\(\int \frac {\sqrt {a+b x^3+c x^6}}{x^{16}} \, dx\) [180]

Optimal result

Integrand size = 20, antiderivative size = 199 \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^{16}} \, dx=\frac {b \left (7 b^2-12 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{384 a^4 x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{15 a x^{15}}+\frac {7 b \left (a+b x^3+c x^6\right )^{3/2}}{120 a^2 x^{12}}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^3+c x^6\right )^{3/2}}{720 a^3 x^9}-\frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{768 a^{9/2}} \] Output:

1/384*b*(-12*a*c+7*b^2)*(b*x^3+2*a)*(c*x^6+b*x^3+a)^(1/2)/a^4/x^6-1/15*(c* 
x^6+b*x^3+a)^(3/2)/a/x^15+7/120*b*(c*x^6+b*x^3+a)^(3/2)/a^2/x^12-1/720*(-3 
2*a*c+35*b^2)*(c*x^6+b*x^3+a)^(3/2)/a^3/x^9-1/768*b*(-12*a*c+7*b^2)*(-4*a* 
c+b^2)*arctanh(1/2*(b*x^3+2*a)/a^(1/2)/(c*x^6+b*x^3+a)^(1/2))/a^(9/2)
 

Mathematica [A] (verified)

Time = 1.08 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^{16}} \, dx=\frac {\sqrt {a+b x^3+c x^6} \left (-384 a^4-48 a^3 b x^3+56 a^2 b^2 x^6-128 a^3 c x^6-70 a b^3 x^9+232 a^2 b c x^9+105 b^4 x^{12}-460 a b^2 c x^{12}+256 a^2 c^2 x^{12}\right )}{5760 a^4 x^{15}}+\frac {\left (7 b^5-40 a b^3 c+48 a^2 b c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{384 a^{9/2}} \] Input:

Integrate[Sqrt[a + b*x^3 + c*x^6]/x^16,x]
 

Output:

(Sqrt[a + b*x^3 + c*x^6]*(-384*a^4 - 48*a^3*b*x^3 + 56*a^2*b^2*x^6 - 128*a 
^3*c*x^6 - 70*a*b^3*x^9 + 232*a^2*b*c*x^9 + 105*b^4*x^12 - 460*a*b^2*c*x^1 
2 + 256*a^2*c^2*x^12))/(5760*a^4*x^15) + ((7*b^5 - 40*a*b^3*c + 48*a^2*b*c 
^2)*ArcTanh[(Sqrt[c]*x^3 - Sqrt[a + b*x^3 + c*x^6])/Sqrt[a]])/(384*a^(9/2) 
)
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {1693, 1167, 27, 1237, 27, 1228, 1152, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^3+c x^6}}{x^{16}} \, dx\)

\(\Big \downarrow \) 1693

\(\displaystyle \frac {1}{3} \int \frac {\sqrt {c x^6+b x^3+a}}{x^{18}}dx^3\)

\(\Big \downarrow \) 1167

\(\displaystyle \frac {1}{3} \left (-\frac {\int \frac {\left (4 c x^3+7 b\right ) \sqrt {c x^6+b x^3+a}}{2 x^{15}}dx^3}{5 a}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{5 a x^{15}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (-\frac {\int \frac {\left (4 c x^3+7 b\right ) \sqrt {c x^6+b x^3+a}}{x^{15}}dx^3}{10 a}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{5 a x^{15}}\right )\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{3} \left (-\frac {-\frac {\int \frac {\left (14 b c x^3+35 b^2-32 a c\right ) \sqrt {c x^6+b x^3+a}}{2 x^{12}}dx^3}{4 a}-\frac {7 b \left (a+b x^3+c x^6\right )^{3/2}}{4 a x^{12}}}{10 a}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{5 a x^{15}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (-\frac {-\frac {\int \frac {\left (14 b c x^3+35 b^2-32 a c\right ) \sqrt {c x^6+b x^3+a}}{x^{12}}dx^3}{8 a}-\frac {7 b \left (a+b x^3+c x^6\right )^{3/2}}{4 a x^{12}}}{10 a}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{5 a x^{15}}\right )\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {1}{3} \left (-\frac {-\frac {-\frac {5 b \left (7 b^2-12 a c\right ) \int \frac {\sqrt {c x^6+b x^3+a}}{x^9}dx^3}{2 a}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^3+c x^6\right )^{3/2}}{3 a x^9}}{8 a}-\frac {7 b \left (a+b x^3+c x^6\right )^{3/2}}{4 a x^{12}}}{10 a}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{5 a x^{15}}\right )\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {1}{3} \left (-\frac {-\frac {-\frac {5 b \left (7 b^2-12 a c\right ) \left (-\frac {\left (b^2-4 a c\right ) \int \frac {1}{x^3 \sqrt {c x^6+b x^3+a}}dx^3}{8 a}-\frac {\left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{4 a x^6}\right )}{2 a}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^3+c x^6\right )^{3/2}}{3 a x^9}}{8 a}-\frac {7 b \left (a+b x^3+c x^6\right )^{3/2}}{4 a x^{12}}}{10 a}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{5 a x^{15}}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{3} \left (-\frac {-\frac {-\frac {5 b \left (7 b^2-12 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 a-x^6}d\frac {b x^3+2 a}{\sqrt {c x^6+b x^3+a}}}{4 a}-\frac {\left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{4 a x^6}\right )}{2 a}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^3+c x^6\right )^{3/2}}{3 a x^9}}{8 a}-\frac {7 b \left (a+b x^3+c x^6\right )^{3/2}}{4 a x^{12}}}{10 a}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{5 a x^{15}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (-\frac {-\frac {-\frac {5 b \left (7 b^2-12 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{8 a^{3/2}}-\frac {\left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{4 a x^6}\right )}{2 a}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^3+c x^6\right )^{3/2}}{3 a x^9}}{8 a}-\frac {7 b \left (a+b x^3+c x^6\right )^{3/2}}{4 a x^{12}}}{10 a}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{5 a x^{15}}\right )\)

