\(\int \frac {x^{14}}{\sqrt {a+b x^3+c x^6}} \, dx\) [204]

Optimal result

Integrand size = 20, antiderivative size = 171 \[ \int \frac {x^{14}}{\sqrt {a+b x^3+c x^6}} \, dx=-\frac {7 b x^6 \sqrt {a+b x^3+c x^6}}{72 c^2}+\frac {x^9 \sqrt {a+b x^3+c x^6}}{12 c}-\frac {\left (5 b \left (21 b^2-44 a c\right )-2 c \left (35 b^2-36 a c\right ) x^3\right ) \sqrt {a+b x^3+c x^6}}{576 c^4}+\frac {\left (35 b^4-120 a b^2 c+48 a^2 c^2\right ) \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{384 c^{9/2}} \] Output:

-7/72*b*x^6*(c*x^6+b*x^3+a)^(1/2)/c^2+1/12*x^9*(c*x^6+b*x^3+a)^(1/2)/c-1/5 
76*(5*b*(-44*a*c+21*b^2)-2*c*(-36*a*c+35*b^2)*x^3)*(c*x^6+b*x^3+a)^(1/2)/c 
^4+1/384*(48*a^2*c^2-120*a*b^2*c+35*b^4)*arctanh(1/2*(2*c*x^3+b)/c^(1/2)/( 
c*x^6+b*x^3+a)^(1/2))/c^(9/2)
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.81 \[ \int \frac {x^{14}}{\sqrt {a+b x^3+c x^6}} \, dx=\frac {\sqrt {a+b x^3+c x^6} \left (-105 b^3+220 a b c+70 b^2 c x^3-72 a c^2 x^3-56 b c^2 x^6+48 c^3 x^9\right )}{576 c^4}+\frac {\left (-35 b^4+120 a b^2 c-48 a^2 c^2\right ) \log \left (b c^4+2 c^5 x^3-2 c^{9/2} \sqrt {a+b x^3+c x^6}\right )}{384 c^{9/2}} \] Input:

Integrate[x^14/Sqrt[a + b*x^3 + c*x^6],x]
 

Output:

(Sqrt[a + b*x^3 + c*x^6]*(-105*b^3 + 220*a*b*c + 70*b^2*c*x^3 - 72*a*c^2*x 
^3 - 56*b*c^2*x^6 + 48*c^3*x^9))/(576*c^4) + ((-35*b^4 + 120*a*b^2*c - 48* 
a^2*c^2)*Log[b*c^4 + 2*c^5*x^3 - 2*c^(9/2)*Sqrt[a + b*x^3 + c*x^6]])/(384* 
c^(9/2))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1693, 1166, 27, 1236, 27, 1225, 1092, 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.

\(\displaystyle \int \frac {x^{14}}{\sqrt {a+b x^3+c x^6}} \, dx\)

\(\Big \downarrow \) 1693

\(\displaystyle \frac {1}{3} \int \frac {x^{12}}{\sqrt {c x^6+b x^3+a}}dx^3\)

\(\Big \downarrow \) 1166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1236

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1225

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

Int[x^14/Sqrt[a + b*x^3 + c*x^6],x]
 

Output:

((x^9*Sqrt[a + b*x^3 + c*x^6])/(4*c) - ((7*b*x^6*Sqrt[a + b*x^3 + c*x^6])/ 
(3*c) - (-1/4*((5*b*(21*b^2 - 44*a*c) - 2*c*(35*b^2 - 36*a*c)*x^3)*Sqrt[a 
+ b*x^3 + c*x^6])/c^2 + (3*(35*b^4 - 120*a*b^2*c + 48*a^2*c^2)*ArcTanh[(b 
+ 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(8*c^(5/2)))/(6*c))/(8*c) 
)/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 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 1166

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)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

rule 1225

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

rule 1236

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

\[\int \frac {x^{14}}{\sqrt {c \,x^{6}+b \,x^{3}+a}}d x\]

Input:

int(x^14/(c*x^6+b*x^3+a)^(1/2),x)
 

Output:

int(x^14/(c*x^6+b*x^3+a)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.77 \[ \int \frac {x^{14}}{\sqrt {a+b x^3+c x^6}} \, dx=\left [\frac {3 \, {\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (48 \, c^{4} x^{9} - 56 \, b c^{3} x^{6} - 105 \, b^{3} c + 220 \, a b c^{2} + 2 \, {\left (35 \, b^{2} c^{2} - 36 \, a c^{3}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{2304 \, c^{5}}, -\frac {3 \, {\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) - 2 \, {\left (48 \, c^{4} x^{9} - 56 \, b c^{3} x^{6} - 105 \, b^{3} c + 220 \, a b c^{2} + 2 \, {\left (35 \, b^{2} c^{2} - 36 \, a c^{3}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{1152 \, c^{5}}\right ] \] Input:

integrate(x^14/(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(c)*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) + 
 4*(48*c^4*x^9 - 56*b*c^3*x^6 - 105*b^3*c + 220*a*b*c^2 + 2*(35*b^2*c^2 - 
36*a*c^3)*x^3)*sqrt(c*x^6 + b*x^3 + a))/c^5, -1/1152*(3*(35*b^4 - 120*a*b^ 
2*c + 48*a^2*c^2)*sqrt(-c)*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*(48*c^4*x^9 - 56*b*c^3*x^6 - 105 
*b^3*c + 220*a*b*c^2 + 2*(35*b^2*c^2 - 36*a*c^3)*x^3)*sqrt(c*x^6 + b*x^3 + 
 a))/c^5]
 

