3.2.85 \(\int x^{14} \sqrt {a+b x^3+c x^6} \, dx\) [185]

Optimal. Leaf size=231 \[ \frac {\left (21 b^4-56 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 c^5}-\frac {b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac {x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c}-\frac {\left (7 b \left (15 b^2-28 a c\right )-6 c \left (21 b^2-20 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{2880 c^4}-\frac {\left (b^2-4 a c\right ) \left (21 b^4-56 a b^2 c+16 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{3072 c^{11/2}} \]

[Out]

-1/20*b*x^6*(c*x^6+b*x^3+a)^(3/2)/c^2+1/18*x^9*(c*x^6+b*x^3+a)^(3/2)/c-1/2880*(7*b*(-28*a*c+15*b^2)-6*c*(-20*a
*c+21*b^2)*x^3)*(c*x^6+b*x^3+a)^(3/2)/c^4-1/3072*(-4*a*c+b^2)*(16*a^2*c^2-56*a*b^2*c+21*b^4)*arctanh(1/2*(2*c*
x^3+b)/c^(1/2)/(c*x^6+b*x^3+a)^(1/2))/c^(11/2)+1/1536*(16*a^2*c^2-56*a*b^2*c+21*b^4)*(2*c*x^3+b)*(c*x^6+b*x^3+
a)^(1/2)/c^5

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Rubi [A]
time = 0.20, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1371, 756, 846, 793, 626, 635, 212} \begin {gather*} -\frac {\left (b^2-4 a c\right ) \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{3072 c^{11/2}}+\frac {\left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 c^5}-\frac {\left (7 b \left (15 b^2-28 a c\right )-6 c x^3 \left (21 b^2-20 a c\right )\right ) \left (a+b x^3+c x^6\right )^{3/2}}{2880 c^4}-\frac {b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac {x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^14*Sqrt[a + b*x^3 + c*x^6],x]

[Out]

((21*b^4 - 56*a*b^2*c + 16*a^2*c^2)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1536*c^5) - (b*x^6*(a + b*x^3 + c*
x^6)^(3/2))/(20*c^2) + (x^9*(a + b*x^3 + c*x^6)^(3/2))/(18*c) - ((7*b*(15*b^2 - 28*a*c) - 6*c*(21*b^2 - 20*a*c
)*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(2880*c^4) - ((b^2 - 4*a*c)*(21*b^4 - 56*a*b^2*c + 16*a^2*c^2)*ArcTanh[(b +
2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(3072*c^(11/2))

