\(\int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^3} \, dx\) [51]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^3} \, dx=\frac {\sqrt {a+x (b+c x)} \left (b x \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-\sqrt {a} \left (\sqrt {a+x (b+c x)}+\sqrt {c} x \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )\right )}{\sqrt {a} \sqrt {x^2 (a+x (b+c x))}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.82, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1967, 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.

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

\(\Big \downarrow \) 1967

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

\(\Big \downarrow \) 1980

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-(Sqrt[a*x^2 + b*x^3 + c*x^4]/x^2) + (x*Sqrt[a + b*x + c*x^2]*(-((b*ArcTan 
h[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/Sqrt[a]) + 2*Sqrt[c]*Arc 
Tanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]))/(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 1967

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*q 
+ 1)), x] - Simp[(n - q)*(p/(m + p*q + 1))   Int[x^(m + n)*(b + 2*c*x^(n - 
q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{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] && LeQ[m + p*q + 1, -(n - q) + 
 1] && NeQ[m + p*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 = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.58

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {c}\, \ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right ) x \sqrt {a}-b x \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right )+b x \ln \left (2\right )-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{2 x \sqrt {a}}\)	\(101\)
risch	\(-\frac {\sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{x^{2}}+\frac {\left (-\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 \sqrt {a}}+\sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{x \sqrt {c \,x^{2}+b x +a}}\)	\(120\)
default	\(\frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (2 x^{2} \sqrt {c \,x^{2}+b x +a}\, c^{\frac {5}{2}}-c^{\frac {3}{2}} \sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b x -2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {3}{2}}+2 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, b x +2 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a \,c^{2} x \right )}{2 x^{2} \sqrt {c \,x^{2}+b x +a}\, a \,c^{\frac {3}{2}}}\)	\(174\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 653, normalized size of antiderivative = 3.77 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^3} \, dx=\left [\frac {2 \, a \sqrt {c} x^{2} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + \sqrt {a} b x^{2} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{4 \, a x^{2}}, -\frac {4 \, a \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - \sqrt {a} b x^{2} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{4 \, a x^{2}}, \frac {\sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + a \sqrt {c} x^{2} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{2 \, a x^{2}}, \frac {\sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) - 2 \, a \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{2 \, a x^{2}}\right ] \] Input:

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

Output:

[1/4*(2*a*sqrt(c)*x^2*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) + sqrt(a)*b*x^2*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)*a)/(a*x^2), -1/4*(4* 
a*sqrt(-c)*x^2*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)*b*x^2*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)*a)/(a*x^2), 1/2*(sqrt(-a)*b*x^2*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)) + a*sqrt(c)*x^2*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(c*x^4 + b*x^3 
+ a*x^2)*a)/(a*x^2), 1/2*(sqrt(-a)*b*x^2*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)) - 2*a*sqrt(-c)*x^2 
*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*sqrt(c*x^4 + b*x^3 + a*x^2)*a)/(a*x^2)]
 

Sympy [F]

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

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

Output:

Integral(sqrt(x**2*(a + b*x + c*x**2))/x**3, x)
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable 
to make series expansion Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^3} \, dx=\frac {-2 \sqrt {c \,x^{2}+b x +a}\, a +\sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b x -\sqrt {a}\, \mathrm {log}\left (x \right ) b x +2 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) a x}{2 a x} \] Input:

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

Output:

( - 2*sqrt(a + b*x + c*x**2)*a + sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x* 
*2) - 2*a - b*x)*b*x - sqrt(a)*log(x)*b*x + 2*sqrt(c)*log( - 2*sqrt(c)*sqr 
t(a + b*x + c*x**2) - b - 2*c*x)*a*x)/(2*a*x)