Sympy [F]

\[ \int \frac {x^{14}}{\sqrt {a+b x^3+c x^6}} \, dx=\int \frac {x^{14}}{\sqrt {a + b x^{3} + c x^{6}}}\, dx \] Input:

integrate(x**14/(c*x**6+b*x**3+a)**(1/2),x)
 

Output:

Integral(x**14/sqrt(a + b*x**3 + c*x**6), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^{14}}{\sqrt {a+b x^3+c x^6}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^14/(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]

\[ \int \frac {x^{14}}{\sqrt {a+b x^3+c x^6}} \, dx=\int { \frac {x^{14}}{\sqrt {c x^{6} + b x^{3} + a}} \,d x } \] Input:

integrate(x^14/(c*x^6+b*x^3+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^14/sqrt(c*x^6 + b*x^3 + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{14}}{\sqrt {a+b x^3+c x^6}} \, dx=\int \frac {x^{14}}{\sqrt {c\,x^6+b\,x^3+a}} \,d x \] Input:

int(x^14/(a + b*x^3 + c*x^6)^(1/2),x)
 

Output:

int(x^14/(a + b*x^3 + c*x^6)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^{14}}{\sqrt {a+b x^3+c x^6}} \, dx=\frac {-288 \sqrt {c \,x^{6}+b \,x^{3}+a}\, a^{2} c^{3}+1160 \sqrt {c \,x^{6}+b \,x^{3}+a}\, a \,b^{2} c^{2}-144 \sqrt {c \,x^{6}+b \,x^{3}+a}\, a b \,c^{3} x^{3}-420 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b^{4} c +140 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b^{3} c^{2} x^{3}-112 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b^{2} c^{3} x^{6}+96 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b \,c^{4} x^{9}-144 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}-\sqrt {c}\, x^{3}\right ) a^{2} b \,c^{2}+360 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}-\sqrt {c}\, x^{3}\right ) a \,b^{3} c -105 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}-\sqrt {c}\, x^{3}\right ) b^{5}+144 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}+\sqrt {c}\, x^{3}\right ) a^{2} b \,c^{2}-360 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}+\sqrt {c}\, x^{3}\right ) a \,b^{3} c +105 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}+\sqrt {c}\, x^{3}\right ) b^{5}+864 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{8}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{2} b \,c^{4}-2160 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{8}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a \,b^{3} c^{3}+630 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{8}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) b^{5} c^{2}+864 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{5}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{3} c^{4}-1728 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{5}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{2} b^{2} c^{3}-450 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{5}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a \,b^{4} c^{2}+315 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{5}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) b^{6} c}{1152 b \,c^{5}} \] Input:

int(x^14/(c*x^6+b*x^3+a)^(1/2),x)
 

Output:

( - 288*sqrt(a + b*x**3 + c*x**6)*a**2*c**3 + 1160*sqrt(a + b*x**3 + c*x** 
6)*a*b**2*c**2 - 144*sqrt(a + b*x**3 + c*x**6)*a*b*c**3*x**3 - 420*sqrt(a 
+ b*x**3 + c*x**6)*b**4*c + 140*sqrt(a + b*x**3 + c*x**6)*b**3*c**2*x**3 - 
 112*sqrt(a + b*x**3 + c*x**6)*b**2*c**3*x**6 + 96*sqrt(a + b*x**3 + c*x** 
6)*b*c**4*x**9 - 144*sqrt(c)*log(sqrt(a + b*x**3 + c*x**6) - sqrt(c)*x**3) 
*a**2*b*c**2 + 360*sqrt(c)*log(sqrt(a + b*x**3 + c*x**6) - sqrt(c)*x**3)*a 
*b**3*c - 105*sqrt(c)*log(sqrt(a + b*x**3 + c*x**6) - sqrt(c)*x**3)*b**5 + 
 144*sqrt(c)*log(sqrt(a + b*x**3 + c*x**6) + sqrt(c)*x**3)*a**2*b*c**2 - 3 
60*sqrt(c)*log(sqrt(a + b*x**3 + c*x**6) + sqrt(c)*x**3)*a*b**3*c + 105*sq 
rt(c)*log(sqrt(a + b*x**3 + c*x**6) + sqrt(c)*x**3)*b**5 + 864*int((sqrt(a 
 + b*x**3 + c*x**6)*x**8)/(a**2 + 2*a*b*x**3 + a*c*x**6 + b**2*x**6 + b*c* 
x**9),x)*a**2*b*c**4 - 2160*int((sqrt(a + b*x**3 + c*x**6)*x**8)/(a**2 + 2 
*a*b*x**3 + a*c*x**6 + b**2*x**6 + b*c*x**9),x)*a*b**3*c**3 + 630*int((sqr 
t(a + b*x**3 + c*x**6)*x**8)/(a**2 + 2*a*b*x**3 + a*c*x**6 + b**2*x**6 + b 
*c*x**9),x)*b**5*c**2 + 864*int((sqrt(a + b*x**3 + c*x**6)*x**5)/(a**2 + 2 
*a*b*x**3 + a*c*x**6 + b**2*x**6 + b*c*x**9),x)*a**3*c**4 - 1728*int((sqrt 
(a + b*x**3 + c*x**6)*x**5)/(a**2 + 2*a*b*x**3 + a*c*x**6 + b**2*x**6 + b* 
c*x**9),x)*a**2*b**2*c**3 - 450*int((sqrt(a + b*x**3 + c*x**6)*x**5)/(a**2 
 + 2*a*b*x**3 + a*c*x**6 + b**2*x**6 + b*c*x**9),x)*a*b**4*c**2 + 315*int( 
(sqrt(a + b*x**3 + c*x**6)*x**5)/(a**2 + 2*a*b*x**3 + a*c*x**6 + b**2*x...