\(\int \frac {1}{(e+f x)^{3/2} (a c+(b c+a d) x+b d x^2)^2} \, dx\) [400]

Optimal result

Integrand size = 31, antiderivative size = 353 \[ \int \frac {1}{(e+f x)^{3/2} \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {f \left (3 a^2 d^2 f^2-2 a b d f (d e+2 c f)+b^2 \left (2 d^2 e^2-2 c d e f+3 c^2 f^2\right )\right )}{(b c-a d)^2 (b e-a f)^2 (d e-c f)^2 \sqrt {e+f x}}-\frac {a b d^2 e-a^2 d^2 f+b^2 c (d e-c f)+b d (2 b d e-(b c+a d) f) x}{(b c-a d)^2 (b e-a f) (d e-c f) \sqrt {e+f x} \left (a c+(b c+a d) x+b d x^2\right )}+\frac {b^{5/2} (4 b d e+3 b c f-7 a d f) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d)^3 (b e-a f)^{5/2}}-\frac {d^{5/2} (4 b d e-7 b c f+3 a d f) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d)^3 (d e-c f)^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 3.26 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.17 \[ \int \frac {1}{(e+f x)^{3/2} \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {-a^3 d^2 f^2 (2 c f+d (e+3 f x))+a^2 b d f \left (4 c^2 f^2+2 c d f^2 x+d^2 \left (2 e^2+e f x-3 f^2 x^2\right )\right )+a b^2 \left (-2 c^3 f^3+2 c^2 d f^3 x+4 c d^2 f^3 x^2+d^3 e \left (-e^2+e f x+2 f^2 x^2\right )\right )-b^3 \left (2 d^3 e^2 x (e+f x)+c^3 f^2 (e+3 f x)+c d^2 e \left (e^2-e f x-2 f^2 x^2\right )+c^2 d f \left (-2 e^2-e f x+3 f^2 x^2\right )\right )}{(b c-a d)^2 (b e-a f)^2 (d e-c f)^2 (a+b x) (c+d x) \sqrt {e+f x}}-\frac {b^{5/2} (4 b d e+3 b c f-7 a d f) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{(b c-a d)^3 (-b e+a f)^{5/2}}-\frac {d^{5/2} (4 b d e-7 b c f+3 a d f) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{(-b c+a d)^3 (-d e+c f)^{5/2}} \] Input:

Integrate[1/((e + f*x)^(3/2)*(a*c + (b*c + a*d)*x + b*d*x^2)^2),x]
 

Output:

(-(a^3*d^2*f^2*(2*c*f + d*(e + 3*f*x))) + a^2*b*d*f*(4*c^2*f^2 + 2*c*d*f^2 
*x + d^2*(2*e^2 + e*f*x - 3*f^2*x^2)) + a*b^2*(-2*c^3*f^3 + 2*c^2*d*f^3*x 
+ 4*c*d^2*f^3*x^2 + d^3*e*(-e^2 + e*f*x + 2*f^2*x^2)) - b^3*(2*d^3*e^2*x*( 
e + f*x) + c^3*f^2*(e + 3*f*x) + c*d^2*e*(e^2 - e*f*x - 2*f^2*x^2) + c^2*d 
*f*(-2*e^2 - e*f*x + 3*f^2*x^2)))/((b*c - a*d)^2*(b*e - a*f)^2*(d*e - c*f) 
^2*(a + b*x)*(c + d*x)*Sqrt[e + f*x]) - (b^(5/2)*(4*b*d*e + 3*b*c*f - 7*a* 
d*f)*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/((b*c - a*d)^3*(- 
(b*e) + a*f)^(5/2)) - (d^(5/2)*(4*b*d*e - 7*b*c*f + 3*a*d*f)*ArcTan[(Sqrt[ 
d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/((-(b*c) + a*d)^3*(-(d*e) + c*f)^(5 
/2))
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {1165, 27, 1198, 1197, 27, 25, 1480, 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.