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 626

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

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*} \int x^{14} \sqrt {a+b x^3+c x^6} \, dx &=\frac {1}{3} \text {Subst}\left (\int x^4 \sqrt {a+b x+c x^2} \, dx,x,x^3\right )\\ &=\frac {x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c}+\frac {\text {Subst}\left (\int x^2 \left (-3 a-\frac {9 b x}{2}\right ) \sqrt {a+b x+c x^2} \, dx,x,x^3\right )}{18 c}\\ &=-\frac {b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac {x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c}+\frac {\text {Subst}\left (\int x \left (9 a b+\frac {3}{4} \left (21 b^2-20 a c\right ) x\right ) \sqrt {a+b x+c x^2} \, dx,x,x^3\right )}{90 c^2}\\ &=-\frac {b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac {x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c}-\frac {\left (7 b \left (15 b^2-28 a c\right )-6 c \left (21 b^2-20 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{2880 c^4}+\frac {\left (21 b^4-56 a b^2 c+16 a^2 c^2\right ) \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^3\right )}{384 c^4}\\ &=\frac {\left (21 b^4-56 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 c^5}-\frac {b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac {x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c}-\frac {\left (7 b \left (15 b^2-28 a c\right )-6 c \left (21 b^2-20 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{2880 c^4}-\frac {\left (\left (b^2-4 a c\right ) \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{3072 c^5}\\ &=\frac {\left (21 b^4-56 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 c^5}-\frac {b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac {x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c}-\frac {\left (7 b \left (15 b^2-28 a c\right )-6 c \left (21 b^2-20 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{2880 c^4}-\frac {\left (\left (b^2-4 a c\right ) \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\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 )}{1536 c^5}\\ &=\frac {\left (21 b^4-56 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 c^5}-\frac {b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac {x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c}-\frac {\left (7 b \left (15 b^2-28 a c\right )-6 c \left (21 b^2-20 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{2880 c^4}-\frac {\left (b^2-4 a c\right ) \left (21 b^4-56 a b^2 c+16 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 )}{3072 c^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.42, size = 206, normalized size = 0.89 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+b x^3+c x^6} \left (315 b^5-210 b^4 c x^3+16 b^2 c^2 x^3 \left (56 a-9 c x^6\right )+168 b^3 c \left (-10 a+c x^6\right )+16 b c^2 \left (113 a^2-34 a c x^6+8 c^2 x^{12}\right )+160 c^3 x^3 \left (-3 a^2+2 a c x^6+8 c^2 x^{12}\right )\right )+15 \left (21 b^6-140 a b^4 c+240 a^2 b^2 c^2-64 a^3 c^3\right ) \log \left (b+2 c x^3-2 \sqrt {c} \sqrt {a+b x^3+c x^6}\right )}{46080 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]*(315*b^5 - 210*b^4*c*x^3 + 16*b^2*c^2*x^3*(56*a - 9*c*x^6) + 168*b^3*c*(-10
*a + c*x^6) + 16*b*c^2*(113*a^2 - 34*a*c*x^6 + 8*c^2*x^12) + 160*c^3*x^3*(-3*a^2 + 2*a*c*x^6 + 8*c^2*x^12)) +
15*(21*b^6 - 140*a*b^4*c + 240*a^2*b^2*c^2 - 64*a^3*c^3)*Log[b + 2*c*x^3 - 2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]])
/(46080*c^(11/2))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{14} \sqrt {c \,x^{6}+b \,x^{3}+a}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]
