\(\int \frac {1}{x^{13} \sqrt {a+b x^3+c x^6}} \, dx\) [213]

Optimal result

Integrand size = 20, antiderivative size = 192 \[ \int \frac {1}{x^{13} \sqrt {a+b x^3+c x^6}} \, dx=-\frac {\sqrt {a+b x^3+c x^6}}{12 a x^{12}}+\frac {7 b \sqrt {a+b x^3+c x^6}}{72 a^2 x^9}-\frac {\left (35 b^2-36 a c\right ) \sqrt {a+b x^3+c x^6}}{288 a^3 x^6}+\frac {5 b \left (21 b^2-44 a c\right ) \sqrt {a+b x^3+c x^6}}{576 a^4 x^3}-\frac {\left (35 b^4-120 a b^2 c+48 a^2 c^2\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{384 a^{9/2}} \] Output:

-1/12*(c*x^6+b*x^3+a)^(1/2)/a/x^12+7/72*b*(c*x^6+b*x^3+a)^(1/2)/a^2/x^9-1/ 
288*(-36*a*c+35*b^2)*(c*x^6+b*x^3+a)^(1/2)/a^3/x^6+5/576*b*(-44*a*c+21*b^2 
)*(c*x^6+b*x^3+a)^(1/2)/a^4/x^3-1/384*(48*a^2*c^2-120*a*b^2*c+35*b^4)*arct 
anh(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 = 0.78 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^{13} \sqrt {a+b x^3+c x^6}} \, dx=\frac {\sqrt {a+b x^3+c x^6} \left (-48 a^3+56 a^2 b x^3-70 a b^2 x^6+72 a^2 c x^6+105 b^3 x^9-220 a b c x^9\right )}{576 a^4 x^{12}}+\frac {\left (35 b^4-120 a b^2 c+48 a^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{192 a^{9/2}} \] Input:

Integrate[1/(x^13*Sqrt[a + b*x^3 + c*x^6]),x]
 

Output:

(Sqrt[a + b*x^3 + c*x^6]*(-48*a^3 + 56*a^2*b*x^3 - 70*a*b^2*x^6 + 72*a^2*c 
*x^6 + 105*b^3*x^9 - 220*a*b*c*x^9))/(576*a^4*x^12) + ((35*b^4 - 120*a*b^2 
*c + 48*a^2*c^2)*ArcTanh[(Sqrt[c]*x^3 - Sqrt[a + b*x^3 + c*x^6])/Sqrt[a]]) 
/(192*a^(9/2))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1693, 1167, 27, 1237, 27, 1237, 27, 1228, 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.

\(\displaystyle \int \frac {1}{x^{13} \sqrt {a+b x^3+c x^6}} \, dx\)

\(\Big \downarrow \) 1693

\(\displaystyle \frac {1}{3} \int \frac {1}{x^{15} \sqrt {c x^6+b x^3+a}}dx^3\)

