\(\int \frac {x^3}{(a+c x^2)^{3/2} (d+e x+f x^2)} \, dx\) [38]

Optimal result

Integrand size = 27, antiderivative size = 430 \[ \int \frac {x^3}{\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\frac {a (c d-a f-c e x)}{c \left (a c e^2+(c d-a f)^2\right ) \sqrt {a+c x^2}}-\frac {\left (2 a d e f-\left (e-\sqrt {e^2-4 d f}\right ) \left (c d^2+a \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right ) \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {\left (2 a d e f-\left (e+\sqrt {e^2-4 d f}\right ) \left (c d^2+a \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right ) \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}} \] Output:

a*(-c*e*x-a*f+c*d)/c/(a*c*e^2+(-a*f+c*d)^2)/(c*x^2+a)^(1/2)-1/2*(2*a*d*e*f 
-(e-(-4*d*f+e^2)^(1/2))*(c*d^2+a*(-d*f+e^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*(2*a*d*e*f-(e+(-4*d*f+ 
e^2)^(1/2))*(c*d^2+a*(-d*f+e^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 = 0.81 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.93 \[ \int \frac {x^3}{\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\frac {-a (-c d+a f+c e x)-\sqrt {a} c \sqrt {a+c x^2} \text {RootSum}\left [c^2 d+2 \sqrt {a} c e \text {$\#$1}-2 c d \text {$\#$1}^2+4 a f \text {$\#$1}^2-2 \sqrt {a} e \text {$\#$1}^3+d \text {$\#$1}^4\&,\frac {\sqrt {a} c d e \log (x)-\sqrt {a} c d e \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )+2 c d^2 \log (x) \text {$\#$1}+2 a e^2 \log (x) \text {$\#$1}-2 a d f \log (x) \text {$\#$1}-2 c d^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}-2 a e^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+2 a d f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}-\sqrt {a} d e \log (x) \text {$\#$1}^2+\sqrt {a} d e \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{\sqrt {a} c e-2 c d \text {$\#$1}+4 a f \text {$\#$1}-3 \sqrt {a} e \text {$\#$1}^2+2 d \text {$\#$1}^3}\&\right ]}{c \left (c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )\right ) \sqrt {a+c x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.85 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {7279, 2009}

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.

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

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-(1/(c*f*Sqrt[a + c*x^2])) - (e*x)/(a*f^2*Sqrt[a + c*x^2]) + (a*f*(c*d^2 + 
 a*(e^2 - d*f)) + c*e*(c*d^2 + a*(e^2 - 2*d*f))*x)/(a*f^2*(a*c*e^2 + (c*d 
- a*f)^2)*Sqrt[a + c*x^2]) - ((2*a*d*e*f - (e - Sqrt[e^2 - 4*d*f])*(c*d^2 
+ a*(e^2 - d*f)))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*S 
qrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(S 
qrt[2]*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 
 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((2*a*d*e*f - (e + Sqrt[e^2 - 4*d*f])*(c 
*d^2 + a*(e^2 - d*f)))*ArcTanh[(2*a*f - c*(e + Sqrt[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]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*( 
e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1575\) vs. \(2(391)=782\).

Time = 2.05 (sec) , antiderivative size = 1576, normalized size of antiderivative = 3.67

method	result	size
default	\(\text {Expression too large to display}\)	\(1576\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27621 vs. \(2 (389) = 778\).

Time = 97.77 (sec) , antiderivative size = 27621, normalized size of antiderivative = 64.23 \[ \int \frac {x^3}{\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - sqrt(a + c*x**2)*a*e - sqrt(a + c*x**2)*c*d*x + int(sqrt(a + c*x**2)/( 
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)*a**2*c*d**2 + int(sqrt(a 
+ c*x**2)/(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)*a*c**2*d**2*x* 
*2 + int((sqrt(a + c*x**2)*x**2)/(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)*a**2*c*d*f - int((sqrt(a + c*x**2)*x**2)/(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)*a**2*c*e**2 + int((sqrt(a + c*x**2)*x**2)/(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)*a*c**2*d*f*x**2 - int((sqrt(a 
+ c*x**2)*x**2)/(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)*a*c**2*e 
**2*x**2)/(a*c*e*f*(a + c*x**2))