\(\displaystyle \int \frac {1}{(e+f x)^{3/2} \left (x (a d+b c)+a c+b d x^2\right )^2} \, dx\)

\(\Big \downarrow \) 1165

\(\displaystyle -\frac {\int \frac {4 b^2 d^2 e^2-b d (b c+a d) f e-3 (b c+a d)^2 f^2+10 a b c d f^2+3 b d f (2 b d e-(b c+a d) f) x}{2 (e+f x)^{3/2} \left (b d x^2+(b c+a d) x+a c\right )}dx}{(b c-a d)^2 (b e-a f) (d e-c f)}-\frac {-a^2 d^2 f+b d x (2 b d e-f (a d+b c))+a b d^2 e+b^2 c (d e-c f)}{\sqrt {e+f x} (b c-a d)^2 (b e-a f) (d e-c f) \left (x (a d+b c)+a c+b d x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {4 b^2 d^2 e^2-b d (b c+a d) f e-3 (b c+a d)^2 f^2+10 a b c d f^2+3 b d f (2 b d e-(b c+a d) f) x}{(e+f x)^{3/2} \left (b d x^2+(b c+a d) x+a c\right )}dx}{2 (b c-a d)^2 (b e-a f) (d e-c f)}-\frac {-a^2 d^2 f+b d x (2 b d e-f (a d+b c))+a b d^2 e+b^2 c (d e-c f)}{\sqrt {e+f x} (b c-a d)^2 (b e-a f) (d e-c f) \left (x (a d+b c)+a c+b d x^2\right )}\)

\(\Big \downarrow \) 1198

\(\displaystyle -\frac {\frac {\int \frac {(d e-c f)^2 (4 d e+3 c f) b^3-a d f \left (5 d^2 e^2-12 c d f e+4 c^2 f^2\right ) b^2-2 a^2 d^2 f^2 (d e+2 c f) b+d f \left (\left (2 d^2 e^2-2 c d f e+3 c^2 f^2\right ) b^2-2 a d f (d e+2 c f) b+3 a^2 d^2 f^2\right ) x b+3 a^3 d^3 f^3}{\sqrt {e+f x} \left (b d x^2+(b c+a d) x+a c\right )}dx}{(b e-a f) (d e-c f)}+\frac {2 f \left (3 a^2 d^2 f^2-2 a b d f (2 c f+d e)+b^2 \left (3 c^2 f^2-2 c d e f+2 d^2 e^2\right )\right )}{\sqrt {e+f x} (b e-a f) (d e-c f)}}{2 (b c-a d)^2 (b e-a f) (d e-c f)}-\frac {-a^2 d^2 f+b d x (2 b d e-f (a d+b c))+a b d^2 e+b^2 c (d e-c f)}{\sqrt {e+f x} (b c-a d)^2 (b e-a f) (d e-c f) \left (x (a d+b c)+a c+b d x^2\right )}\)

\(\Big \downarrow \) 1197

\(\displaystyle -\frac {\frac {2 \int \frac {f \left ((2 b d e-b c f-a d f) \left (d^2 e^2 b^2-3 c^2 f^2 b^2-c d e f b^2+7 a c d f^2 b-a d^2 e f b-3 a^2 d^2 f^2\right )+b d \left (\left (2 d^2 e^2-2 c d f e+3 c^2 f^2\right ) b^2-2 a d f (d e+2 c f) b+3 a^2 d^2 f^2\right ) (e+f x)\right )}{b d (e+f x)^2-(2 b d e-b c f-a d f) (e+f x)+(b e-a f) (d e-c f)}d\sqrt {e+f x}}{(b e-a f) (d e-c f)}+\frac {2 f \left (3 a^2 d^2 f^2-2 a b d f (2 c f+d e)+b^2 \left (3 c^2 f^2-2 c d e f+2 d^2 e^2\right )\right )}{\sqrt {e+f x} (b e-a f) (d e-c f)}}{2 (b c-a d)^2 (b e-a f) (d e-c f)}-\frac {-a^2 d^2 f+b d x (2 b d e-f (a d+b c))+a b d^2 e+b^2 c (d e-c f)}{\sqrt {e+f x} (b c-a d)^2 (b e-a f) (d e-c f) \left (x (a d+b c)+a c+b d x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {2 f \int -\frac {(2 b d e-b c f-a d f) \left (-\left (\left (d^2 e^2-c d f e-3 c^2 f^2\right ) b^2\right )+a d f (d e-7 c f) b+3 a^2 d^2 f^2\right )-b d \left (\left (2 d^2 e^2-2 c d f e+3 c^2 f^2\right ) b^2-2 a d f (d e+2 c f) b+3 a^2 d^2 f^2\right ) (e+f x)}{b d (e+f x)^2-(2 b d e-b c f-a d f) (e+f x)+(b e-a f) (d e-c f)}d\sqrt {e+f x}}{(b e-a f) (d e-c f)}+\frac {2 f \left (3 a^2 d^2 f^2-2 a b d f (2 c f+d e)+b^2 \left (3 c^2 f^2-2 c d e f+2 d^2 e^2\right )\right )}{\sqrt {e+f x} (b e-a f) (d e-c f)}}{2 (b c-a d)^2 (b e-a f) (d e-c f)}-\frac {-a^2 d^2 f+b d x (2 b d e-f (a d+b c))+a b d^2 e+b^2 c (d e-c f)}{\sqrt {e+f x} (b c-a d)^2 (b e-a f) (d e-c f) \left (x (a d+b c)+a c+b d x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {2 f \left (3 a^2 d^2 f^2-2 a b d f (2 c f+d e)+b^2 \left (3 c^2 f^2-2 c d e f+2 d^2 e^2\right )\right )}{\sqrt {e+f x} (b e-a f) (d e-c f)}-\frac {2 f \int \frac {(2 b d e-b c f-a d f) \left (-\left (\left (d^2 e^2-c d f e-3 c^2 f^2\right ) b^2\right )+a d f (d e-7 c f) b+3 a^2 d^2 f^2\right )-b d \left (\left (2 d^2 e^2-2 c d f e+3 c^2 f^2\right ) b^2-2 a d f (d e+2 c f) b+3 a^2 d^2 f^2\right ) (e+f x)}{b d (e+f x)^2-(2 b d e-b c f-a d f) (e+f x)+(b e-a f) (d e-c f)}d\sqrt {e+f x}}{(b e-a f) (d e-c f)}}{2 (b c-a d)^2 (b e-a f) (d e-c f)}-\frac {-a^2 d^2 f+b d x (2 b d e-f (a d+b c))+a b d^2 e+b^2 c (d e-c f)}{\sqrt {e+f x} (b c-a d)^2 (b e-a f) (d e-c f) \left (x (a d+b c)+a c+b d x^2\right )}\)

