\(\int \frac {(e+f x)^{3/2}}{(a c+(b c+a d) x+b d x^2)^2} \, dx\) [397]

Optimal result

Integrand size = 31, antiderivative size = 220 \[ \int \frac {(e+f x)^{3/2}}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {\sqrt {e+f x} (b c e+a d e-2 a c f+(2 b d e-(b c+a d) f) x)}{(b c-a d)^2 \left (a c+(b c+a d) x+b d x^2\right )}+\frac {\sqrt {b e-a f} (4 b d e-3 b c f-a d f) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{\sqrt {b} (b c-a d)^3}-\frac {\sqrt {d e-c f} (4 b d e-b c f-3 a d f) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (b c-a d)^3} \] Output:

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

Mathematica [A] (verified)

Time = 1.40 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.96 \[ \int \frac {(e+f x)^{3/2}}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {\sqrt {e+f x} (b (-c e-2 d e x+c f x)+a (-d e+2 c f+d f x))}{(b c-a d)^2 (a+b x) (c+d x)}+\frac {\sqrt {-b e+a f} (4 b d e-3 b c f-a d f) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{\sqrt {b} (b c-a d)^3}+\frac {\sqrt {-d e+c f} (4 b d e-b c f-3 a d f) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{\sqrt {d} (-b c+a d)^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {1164, 27, 1197, 27, 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.

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

\(\Big \downarrow \) 1164

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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 1164

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[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 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 = 1.44 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.04

method	result	size
derivativedivides	\(2 f^{3} \left (\frac {\left (a f -b e \right ) \left (\frac {\left (\frac {1}{2} a d f -\frac {1}{2} b c f \right ) \sqrt {f x +e}}{b \left (f x +e \right )+a f -b e}+\frac {\left (a d f +3 b c f -4 b d e \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}\right )}{f^{3} \left (a d -b c \right )^{3}}-\frac {\left (c f -d e \right ) \left (\frac {\left (-\frac {1}{2} a d f +\frac {1}{2} b c f \right ) \sqrt {f x +e}}{d \left (f x +e \right )+c f -d e}+\frac {\left (3 a d f +b c f -4 b d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}\right )}{f^{3} \left (a d -b c \right )^{3}}\right )\)	\(228\)
default	\(2 f^{3} \left (\frac {\left (a f -b e \right ) \left (\frac {\left (\frac {1}{2} a d f -\frac {1}{2} b c f \right ) \sqrt {f x +e}}{b \left (f x +e \right )+a f -b e}+\frac {\left (a d f +3 b c f -4 b d e \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}\right )}{f^{3} \left (a d -b c \right )^{3}}-\frac {\left (c f -d e \right ) \left (\frac {\left (-\frac {1}{2} a d f +\frac {1}{2} b c f \right ) \sqrt {f x +e}}{d \left (f x +e \right )+c f -d e}+\frac {\left (3 a d f +b c f -4 b d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}\right )}{f^{3} \left (a d -b c \right )^{3}}\right )\)	\(228\)
pseudoelliptic	\(\frac {\left (\left (3 c f -4 d e \right ) b +a d f \right ) \left (a f -b e \right ) \left (d x +c \right ) \sqrt {\left (c f -d e \right ) d}\, \left (b x +a \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+2 \sqrt {\left (a f -b e \right ) b}\, \left (-\frac {3 \left (c f -d e \right ) \left (d x +c \right ) \left (\frac {\left (c f -4 d e \right ) b}{3}+a d f \right ) \left (b x +a \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2}+\sqrt {f x +e}\, \left (a d -b c \right ) \left (\left (-d e x -\frac {c \left (-f x +e \right )}{2}\right ) b +\left (\frac {\left (f x -e \right ) d}{2}+c f \right ) a \right ) \sqrt {\left (c f -d e \right ) d}\right )}{\left (b x +a \right ) \left (a d -b c \right )^{3} \sqrt {\left (a f -b e \right ) b}\, \left (d x +c \right ) \sqrt {\left (c f -d e \right ) d}}\)	\(256\)

Input:

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

Output:

2*f^3*((a*f-b*e)/f^3/(a*d-b*c)^3*((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*(a*d*f+3*b*c*f-4*b*d*e)/((a*f-b*e)*b)^(1/2)*arctan(b*( 
f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2)))-(c*f-d*e)/f^3/(a*d-b*c)^3*((-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+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. 519 vs. \(2 (198) = 396\).

