\(\int \frac {f+g x}{(d+e x)^{5/2} (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\) [239]

Optimal result

Integrand size = 46, antiderivative size = 382 \[ \int \frac {f+g x}{(d+e x)^{5/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 c^2 (c e f+c d g-b e g) \sqrt {d+e x}}{e^2 (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e^2 (2 c d-b e)^2 (d+e x)^{7/2}}-\frac {(11 c e f+c d g-6 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{12 e^2 (2 c d-b e)^3 (d+e x)^{5/2}}-\frac {c (19 c e f+9 c d g-14 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e^2 (2 c d-b e)^4 (d+e x)^{3/2}}-\frac {5 c^2 (7 c e f+5 c d g-6 b e g) \text {arctanh}\left (\frac {\sqrt {2 c d-b e} \sqrt {d+e x}}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 e^2 (2 c d-b e)^{9/2}} \] Output:

2*c^2*(-b*e*g+c*d*g+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)-1/3*(-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)^2/(e*x+d)^(7/2)-1/12*(-6*b*e*g+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)^3/(e*x+d)^(5/2)-1/8*c*( 
-14*b*e*g+9*c*d*g+19*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)-5/8*c^2*(-6*b*e*g+5*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 = 2.05 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.89 \[ \int \frac {f+g x}{(d+e x)^{5/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {c^2 (d+e x)^{3/2} \left (\frac {(-b e+c (d-e x)) \left (4 b^3 e^3 (2 e f+d g+3 e g x)+b c^2 e \left (-13 d^3 g+d e^2 x (126 f-185 g x)+5 e^3 x^2 (7 f-18 g x)+d^2 e (187 f-12 g x)\right )+c^3 \left (49 d^4 g+105 e^4 f x^3-85 d^3 e (f-g x)+5 d e^3 x^2 (49 f+15 g x)+7 d^2 e^2 x (17 f+25 g x)\right )-2 b^2 c e^2 \left (20 d^2 g+e^2 x (7 f+15 g x)+d e (31 f+59 g x)\right )\right )}{c^2 (-2 c d+b e)^4 (d+e x)^3}+\frac {15 (7 c e f+5 c d g-6 b e g) (-b e+c (d-e x))^{3/2} \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{9/2}}\right )}{24 e^2 ((d+e x) (-b e+c (d-e x)))^{3/2}} \] Input:

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

Output:

(c^2*(d + e*x)^(3/2)*(((-(b*e) + c*(d - e*x))*(4*b^3*e^3*(2*e*f + d*g + 3* 
e*g*x) + b*c^2*e*(-13*d^3*g + d*e^2*x*(126*f - 185*g*x) + 5*e^3*x^2*(7*f - 
 18*g*x) + d^2*e*(187*f - 12*g*x)) + c^3*(49*d^4*g + 105*e^4*f*x^3 - 85*d^ 
3*e*(f - g*x) + 5*d*e^3*x^2*(49*f + 15*g*x) + 7*d^2*e^2*x*(17*f + 25*g*x)) 
 - 2*b^2*c*e^2*(20*d^2*g + e^2*x*(7*f + 15*g*x) + d*e*(31*f + 59*g*x))))/( 
c^2*(-2*c*d + b*e)^4*(d + e*x)^3) + (15*(7*c*e*f + 5*c*d*g - 6*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)^(9/2)))/(24*e^2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2) 
)
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 379, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1220, 1135, 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.

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

\(\Big \downarrow \) 1220

\(\displaystyle \frac {(-6 b e g+5 c d g+7 c e f) \int \frac {1}{(d+e x)^{3/2} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{6 e (2 c d-b e)}-\frac {e f-d g}{3 e^2 (d+e x)^{5/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {(-6 b e g+5 c d g+7 c e f) \left (\frac {5 c \int \frac {1}{\sqrt {d+e x} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{4 (2 c d-b e)}-\frac {1}{2 e (d+e x)^{3/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{6 e (2 c d-b e)}-\frac {e f-d g}{3 e^2 (d+e x)^{5/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {(-6 b e g+5 c d g+7 c e f) \left (\frac {5 c \left (\frac {3 c \int \frac {\sqrt {d+e x}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{4 (2 c d-b e)}-\frac {1}{2 e (d+e x)^{3/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{6 e (2 c d-b e)}-\frac {e f-d g}{3 e^2 (d+e x)^{5/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\)

\(\Big \downarrow \) 1132

\(\displaystyle \frac {(-6 b e g+5 c d g+7 c e f) \left (\frac {5 c \left (\frac {3 c \left (\frac {\int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 c d-b e}+\frac {2 \sqrt {d+e x}}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{4 (2 c d-b e)}-\frac {1}{2 e (d+e x)^{3/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{6 e (2 c d-b e)}-\frac {e f-d g}{3 e^2 (d+e x)^{5/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\)

