\(\int \frac {f+g x}{(d+e x)^{3/2} (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\) [248]

Optimal result

Integrand size = 46, antiderivative size = 459 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 c^2 (c e f+c d g-b e g) (d+e x)^{3/2}}{3 e^2 (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {2 c^2 (4 c e f+2 c d g-3 b e g) \sqrt {d+e x}}{e^2 (2 c d-b e)^5 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e^2 (2 c d-b e)^3 (d+e x)^{7/2}}-\frac {(17 c e f-5 c d g-6 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{12 e^2 (2 c d-b e)^4 (d+e x)^{5/2}}-\frac {c (41 c e f+3 c d g-22 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e^2 (2 c d-b e)^5 (d+e x)^{3/2}}-\frac {35 c^2 (3 c e f+c d g-2 b e g) \text {arctanh}\left (\frac {\sqrt {2 c d-b e} \sqrt {d+e x}}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 e^2 (2 c d-b e)^{11/2}} \] Output:

2/3*c^2*(-b*e*g+c*d*g+c*e*f)*(e*x+d)^(3/2)/e^2/(-b*e+2*c*d)^4/(d*(-b*e+c*d 
)-b*e^2*x-c*e^2*x^2)^(3/2)+2*c^2*(-3*b*e*g+2*c*d*g+4*c*e*f)*(e*x+d)^(1/2)/ 
e^2/(-b*e+2*c*d)^5/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)-1/3*(-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)^(7/2)-1/1 
2*(-6*b*e*g-5*c*d*g+17*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)^(5/2)-1/8*c*(-22*b*e*g+3*c*d*g+41*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)^5/(e*x+d)^(3/2)-35/8*c^2*(-2* 
b*e*g+c*d*g+3*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)^(11/2)
 

Mathematica [A] (verified)

Time = 3.22 (sec) , antiderivative size = 438, normalized size of antiderivative = 0.95 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {c^2 (d+e x)^{5/2} \left (\frac {(-b e+c (d-e x)) \left (-4 b^4 e^4 (2 e f+d g+3 e g x)+2 b^3 c e^3 \left (28 d^2 g+3 e^2 x (3 f+7 g x)+d e (41 f+81 g x)\right )+c^4 \left (171 d^5 g-315 e^5 f x^4+14 d^2 e^3 x^2 (27 f-10 g x)-105 d e^4 x^3 (4 f+g x)+18 d^3 e^2 x (34 f+7 g x)+d^4 e (f+204 g x)\right )+b^2 c^2 e^2 \left (103 d^3 g+7 e^3 x^2 (-9 f+40 g x)+3 d e^2 x (-78 f+217 g x)+d^2 e (-363 f+282 g x)\right )-2 b c^3 e \left (163 d^4 g+14 d e^3 x^2 (36 f-5 g x)-105 e^4 x^3 (-2 f+g x)+6 d^2 e^2 x (45 f+49 g x)+d^3 e (-152 f+294 g x)\right )\right )}{c^2 (2 c d-b e)^5 (d+e x)^3}-\frac {105 (3 c e f+c d g-2 b e g) (-b e+c (d-e x))^{5/2} \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{11/2}}\right )}{24 e^2 ((d+e x) (-b e+c (d-e x)))^{5/2}} \] 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) 
^(5/2)),x]
 

Output:

(c^2*(d + e*x)^(5/2)*(((-(b*e) + c*(d - e*x))*(-4*b^4*e^4*(2*e*f + d*g + 3 
*e*g*x) + 2*b^3*c*e^3*(28*d^2*g + 3*e^2*x*(3*f + 7*g*x) + d*e*(41*f + 81*g 
*x)) + c^4*(171*d^5*g - 315*e^5*f*x^4 + 14*d^2*e^3*x^2*(27*f - 10*g*x) - 1 
05*d*e^4*x^3*(4*f + g*x) + 18*d^3*e^2*x*(34*f + 7*g*x) + d^4*e*(f + 204*g* 
x)) + b^2*c^2*e^2*(103*d^3*g + 7*e^3*x^2*(-9*f + 40*g*x) + 3*d*e^2*x*(-78* 
f + 217*g*x) + d^2*e*(-363*f + 282*g*x)) - 2*b*c^3*e*(163*d^4*g + 14*d*e^3 
*x^2*(36*f - 5*g*x) - 105*e^4*x^3*(-2*f + g*x) + 6*d^2*e^2*x*(45*f + 49*g* 
x) + d^3*e*(-152*f + 294*g*x))))/(c^2*(2*c*d - b*e)^5*(d + e*x)^3) - (105* 
(3*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)^(11/2)))/(24*e^2*((d + e 
*x)*(-(b*e) + c*(d - e*x)))^(5/2))
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 452, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1220, 1135, 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.

