\(\int \frac {x^{14}}{\sqrt {a+b x^3+c x^6}} \, dx\) [219]

Optimal result

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

[Out]

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)-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-1/576*(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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1371, 756, 846, 793, 635, 212} \[ \int \frac {x^{14}}{\sqrt {a+b x^3+c x^6}} \, dx=\frac {\left (48 a^2 c^2-120 a b^2 c+35 b^4\right ) \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{384 c^{9/2}}-\frac {\left (5 b \left (21 b^2-44 a c\right )-2 c x^3 \left (35 b^2-36 a c\right )\right ) \sqrt {a+b x^3+c x^6}}{576 c^4}-\frac {7 b x^6 \sqrt {a+b x^3+c x^6}}{72 c^2}+\frac {x^9 \sqrt {a+b x^3+c x^6}}{12 c} \]

[In]

Int[x^14/Sqrt[a + b*x^3 + c*x^6],x]

[Out]

(-7*b*x^6*Sqrt[a + b*x^3 + c*x^6])/(72*c^2) + (x^9*Sqrt[a + b*x^3 + c*x^6])/(12*c) - ((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])/(576*c^4) + ((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])])/(384*c^(9/2))

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 756

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 793

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] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 846

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 1371

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(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] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {x^4}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right ) \\ & = \frac {x^9 \sqrt {a+b x^3+c x^6}}{12 c}+\frac {\text {Subst}\left (\int \frac {x^2 \left (-3 a-\frac {7 b x}{2}\right )}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{12 c} \\ & = -\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 {\text {Subst}\left (\int \frac {x \left (7 a b+\frac {1}{4} \left (35 b^2-36 a c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{36 c^2} \\ & = -\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 {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{384 c^4} \\ & = -\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 {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^3}{\sqrt {a+b x^3+c x^6}}\right )}{192 c^4} \\ & = -\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 ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{384 c^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.81 \[ \int \frac {x^{14}}{\sqrt {a+b x^3+c x^6}} \, dx=\frac {\sqrt {a+b x^3+c x^6} \left (-105 b^3+220 a b c+70 b^2 c x^3-72 a c^2 x^3-56 b c^2 x^6+48 c^3 x^9\right )}{576 c^4}+\frac {\left (-35 b^4+120 a b^2 c-48 a^2 c^2\right ) \log \left (b c^4+2 c^5 x^3-2 c^{9/2} \sqrt {a+b x^3+c x^6}\right )}{384 c^{9/2}} \]

[In]

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

[Out]

(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))/(57
6*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]])/(38
4*c^(9/2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.77 \[ \int \frac {x^{14}}{\sqrt {a+b x^3+c x^6}} \, dx=\left [\frac {3 \, {\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (48 \, c^{4} x^{9} - 56 \, b c^{3} x^{6} - 105 \, b^{3} c + 220 \, a b c^{2} + 2 \, {\left (35 \, b^{2} c^{2} - 36 \, a c^{3}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{2304 \, c^{5}}, -\frac {3 \, {\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) - 2 \, {\left (48 \, c^{4} x^{9} - 56 \, b c^{3} x^{6} - 105 \, b^{3} c + 220 \, a b c^{2} + 2 \, {\left (35 \, b^{2} c^{2} - 36 \, a c^{3}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{1152 \, c^{5}}\right ] \]

[In]

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

[Out]

[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]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x^14/(a + b*x^3 + c*x^6)^(1/2),x)

[Out]

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