\(\int \frac {x^2}{(d+e x) \sqrt {a+b x+c x^2}} \, dx\) [32]

Optimal result

Integrand size = 25, antiderivative size = 157 \[ \int \frac {x^2}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {a+b x+c x^2}}{c e}-\frac {(2 c d+b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{3/2} e^2}+\frac {d^2 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^2 \sqrt {c d^2-b d e+a e^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.44 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.94 \[ \int \frac {x^2}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=-\frac {-\frac {2 e \sqrt {a+x (b+c x)}}{c}+\frac {4 d^2 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{\sqrt {-c d^2+e (b d-a e)}}+\frac {(2 c d+b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{c^{3/2}}}{2 e^2} \] Input:

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

Output:

-1/2*((-2*e*Sqrt[a + x*(b + c*x)])/c + (4*d^2*ArcTan[(Sqrt[c]*(d + e*x) - 
e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]])/Sqrt[-(c*d^2) + 
e*(b*d - a*e)] + ((2*c*d + b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x* 
(b + c*x)])])/c^(3/2))/e^2
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1267, 27, 1269, 1092, 219, 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.

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

\(\Big \downarrow \) 1267

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

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 1267

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b 
*x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m 
 + n + 2*p + 1))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m 
 + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d 
+ e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) 
 - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
, m, p}, x] && IGtQ[n, 1] && IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [A] (verified)

Time = 1.43 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.46

method	result	size
risch	\(\frac {\sqrt {c \,x^{2}+b x +a}}{c e}-\frac {\frac {\left (b e +2 c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}+\frac {2 c \,d^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{2 c e}\)	\(230\)
default	\(\frac {\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}}{e}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{3} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {d \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e^{2} \sqrt {c}}\)	\(247\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (137) = 274\).

Time = 11.80 (sec) , antiderivative size = 1306, normalized size of antiderivative = 8.32 \[ \int \frac {x^2}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(2*sqrt(c*d^2 - b*d*e + a*e^2)*c^2*d^2*log((8*a*b*d*e - 8*a^2*e^2 - ( 
b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqr 
t(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e 
)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e 
*x + d^2)) + (2*c^2*d^3 - b*c*d^2*e + a*b*e^3 - (b^2 - 2*a*c)*d*e^2)*sqrt( 
c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sq 
rt(c) - 4*a*c) + 4*(c^2*d^2*e - b*c*d*e^2 + a*c*e^3)*sqrt(c*x^2 + b*x + a) 
)/(c^3*d^2*e^2 - b*c^2*d*e^3 + a*c^2*e^4), 1/4*(4*sqrt(-c*d^2 + b*d*e - a* 
e^2)*c^2*d^2*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a 
)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 
- b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)) + (2*c^2*d^3 
- b*c*d^2*e + a*b*e^3 - (b^2 - 2*a*c)*d*e^2)*sqrt(c)*log(-8*c^2*x^2 - 8*b* 
c*x - b^2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*(c^2* 
d^2*e - b*c*d*e^2 + a*c*e^3)*sqrt(c*x^2 + b*x + a))/(c^3*d^2*e^2 - b*c^2*d 
*e^3 + a*c^2*e^4), 1/2*(sqrt(c*d^2 - b*d*e + a*e^2)*c^2*d^2*log((8*a*b*d*e 
 - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)* 
e^2)*x^2 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a* 
e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/ 
(e^2*x^2 + 2*d*e*x + d^2)) + (2*c^2*d^3 - b*c*d^2*e + a*b*e^3 - (b^2 - 2*a 
*c)*d*e^2)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-...
 

Sympy [F]

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

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

Output:

Integral(x**2/((d + e*x)*sqrt(a + b*x + c*x**2)), x)
 

Maxima [F(-2)]

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

integrate(x^2/(e*x+d)/(c*x^2+b*x+a)^(1/2),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((b/e-(2*c*d)/e^2)^2>0)', see `as 
sume?` for
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^2}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 4437, normalized size of antiderivative = 28.26 \[ \int \frac {x^2}{(d+e x) \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a* 
e**2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d 
**2)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((2*sqrt(c)*sqrt(a + b*x + c*x**2)* 
e + b*e + 2*c*e*x)/sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sq 
rt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d 
*e - 8*c**2*d**2))*b*c**2*d**2*e + 4*sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + 
c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b 
**2*e**2 + 8*b*c*d*e - 8*c**2*d**2)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((2* 
sqrt(c)*sqrt(a + b*x + c*x**2)*e + b*e + 2*c*e*x)/sqrt(4*sqrt(c)*sqrt(a*e* 
*2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*c*d - 4 
*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2))*c**3*d**3 - 4*sqrt(c)*sq 
rt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b 
*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2)*ata 
n((2*sqrt(c)*sqrt(a + b*x + c*x**2)*e + b*e + 2*c*e*x)/sqrt(4*sqrt(c)*sqrt 
(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*c* 
d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2))*a*c**2*d**2*e**2 + 
4*sqrt(c)*sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqr 
t(a*e**2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c* 
*2*d**2)*atan((2*sqrt(c)*sqrt(a + b*x + c*x**2)*e + b*e + 2*c*e*x)/sqrt(4* 
sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b*d...