\(\Big \downarrow \) 1136

\(\displaystyle \frac {(-6 b e g+5 c d g+7 c e f) \left (\frac {5 c \left (\frac {3 c \left (\frac {2 e \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}}{2 c d-b e}+\frac {2 \sqrt {d+e x}}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{4 (2 c d-b e)}-\frac {1}{2 e (d+e x)^{3/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{6 e (2 c d-b e)}-\frac {e f-d g}{3 e^2 (d+e x)^{5/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (\frac {5 c \left (\frac {3 c \left (\frac {2 \sqrt {d+e x}}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e (2 c d-b e)^{3/2}}\right )}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{4 (2 c d-b e)}-\frac {1}{2 e (d+e x)^{3/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (-6 b e g+5 c d g+7 c e f)}{6 e (2 c d-b e)}-\frac {e f-d g}{3 e^2 (d+e x)^{5/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\)

Input:

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

Output:

-1/3*(e*f - d*g)/(e^2*(2*c*d - b*e)*(d + e*x)^(5/2)*Sqrt[d*(c*d - b*e) - b 
*e^2*x - c*e^2*x^2]) + ((7*c*e*f + 5*c*d*g - 6*b*e*g)*(-1/2*1/(e*(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*(-( 
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]/(Sqr 
t[2*c*d - b*e]*Sqrt[d + e*x])])/(e*(2*c*d - b*e)^(3/2))))/(2*(2*c*d - b*e) 
)))/(4*(2*c*d - b*e))))/(6*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. \(1215\) vs. \(2(350)=700\).

Time = 1.66 (sec) , antiderivative size = 1216, normalized size of antiderivative = 3.18

method	result	size
default	\(\text {Expression too large to display}\)	\(1216\)

Input:

int((g*x+f)/(e*x+d)^(5/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x,method= 
_RETURNVERBOSE)
 

Output:

1/24/(e*x+d)^(7/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-126*(b*e-2*c*d)^(1/2 
)*b*c^2*d*e^3*f*x+270*(-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-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+90*(b*e-2*c*d)^(1/2) 
*b*c^2*e^4*g*x^3-75*(b*e-2*c*d)^(1/2)*c^3*d*e^3*g*x^3+30*(b*e-2*c*d)^(1/2) 
*b^2*c*e^4*g*x^2-35*(b*e-2*c*d)^(1/2)*b*c^2*e^4*f*x^2-175*(b*e-2*c*d)^(1/2 
)*c^3*d^2*e^2*g*x^2+270*(-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^ 
3-75*(-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 
))*c^3*d^4*g-12*(b*e-2*c*d)^(1/2)*b^3*e^4*g*x-4*(b*e-2*c*d)^(1/2)*b^3*d*e^ 
3*g+85*(b*e-2*c*d)^(1/2)*c^3*d^3*e*f+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^3*e*g+118*(b*e-2*c*d)^(1/2)* 
b^2*c*d*e^3*g*x+12*(b*e-2*c*d)^(1/2)*b*c^2*d^2*e^2*g*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*e^4*g*x^3-75 
*(-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))*c 
^3*d*e^3*g*x^3-225*(-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^2*e^2*g*x^2-315*(-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*e^3*f*x^2-225*(-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^3*e*g*x-315*( 
-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))*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1402 vs. \(2 (350) = 700\).

Time = 0.67 (sec) , antiderivative size = 2834, normalized size of antiderivative = 7.42 \[ \int \frac {f+g x}{(d+e x)^{5/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)/(e*x+d)^(5/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, 
algorithm="fricas")
 

Output:

