\(\int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx\) [176]

Optimal result

Integrand size = 20, antiderivative size = 112 \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx=-\frac {\sqrt {a+b x^3+c x^6}}{3 x^3}-\frac {b \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{6 \sqrt {a}}+\frac {1}{3} \sqrt {c} \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right ) \] Output:

-1/3*(c*x^6+b*x^3+a)^(1/2)/x^3-1/6*b*arctanh(1/2*(b*x^3+2*a)/a^(1/2)/(c*x^ 
6+b*x^3+a)^(1/2))/a^(1/2)+1/3*c^(1/2)*arctanh(1/2*(2*c*x^3+b)/c^(1/2)/(c*x 
^6+b*x^3+a)^(1/2))
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx=\frac {1}{3} \left (-\frac {\sqrt {a+b x^3+c x^6}}{x^3}+\frac {b \text {arctanh}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{\sqrt {a}}-\sqrt {c} \log \left (b+2 c x^3-2 \sqrt {c} \sqrt {a+b x^3+c x^6}\right )\right ) \] Input:

Integrate[Sqrt[a + b*x^3 + c*x^6]/x^4,x]
 

Output:

(-(Sqrt[a + b*x^3 + c*x^6]/x^3) + (b*ArcTanh[(Sqrt[c]*x^3 - Sqrt[a + b*x^3 
 + c*x^6])/Sqrt[a]])/Sqrt[a] - Sqrt[c]*Log[b + 2*c*x^3 - 2*Sqrt[c]*Sqrt[a 
+ b*x^3 + c*x^6]])/3
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1693, 1161, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx\)

\(\Big \downarrow \) 1693

\(\displaystyle \frac {1}{3} \int \frac {\sqrt {c x^6+b x^3+a}}{x^6}dx^3\)

\(\Big \downarrow \) 1161

\(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \frac {2 c x^3+b}{x^3 \sqrt {c x^6+b x^3+a}}dx^3-\frac {\sqrt {a+b x^3+c x^6}}{x^3}\right )\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (2 c \int \frac {1}{\sqrt {c x^6+b x^3+a}}dx^3+b \int \frac {1}{x^3 \sqrt {c x^6+b x^3+a}}dx^3\right )-\frac {\sqrt {a+b x^3+c x^6}}{x^3}\right )\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (4 c \int \frac {1}{4 c-x^6}d\frac {2 c x^3+b}{\sqrt {c x^6+b x^3+a}}+b \int \frac {1}{x^3 \sqrt {c x^6+b x^3+a}}dx^3\right )-\frac {\sqrt {a+b x^3+c x^6}}{x^3}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (b \int \frac {1}{x^3 \sqrt {c x^6+b x^3+a}}dx^3+2 \sqrt {c} \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )\right )-\frac {\sqrt {a+b x^3+c x^6}}{x^3}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (2 \sqrt {c} \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )-2 b \int \frac {1}{4 a-x^6}d\frac {b x^3+2 a}{\sqrt {c x^6+b x^3+a}}\right )-\frac {\sqrt {a+b x^3+c x^6}}{x^3}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (2 \sqrt {c} \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )-\frac {b \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{\sqrt {a}}\right )-\frac {\sqrt {a+b x^3+c x^6}}{x^3}\right )\)

Input:

Int[Sqrt[a + b*x^3 + c*x^6]/x^4,x]
 

Output:

(-(Sqrt[a + b*x^3 + c*x^6]/x^3) + (-((b*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*S 
qrt[a + b*x^3 + c*x^6])])/Sqrt[a]) + 2*Sqrt[c]*ArcTanh[(b + 2*c*x^3)/(2*Sq 
rt[c]*Sqrt[a + b*x^3 + c*x^6])])/2)/3
 

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 1161

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si 
mp[p/(e*(m + 1))   Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 
 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || 
 LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, 
 c, d, e, m, p, 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 1693

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, 
x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ 
[Simplify[(m + 1)/n]]
 

Maple [F]

\[\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{x^{4}}d x\]

Input:

int((c*x^6+b*x^3+a)^(1/2)/x^4,x)
 