\(\Big \downarrow \) 1167

\(\displaystyle \frac {1}{3} \left (-\frac {\int \frac {6 c x^3+7 b}{2 x^{12} \sqrt {c x^6+b x^3+a}}dx^3}{4 a}-\frac {\sqrt {a+b x^3+c x^6}}{4 a x^{12}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (-\frac {\int \frac {6 c x^3+7 b}{x^{12} \sqrt {c x^6+b x^3+a}}dx^3}{8 a}-\frac {\sqrt {a+b x^3+c x^6}}{4 a x^{12}}\right )\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{3} \left (-\frac {-\frac {\int \frac {28 b c x^3+35 b^2-36 a c}{2 x^9 \sqrt {c x^6+b x^3+a}}dx^3}{3 a}-\frac {7 b \sqrt {a+b x^3+c x^6}}{3 a x^9}}{8 a}-\frac {\sqrt {a+b x^3+c x^6}}{4 a x^{12}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (-\frac {-\frac {\int \frac {28 b c x^3+35 b^2-36 a c}{x^9 \sqrt {c x^6+b x^3+a}}dx^3}{6 a}-\frac {7 b \sqrt {a+b x^3+c x^6}}{3 a x^9}}{8 a}-\frac {\sqrt {a+b x^3+c x^6}}{4 a x^{12}}\right )\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{3} \left (-\frac {-\frac {-\frac {\int \frac {2 c \left (35 b^2-36 a c\right ) x^3+5 b \left (21 b^2-44 a c\right )}{2 x^6 \sqrt {c x^6+b x^3+a}}dx^3}{2 a}-\frac {\left (35 b^2-36 a c\right ) \sqrt {a+b x^3+c x^6}}{2 a x^6}}{6 a}-\frac {7 b \sqrt {a+b x^3+c x^6}}{3 a x^9}}{8 a}-\frac {\sqrt {a+b x^3+c x^6}}{4 a x^{12}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (-\frac {-\frac {-\frac {\int \frac {2 c \left (35 b^2-36 a c\right ) x^3+5 b \left (21 b^2-44 a c\right )}{x^6 \sqrt {c x^6+b x^3+a}}dx^3}{4 a}-\frac {\left (35 b^2-36 a c\right ) \sqrt {a+b x^3+c x^6}}{2 a x^6}}{6 a}-\frac {7 b \sqrt {a+b x^3+c x^6}}{3 a x^9}}{8 a}-\frac {\sqrt {a+b x^3+c x^6}}{4 a x^{12}}\right )\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {1}{3} \left (-\frac {-\frac {-\frac {-\frac {3 \left (48 a^2 c^2-120 a b^2 c+35 b^4\right ) \int \frac {1}{x^3 \sqrt {c x^6+b x^3+a}}dx^3}{2 a}-\frac {5 b \left (21 b^2-44 a c\right ) \sqrt {a+b x^3+c x^6}}{a x^3}}{4 a}-\frac {\left (35 b^2-36 a c\right ) \sqrt {a+b x^3+c x^6}}{2 a x^6}}{6 a}-\frac {7 b \sqrt {a+b x^3+c x^6}}{3 a x^9}}{8 a}-\frac {\sqrt {a+b x^3+c x^6}}{4 a x^{12}}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{3} \left (-\frac {-\frac {-\frac {\frac {3 \left (48 a^2 c^2-120 a b^2 c+35 b^4\right ) \int \frac {1}{4 a-x^6}d\frac {b x^3+2 a}{\sqrt {c x^6+b x^3+a}}}{a}-\frac {5 b \left (21 b^2-44 a c\right ) \sqrt {a+b x^3+c x^6}}{a x^3}}{4 a}-\frac {\left (35 b^2-36 a c\right ) \sqrt {a+b x^3+c x^6}}{2 a x^6}}{6 a}-\frac {7 b \sqrt {a+b x^3+c x^6}}{3 a x^9}}{8 a}-\frac {\sqrt {a+b x^3+c x^6}}{4 a x^{12}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (-\frac {-\frac {-\frac {\frac {3 \left (48 a^2 c^2-120 a b^2 c+35 b^4\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{2 a^{3/2}}-\frac {5 b \left (21 b^2-44 a c\right ) \sqrt {a+b x^3+c x^6}}{a x^3}}{4 a}-\frac {\left (35 b^2-36 a c\right ) \sqrt {a+b x^3+c x^6}}{2 a x^6}}{6 a}-\frac {7 b \sqrt {a+b x^3+c x^6}}{3 a x^9}}{8 a}-\frac {\sqrt {a+b x^3+c x^6}}{4 a x^{12}}\right )\)

Input:

Int[1/(x^13*Sqrt[a + b*x^3 + c*x^6]),x]
 

Output:

(-1/4*Sqrt[a + b*x^3 + c*x^6]/(a*x^12) - ((-7*b*Sqrt[a + b*x^3 + c*x^6])/( 
3*a*x^9) - (-1/2*((35*b^2 - 36*a*c)*Sqrt[a + b*x^3 + c*x^6])/(a*x^6) - ((- 
5*b*(21*b^2 - 44*a*c)*Sqrt[a + b*x^3 + c*x^6])/(a*x^3) + (3*(35*b^4 - 120* 
a*b^2*c + 48*a^2*c^2)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c* 
x^6])])/(2*a^(3/2)))/(4*a))/(6*a))/(8*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 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]

\[\int \frac {1}{x^{13} \sqrt {c \,x^{6}+b \,x^{3}+a}}d x\]

Input:

int(1/x^13/(c*x^6+b*x^3+a)^(1/2),x)
 

Output:

int(1/x^13/(c*x^6+b*x^3+a)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.70 \[ \int \frac {1}{x^{13} \sqrt {a+b x^3+c x^6}} \, dx=\left [\frac {3 \, {\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} \sqrt {a} x^{12} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \, {\left (5 \, {\left (21 \, a b^{3} - 44 \, a^{2} b c\right )} x^{9} + 56 \, a^{3} b x^{3} - 2 \, {\left (35 \, a^{2} b^{2} - 36 \, a^{3} c\right )} x^{6} - 48 \, a^{4}\right )} \sqrt {c x^{6} + b x^{3} + a}}{2304 \, a^{5} x^{12}}, \frac {3 \, {\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} \sqrt {-a} x^{12} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \, {\left (5 \, {\left (21 \, a b^{3} - 44 \, a^{2} b c\right )} x^{9} + 56 \, a^{3} b x^{3} - 2 \, {\left (35 \, a^{2} b^{2} - 36 \, a^{3} c\right )} x^{6} - 48 \, a^{4}\right )} \sqrt {c x^{6} + b x^{3} + a}}{1152 \, a^{5} x^{12}}\right ] \] Input:

integrate(1/x^13/(c*x^6+b*x^3+a)^(1/2),x, algorithm="fricas")
 

Output:

[1/2304*(3*(35*b^4 - 120*a*b^2*c + 48*a^2*c^2)*sqrt(a)*x^12*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*(5*(21*a*b^3 - 44*a^2*b*c)*x^9 + 56*a^3*b*x^3 - 2*(35*a^2* 
b^2 - 36*a^3*c)*x^6 - 48*a^4)*sqrt(c*x^6 + b*x^3 + a))/(a^5*x^12), 1/1152* 
(3*(35*b^4 - 120*a*b^2*c + 48*a^2*c^2)*sqrt(-a)*x^12*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*(5*(21 
*a*b^3 - 44*a^2*b*c)*x^9 + 56*a^3*b*x^3 - 2*(35*a^2*b^2 - 36*a^3*c)*x^6 - 
48*a^4)*sqrt(c*x^6 + b*x^3 + a))/(a^5*x^12)]
 

Sympy [F]

\[ \int \frac {1}{x^{13} \sqrt {a+b x^3+c x^6}} \, dx=\int \frac {1}{x^{13} \sqrt {a + b x^{3} + c x^{6}}}\, dx \] Input:

integrate(1/x**13/(c*x**6+b*x**3+a)**(1/2),x)
 

Output:

Integral(1/(x**13*sqrt(a + b*x**3 + c*x**6)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^{13} \sqrt {a+b x^3+c x^6}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/x^13/(c*x^6+b*x^3+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(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [F]

\[ \int \frac {1}{x^{13} \sqrt {a+b x^3+c x^6}} \, dx=\int { \frac {1}{\sqrt {c x^{6} + b x^{3} + a} x^{13}} \,d x } \] Input:

integrate(1/x^13/(c*x^6+b*x^3+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^13), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^{13} \sqrt {a+b x^3+c x^6}} \, dx=\int \frac {1}{x^{13}\,\sqrt {c\,x^6+b\,x^3+a}} \,d x \] Input:

int(1/(x^13*(a + b*x^3 + c*x^6)^(1/2)),x)
 

Output:

int(1/(x^13*(a + b*x^3 + c*x^6)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^{13} \sqrt {a+b x^3+c x^6}} \, dx=\text {too large to display} \] Input:

int(1/x^13/(c*x^6+b*x^3+a)^(1/2),x)
 

Output:

( - 294912*sqrt(a + b*x**3 + c*x**6)*a**9*b**4*c**7 - 393216*sqrt(a + b*x* 
*3 + c*x**6)*a**9*b**3*c**8*x**3 + 589824*sqrt(a + b*x**3 + c*x**6)*a**9*b 
**2*c**9*x**6 + 1179648*sqrt(a + b*x**3 + c*x**6)*a**9*c**11*x**12 - 86016 
0*sqrt(a + b*x**3 + c*x**6)*a**8*b**6*c**6 + 1130496*sqrt(a + b*x**3 + c*x 
**6)*a**8*b**5*c**7*x**3 - 245760*sqrt(a + b*x**3 + c*x**6)*a**8*b**4*c**8 
*x**6 - 688128*sqrt(a + b*x**3 + c*x**6)*a**8*b**3*c**9*x**9 - 2949120*sqr 
t(a + b*x**3 + c*x**6)*a**8*b**2*c**10*x**12 - 325632*sqrt(a + b*x**3 + c* 
x**6)*a**7*b**8*c**5 + 569344*sqrt(a + b*x**3 + c*x**6)*a**7*b**7*c**6*x** 
3 + 528384*sqrt(a + b*x**3 + c*x**6)*a**7*b**6*c**7*x**6 - 860160*sqrt(a + 
 b*x**3 + c*x**6)*a**7*b**5*c**8*x**9 - 540672*sqrt(a + b*x**3 + c*x**6)*a 
**7*b**4*c**9*x**12 + 483840*sqrt(a + b*x**3 + c*x**6)*a**6*b**10*c**4 + 1 
729536*sqrt(a + b*x**3 + c*x**6)*a**6*b**9*c**5*x**3 - 2247680*sqrt(a + b* 
x**3 + c*x**6)*a**6*b**8*c**6*x**6 - 1845248*sqrt(a + b*x**3 + c*x**6)*a** 
6*b**7*c**7*x**9 + 3317760*sqrt(a + b*x**3 + c*x**6)*a**6*b**6*c**8*x**12 
+ 99840*sqrt(a + b*x**3 + c*x**6)*a**5*b**12*c**3 - 846080*sqrt(a + b*x**3 
 + c*x**6)*a**5*b**11*c**4*x**3 - 2465280*sqrt(a + b*x**3 + c*x**6)*a**5*b 
**10*c**5*x**6 + 668160*sqrt(a + b*x**3 + c*x**6)*a**5*b**9*c**6*x**9 - 23 
8080*sqrt(a + b*x**3 + c*x**6)*a**5*b**8*c**7*x**12 - 67200*sqrt(a + b*x** 
3 + c*x**6)*a**4*b**14*c**2 - 12160*sqrt(a + b*x**3 + c*x**6)*a**4*b**13*c 
**3*x**3 + 751360*sqrt(a + b*x**3 + c*x**6)*a**4*b**12*c**4*x**6 + 6515...