Input:

Int[Sqrt[a + b*x^3 + c*x^6]/x^16,x]
 

Output:

(-1/5*(a + b*x^3 + c*x^6)^(3/2)/(a*x^15) - ((-7*b*(a + b*x^3 + c*x^6)^(3/2 
))/(4*a*x^12) - (-1/3*((35*b^2 - 32*a*c)*(a + b*x^3 + c*x^6)^(3/2))/(a*x^9 
) - (5*b*(7*b^2 - 12*a*c)*(-1/4*((2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(a 
*x^6) + ((b^2 - 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c 
*x^6])])/(8*a^(3/2))))/(2*a))/(8*a))/(10*a))/3
 

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 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 0]
 

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 1167

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d 
^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2))   Int[ 
(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, 
 x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m 
, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp 
lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* 
x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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]

\[\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{x^{16}}d x\]

Input:

int((c*x^6+b*x^3+a)^(1/2)/x^16,x)
 

Output:

int((c*x^6+b*x^3+a)^(1/2)/x^16,x)
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.95 \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^{16}} \, dx=\left [\frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt {a} x^{15} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \, {\left ({\left (105 \, a b^{4} - 460 \, a^{2} b^{2} c + 256 \, a^{3} c^{2}\right )} x^{12} - 2 \, {\left (35 \, a^{2} b^{3} - 116 \, a^{3} b c\right )} x^{9} - 48 \, a^{4} b x^{3} + 8 \, {\left (7 \, a^{3} b^{2} - 16 \, a^{4} c\right )} x^{6} - 384 \, a^{5}\right )} \sqrt {c x^{6} + b x^{3} + a}}{23040 \, a^{5} x^{15}}, \frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt {-a} x^{15} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (105 \, a b^{4} - 460 \, a^{2} b^{2} c + 256 \, a^{3} c^{2}\right )} x^{12} - 2 \, {\left (35 \, a^{2} b^{3} - 116 \, a^{3} b c\right )} x^{9} - 48 \, a^{4} b x^{3} + 8 \, {\left (7 \, a^{3} b^{2} - 16 \, a^{4} c\right )} x^{6} - 384 \, a^{5}\right )} \sqrt {c x^{6} + b x^{3} + a}}{11520 \, a^{5} x^{15}}\right ] \] Input:

integrate((c*x^6+b*x^3+a)^(1/2)/x^16,x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