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

\(\Big \downarrow \) 1220

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

\(\Big \downarrow \) 1135

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

\(\Big \downarrow \) 1132

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

\(\Big \downarrow \) 1135

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

\(\Big \downarrow \) 1132

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

\(\Big \downarrow \) 1136

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

\(\Big \downarrow \) 221

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

Input:

Int[(f + g*x)/((d + e*x)^(3/2)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2) 
),x]
 

Output:

-1/3*(e*f - d*g)/(e^2*(2*c*d - b*e)*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2 
*x - c*e^2*x^2)^(3/2)) + ((3*c*e*f + c*d*g - 2*b*e*g)*(-1/2*1/(e*(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*((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*(2*c*d - b*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 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. \(2002\) vs. \(2(421)=842\).

Time = 1.58 (sec) , antiderivative size = 2003, normalized size of antiderivative = 4.36

method	result	size
default	\(\text {Expression too large to display}\)	\(2003\)

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)^(5/2),x,method= 
_RETURNVERBOSE)
 

Output:

-1/24*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-945*arctan((-c*e*x-b*e+c*d)^(1/2) 
/(b*e-2*c*d)^(1/2))*b*c^3*d*e^4*f*x^2*(-c*e*x-b*e+c*d)^(1/2)+162*(b*e-2*c* 
d)^(1/2)*b^3*c*d*e^4*g*x+282*(b*e-2*c*d)^(1/2)*b^2*c^2*d^2*e^3*g*x-234*(b* 
e-2*c*d)^(1/2)*b^2*c^2*d*e^4*f*x-588*(b*e-2*c*d)^(1/2)*b*c^3*d^3*e^2*g*x-5 
40*(b*e-2*c*d)^(1/2)*b*c^3*d^2*e^3*f*x+140*(b*e-2*c*d)^(1/2)*b*c^3*d*e^4*g 
*x^3-588*(b*e-2*c*d)^(1/2)*b*c^3*d^2*e^3*g*x^2-1008*(b*e-2*c*d)^(1/2)*b*c^ 
3*d*e^4*f*x^2-315*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^3*d 
^3*e^2*f*(-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))*c^4*d*e^4*g*x^4*(-c*e*x-b*e+c*d)^(1/2)+210*arctan((-c*e*x-b*e+c 
*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c^2*e^5*g*x^3*(-c*e*x-b*e+c*d)^(1/2)+82*( 
b*e-2*c*d)^(1/2)*b^3*c*d*e^4*f+103*(b*e-2*c*d)^(1/2)*b^2*c^2*d^3*e^2*g+210 
*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^3*e^5*g*x^4*(-c*e*x- 
b*e+c*d)^(1/2)-315*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^3* 
d^4*e*g*(-c*e*x-b*e+c*d)^(1/2)+651*(b*e-2*c*d)^(1/2)*b^2*c^2*d*e^4*g*x^2+6 
30*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c^2*d^2*e^3*g*x*(- 
c*e*x-b*e+c*d)^(1/2)-735*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))* 
b*c^3*d^3*e^2*g*x*(-c*e*x-b*e+c*d)^(1/2)+315*arctan((-c*e*x-b*e+c*d)^(1/2) 
/(b*e-2*c*d)^(1/2))*b*c^3*d*e^4*g*x^3*(-c*e*x-b*e+c*d)^(1/2)+630*arctan((- 
c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c^2*d*e^4*g*x^2*(-c*e*x-b*e+c* 
d)^(1/2)-315*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^3*d^2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2028 vs. \(2 (421) = 842\).

Time = 3.37 (sec) , antiderivative size = 4086, normalized size of antiderivative = 8.90 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \] 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)^(5/2),x, 
algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {f + g x}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \] 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)**(5/ 
2),x)
 

Output:

Integral((f + g*x)/((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(d + e*x)**(3/ 
2)), x)
 

Maxima [F]

\[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int { \frac {g x + f}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] 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)^(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)*(e*x + d 
)^(3/2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1067 vs. \(2 (421) = 842\).