Time = 0.26 (sec) , antiderivative size = 2116, normalized size of antiderivative = 9.62 \[ \int \frac {(e+f x)^{3/2}}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*((4*a*b*c*d*e + (4*b^2*d^2*e - (3*b^2*c*d + a*b*d^2)*f)*x^2 - (3*a*b* 
c^2 + a^2*c*d)*f + (4*(b^2*c*d + a*b*d^2)*e - (3*b^2*c^2 + 4*a*b*c*d + a^2 
*d^2)*f)*x)*sqrt((b*e - a*f)/b)*log((b*f*x + 2*b*e - a*f + 2*sqrt(f*x + e) 
*b*sqrt((b*e - a*f)/b))/(b*x + a)) + (4*a*b*c*d*e + (4*b^2*d^2*e - (b^2*c* 
d + 3*a*b*d^2)*f)*x^2 - (a*b*c^2 + 3*a^2*c*d)*f + (4*(b^2*c*d + a*b*d^2)*e 
 - (b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*f)*x)*sqrt((d*e - c*f)/d)*log((d*f*x 
+ 2*d*e - c*f - 2*sqrt(f*x + e)*d*sqrt((d*e - c*f)/d))/(d*x + c)) - 2*((b^ 
2*c^2 - a^2*d^2)*e - 2*(a*b*c^2 - a^2*c*d)*f + (2*(b^2*c*d - a*b*d^2)*e - 
(b^2*c^2 - a^2*d^2)*f)*x)*sqrt(f*x + e))/(a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3* 
a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 
 - a^3*b*d^4)*x^2 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x) 
, 1/2*(2*(4*a*b*c*d*e + (4*b^2*d^2*e - (3*b^2*c*d + a*b*d^2)*f)*x^2 - (3*a 
*b*c^2 + a^2*c*d)*f + (4*(b^2*c*d + a*b*d^2)*e - (3*b^2*c^2 + 4*a*b*c*d + 
a^2*d^2)*f)*x)*sqrt(-(b*e - a*f)/b)*arctan(-sqrt(f*x + e)*b*sqrt(-(b*e - a 
*f)/b)/(b*e - a*f)) + (4*a*b*c*d*e + (4*b^2*d^2*e - (b^2*c*d + 3*a*b*d^2)* 
f)*x^2 - (a*b*c^2 + 3*a^2*c*d)*f + (4*(b^2*c*d + a*b*d^2)*e - (b^2*c^2 + 4 
*a*b*c*d + 3*a^2*d^2)*f)*x)*sqrt((d*e - c*f)/d)*log((d*f*x + 2*d*e - c*f - 
 2*sqrt(f*x + e)*d*sqrt((d*e - c*f)/d))/(d*x + c)) - 2*((b^2*c^2 - a^2*d^2 
)*e - 2*(a*b*c^2 - a^2*c*d)*f + (2*(b^2*c*d - a*b*d^2)*e - (b^2*c^2 - a^2* 
d^2)*f)*x)*sqrt(f*x + e))/(a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((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. 425 vs. \(2 (198) = 396\).

Time = 0.42 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.93 \[ \int \frac {(e+f x)^{3/2}}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {{\left (4 \, b^{2} d e^{2} - 3 \, b^{2} c e f - 5 \, a b d e f + 3 \, a b c f^{2} + a^{2} d f^{2}\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-b^{2} e + a b f}} + \frac {{\left (4 \, b d^{2} e^{2} - 5 \, b c d e f - 3 \, a d^{2} e f + b c^{2} f^{2} + 3 \, a c d f^{2}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-d^{2} e + c d f}} - \frac {2 \, {\left (f x + e\right )}^{\frac {3}{2}} b d e f - 2 \, \sqrt {f x + e} b d e^{2} f - {\left (f x + e\right )}^{\frac {3}{2}} b c f^{2} - {\left (f x + e\right )}^{\frac {3}{2}} a d f^{2} + 2 \, \sqrt {f x + e} b c e f^{2} + 2 \, \sqrt {f x + e} a d e f^{2} - 2 \, \sqrt {f x + e} a c f^{3}}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left ({\left (f x + e\right )}^{2} b d - 2 \, {\left (f x + e\right )} b d e + b d e^{2} + {\left (f x + e\right )} b c f + {\left (f x + e\right )} a d f - b c e f - a d e f + a c f^{2}\right )}} \] Input:

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

Output:

-(4*b^2*d*e^2 - 3*b^2*c*e*f - 5*a*b*d*e*f + 3*a*b*c*f^2 + a^2*d*f^2)*arcta 
n(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2* 
b*c*d^2 - a^3*d^3)*sqrt(-b^2*e + a*b*f)) + (4*b*d^2*e^2 - 5*b*c*d*e*f - 3* 
a*d^2*e*f + b*c^2*f^2 + 3*a*c*d*f^2)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + 
c*d*f))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-d^2*e + 
 c*d*f)) - (2*(f*x + e)^(3/2)*b*d*e*f - 2*sqrt(f*x + e)*b*d*e^2*f - (f*x + 
 e)^(3/2)*b*c*f^2 - (f*x + e)^(3/2)*a*d*f^2 + 2*sqrt(f*x + e)*b*c*e*f^2 + 
2*sqrt(f*x + e)*a*d*e*f^2 - 2*sqrt(f*x + e)*a*c*f^3)/((b^2*c^2 - 2*a*b*c*d 
 + a^2*d^2)*((f*x + e)^2*b*d - 2*(f*x + e)*b*d*e + b*d*e^2 + (f*x + e)*b*c 
*f + (f*x + e)*a*d*f - b*c*e*f - a*d*e*f + a*c*f^2))
 