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

Int[1/((e + f*x)^(3/2)*(a*c + (b*c + a*d)*x + b*d*x^2)^2),x]
 

Output:

-((a*b*d^2*e - a^2*d^2*f + b^2*c*(d*e - c*f) + b*d*(2*b*d*e - (b*c + a*d)* 
f)*x)/((b*c - a*d)^2*(b*e - a*f)*(d*e - c*f)*Sqrt[e + f*x]*(a*c + (b*c + a 
*d)*x + b*d*x^2))) - ((2*f*(3*a^2*d^2*f^2 - 2*a*b*d*f*(d*e + 2*c*f) + b^2* 
(2*d^2*e^2 - 2*c*d*e*f + 3*c^2*f^2)))/((b*e - a*f)*(d*e - c*f)*Sqrt[e + f* 
x]) - (2*f*((b^(5/2)*(d*e - c*f)^2*(4*b*d*e + 3*b*c*f - 7*a*d*f)*ArcTanh[( 
Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/((b*c - a*d)*f*Sqrt[b*e - a*f]) - 
 (d^(5/2)*(b*e - a*f)^2*(4*b*d*e - 7*b*c*f + 3*a*d*f)*ArcTanh[(Sqrt[d]*Sqr 
t[e + f*x])/Sqrt[d*e - c*f]])/((b*c - a*d)*f*Sqrt[d*e - c*f])))/((b*e - a* 
f)*(d*e - c*f)))/(2*(b*c - a*d)^2*(b*e - a*f)*(d*e - c*f))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 1165

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) 
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 
2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d 
+ e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p 
+ 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + 
 b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] 
 && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - 
b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr 
eeQ[{a, b, c, d, e, f, g}, x]
 

rule 1198

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

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 1.68 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.76

