x))^(3/2)*ArcTan[Sqrt[c*d - b*e - c*e*x]/Sqrt[-2*c*d + b*e]])/(-2 
*c*d + b*e)^(7/2)))/(4*e^2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2))
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1220, 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.

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

\(\Big \downarrow \) 1220

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

\(\Big \downarrow \) 1135

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

\(\Big \downarrow \) 1132

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

\(\Big \downarrow \) 1136

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/2*(e*f - d*g)/(e^2*(2*c*d - b*e)*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b 
*e^2*x - c*e^2*x^2]) + ((5*c*e*f + 3*c*d*g - 4*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*Sqr 
t[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]*Sq 
rt[d + e*x])])/(e*(2*c*d - b*e)^(3/2))))/(2*(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. \(815\) vs. \(2(277)=554\).

Time = 1.63 (sec) , antiderivative size = 816, normalized size of antiderivative = 2.69

method	result	size
default	\(-\frac {\sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (12 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, b c \,e^{3} g \,x^{2}-9 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d \,e^{2} g \,x^{2}-15 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} e^{3} f \,x^{2}+12 \sqrt {b e -2 c d}\, b c \,e^{3} g \,x^{2}-9 \sqrt {b e -2 c d}\, c^{2} d \,e^{2} g \,x^{2}-15 \sqrt {b e -2 c d}\, c^{2} e^{3} f \,x^{2}+24 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, b c d \,e^{2} g x -18 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e g x -30 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d \,e^{2} f x +4 \sqrt {b e -2 c d}\, b^{2} e^{3} g x +13 \sqrt {b e -2 c d}\, b c d \,e^{2} g x -5 \sqrt {b e -2 c d}\, b c \,e^{3} f x -12 \sqrt {b e -2 c d}\, c^{2} d^{2} e g x -20 \sqrt {b e -2 c d}\, c^{2} d \,e^{2} f x +12 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, b c \,d^{2} e g -9 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d^{3} g -15 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e f +2 \sqrt {b e -2 c d}\, b^{2} d \,e^{2} g +2 \sqrt {b e -2 c d}\, b^{2} e^{3} f +9 \sqrt {b e -2 c d}\, b c \,d^{2} e g -13 \sqrt {b e -2 c d}\, b c d \,e^{2} f -11 \sqrt {b e -2 c d}\, c^{2} d^{3} g +3 \sqrt {b e -2 c d}\, c^{2} d^{2} e f \right )}{4 \left (e x +d \right )^{\frac {5}{2}} \left (c e x +b e -c d \right ) e^{2} \left (b e -2 c d \right )^{\frac {7}{2}}}\)	\(816\)

Input:

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

Output:

-1/4/(e*x+d)^(5/2)*(-(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))*(-c*e*x-b*e+c*d)^(1/2)*b*c*e^3*g*x^2-9*arct 
an((-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*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*e^3*f*x^2+12*(b*e-2*c*d)^(1/2)*b*c*e^3*g*x^2-9*(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+24*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-18 
*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*g*x-30*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*f*x+4*(b*e-2*c*d)^(1/2)*b^2*e^3*g*x+13*(b*e-2*c*d) 
^(1/2)*b*c*d*e^2*g*x-5*(b*e-2*c*d)^(1/2)*b*c*e^3*f*x-12*(b*e-2*c*d)^(1/2)* 
c^2*d^2*e*g*x-20*(b*e-2*c*d)^(1/2)*c^2*d*e^2*f*x+12*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-9*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+2*(b*e-2*c*d)^(1/2)*b^2*d*e^2*g+2*(b*e-2*c*d)^(1/2)*b^2*e^3*f+9 
*(b*e-2*c*d)^(1/2)*b*c*d^2*e*g-13*(b*e-2*c*d)^(1/2)*b*c*d*e^2*f-11*(b*e-2* 
c*d)^(1/2)*c^2*d^3*g+3*(b*e-2*c*d)^(1/2)*c^2*d^2*e*f)/(c*e*x+b*e-c*d)/e^2/ 
(b*e-2*c*d)^(7/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 974 vs. \(2 (277) = 554\).