Time = 0.39 (sec) , antiderivative size = 1067, normalized size of antiderivative = 2.32 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \] 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)^(5/2),x, 
algorithm="giac")
 

Output:

1/24*(105*(3*c^3*e*f + c^3*d*g - 2*b*c^2*e*g)*arctan(sqrt(-(e*x + d)*c + 2 
*c*d - b*e)/sqrt(-2*c*d + b*e))/((32*c^5*d^5*e - 80*b*c^4*d^4*e^2 + 80*b^2 
*c^3*d^3*e^3 - 40*b^3*c^2*d^2*e^4 + 10*b^4*c*d*e^5 - b^5*e^6)*sqrt(-2*c*d 
+ b*e)) + (256*c^7*d^4*e*f - 512*b*c^6*d^3*e^2*f + 384*b^2*c^5*d^2*e^3*f - 
 128*b^3*c^4*d*e^4*f + 16*b^4*c^3*e^5*f + 256*c^7*d^5*g - 768*b*c^6*d^4*e* 
g + 896*b^2*c^5*d^3*e^2*g - 512*b^3*c^4*d^2*e^3*g + 144*b^4*c^3*d*e^4*g - 
16*b^5*c^2*e^5*g - 1152*((e*x + d)*c - 2*c*d + b*e)*c^6*d^3*e*f + 1728*((e 
*x + d)*c - 2*c*d + b*e)*b*c^5*d^2*e^2*f - 864*((e*x + d)*c - 2*c*d + b*e) 
*b^2*c^4*d*e^3*f + 144*((e*x + d)*c - 2*c*d + b*e)*b^3*c^3*e^4*f - 384*((e 
*x + d)*c - 2*c*d + b*e)*c^6*d^4*g + 1344*((e*x + d)*c - 2*c*d + b*e)*b*c^ 
5*d^3*e*g - 1440*((e*x + d)*c - 2*c*d + b*e)*b^2*c^4*d^2*e^2*g + 624*((e*x 
 + d)*c - 2*c*d + b*e)*b^3*c^3*d*e^3*g - 96*((e*x + d)*c - 2*c*d + b*e)*b^ 
4*c^2*e^4*g - 2772*((e*x + d)*c - 2*c*d + b*e)^2*c^5*d^2*e*f + 2772*((e*x 
+ d)*c - 2*c*d + b*e)^2*b*c^4*d*e^2*f - 693*((e*x + d)*c - 2*c*d + b*e)^2* 
b^2*c^3*e^3*f - 924*((e*x + d)*c - 2*c*d + b*e)^2*c^5*d^3*g + 2772*((e*x + 
 d)*c - 2*c*d + b*e)^2*b*c^4*d^2*e*g - 2079*((e*x + d)*c - 2*c*d + b*e)^2* 
b^2*c^3*d*e^2*g + 462*((e*x + d)*c - 2*c*d + b*e)^2*b^3*c^2*e^3*g - 1680*( 
(e*x + d)*c - 2*c*d + b*e)^3*c^4*d*e*f + 840*((e*x + d)*c - 2*c*d + b*e)^3 
*b*c^3*e^2*f - 560*((e*x + d)*c - 2*c*d + b*e)^3*c^4*d^2*g + 1400*((e*x + 
d)*c - 2*c*d + b*e)^3*b*c^3*d*e*g - 560*((e*x + d)*c - 2*c*d + b*e)^3*b...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {f+g\,x}{{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}} \,d x \] 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)^(5/2) 
),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)^(5/2) 
), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 2504, normalized size of antiderivative = 5.46 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] 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)^(5/2),x)
 

Output:

(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**2*c**2*d**3*e**2*g + 630*sqrt(b*e - 2*c*d)*s 
qrt( - 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**2*d**2*e**3*g*x + 630*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**2*d*e**4*g* 
x**2 + 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**2*c**2*e**5*g*x**3 - 315*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**3*d**4*e*g - 315*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**3*d**3*e**2*f - 
735*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**3*d**3*e**2*g*x - 945*sqrt(b*e - 2*c*d)*sqr 
t( - b*e + c*d - c*e*x)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d)) 
*b*c**3*d**2*e**3*f*x - 315*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))*b*c**3*d**2*e**3*g*x**2 
- 945*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**3*d*e**4*f*x**2 + 315*sqrt(b*e - 2*c*d)*s 
qrt( - b*e + c*d - c*e*x)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d 
))*b*c**3*d*e**4*g*x**3 - 315*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**3*e**5*f*x**3 ...