\(\int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^2} \, dx\) [50]

Optimal result

Integrand size = 24, antiderivative size = 173 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^2} \, dx=\frac {\sqrt {a x^2+b x^3+c x^4}}{x}-\frac {\sqrt {a} x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a x^2+b x^3+c x^4}}+\frac {b x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} \sqrt {a x^2+b x^3+c x^4}} \] Output:

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

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^2} \, dx=\frac {x \sqrt {a+x (b+c x)} \left (2 \sqrt {c} \sqrt {a+x (b+c x)}+4 \sqrt {a} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-b \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{2 \sqrt {c} \sqrt {x^2 (a+x (b+c x))}} \] Input:

Integrate[Sqrt[a*x^2 + b*x^3 + c*x^4]/x^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.81, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1968, 1980, 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.

\(\displaystyle \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^2} \, dx\)

\(\Big \downarrow \) 1968

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

\(\Big \downarrow \) 1980

\(\displaystyle \frac {x \sqrt {a+b x+c x^2} \int \frac {2 a+b x}{x \sqrt {c x^2+b x+a}}dx}{2 \sqrt {a x^2+b x^3+c x^4}}+\frac {\sqrt {a x^2+b x^3+c x^4}}{x}\)

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

Int[Sqrt[a*x^2 + b*x^3 + c*x^4]/x^2,x]
 

Output:

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

Defintions of rubi rules used

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 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]
 

rule 1968

Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_ 
), x_Symbol] :> Simp[x^(m + 1)*((a*x^q + b*x^n + c*x^(2*n - q))^p/(m + p*(2 
*n - q) + 1)), x] + Simp[(n - q)*(p/(m + p*(2*n - q) + 1))   Int[x^(m + q)* 
(2*a + b*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; Free 
Q[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] &&  !IntegerQ[p] && NeQ[b 
^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && RationalQ[m, q] && GtQ[m + p*q 
 + 1, -(n - q)] && NeQ[m + p*(2*n - q) + 1, 0]
 

rule 1980

Int[((A_) + (B_.)*(x_)^(j_.))/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c 
_.)*(x_)^(r_.)], x_Symbol] :> Simp[x^(q/2)*(Sqrt[a + b*x^(n - q) + c*x^(2*( 
n - q))]/Sqrt[a*x^q + b*x^n + c*x^(2*n - q)])   Int[(A + B*x^(n - q))/(x^(q 
/2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]), x], x] /; FreeQ[{a, b, c, A, 
B, n, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q] && PosQ[n - q] && EqQ[n, 3 
] && EqQ[q, 2]
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.53

method	result	size
pseudoelliptic	\(\frac {\left (\left (-\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right )+\ln \left (2\right )\right ) \sqrt {a}+\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {c}+\frac {\ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right ) b}{2}}{\sqrt {c}}\)	\(91\)
default	\(-\frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (2 \sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c}-b \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right )-2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}\right )}{2 x \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}\)	\(126\)

Input:

int((c*x^4+b*x^3+a*x^2)^(1/2)/x^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 638, normalized size of antiderivative = 3.69 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^2} \, dx=\left [\frac {b \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 2 \, \sqrt {a} c x \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} c}{4 \, c x}, -\frac {b \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - \sqrt {a} c x \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) - 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} c}{2 \, c x}, \frac {4 \, \sqrt {-a} c x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + b \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} c}{4 \, c x}, \frac {2 \, \sqrt {-a} c x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) - b \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} c}{2 \, c x}\right ] \] Input:

integrate((c*x^4+b*x^3+a*x^2)^(1/2)/x^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((c*x**4+b*x**3+a*x**2)**(1/2)/x**2,x)
 

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^2} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{3} + a x^{2}}}{x^{2}} \,d x } \] Input:

integrate((c*x^4+b*x^3+a*x^2)^(1/2)/x^2,x, algorithm="maxima")
 

Output:

integrate(sqrt(c*x^4 + b*x^3 + a*x^2)/x^2, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c*x^4+b*x^3+a*x^2)^(1/2)/x^2,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Degree mismatch inside factorisatio 
n over extensionNot implemented, e.g. for multivariate mod/approx polynomi 
alsError:
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^2} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^3+a\,x^2}}{x^2} \,d x \] Input:

int((a*x^2 + b*x^3 + c*x^4)^(1/2)/x^2,x)
 

Output:

int((a*x^2 + b*x^3 + c*x^4)^(1/2)/x^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 828, normalized size of antiderivative = 4.79 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^2} \, dx =\text {Too large to display} \] Input:

int((c*x^4+b*x^3+a*x^2)^(1/2)/x^2,x)
 

Output:

( - 2*sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)*atan((2*sqrt(c)*sqr 
t(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)) 
*b*c - 4*sqrt(c)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)*atan((2*sqrt(c)* 
sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b** 
2))*a*c - sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2)*log( - sqrt(4*s 
qrt(c)*sqrt(a)*b + 4*a*c + b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 
2*c*x)*b*c + sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2)*log(sqrt(4*s 
qrt(c)*sqrt(a)*b + 4*a*c + b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 
2*c*x)*b*c + 2*sqrt(c)*sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2)*log( - sqr 
t(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 
 b + 2*c*x)*a*c - 2*sqrt(c)*sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2)*log(s 
qrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) 
 + b + 2*c*x)*a*c + 8*sqrt(a + b*x + c*x**2)*a*c**2 - 2*sqrt(a + b*x + c*x 
**2)*b**2*c + 4*sqrt(a)*log( - sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2) + 
2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)*a*c**2 - sqrt(a)*log( - sqrt 
(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 
b + 2*c*x)*b**2*c + 4*sqrt(a)*log(sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2) 
 + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)*a*c**2 - sqrt(a)*log(sqrt 
(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 
b + 2*c*x)*b**2*c - 4*sqrt(a)*log(4*sqrt(c)*sqrt(a + b*x + c*x**2)*b + ...