\(\int \frac {x}{(a+c x^2)^{3/2} (d+e x+f x^2)} \, dx\) [40]

Optimal result

Integrand size = 25, antiderivative size = 411 \[ \int \frac {x}{\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=-\frac {c d-a f-c e x}{\left (a c e^2+(c d-a f)^2\right ) \sqrt {a+c x^2}}+\frac {f \left (2 c d e-(c d-a f) \left (e-\sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right ) \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}-\frac {f \left (2 c d e-(c d-a f) \left (e+\sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right ) \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.66 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.80 \[ \int \frac {x}{\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\frac {-c d+a f+c e x-\sqrt {a+c x^2} \text {RootSum}\left [a^2 f+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {-a c d f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+a^2 f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )-2 c^{3/2} d e \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c d f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{a \sqrt {c} e+4 c d \text {$\#$1}-2 a f \text {$\#$1}-3 \sqrt {c} e \text {$\#$1}^2+2 f \text {$\#$1}^3}\&\right ]}{\left (c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )\right ) \sqrt {a+c x^2}} \] Input:

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

Output:

(-(c*d) + a*f + c*e*x - Sqrt[a + c*x^2]*RootSum[a^2*f + 2*a*Sqrt[c]*e*#1 + 
 4*c*d*#1^2 - 2*a*f*#1^2 - 2*Sqrt[c]*e*#1^3 + f*#1^4 & , (-(a*c*d*f*Log[-( 
Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]) + a^2*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + 
c*x^2] - #1] - 2*c^(3/2)*d*e*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 + 
 c*d*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2 - a*f^2*Log[-(Sqrt[c] 
*x) + Sqrt[a + c*x^2] - #1]*#1^2)/(a*Sqrt[c]*e + 4*c*d*#1 - 2*a*f*#1 - 3*S 
qrt[c]*e*#1^2 + 2*f*#1^3) & ])/((c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))*Sq 
rt[a + c*x^2])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 395, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1350, 27, 1367, 488, 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.

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

\(\Big \downarrow \) 1350

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1367

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-((c*d - a*f - c*e*x)/((a*c*e^2 + (c*d - a*f)^2)*Sqrt[a + c*x^2])) - (-((f 
*(2*c*d*e - (c*d - a*f)*(e - Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - S 
qrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 
 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*( 
e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])])) + (f*(2*c*d*e - (c*d - a*f)*(e + Sqr 
t[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sq 
rt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sq 
rt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f 
])]))/(a*c*e^2 + (c*d - a*f)^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 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 488

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

rule 1350

Int[((g_.) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f 
_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(a + c*x^2)^(p + 1)*((d + e*x + f*x^2)^ 
(q + 1)/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)))*(g*c*(2*a*c*e) + ((-a 
)*h)*(2*c^2*d - c*(2*a*f)) + c*(g*(2*c^2*d - c*(2*a*f)) - h*(-2*a*c*e))*x), 
 x] + Simp[1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1))   Int[(a + c*x^2) 
^(p + 1)*(d + e*x + f*x^2)^q*Simp[(-2*g*c)*((c*d - a*f)^2 - ((-a)*e)*(c*e)) 
*(p + 1) + (2*(g*c*(c*d - a*f) - a*((-h)*c*e)))*(a*f*(p + 1) - c*d*(p + 2)) 
 - e*((g*c)*(2*a*c*e) + ((-a)*h)*(2*c^2*d - c*((Plus[2])*a*f)))*(p + q + 2) 
 - (2*f*((g*c)*(2*a*c*e) + ((-a)*h)*(2*c^2*d + (-c)*((Plus[2])*a*f)))*(p + 
q + 2) - (2*(g*c*(c*d - a*f) - a*((-h)*c*e)))*((-c)*e*(2*p + q + 4)))*x - c 
*f*(2*(g*c*(c*d - a*f) - a*((-h)*c*e)))*(2*p + 2*q + 5)*x^2, x], x], x] /; 
FreeQ[{a, c, d, e, f, g, h, q}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && 
NeQ[a*c*e^2 + (c*d - a*f)^2, 0] &&  !( !IntegerQ[p] && ILtQ[q, -1])
 

rule 1367

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1489\) vs. \(2(372)=744\).

Time = 2.18 (sec) , antiderivative size = 1490, normalized size of antiderivative = 3.63

method	result	size
default	\(\text {Expression too large to display}\)	\(1490\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26234 vs. \(2 (368) = 736\).

Time = 95.20 (sec) , antiderivative size = 26234, normalized size of antiderivative = 63.83 \[ \int \frac {x}{\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x/(c*x^2+a)^(3/2)/(f*x^2+e*x+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(4*d*f-e^2>0)', see `assume?` for 
 more deta
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x}{\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\int \frac {\sqrt {c \,x^{2}+a}\, x}{c^{2} f \,x^{6}+c^{2} e \,x^{5}+2 a c f \,x^{4}+c^{2} d \,x^{4}+2 a c e \,x^{3}+a^{2} f \,x^{2}+2 a c d \,x^{2}+a^{2} e x +a^{2} d}d x \] Input:

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

Output:

int((sqrt(a + c*x**2)*x)/(a**2*d + a**2*e*x + a**2*f*x**2 + 2*a*c*d*x**2 + 
 2*a*c*e*x**3 + 2*a*c*f*x**4 + c**2*d*x**4 + c**2*e*x**5 + c**2*f*x**6),x)