\(\int \frac {1}{(e+f x)^{3/2} (a c+(b c+a d) x+b d x^2)} \, dx\) [394]

Optimal result

Integrand size = 31, antiderivative size = 147 \[ \int \frac {1}{(e+f x)^{3/2} \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {2 f}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d) (b e-a f)^{3/2}}+\frac {2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d) (d e-c f)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(e+f x)^{3/2} \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {2 f}{(b e-a f) (d e-c f) \sqrt {e+f x}}+\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{(-b c+a d) (-b e+a f)^{3/2}}+\frac {2 d^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{(b c-a d) (-d e+c f)^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.31, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1147, 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.

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

\(\Big \downarrow \) 1147

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

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 1147

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

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 = 1.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.02

method	result	size
pseudoelliptic	\(-\frac {2 f}{\left (c f -d e \right ) \left (a f -b e \right ) \sqrt {f x +e}}+\frac {2 b^{2} \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{\left (a f -b e \right ) \left (a d -b c \right ) \sqrt {\left (a f -b e \right ) b}}-\frac {2 d^{2} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) \left (a d -b c \right ) \sqrt {\left (c f -d e \right ) d}}\)	\(150\)
derivativedivides	\(2 f \left (-\frac {d^{2} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) f \left (a d -b c \right ) \sqrt {\left (c f -d e \right ) d}}-\frac {1}{\left (c f -d e \right ) \left (a f -b e \right ) \sqrt {f x +e}}+\frac {b^{2} \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{\left (a f -b e \right ) f \left (a d -b c \right ) \sqrt {\left (a f -b e \right ) b}}\right )\)	\(157\)
default	\(2 f \left (-\frac {d^{2} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) f \left (a d -b c \right ) \sqrt {\left (c f -d e \right ) d}}-\frac {1}{\left (c f -d e \right ) \left (a f -b e \right ) \sqrt {f x +e}}+\frac {b^{2} \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{\left (a f -b e \right ) f \left (a d -b c \right ) \sqrt {\left (a f -b e \right ) b}}\right )\)	\(157\)

Input:

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

Output:

-2*f/(c*f-d*e)/(a*f-b*e)/(f*x+e)^(1/2)+2/(a*f-b*e)*b^2/(a*d-b*c)/((a*f-b*e 
)*b)^(1/2)*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))-2/(c*f-d*e)*d^2/(a* 
d-b*c)/((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. 281 vs. \(2 (125) = 250\).

Time = 0.73 (sec) , antiderivative size = 1243, normalized size of antiderivative = 8.46 \[ \int \frac {1}{(e+f x)^{3/2} \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (122) = 244\).

Time = 4.37 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.03 \[ \int \frac {1}{(e+f x)^{3/2} \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\begin {cases} \frac {2 \left (\frac {b f \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {a f - b e}{b}}} \right )}}{\sqrt {\frac {a f - b e}{b}} \left (a d - b c\right ) \left (a f - b e\right )} - \frac {d f \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{\sqrt {\frac {c f - d e}{d}} \left (a d - b c\right ) \left (c f - d e\right )} - \frac {f^{2}}{\sqrt {e + f x} \left (a f - b e\right ) \left (c f - d e\right )}\right )}{f} & \text {for}\: f \neq 0 \\\frac {- \frac {2 b d \left (\begin {cases} \frac {x + \frac {a d + b c}{2 b d}}{a d} & \text {for}\: b = 0 \\- \frac {x + \frac {a d + b c}{2 b d}}{b c} & \text {for}\: d = 0 \\- \frac {\log {\left (a d - b c - 2 b d \left (x + \frac {a d + b c}{2 b d}\right ) \right )}}{2 b d} & \text {otherwise} \end {cases}\right )}{a d - b c} - \frac {2 b d \left (\begin {cases} \frac {x + \frac {a d + b c}{2 b d}}{a d} & \text {for}\: b = 0 \\- \frac {x + \frac {a d + b c}{2 b d}}{b c} & \text {for}\: d = 0 \\\frac {\log {\left (a d - b c + 2 b d \left (x + \frac {a d + b c}{2 b d}\right ) \right )}}{2 b d} & \text {otherwise} \end {cases}\right )}{a d - b c}}{e^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((2*(b*f*atan(sqrt(e + f*x)/sqrt((a*f - b*e)/b))/(sqrt((a*f - b*e 
)/b)*(a*d - b*c)*(a*f - b*e)) - d*f*atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d) 
)/(sqrt((c*f - d*e)/d)*(a*d - b*c)*(c*f - d*e)) - f**2/(sqrt(e + f*x)*(a*f 
 - b*e)*(c*f - d*e)))/f, Ne(f, 0)), ((-2*b*d*Piecewise(((x + (a*d + b*c)/( 
2*b*d))/(a*d), Eq(b, 0)), (-(x + (a*d + b*c)/(2*b*d))/(b*c), Eq(d, 0)), (- 
log(a*d - b*c - 2*b*d*(x + (a*d + b*c)/(2*b*d)))/(2*b*d), True))/(a*d - b* 
c) - 2*b*d*Piecewise(((x + (a*d + b*c)/(2*b*d))/(a*d), Eq(b, 0)), (-(x + ( 
a*d + b*c)/(2*b*d))/(b*c), Eq(d, 0)), (log(a*d - b*c + 2*b*d*(x + (a*d + b 
*c)/(2*b*d)))/(2*b*d), True))/(a*d - b*c))/e**(3/2), True))
 

