\(\int \frac {1}{(a+b x+c x^2)^{3/2} (d-f x^2)} \, dx\) [73]

Optimal result

Integrand size = 25, antiderivative size = 310 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=-\frac {2 \left (b \left (b^2 f-c (c d+3 a f)\right )-c \left (2 c^2 d-b^2 f+2 a c f\right ) x\right )}{\left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}+\frac {f \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2}}+\frac {f \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2}} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.24 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.50 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\frac {-4 b^3 f+4 b c (c d+3 a f)-4 b^2 c f x+8 c^2 (c d+a f) x+\left (b^2-4 a c\right ) f \sqrt {a+x (b+c x)} \text {RootSum}\left [c^2 d-b^2 f+4 \sqrt {a} b f \text {$\#$1}-2 c d \text {$\#$1}^2-4 a f \text {$\#$1}^2+d \text {$\#$1}^4\&,\frac {-c^2 d \log (x)-b^2 f \log (x)-a c f \log (x)+c^2 d \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )+b^2 f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )+a c f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )+2 \sqrt {a} b f \log (x) \text {$\#$1}-2 \sqrt {a} b f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+c d \log (x) \text {$\#$1}^2+a f \log (x) \text {$\#$1}^2-c d \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-a f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} b f+c d \text {$\#$1}+2 a f \text {$\#$1}-d \text {$\#$1}^3}\&\right ]}{2 \left (b^2-4 a c\right ) \left (-c^2 d^2-2 a c d f+f \left (b^2 d-a^2 f\right )\right ) \sqrt {a+x (b+c x)}} \] Input:

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

Output:

(-4*b^3*f + 4*b*c*(c*d + 3*a*f) - 4*b^2*c*f*x + 8*c^2*(c*d + a*f)*x + (b^2 
 - 4*a*c)*f*Sqrt[a + x*(b + c*x)]*RootSum[c^2*d - b^2*f + 4*Sqrt[a]*b*f*#1 
 - 2*c*d*#1^2 - 4*a*f*#1^2 + d*#1^4 & , (-(c^2*d*Log[x]) - b^2*f*Log[x] - 
a*c*f*Log[x] + c^2*d*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + b^2*f* 
Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + a*c*f*Log[-Sqrt[a] + Sqrt[a 
 + b*x + c*x^2] - x*#1] + 2*Sqrt[a]*b*f*Log[x]*#1 - 2*Sqrt[a]*b*f*Log[-Sqr 
t[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 + c*d*Log[x]*#1^2 + a*f*Log[x]*#1^ 
2 - c*d*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1^2 - a*f*Log[-Sqrt[ 
a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1^2)/(-(Sqrt[a]*b*f) + c*d*#1 + 2*a*f* 
#1 - d*#1^3) & ])/(2*(b^2 - 4*a*c)*(-(c^2*d^2) - 2*a*c*d*f + f*(b^2*d - a^ 
2*f))*Sqrt[a + x*(b + c*x)])
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1306, 27, 1366, 25, 27, 1154, 219}

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.

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

\(\Big \downarrow \) 1306

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1366

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1306

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (f_.)*(x_)^2)^(q_), x 
_Symbol] :> Simp[(b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(2*a*f 
))*x)*(a + b*x + c*x^2)^(p + 1)*((d + f*x^2)^(q + 1)/((b^2 - 4*a*c)*(b^2*d* 
f + (c*d - a*f)^2)*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - 
a*f)^2)*(p + 1))   Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q*Simp[2*c*(b^ 
2*d*f + (c*d - a*f)^2)*(p + 1) - (2*c^2*d + b^2*f - c*(2*a*f))*(a*f*(p + 1) 
 - c*d*(p + 2)) + (2*f*(b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + 
 b^2*f - c*(2*a*f))*(b*f*(p + 1)))*x + c*f*(2*c^2*d + b^2*f - c*(2*a*f))*(2 
*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, q}, x] && NeQ[b^2 - 
4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] &&  !( !IntegerQ 
[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]
 

rule 1366

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(902\) vs. \(2(252)=504\).

Time = 2.44 (sec) , antiderivative size = 903, normalized size of antiderivative = 2.91

method	result	size
default	\(-\frac {\frac {f}{\left (b \sqrt {d f}+a f +c d \right ) \sqrt {c \left (x -\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+a f +c d}{f}}}-\frac {\left (2 c \sqrt {d f}+f b \right ) \left (2 c \left (x -\frac {\sqrt {d f}}{f}\right )+\frac {2 c \sqrt {d f}+f b}{f}\right )}{\left (b \sqrt {d f}+a f +c d \right ) \left (\frac {4 c \left (b \sqrt {d f}+a f +c d \right )}{f}-\frac {\left (2 c \sqrt {d f}+f b \right )^{2}}{f^{2}}\right ) \sqrt {c \left (x -\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+a f +c d}{f}}}-\frac {f \ln \left (\frac {\frac {2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x -\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+a f +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{\left (b \sqrt {d f}+a f +c d \right ) \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}}}{2 \sqrt {d f}}+\frac {\frac {f}{\left (-b \sqrt {d f}+a f +c d \right ) \sqrt {c \left (x +\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+a f +c d}{f}}}-\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (2 c \left (x +\frac {\sqrt {d f}}{f}\right )+\frac {-2 c \sqrt {d f}+f b}{f}\right )}{\left (-b \sqrt {d f}+a f +c d \right ) \left (\frac {4 c \left (-b \sqrt {d f}+a f +c d \right )}{f}-\frac {\left (-2 c \sqrt {d f}+f b \right )^{2}}{f^{2}}\right ) \sqrt {c \left (x +\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+a f +c d}{f}}}-\frac {f \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x +\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+a f +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{\left (-b \sqrt {d f}+a f +c d \right ) \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}}}{2 \sqrt {d f}}\)	\(903\)