Output:

int((c*x^6+b*x^3+a)^(1/2)/x^4,x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 601, normalized size of antiderivative = 5.37 \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx=\left [\frac {2 \, a \sqrt {c} x^{3} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) + \sqrt {a} b x^{3} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 4 \, \sqrt {c x^{6} + b x^{3} + a} a}{12 \, a x^{3}}, -\frac {4 \, a \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) - \sqrt {a} b x^{3} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \, \sqrt {c x^{6} + b x^{3} + a} a}{12 \, a x^{3}}, \frac {\sqrt {-a} b x^{3} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + a \sqrt {c} x^{3} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) - 2 \, \sqrt {c x^{6} + b x^{3} + a} a}{6 \, a x^{3}}, \frac {\sqrt {-a} b x^{3} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) - 2 \, a \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) - 2 \, \sqrt {c x^{6} + b x^{3} + a} a}{6 \, a x^{3}}\right ] \] Input:

integrate((c*x^6+b*x^3+a)^(1/2)/x^4,x, algorithm="fricas")
 

Output:

[1/12*(2*a*sqrt(c)*x^3*log(-8*c^2*x^6 - 8*b*c*x^3 - b^2 - 4*sqrt(c*x^6 + b 
*x^3 + a)*(2*c*x^3 + b)*sqrt(c) - 4*a*c) + sqrt(a)*b*x^3*log(-((b^2 + 4*a* 
c)*x^6 + 8*a*b*x^3 - 4*sqrt(c*x^6 + b*x^3 + a)*(b*x^3 + 2*a)*sqrt(a) + 8*a 
^2)/x^6) - 4*sqrt(c*x^6 + b*x^3 + a)*a)/(a*x^3), -1/12*(4*a*sqrt(-c)*x^3*a 
rctan(1/2*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)*sqrt(-c)/(c^2*x^6 + b*c*x^ 
3 + a*c)) - sqrt(a)*b*x^3*log(-((b^2 + 4*a*c)*x^6 + 8*a*b*x^3 - 4*sqrt(c*x 
^6 + b*x^3 + a)*(b*x^3 + 2*a)*sqrt(a) + 8*a^2)/x^6) + 4*sqrt(c*x^6 + b*x^3 
 + a)*a)/(a*x^3), 1/6*(sqrt(-a)*b*x^3*arctan(1/2*sqrt(c*x^6 + b*x^3 + a)*( 
b*x^3 + 2*a)*sqrt(-a)/(a*c*x^6 + a*b*x^3 + a^2)) + a*sqrt(c)*x^3*log(-8*c^ 
2*x^6 - 8*b*c*x^3 - b^2 - 4*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)*sqrt(c) 
- 4*a*c) - 2*sqrt(c*x^6 + b*x^3 + a)*a)/(a*x^3), 1/6*(sqrt(-a)*b*x^3*arcta 
n(1/2*sqrt(c*x^6 + b*x^3 + a)*(b*x^3 + 2*a)*sqrt(-a)/(a*c*x^6 + a*b*x^3 + 
a^2)) - 2*a*sqrt(-c)*x^3*arctan(1/2*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)* 
sqrt(-c)/(c^2*x^6 + b*c*x^3 + a*c)) - 2*sqrt(c*x^6 + b*x^3 + a)*a)/(a*x^3) 
]
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx=\int \frac {\sqrt {a + b x^{3} + c x^{6}}}{x^{4}}\, dx \] Input:

integrate((c*x**6+b*x**3+a)**(1/2)/x**4,x)
 

Output:

Integral(sqrt(a + b*x**3 + c*x**6)/x**4, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^6+b*x^3+a)^(1/2)/x^4,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*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [F]

\[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx=\int { \frac {\sqrt {c x^{6} + b x^{3} + a}}{x^{4}} \,d x } \] Input:

integrate((c*x^6+b*x^3+a)^(1/2)/x^4,x, algorithm="giac")
 

Output:

integrate(sqrt(c*x^6 + b*x^3 + a)/x^4, x)
 

Mupad [B] (verification not implemented)