Maxima [F(-2)]

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

integrate(1/(f*x+e)^(3/2)/(a*c+(a*d+b*c)*x+b*d*x^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 [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(e+f x)^{3/2} \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {2 \, b^{2} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{{\left (b^{2} c e - a b d e - a b c f + a^{2} d f\right )} \sqrt {-b^{2} e + a b f}} - \frac {2 \, d^{2} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (b c d e - a d^{2} e - b c^{2} f + a c d f\right )} \sqrt {-d^{2} e + c d f}} - \frac {2 \, f}{{\left (b d e^{2} - b c e f - a d e f + a c f^{2}\right )} \sqrt {f x + e}} \] Input:

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

Output:

2*b^2*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^2*c*e - a*b*d*e - a 
*b*c*f + a^2*d*f)*sqrt(-b^2*e + a*b*f)) - 2*d^2*arctan(sqrt(f*x + e)*d/sqr 
t(-d^2*e + c*d*f))/((b*c*d*e - a*d^2*e - b*c^2*f + a*c*d*f)*sqrt(-d^2*e + 
c*d*f)) - 2*f/((b*d*e^2 - b*c*e*f - a*d*e*f + a*c*f^2)*sqrt(f*x + e))
 

Mupad [B] (verification not implemented)

Time = 8.18 (sec) , antiderivative size = 26252, normalized size of antiderivative = 178.59 \[ \int \frac {1}{(e+f x)^{3/2} \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