Input:

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

Output:

-1/2/(d*f)^(1/2)*(1/(b*(d*f)^(1/2)+a*f+c*d)*f/(c*(x-(d*f)^(1/2)/f)^2+(2*c* 
(d*f)^(1/2)+f*b)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2)-(2*c 
*(d*f)^(1/2)+f*b)/(b*(d*f)^(1/2)+a*f+c*d)*(2*c*(x-(d*f)^(1/2)/f)+(2*c*(d*f 
)^(1/2)+f*b)/f)/(4*c*(b*(d*f)^(1/2)+a*f+c*d)/f-(2*c*(d*f)^(1/2)+f*b)^2/f^2 
)/(c*(x-(d*f)^(1/2)/f)^2+(2*c*(d*f)^(1/2)+f*b)/f*(x-(d*f)^(1/2)/f)+(b*(d*f 
)^(1/2)+a*f+c*d)/f)^(1/2)-1/(b*(d*f)^(1/2)+a*f+c*d)*f/((b*(d*f)^(1/2)+a*f+ 
c*d)/f)^(1/2)*ln((2*(b*(d*f)^(1/2)+a*f+c*d)/f+(2*c*(d*f)^(1/2)+f*b)/f*(x-( 
d*f)^(1/2)/f)+2*((b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2)*(c*(x-(d*f)^(1/2)/f)^2+( 
2*c*(d*f)^(1/2)+f*b)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2)) 
/(x-(d*f)^(1/2)/f)))+1/2/(d*f)^(1/2)*(f/(-b*(d*f)^(1/2)+a*f+c*d)/(c*(x+(d* 
f)^(1/2)/f)^2+1/f*(-2*c*(d*f)^(1/2)+f*b)*(x+(d*f)^(1/2)/f)+1/f*(-b*(d*f)^( 
1/2)+a*f+c*d))^(1/2)-(-2*c*(d*f)^(1/2)+f*b)/(-b*(d*f)^(1/2)+a*f+c*d)*(2*c* 
(x+(d*f)^(1/2)/f)+1/f*(-2*c*(d*f)^(1/2)+f*b))/(4*c/f*(-b*(d*f)^(1/2)+a*f+c 
*d)-1/f^2*(-2*c*(d*f)^(1/2)+f*b)^2)/(c*(x+(d*f)^(1/2)/f)^2+1/f*(-2*c*(d*f) 
^(1/2)+f*b)*(x+(d*f)^(1/2)/f)+1/f*(-b*(d*f)^(1/2)+a*f+c*d))^(1/2)-f/(-b*(d 
*f)^(1/2)+a*f+c*d)/(1/f*(-b*(d*f)^(1/2)+a*f+c*d))^(1/2)*ln((2/f*(-b*(d*f)^ 
(1/2)+a*f+c*d)+1/f*(-2*c*(d*f)^(1/2)+f*b)*(x+(d*f)^(1/2)/f)+2*(1/f*(-b*(d* 
f)^(1/2)+a*f+c*d))^(1/2)*(c*(x+(d*f)^(1/2)/f)^2+1/f*(-2*c*(d*f)^(1/2)+f*b) 
*(x+(d*f)^(1/2)/f)+1/f*(-b*(d*f)^(1/2)+a*f+c*d))^(1/2))/(x+(d*f)^(1/2)/f)) 
)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 17397 vs. \(2 (252) = 504\).

Time = 12.17 (sec) , antiderivative size = 17397, normalized size of antiderivative = 56.12 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=- \int \frac {1}{- a d \sqrt {a + b x + c x^{2}} + a f x^{2} \sqrt {a + b x + c x^{2}} - b d x \sqrt {a + b x + c x^{2}} + b f x^{3} \sqrt {a + b x + c x^{2}} - c d x^{2} \sqrt {a + b x + c x^{2}} + c f x^{4} \sqrt {a + b x + c x^{2}}}\, dx \] Input:

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

Output:

-Integral(1/(-a*d*sqrt(a + b*x + c*x**2) + a*f*x**2*sqrt(a + b*x + c*x**2) 
 - b*d*x*sqrt(a + b*x + c*x**2) + b*f*x**3*sqrt(a + b*x + c*x**2) - c*d*x* 
*2*sqrt(a + b*x + c*x**2) + c*f*x**4*sqrt(a + b*x + c*x**2)), x)
 

Maxima [F(-2)]

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

integrate(1/(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),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*sqrt(4*d*f))/(2*f^2)>0)', se 
e `assume?
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.99 (sec) , antiderivative size = 25455, normalized size of antiderivative = 82.11 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

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