method	result	size
derivativedivides	\(2 f^{3} \left (-\frac {b^{3} \left (\frac {\left (\frac {1}{2} a d f -\frac {1}{2} b c f \right ) \sqrt {f x +e}}{b \left (f x +e \right )+a f -b e}+\frac {\left (7 a d f -3 b c f -4 b d e \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}\right )}{\left (a d -b c \right )^{3} f^{3} \left (a f -b e \right )^{2}}-\frac {1}{\left (c f -d e \right )^{2} \left (a f -b e \right )^{2} \sqrt {f x +e}}-\frac {d^{3} \left (\frac {\left (\frac {1}{2} a d f -\frac {1}{2} b c f \right ) \sqrt {f x +e}}{d \left (f x +e \right )+c f -d e}+\frac {\left (3 a d f -7 b c f +4 b d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}\right )}{\left (a d -b c \right )^{3} f^{3} \left (c f -d e \right )^{2}}\right )\)	\(270\)
default	\(2 f^{3} \left (-\frac {b^{3} \left (\frac {\left (\frac {1}{2} a d f -\frac {1}{2} b c f \right ) \sqrt {f x +e}}{b \left (f x +e \right )+a f -b e}+\frac {\left (7 a d f -3 b c f -4 b d e \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}\right )}{\left (a d -b c \right )^{3} f^{3} \left (a f -b e \right )^{2}}-\frac {1}{\left (c f -d e \right )^{2} \left (a f -b e \right )^{2} \sqrt {f x +e}}-\frac {d^{3} \left (\frac {\left (\frac {1}{2} a d f -\frac {1}{2} b c f \right ) \sqrt {f x +e}}{d \left (f x +e \right )+c f -d e}+\frac {\left (3 a d f -7 b c f +4 b d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}\right )}{\left (a d -b c \right )^{3} f^{3} \left (c f -d e \right )^{2}}\right )\)	\(270\)
pseudoelliptic	\(-\frac {2 \left (\frac {7 \left (\left (a d -\frac {3 b c}{7}\right ) f -\frac {4 b d e}{7}\right ) \sqrt {\left (c f -d e \right ) d}\, b^{3} \left (c f -d e \right )^{2} \left (d x +c \right ) \sqrt {f x +e}\, \left (b x +a \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2}+\left (\frac {3 d^{3} \left (a f -b e \right )^{2} \left (\left (a d -\frac {7 b c}{3}\right ) f +\frac {4 b d e}{3}\right ) \left (d x +c \right ) \sqrt {f x +e}\, \left (b x +a \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2}+\left (a d -b c \right ) \sqrt {\left (c f -d e \right ) d}\, \left (\left (\frac {3 c^{2} x \left (d x +c \right ) b^{3}}{2}+a c \left (d x +c \right ) \left (-2 d x +c \right ) b^{2}-2 d \left (-\frac {3}{4} d^{2} x^{2}+\frac {1}{2} c d x +c^{2}\right ) a^{2} b +a^{3} d^{2} \left (\frac {3 d x}{2}+c \right )\right ) f^{3}+\frac {\left (c \left (d x +c \right ) \left (-2 d x +c \right ) b^{3}-2 a \,b^{2} d^{3} x^{2}-a^{2} b \,d^{3} x +a^{3} d^{3}\right ) e \,f^{2}}{2}-d \,e^{2} \left (\left (-d^{2} x^{2}+\frac {1}{2} c d x +c^{2}\right ) b^{2}+\frac {a b \,d^{2} x}{2}+a^{2} d^{2}\right ) b f +\frac {d^{2} e^{3} b^{2} \left (\left (2 d x +c \right ) b +a d \right )}{2}\right )\right ) \sqrt {\left (a f -b e \right ) b}\right )}{\sqrt {f x +e}\, \sqrt {\left (c f -d e \right ) d}\, \sqrt {\left (a f -b e \right ) b}\, \left (c f -d e \right )^{2} \left (a f -b e \right )^{2} \left (b x +a \right ) \left (a d -b c \right )^{3} \left (d x +c \right )}\)	\(460\)

Input:

int(1/(f*x+e)^(3/2)/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x,method=_RETURNVERBOSE)
 

Output:

2*f^3*(-b^3/(a*d-b*c)^3/f^3/(a*f-b*e)^2*((1/2*a*d*f-1/2*b*c*f)*(f*x+e)^(1/ 
2)/(b*(f*x+e)+a*f-b*e)+1/2*(7*a*d*f-3*b*c*f-4*b*d*e)/((a*f-b*e)*b)^(1/2)*a 
rctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2)))-1/(c*f-d*e)^2/(a*f-b*e)^2/(f*x 
+e)^(1/2)-d^3/(a*d-b*c)^3/f^3/(c*f-d*e)^2*((1/2*a*d*f-1/2*b*c*f)*(f*x+e)^( 
1/2)/(d*(f*x+e)+c*f-d*e)+1/2*(3*a*d*f-7*b*c*f+4*b*d*e)/((c*f-d*e)*d)^(1/2) 
*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2870 vs. \(2 (329) = 658\).