[1/23040*(15*(7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*sqrt(a)*x^15*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*((105*a*b^4 - 460*a^2*b^2*c + 256*a^3*c^2)*x^12 - 2*(35* 
a^2*b^3 - 116*a^3*b*c)*x^9 - 48*a^4*b*x^3 + 8*(7*a^3*b^2 - 16*a^4*c)*x^6 - 
 384*a^5)*sqrt(c*x^6 + b*x^3 + a))/(a^5*x^15), 1/11520*(15*(7*b^5 - 40*a*b 
^3*c + 48*a^2*b*c^2)*sqrt(-a)*x^15*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)) + 2*((105*a*b^4 - 460*a^2*b^ 
2*c + 256*a^3*c^2)*x^12 - 2*(35*a^2*b^3 - 116*a^3*b*c)*x^9 - 48*a^4*b*x^3 
+ 8*(7*a^3*b^2 - 16*a^4*c)*x^6 - 384*a^5)*sqrt(c*x^6 + b*x^3 + a))/(a^5*x^ 
15)]
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^{16}} \, dx=\int \frac {\sqrt {a + b x^{3} + c x^{6}}}{x^{16}}\, dx \] Input:

integrate((c*x**6+b*x**3+a)**(1/2)/x**16,x)
 

Output:

Integral(sqrt(a + b*x**3 + c*x**6)/x**16, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^{16}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^6+b*x^3+a)^(1/2)/x^16,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]

\[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^{16}} \, dx=\int { \frac {\sqrt {c x^{6} + b x^{3} + a}}{x^{16}} \,d x } \] Input:

integrate((c*x^6+b*x^3+a)^(1/2)/x^16,x, algorithm="giac")
 

Output:

integrate(sqrt(c*x^6 + b*x^3 + a)/x^16, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^{16}} \, dx=\int \frac {\sqrt {c\,x^6+b\,x^3+a}}{x^{16}} \,d x \] Input:

int((a + b*x^3 + c*x^6)^(1/2)/x^16,x)
 

Output:

int((a + b*x^3 + c*x^6)^(1/2)/x^16, x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^3+c x^6}}{x^{16}} \, dx=\text {too large to display} \] Input:

int((c*x^6+b*x^3+a)^(1/2)/x^16,x)
 

Output:

( - 7077888*sqrt(a + b*x**3 + c*x**6)*a**10*b**2*c**8 - 11796480*sqrt(a + 
b*x**3 + c*x**6)*a**9*b**4*c**7 - 5308416*sqrt(a + b*x**3 + c*x**6)*a**9*b 
**3*c**8*x**3 + 3538944*sqrt(a + b*x**3 + c*x**6)*a**9*b**2*c**9*x**6 - 17 
694720*sqrt(a + b*x**3 + c*x**6)*a**9*b*c**10*x**9 + 35389440*sqrt(a + b*x 
**3 + c*x**6)*a**9*c**11*x**12 + 20643840*sqrt(a + b*x**3 + c*x**6)*a**8*b 
**6*c**6 + 5898240*sqrt(a + b*x**3 + c*x**6)*a**8*b**5*c**7*x**3 - 7569408 
*sqrt(a + b*x**3 + c*x**6)*a**8*b**4*c**8*x**6 - 3833856*sqrt(a + b*x**3 + 
 c*x**6)*a**8*b**3*c**9*x**9 + 34209792*sqrt(a + b*x**3 + c*x**6)*a**8*b** 
2*c**10*x**12 + 21159936*sqrt(a + b*x**3 + c*x**6)*a**7*b**8*c**5 - 915456 
0*sqrt(a + b*x**3 + c*x**6)*a**7*b**7*c**6*x**3 + 15646720*sqrt(a + b*x**3 
 + c*x**6)*a**7*b**6*c**7*x**6 + 44974080*sqrt(a + b*x**3 + c*x**6)*a**7*b 
**5*c**8*x**9 - 67829760*sqrt(a + b*x**3 + c*x**6)*a**7*b**4*c**9*x**12 - 
19031040*sqrt(a + b*x**3 + c*x**6)*a**6*b**10*c**4 + 10002432*sqrt(a + b*x 
**3 + c*x**6)*a**6*b**9*c**5*x**3 + 7923712*sqrt(a + b*x**3 + c*x**6)*a**6 
*b**8*c**6*x**6 - 38184960*sqrt(a + b*x**3 + c*x**6)*a**6*b**7*c**7*x**9 - 
 34037760*sqrt(a + b*x**3 + c*x**6)*a**6*b**6*c**8*x**12 + 3386880*sqrt(a 
+ b*x**3 + c*x**6)*a**5*b**12*c**3 + 1117440*sqrt(a + b*x**3 + c*x**6)*a** 
5*b**11*c**4*x**3 - 22674944*sqrt(a + b*x**3 + c*x**6)*a**5*b**10*c**5*x** 
6 - 18344448*sqrt(a + b*x**3 + c*x**6)*a**5*b**9*c**6*x**9 + 28302336*sqrt 
(a + b*x**3 + c*x**6)*a**5*b**8*c**7*x**12 - 7653120*sqrt(a + b*x**3 + ...