(atan((((-d^3*(c*f - d*e)^3)^(1/2)*((e + f*x)^(1/2)*(16*b^8*d^8*e^8*f^2 + 
8*a^3*b^5*c^5*d^3*f^10 + 8*a^5*b^3*c^3*d^5*f^10 + 104*a^2*b^6*d^8*e^6*f^4 
- 88*a^3*b^5*d^8*e^5*f^5 + 40*a^4*b^4*d^8*e^4*f^6 - 8*a^5*b^3*d^8*e^3*f^7 
+ 104*b^8*c^2*d^6*e^6*f^4 - 88*b^8*c^3*d^5*e^5*f^5 + 40*b^8*c^4*d^4*e^4*f^ 
6 - 8*b^8*c^5*d^3*e^3*f^7 - 64*a*b^7*d^8*e^7*f^3 - 64*b^8*c*d^7*e^7*f^3 + 
240*a*b^7*c*d^7*e^6*f^4 - 360*a*b^7*c^2*d^6*e^5*f^5 + 280*a*b^7*c^3*d^5*e^ 
4*f^6 - 120*a*b^7*c^4*d^4*e^3*f^7 + 24*a*b^7*c^5*d^3*e^2*f^8 - 360*a^2*b^6 
*c*d^7*e^5*f^5 - 24*a^2*b^6*c^5*d^3*e*f^9 + 280*a^3*b^5*c*d^7*e^4*f^6 - 40 
*a^3*b^5*c^4*d^4*e*f^9 - 120*a^4*b^4*c*d^7*e^3*f^7 - 40*a^4*b^4*c^3*d^5*e* 
f^9 + 24*a^5*b^3*c*d^7*e^2*f^8 - 24*a^5*b^3*c^2*d^6*e*f^9 + 480*a^2*b^6*c^ 
2*d^6*e^4*f^6 - 320*a^2*b^6*c^3*d^5*e^3*f^7 + 120*a^2*b^6*c^4*d^4*e^2*f^8 
- 320*a^3*b^5*c^2*d^6*e^3*f^7 + 160*a^3*b^5*c^3*d^5*e^2*f^8 + 120*a^4*b^4* 
c^2*d^6*e^2*f^8) + ((-d^3*(c*f - d*e)^3)^(1/2)*(8*a^5*b^4*c^6*d^3*f^12 - 8 
*a^4*b^5*c^7*d^2*f^12 - ((e + f*x)^(1/2)*(-d^3*(c*f - d*e)^3)^(1/2)*(8*a^6 
*b^4*c^7*d^3*f^13 - 8*a^5*b^5*c^8*d^2*f^13 + 8*a^7*b^3*c^6*d^4*f^13 - 8*a^ 
8*b^2*c^5*d^5*f^13 + 16*a^2*b^8*d^10*e^11*f^2 - 88*a^3*b^7*d^10*e^10*f^3 + 
 200*a^4*b^6*d^10*e^9*f^4 - 240*a^5*b^5*d^10*e^8*f^5 + 160*a^6*b^4*d^10*e^ 
7*f^6 - 56*a^7*b^3*d^10*e^6*f^7 + 8*a^8*b^2*d^10*e^5*f^8 + 16*b^10*c^2*d^8 
*e^11*f^2 - 88*b^10*c^3*d^7*e^10*f^3 + 200*b^10*c^4*d^6*e^9*f^4 - 240*b^10 
*c^5*d^5*e^8*f^5 + 160*b^10*c^6*d^4*e^7*f^6 - 56*b^10*c^7*d^3*e^6*f^7 +...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 528, normalized size of antiderivative = 3.59 \[ \int \frac {1}{(e+f x)^{3/2} \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {2 \sqrt {b}\, \sqrt {f x +e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, b}{\sqrt {b}\, \sqrt {a f -b e}}\right ) b \,c^{2} f^{2}-4 \sqrt {b}\, \sqrt {f x +e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, b}{\sqrt {b}\, \sqrt {a f -b e}}\right ) b c d e f +2 \sqrt {b}\, \sqrt {f x +e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, b}{\sqrt {b}\, \sqrt {a f -b e}}\right ) b \,d^{2} e^{2}-2 \sqrt {d}\, \sqrt {f x +e}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) a^{2} d \,f^{2}+4 \sqrt {d}\, \sqrt {f x +e}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) a b d e f -2 \sqrt {d}\, \sqrt {f x +e}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) b^{2} d \,e^{2}-2 a^{2} c d \,f^{3}+2 a^{2} d^{2} e \,f^{2}+2 a b \,c^{2} f^{3}-2 a b \,d^{2} e^{2} f -2 b^{2} c^{2} e \,f^{2}+2 b^{2} c d \,e^{2} f}{\sqrt {f x +e}\, \left (a^{3} c^{2} d \,f^{4}-2 a^{3} c \,d^{2} e \,f^{3}+a^{3} d^{3} e^{2} f^{2}-a^{2} b \,c^{3} f^{4}+3 a^{2} b c \,d^{2} e^{2} f^{2}-2 a^{2} b \,d^{3} e^{3} f +2 a \,b^{2} c^{3} e \,f^{3}-3 a \,b^{2} c^{2} d \,e^{2} f^{2}+a \,b^{2} d^{3} e^{4}-b^{3} c^{3} e^{2} f^{2}+2 b^{3} c^{2} d \,e^{3} f -b^{3} c \,d^{2} e^{4}\right )} \] Input:

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

Output:

(2*(sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)* 
sqrt(a*f - b*e)))*b*c**2*f**2 - 2*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*at 
an((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*b*c*d*e*f + sqrt(b)*sqrt(e 
 + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))* 
b*d**2*e**2 - sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d) 
/(sqrt(d)*sqrt(c*f - d*e)))*a**2*d*f**2 + 2*sqrt(d)*sqrt(e + f*x)*sqrt(c*f 
 - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b*d*e*f - sqrt 
(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f 
 - d*e)))*b**2*d*e**2 - a**2*c*d*f**3 + a**2*d**2*e*f**2 + a*b*c**2*f**3 - 
 a*b*d**2*e**2*f - b**2*c**2*e*f**2 + b**2*c*d*e**2*f))/(sqrt(e + f*x)*(a* 
*3*c**2*d*f**4 - 2*a**3*c*d**2*e*f**3 + a**3*d**3*e**2*f**2 - a**2*b*c**3* 
f**4 + 3*a**2*b*c*d**2*e**2*f**2 - 2*a**2*b*d**3*e**3*f + 2*a*b**2*c**3*e* 
f**3 - 3*a*b**2*c**2*d*e**2*f**2 + a*b**2*d**3*e**4 - b**3*c**3*e**2*f**2 
+ 2*b**3*c**2*d*e**3*f - b**3*c*d**2*e**4))