Time = 75.96 (sec) , antiderivative size = 11596, normalized size of antiderivative = 32.85 \[ \int \frac {1}{(e+f x)^{3/2} \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(f*x+e)^(3/2)/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="fricas 
")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{(e+f x)^{3/2} \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\int \frac {1}{\left (a + b x\right )^{2} \left (c + d x\right )^{2} \left (e + f x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(f*x+e)**(3/2)/(a*c+(a*d+b*c)*x+b*d*x**2)**2,x)
 

Output:

Integral(1/((a + b*x)**2*(c + d*x)**2*(e + f*x)**(3/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(e+f x)^{3/2} \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(f*x+e)^(3/2)/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="maxima 
")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1245 vs. \(2 (329) = 658\).

Time = 0.42 (sec) , antiderivative size = 1245, normalized size of antiderivative = 3.53 \[ \int \frac {1}{(e+f x)^{3/2} \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(f*x+e)^(3/2)/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="giac")
 

Output:

-(4*b^4*d*e + 3*b^4*c*f - 7*a*b^3*d*f)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e 
+ a*b*f))/((b^5*c^3*e^2 - 3*a*b^4*c^2*d*e^2 + 3*a^2*b^3*c*d^2*e^2 - a^3*b^ 
2*d^3*e^2 - 2*a*b^4*c^3*e*f + 6*a^2*b^3*c^2*d*e*f - 6*a^3*b^2*c*d^2*e*f + 
2*a^4*b*d^3*e*f + a^2*b^3*c^3*f^2 - 3*a^3*b^2*c^2*d*f^2 + 3*a^4*b*c*d^2*f^ 
2 - a^5*d^3*f^2)*sqrt(-b^2*e + a*b*f)) + (4*b*d^4*e - 7*b*c*d^3*f + 3*a*d^ 
4*f)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((b^3*c^3*d^2*e^2 - 3*a* 
b^2*c^2*d^3*e^2 + 3*a^2*b*c*d^4*e^2 - a^3*d^5*e^2 - 2*b^3*c^4*d*e*f + 6*a* 
b^2*c^3*d^2*e*f - 6*a^2*b*c^2*d^3*e*f + 2*a^3*c*d^4*e*f + b^3*c^5*f^2 - 3* 
a*b^2*c^4*d*f^2 + 3*a^2*b*c^3*d^2*f^2 - a^3*c^2*d^3*f^2)*sqrt(-d^2*e + c*d 
*f)) - (2*(f*x + e)^2*b^3*d^3*e^2*f - 2*(f*x + e)*b^3*d^3*e^3*f - 2*(f*x + 
 e)^2*b^3*c*d^2*e*f^2 - 2*(f*x + e)^2*a*b^2*d^3*e*f^2 + 3*(f*x + e)*b^3*c* 
d^2*e^2*f^2 + 3*(f*x + e)*a*b^2*d^3*e^2*f^2 + 3*(f*x + e)^2*b^3*c^2*d*f^3 
- 4*(f*x + e)^2*a*b^2*c*d^2*f^3 + 3*(f*x + e)^2*a^2*b*d^3*f^3 - 7*(f*x + e 
)*b^3*c^2*d*e*f^3 + 8*(f*x + e)*a*b^2*c*d^2*e*f^3 - 7*(f*x + e)*a^2*b*d^3* 
e*f^3 + 2*b^3*c^2*d*e^2*f^3 - 4*a*b^2*c*d^2*e^2*f^3 + 2*a^2*b*d^3*e^2*f^3 
+ 3*(f*x + e)*b^3*c^3*f^4 - 2*(f*x + e)*a*b^2*c^2*d*f^4 - 2*(f*x + e)*a^2* 
b*c*d^2*f^4 + 3*(f*x + e)*a^3*d^3*f^4 - 2*b^3*c^3*e*f^4 + 2*a*b^2*c^2*d*e* 
f^4 + 2*a^2*b*c*d^2*e*f^4 - 2*a^3*d^3*e*f^4 + 2*a*b^2*c^3*f^5 - 4*a^2*b*c^ 
2*d*f^5 + 2*a^3*c*d^2*f^5)/((b^4*c^2*d^2*e^4 - 2*a*b^3*c*d^3*e^4 + a^2*b^2 
*d^4*e^4 - 2*b^4*c^3*d*e^3*f + 2*a*b^3*c^2*d^2*e^3*f + 2*a^2*b^2*c*d^3*...
 