Time = 0.32 (sec) , antiderivative size = 1978, normalized size of antiderivative = 6.53 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/8*(3*((5*c^3*e^5*f + (3*c^3*d*e^4 - 4*b*c^2*e^5)*g)*x^4 + (5*(2*c^3*d* 
e^4 + b*c^2*e^5)*f + (6*c^3*d^2*e^3 - 5*b*c^2*d*e^4 - 4*b^2*c*e^5)*g)*x^3 
+ 3*(5*b*c^2*d*e^4*f + (3*b*c^2*d^2*e^3 - 4*b^2*c*d*e^4)*g)*x^2 - 5*(c^3*d 
^4*e - b*c^2*d^3*e^2)*f - (3*c^3*d^5 - 7*b*c^2*d^4*e + 4*b^2*c*d^3*e^2)*g 
- (5*(2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3)*f + (6*c^3*d^4*e - 17*b*c^2*d^3*e^2 
 + 12*b^2*c*d^2*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*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 + (6* 
c^3*d^2*e^2 - 11*b*c^2*d*e^3 + 4*b^2*c*e^4)*g)*x^2 - (6*c^3*d^3*e - 29*b*c 
^2*d^2*e^2 + 17*b^2*c*d*e^3 - 2*b^3*e^4)*f + (22*c^3*d^4 - 29*b*c^2*d^3*e 
+ 5*b^2*c*d^2*e^2 + 2*b^3*d*e^3)*g + (5*(8*c^3*d^2*e^2 - 2*b*c^2*d*e^3 - b 
^2*c*e^4)*f + (24*c^3*d^3*e - 38*b*c^2*d^2*e^2 + 5*b^2*c*d*e^3 + 4*b^3*e^4 
)*g)*x)*sqrt(e*x + d))/(16*c^5*d^8*e^2 - 48*b*c^4*d^7*e^3 + 56*b^2*c^3*d^6 
*e^4 - 32*b^3*c^2*d^5*e^5 + 9*b^4*c*d^4*e^6 - b^5*d^3*e^7 - (16*c^5*d^4*e^ 
6 - 32*b*c^4*d^3*e^7 + 24*b^2*c^3*d^2*e^8 - 8*b^3*c^2*d*e^9 + b^4*c*e^10)* 
x^4 - (32*c^5*d^5*e^5 - 48*b*c^4*d^4*e^6 + 16*b^2*c^3*d^3*e^7 + 8*b^3*c^2* 
d^2*e^8 - 6*b^4*c*d*e^9 + b^5*e^10)*x^3 - 3*(16*b*c^4*d^5*e^5 - 32*b^2*c^3 
*d^4*e^6 + 24*b^3*c^2*d^3*e^7 - 8*b^4*c*d^2*e^8 + b^5*d*e^9)*x^2 + (32*c^5 
*d^7*e^3 - 112*b*c^4*d^6*e^4 + 144*b^2*c^3*d^5*e^5 - 88*b^3*c^2*d^4*e^6...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.54 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {\frac {3 \, {\left (5 \, c^{2} e f + 3 \, c^{2} d g - 4 \, b c e g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (8 \, c^{3} d^{3} e - 12 \, b c^{2} d^{2} e^{2} + 6 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} \sqrt {-2 \, c d + b e}} + \frac {8 \, {\left (c^{2} e f + c^{2} d g - b c e g\right )}}{{\left (8 \, c^{3} d^{3} e - 12 \, b c^{2} d^{2} e^{2} + 6 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}} - \frac {18 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d e f - 9 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} e^{2} f - 2 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g - 7 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} d e g + 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c e^{2} g - 7 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} e f - {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d g + 4 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c e g}{{\left (8 \, c^{3} d^{3} e - 12 \, b c^{2} d^{2} e^{2} + 6 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} {\left (e x + d\right )}^{2} c^{2}}}{4 \, e} \] Input:

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

Output:

1/4*(3*(5*c^2*e*f + 3*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))/((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)) + 8*(c^2*e*f + c^2*d*g - b*c*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)*sqrt(-(e*x + d)*c + 
 2*c*d - b*e)) - (18*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d*e*f - 9*sqrt(- 
(e*x + d)*c + 2*c*d - b*e)*b*c^2*e^2*f - 2*sqrt(-(e*x + d)*c + 2*c*d - b*e 
)*c^3*d^2*g - 7*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 - 7*(-(e*x + d)*c + 2*c*d - b*e)^(3/2) 
*c^2*e*f - (-(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)/((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)^2*c^2))/e
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 992, normalized size of antiderivative = 3.27 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(12*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 + 24*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 + 12*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*g*x**2 - 9*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) 
)*c**2*d**3*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))*c**2*d**2*e*f - 18*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**2*e*g*x - 30*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*e**2*f*x - 9*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))*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 + 2*b**3*d*e**3*g + 2*b**3*e**4*f + 4*b**3*e**4*g*x + 5*b**2*c*d** 
2*e**2*g - 17*b**2*c*d*e**3*f + 5*b**2*c*d*e**3*g*x - 5*b**2*c*e**4*f*x + 
12*b**2*c*e**4*g*x**2 - 29*b*c**2*d**3*e*g + 29*b*c**2*d**2*e**2*f - 38*b* 
c**2*d**2*e**2*g*x - 10*b*c**2*d*e**3*f*x - 33*b*c**2*d*e**3*g*x**2 - 15*b 
*c**2*e**4*f*x**2 + 22*c**3*d**4*g - 6*c**3*d**3*e*f + 24*c**3*d**3*e*g*x 
+ 40*c**3*d**2*e**2*f*x + 18*c**3*d**2*e**2*g*x**2 + 30*c**3*d*e**3*f*x...