Time = 20.30 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx=\frac {\sqrt {c}\,\ln \left (\sqrt {c\,x^6+b\,x^3+a}+\frac {c\,x^3+\frac {b}{2}}{\sqrt {c}}\right )}{3}-\frac {\sqrt {c\,x^6+b\,x^3+a}}{3\,x^3}-\frac {b\,\ln \left (\frac {b}{2}+\frac {a}{x^3}+\frac {\sqrt {a}\,\sqrt {c\,x^6+b\,x^3+a}}{x^3}\right )}{6\,\sqrt {a}} \] Input:

int((a + b*x^3 + c*x^6)^(1/2)/x^4,x)
 

Output:

(c^(1/2)*log((a + b*x^3 + c*x^6)^(1/2) + (b/2 + c*x^3)/c^(1/2)))/3 - (a + 
b*x^3 + c*x^6)^(1/2)/(3*x^3) - (b*log(b/2 + a/x^3 + (a^(1/2)*(a + b*x^3 + 
c*x^6)^(1/2))/x^3))/(6*a^(1/2))
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^4} \, dx =\text {Too large to display} \] Input:

int((c*x^6+b*x^3+a)^(1/2)/x^4,x)
 

Output:

( - 2*sqrt(a + b*x**3 + c*x**6)*a + sqrt(a)*log(sqrt(a + b*x**3 + c*x**6) 
- sqrt(a))*b*x**3 - sqrt(a)*log(sqrt(a + b*x**3 + c*x**6) + sqrt(a))*b*x** 
3 - 2*sqrt(c)*log(sqrt(a + b*x**3 + c*x**6) - sqrt(c)*x**3)*a*x**3 + 2*sqr 
t(c)*log(sqrt(a + b*x**3 + c*x**6) + sqrt(c)*x**3)*a*x**3 - 18*int((sqrt(a 
 + b*x**3 + c*x**6)*x**5)/(3*a**3*b*c + 3*a**3*c**2*x**3 + a**2*b**3 + 7*a 
**2*b**2*c*x**3 + 9*a**2*b*c**2*x**6 + 3*a**2*c**3*x**9 + 2*a*b**4*x**3 + 
6*a*b**3*c*x**6 + 7*a*b**2*c**2*x**9 + 3*a*b*c**3*x**12 + b**5*x**6 + 2*b* 
*4*c*x**9 + b**3*c**2*x**12),x)*a**3*c**3*x**3 - 15*int((sqrt(a + b*x**3 + 
 c*x**6)*x**5)/(3*a**3*b*c + 3*a**3*c**2*x**3 + a**2*b**3 + 7*a**2*b**2*c* 
x**3 + 9*a**2*b*c**2*x**6 + 3*a**2*c**3*x**9 + 2*a*b**4*x**3 + 6*a*b**3*c* 
x**6 + 7*a*b**2*c**2*x**9 + 3*a*b*c**3*x**12 + b**5*x**6 + 2*b**4*c*x**9 + 
 b**3*c**2*x**12),x)*a**2*b**2*c**2*x**3 - 3*int((sqrt(a + b*x**3 + c*x**6 
)*x**5)/(3*a**3*b*c + 3*a**3*c**2*x**3 + a**2*b**3 + 7*a**2*b**2*c*x**3 + 
9*a**2*b*c**2*x**6 + 3*a**2*c**3*x**9 + 2*a*b**4*x**3 + 6*a*b**3*c*x**6 + 
7*a*b**2*c**2*x**9 + 3*a*b*c**3*x**12 + b**5*x**6 + 2*b**4*c*x**9 + b**3*c 
**2*x**12),x)*a*b**4*c*x**3 - 27*int((sqrt(a + b*x**3 + c*x**6)*x**2)/(3*a 
**3*b*c + 3*a**3*c**2*x**3 + a**2*b**3 + 7*a**2*b**2*c*x**3 + 9*a**2*b*c** 
2*x**6 + 3*a**2*c**3*x**9 + 2*a*b**4*x**3 + 6*a*b**3*c*x**6 + 7*a*b**2*c** 
2*x**9 + 3*a*b*c**3*x**12 + b**5*x**6 + 2*b**4*c*x**9 + b**3*c**2*x**12),x 
)*a**3*b*c**2*x**3 - 9*int((sqrt(a + b*x**3 + c*x**6)*x**2)/(3*a**3*b*c...