Mupad [B] (verification not implemented)

Time = 32.86 (sec) , antiderivative size = 172959, normalized size of antiderivative = 489.97 \[ \int \frac {1}{(e+f x)^{3/2} \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

int(1/((e + f*x)^(3/2)*(a*c + x*(a*d + b*c) + b*d*x^2)^2),x)
 

Output:

(atan(((((e + f*x)^(1/2)*(18*a^6*b^15*c^18*d^3*f^20 - 192*a^7*b^14*c^17*d^ 
4*f^20 + 872*a^8*b^13*c^16*d^5*f^20 - 2208*a^9*b^12*c^15*d^6*f^20 + 3518*a 
^10*b^11*c^14*d^7*f^20 - 4000*a^11*b^10*c^13*d^8*f^20 + 3984*a^12*b^9*c^12 
*d^9*f^20 - 4000*a^13*b^8*c^11*d^10*f^20 + 3518*a^14*b^7*c^10*d^11*f^20 - 
2208*a^15*b^6*c^9*d^12*f^20 + 872*a^16*b^5*c^8*d^13*f^20 - 192*a^17*b^4*c^ 
7*d^14*f^20 + 18*a^18*b^3*c^6*d^15*f^20 + 64*a^6*b^15*d^21*e^18*f^2 - 576* 
a^7*b^14*d^21*e^17*f^3 + 2228*a^8*b^13*d^21*e^16*f^4 - 4768*a^9*b^12*d^21* 
e^15*f^5 + 5960*a^10*b^11*d^21*e^14*f^6 - 3976*a^11*b^10*d^21*e^13*f^7 + 5 
78*a^12*b^9*d^21*e^12*f^8 + 1004*a^13*b^8*d^21*e^11*f^9 - 442*a^14*b^7*d^2 
1*e^10*f^10 - 320*a^15*b^6*d^21*e^9*f^11 + 362*a^16*b^5*d^21*e^8*f^12 - 13 
2*a^17*b^4*d^21*e^7*f^13 + 18*a^18*b^3*d^21*e^6*f^14 + 64*b^21*c^6*d^15*e^ 
18*f^2 - 576*b^21*c^7*d^14*e^17*f^3 + 2228*b^21*c^8*d^13*e^16*f^4 - 4768*b 
^21*c^9*d^12*e^15*f^5 + 5960*b^21*c^10*d^11*e^14*f^6 - 3976*b^21*c^11*d^10 
*e^13*f^7 + 578*b^21*c^12*d^9*e^12*f^8 + 1004*b^21*c^13*d^8*e^11*f^9 - 442 
*b^21*c^14*d^7*e^10*f^10 - 320*b^21*c^15*d^6*e^9*f^11 + 362*b^21*c^16*d^5* 
e^8*f^12 - 132*b^21*c^17*d^4*e^7*f^13 + 18*b^21*c^18*d^3*e^6*f^14 - 384*a* 
b^20*c^5*d^16*e^18*f^2 + 2880*a*b^20*c^6*d^15*e^17*f^3 - 8032*a*b^20*c^7*d 
^14*e^16*f^4 + 7264*a*b^20*c^8*d^13*e^15*f^5 + 11920*a*b^20*c^9*d^12*e^14* 
f^6 - 39704*a*b^20*c^10*d^11*e^13*f^7 + 44752*a*b^20*c^11*d^10*e^12*f^8 - 
19988*a*b^20*c^12*d^9*e^11*f^9 - 4856*a*b^20*c^13*d^8*e^10*f^10 + 9220*...
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 6383, normalized size of antiderivative = 18.08 \[ \int \frac {1}{(e+f x)^{3/2} \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(1/(f*x+e)^(3/2)/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x)
 

Output:

( - 7*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b 
)*sqrt(a*f - b*e)))*a**2*b**2*c**4*d*f**4 + 21*sqrt(b)*sqrt(e + f*x)*sqrt( 
a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*c** 
3*d**2*e*f**3 - 7*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x 
)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*c**3*d**2*f**4*x - 21*sqrt(b)*sq 
rt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e 
)))*a**2*b**2*c**2*d**3*e**2*f**2 + 21*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b* 
e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*c**2*d**3*e 
*f**3*x + 7*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/( 
sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*c*d**4*e**3*f - 21*sqrt(b)*sqrt(e + f* 
x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2* 
b**2*c*d**4*e**2*f**2*x + 7*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sq 
rt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*d**5*e**3*f*x + 3*sqrt 
(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f 
 - b*e)))*a*b**3*c**5*f**4 - 5*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan( 
(sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**3*c**4*d*e*f**3 - 4*sqrt 
(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f 
 - b*e)))*a*b**3*c**4*d*f**4*x - 3*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*a 
tan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**3*c**3*d**2*e**2*f** 
2 + 16*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sq...