\(\int \frac {x}{(a+b x+c x^2)^{3/2} (d-f x^2)} \, dx\) [72]

Optimal result

Integrand size = 26, antiderivative size = 299 \[ \int \frac {x}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=-\frac {2 \left (a \left (2 c^2 d-b^2 f+2 a c f\right )+b c (c d-a f) x\right )}{\left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}-\frac {\sqrt {f} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2}}+\frac {\sqrt {f} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2}} \] Output:

(-2*a*(2*a*c*f-b^2*f+2*c^2*d)-2*b*c*(-a*f+c*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^(1/2)*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))/(c*d-b*d^(1/2)*f^(1/2)+a*f)^(3/2)+1/2*f^(1/2)*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))/(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 = 0.98 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.36 \[ \int \frac {x}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\frac {-8 a^2 c f-4 b c^2 d x+4 a \left (-2 c^2 d+b^2 f+b c f x\right )-\left (b^2-4 a c\right ) f \sqrt {a+x (b+c x)} \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b^2 d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 b \sqrt {c} d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 \left (b^2-4 a c\right ) \left (-c^2 d^2-2 a c d f+f \left (b^2 d-a^2 f\right )\right ) \sqrt {a+x (b+c x)}} \] Input:

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

Output:

(-8*a^2*c*f - 4*b*c^2*d*x + 4*a*(-2*c^2*d + b^2*f + b*c*f*x) - (b^2 - 4*a* 
c)*f*Sqrt[a + x*(b + c*x)]*RootSum[b^2*d - a^2*f - 4*b*Sqrt[c]*d*#1 + 4*c* 
d*#1^2 + 2*a*f*#1^2 - f*#1^4 & , (b^2*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + 
c*x^2] - #1] + a*c*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] + a^2* 
f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - 2*b*Sqrt[c]*d*Log[-(Sqr 
t[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - c*d*Log[-(Sqrt[c]*x) + Sqrt[a + 
 b*x + c*x^2] - #1]*#1^2 - a*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - 
#1]*#1^2)/(b*Sqrt[c]*d - 2*c*d*#1 - a*f*#1 + f*#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 = 0.50 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.21, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1351, 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.

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

\(\Big \downarrow \) 1351

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1366

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(-2*(a*(2*c^2*d - b^2*f + 2*a*c*f) + b*c*(c*d - a*f)*x))/((b^2 - 4*a*c)*(b 
^2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x + c*x^2]) + (f*(((c*d + b*Sqrt[d]*Sqr 
t[f] + a*f)*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[ 
f]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) - ((c*d - b*Sqrt[d]*Sqrt[f] + a*f) 
*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[f]*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 1351

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f 
_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(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)))*((g*c)*((-b)*(c* 
d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(2*a*f)) + c*(g*(2*c^2*d + b^2 
*f - c*(2*a*f)) - h*(b*c*d + a*b*f))*x), 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*S 
imp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d)*((-b)*f))*(p + 1) + (b^2*(g*f) - b 
*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f* 
((g*c)*((-b)*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(2*a*f)))*(p + 
 q + 2) - (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(b*f*(p + 1 
)))*x - c*f*(b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(2*p + 2* 
q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, 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])
 

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. \(898\) vs. \(2(242)=484\).

Time = 2.44 (sec) , antiderivative size = 899, normalized size of antiderivative = 3.01

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 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 f}\)	\(899\)

Input:

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

Output:

-1/2/f*(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/f*(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. 17258 vs. \(2 (241) = 482\).

Time = 13.99 (sec) , antiderivative size = 17258, normalized size of antiderivative = 57.72 \[ \int \frac {x}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.99 (sec) , antiderivative size = 25257, normalized size of antiderivative = 84.47 \[ \int \frac {x}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 8*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)* 
sqrt(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 
*sqrt(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 
**3*b*c*f**2 + 2*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 + s 
qrt(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*sqrt(d)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d...