[1/48*(15*((7*c^4*e^6*f + (5*c^4*d*e^5 - 6*b*c^3*e^6)*g)*x^5 + (7*(3*c^4*d 
*e^5 + b*c^3*e^6)*f + (15*c^4*d^2*e^4 - 13*b*c^3*d*e^5 - 6*b^2*c^2*e^6)*g) 
*x^4 + 2*(7*(c^4*d^2*e^4 + 2*b*c^3*d*e^5)*f + (5*c^4*d^3*e^3 + 4*b*c^3*d^2 
*e^4 - 12*b^2*c^2*d*e^5)*g)*x^3 - 2*(7*(c^4*d^3*e^3 - 3*b*c^3*d^2*e^4)*f + 
 (5*c^4*d^4*e^2 - 21*b*c^3*d^3*e^3 + 18*b^2*c^2*d^2*e^4)*g)*x^2 - 7*(c^4*d 
^5*e - b*c^3*d^4*e^2)*f - (5*c^4*d^6 - 11*b*c^3*d^5*e + 6*b^2*c^2*d^4*e^2) 
*g - (7*(3*c^4*d^4*e^2 - 4*b*c^3*d^3*e^3)*f + (15*c^4*d^5*e - 38*b*c^3*d^4 
*e^2 + 24*b^2*c^2*d^3*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*sq 
rt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(15*(7*(2*c^4*d*e^4 - b*c^3*e^5)* 
f + (10*c^4*d^2*e^3 - 17*b*c^3*d*e^4 + 6*b^2*c^2*e^5)*g)*x^3 + 5*(7*(14*c^ 
4*d^2*e^3 - 5*b*c^3*d*e^4 - b^2*c^2*e^5)*f + (70*c^4*d^3*e^2 - 109*b*c^3*d 
^2*e^3 + 25*b^2*c^2*d*e^4 + 6*b^3*c*e^5)*g)*x^2 - (170*c^4*d^4*e - 459*b*c 
^3*d^3*e^2 + 311*b^2*c^2*d^2*e^3 - 78*b^3*c*d*e^4 + 8*b^4*e^5)*f + (98*c^4 
*d^5 - 75*b*c^3*d^4*e - 67*b^2*c^2*d^3*e^2 + 48*b^3*c*d^2*e^3 - 4*b^4*d*e^ 
4)*g + (7*(34*c^4*d^3*e^2 + 19*b*c^3*d^2*e^3 - 22*b^2*c^2*d*e^4 + 2*b^3*c* 
e^5)*f + (170*c^4*d^4*e - 109*b*c^3*d^3*e^2 - 224*b^2*c^2*d^2*e^3 + 142*b^ 
3*c*d*e^4 - 12*b^4*e^5)*g)*x)*sqrt(e*x + d))/(32*c^6*d^10*e^2 - 112*b*c^5* 
d^9*e^3 + 160*b^2*c^4*d^8*e^4 - 120*b^3*c^3*d^7*e^5 + 50*b^4*c^2*d^6*e^...
 

Sympy [F]

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

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

Maxima [F]

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

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 776 vs. \(2 (350) = 700\).

Time = 0.35 (sec) , antiderivative size = 776, normalized size of antiderivative = 2.03 \[ \int \frac {f+g x}{(d+e x)^{5/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate((g*x+f)/(e*x+d)^(5/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, 
algorithm="giac")
 

Output:

1/24*(15*(7*c^3*e*f + 5*c^3*d*g - 6*b*c^2*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)) + 48*(c^3*e*f 
 + c^3*d*g - b*c^2*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)*sqrt(-(e*x + d)*c + 2*c*d - b*e)) - (348*s 
qrt(-(e*x + d)*c + 2*c*d - b*e)*c^5*d^2*e*f - 348*sqrt(-(e*x + d)*c + 2*c* 
d - b*e)*b*c^4*d*e^2*f + 87*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^3*e^3*f 
 + 84*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^5*d^3*g - 300*sqrt(-(e*x + d)*c + 
 2*c*d - b*e)*b*c^4*d^2*e*g + 237*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^3 
*d*e^2*g - 54*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^3*c^2*e^3*g - 272*(-(e*x 
+ d)*c + 2*c*d - b*e)^(3/2)*c^4*d*e*f + 136*(-(e*x + d)*c + 2*c*d - b*e)^( 
3/2)*b*c^3*e^2*f - 112*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^4*d^2*g + 248* 
(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^3*d*e*g - 96*(-(e*x + d)*c + 2*c*d 
- b*e)^(3/2)*b^2*c^2*e^2*g + 57*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + 
 d)*c + 2*c*d - b*e)*c^3*e*f + 27*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x 
 + d)*c + 2*c*d - b*e)*c^3*d*g - 42*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e 
*x + d)*c + 2*c*d - b*e)*b*c^2*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)^3*c^3))/e
 

Mupad [F(-1)]

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

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1500, normalized size of antiderivative = 3.93 \[ \int \frac {f+g x}{(d+e x)^{5/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 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**3*e*g - 270*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 - 270*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))*b*c**2*d*e**3*g*x**2 - 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*e**4*g*x**3 + 75*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 
**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 + 225*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**3*d**3*e*g*x + 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))*c**3*d**2*e**2*f*x + 225*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**3*d**2*e**2*g*x**2 + 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))*c**3* 
d*e**3*f*x**2 + 75*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*e**3*g*x**3 + 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*e**4*f*x**3 + 4*b**4*d*e**4*g + 8*b**4*e**5*f + 12*b...