Mupad [B] (verification not implemented)

Time = 7.57 (sec) , antiderivative size = 11194, normalized size of antiderivative = 50.88 \[ \int \frac {(e+f x)^{3/2}}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((2*(e + f*x)^(1/2)*(a*c*f^3 - a*d*e*f^2 - b*c*e*f^2 + b*d*e^2*f))/(a^2*d^ 
2 + b^2*c^2 - 2*a*b*c*d) + (f*(e + f*x)^(3/2)*(a*d*f + b*c*f - 2*b*d*e))/( 
a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/((e + f*x)*(a*d*f + b*c*f - 2*b*d*e) + b*d 
*(e + f*x)^2 + a*c*f^2 + b*d*e^2 - a*d*e*f - b*c*e*f) - (atan((((-b*(a*f - 
 b*e))^(1/2)*((2*(e + f*x)^(1/2)*(a^4*b*d^5*f^6 + b^5*c^4*d*f^6 + 32*b^5*d 
^5*e^4*f^2 + 18*a^2*b^3*c^2*d^3*f^6 + 42*a^2*b^3*d^5*e^2*f^4 + 42*b^5*c^2* 
d^3*e^2*f^4 + 6*a*b^4*c^3*d^2*f^6 + 6*a^3*b^2*c*d^4*f^6 - 64*a*b^4*d^5*e^3 
*f^3 - 10*a^3*b^2*d^5*e*f^5 - 64*b^5*c*d^4*e^3*f^3 - 10*b^5*c^3*d^2*e*f^5 
+ 108*a*b^4*c*d^4*e^2*f^4 - 54*a*b^4*c^2*d^3*e*f^5 - 54*a^2*b^3*c*d^4*e*f^ 
5))/(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3 
) - ((-b*(a*f - b*e))^(1/2)*((120*a^3*b^6*c^5*d^4*f^5 - 48*a^2*b^7*c^6*d^3 
*f^5 - 160*a^4*b^5*c^4*d^5*f^5 + 120*a^5*b^4*c^3*d^6*f^5 - 48*a^6*b^3*c^2* 
d^7*f^5 + 8*a^6*b^3*d^9*e^2*f^3 + 8*b^9*c^6*d^3*e^2*f^3 + 8*a*b^8*c^7*d^2* 
f^5 + 8*a^7*b^2*c*d^8*f^5 - 8*a^7*b^2*d^9*e*f^4 - 8*b^9*c^7*d^2*e*f^4 + 40 
*a*b^8*c^6*d^3*e*f^4 + 40*a^6*b^3*c*d^8*e*f^4 - 48*a*b^8*c^5*d^4*e^2*f^3 - 
 72*a^2*b^7*c^5*d^4*e*f^4 + 40*a^3*b^6*c^4*d^5*e*f^4 + 40*a^4*b^5*c^3*d^6* 
e*f^4 - 48*a^5*b^4*c*d^8*e^2*f^3 - 72*a^5*b^4*c^2*d^7*e*f^4 + 120*a^2*b^7* 
c^4*d^5*e^2*f^3 - 160*a^3*b^6*c^3*d^6*e^2*f^3 + 120*a^4*b^5*c^2*d^7*e^2*f^ 
3)/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b 
^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) + ((e + f*x)^(1/2)*(-b*(a*f...
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 1273, normalized size of antiderivative = 5.79 \[ \int \frac {(e+f x)^{3/2}}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e))) 
*a**2*c*d**2*f + sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*s 
qrt(a*f - b*e)))*a**2*d**3*f*x + 3*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + 
f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b*c**2*d*f - 4*sqrt(b)*sqrt(a*f - b*e 
)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b*c*d**2*e + 4*sqrt( 
b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b*c 
*d**2*f*x - 4*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt 
(a*f - b*e)))*a*b*d**3*e*x + sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b 
)/(sqrt(b)*sqrt(a*f - b*e)))*a*b*d**3*f*x**2 + 3*sqrt(b)*sqrt(a*f - b*e)*a 
tan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*b**2*c**2*d*f*x - 4*sqrt( 
b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*b**2* 
c*d**2*e*x + 3*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqr 
t(a*f - b*e)))*b**2*c*d**2*f*x**2 - 4*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e 
 + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*b**2*d**3*e*x**2 - 3*sqrt(d)*sqrt(c* 
f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c*d*f - 
3*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)) 
)*a**2*b*d**2*f*x - sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d 
)*sqrt(c*f - d*e)))*a*b**2*c**2*f + 4*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e 
 + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b**2*c*d*e - 4*sqrt(d)*sqrt(c*f - 
d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b**2*c*d*f*x +...