time = 0.39, size = 451, normalized size = 1.95 \begin {gather*} \left [-\frac {15 \, {\left (21 \, b^{6} - 140 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\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 (1280 \, c^{6} x^{15} + 128 \, b c^{5} x^{12} - 16 \, {\left (9 \, b^{2} c^{4} - 20 \, a c^{5}\right )} x^{9} + 8 \, {\left (21 \, b^{3} c^{3} - 68 \, a b c^{4}\right )} x^{6} + 315 \, b^{5} c - 1680 \, a b^{3} c^{2} + 1808 \, a^{2} b c^{3} - 2 \, {\left (105 \, b^{4} c^{2} - 448 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{92160 \, c^{6}}, \frac {15 \, {\left (21 \, b^{6} - 140 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\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 (1280 \, c^{6} x^{15} + 128 \, b c^{5} x^{12} - 16 \, {\left (9 \, b^{2} c^{4} - 20 \, a c^{5}\right )} x^{9} + 8 \, {\left (21 \, b^{3} c^{3} - 68 \, a b c^{4}\right )} x^{6} + 315 \, b^{5} c - 1680 \, a b^{3} c^{2} + 1808 \, a^{2} b c^{3} - 2 \, {\left (105 \, b^{4} c^{2} - 448 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{46080 \, c^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/92160*(15*(21*b^6 - 140*a*b^4*c + 240*a^2*b^2*c^2 - 64*a^3*c^3)*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*(1280*c^6*x^15 + 128*b*c^5*x^12 - 16*(9*b^2*c^4
- 20*a*c^5)*x^9 + 8*(21*b^3*c^3 - 68*a*b*c^4)*x^6 + 315*b^5*c - 1680*a*b^3*c^2 + 1808*a^2*b*c^3 - 2*(105*b^4*c
^2 - 448*a*b^2*c^3 + 240*a^2*c^4)*x^3)*sqrt(c*x^6 + b*x^3 + a))/c^6, 1/46080*(15*(21*b^6 - 140*a*b^4*c + 240*a
^2*b^2*c^2 - 64*a^3*c^3)*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*(1280*c^6*x^15 + 128*b*c^5*x^12 - 16*(9*b^2*c^4 - 20*a*c^5)*x^9 + 8*(21*b^3*c^3 - 68*a*b*c^4)*x^6
+ 315*b^5*c - 1680*a*b^3*c^2 + 1808*a^2*b*c^3 - 2*(105*b^4*c^2 - 448*a*b^2*c^3 + 240*a^2*c^4)*x^3)*sqrt(c*x^6
+ b*x^3 + a))/c^6]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{14} \sqrt {a + b x^{3} + c x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 2.94, size = 543, normalized size = 2.35 \begin {gather*} \frac {x^9\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{18\,c}-\frac {b\,\left (\frac {x^6\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{5\,c}+\frac {7\,b\,\left (\frac {a\,\left (\left (\frac {b}{4\,c}+\frac {x^3}{2}\right )\,\sqrt {c\,x^6+b\,x^3+a}+\frac {\ln \left (\sqrt {c\,x^6+b\,x^3+a}+\frac {c\,x^3+\frac {b}{2}}{\sqrt {c}}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}-\frac {x^3\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{4\,c}+\frac {5\,b\,\left (\frac {\left (8\,c\,\left (c\,x^6+a\right )-3\,b^2+2\,b\,c\,x^3\right )\,\sqrt {c\,x^6+b\,x^3+a}}{24\,c^2}+\frac {\ln \left (2\,\sqrt {c\,x^6+b\,x^3+a}+\frac {2\,c\,x^3+b}{\sqrt {c}}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}\right )}{8\,c}\right )}{10\,c}-\frac {2\,a\,\left (\frac {\left (8\,c\,\left (c\,x^6+a\right )-3\,b^2+2\,b\,c\,x^3\right )\,\sqrt {c\,x^6+b\,x^3+a}}{24\,c^2}+\frac {\ln \left (2\,\sqrt {c\,x^6+b\,x^3+a}+\frac {2\,c\,x^3+b}{\sqrt {c}}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}\right )}{5\,c}\right )}{4\,c}+\frac {a\,\left (\frac {a\,\left (\left (\frac {b}{4\,c}+\frac {x^3}{2}\right )\,\sqrt {c\,x^6+b\,x^3+a}+\frac {\ln \left (\sqrt {c\,x^6+b\,x^3+a}+\frac {c\,x^3+\frac {b}{2}}{\sqrt {c}}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}-\frac {x^3\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{4\,c}+\frac {5\,b\,\left (\frac {\left (8\,c\,\left (c\,x^6+a\right )-3\,b^2+2\,b\,c\,x^3\right )\,\sqrt {c\,x^6+b\,x^3+a}}{24\,c^2}+\frac {\ln \left (2\,\sqrt {c\,x^6+b\,x^3+a}+\frac {2\,c\,x^3+b}{\sqrt {c}}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}\right )}{8\,c}\right )}{6\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14*(a + b*x^3 + c*x^6)^(1/2),x)

[Out]

(x^9*(a + b*x^3 + c*x^6)^(3/2))/(18*c) - (b*((x^6*(a + b*x^3 + c*x^6)^(3/2))/(5*c) + (7*b*((a*((b/(4*c) + x^3/
2)*(a + b*x^3 + c*x^6)^(1/2) + (log((a + b*x^3 + c*x^6)^(1/2) + (b/2 + c*x^3)/c^(1/2))*(a*c - b^2/4))/(2*c^(3/
2))))/(4*c) - (x^3*(a + b*x^3 + c*x^6)^(3/2))/(4*c) + (5*b*(((8*c*(a + c*x^6) - 3*b^2 + 2*b*c*x^3)*(a + b*x^3
+ c*x^6)^(1/2))/(24*c^2) + (log(2*(a + b*x^3 + c*x^6)^(1/2) + (b + 2*c*x^3)/c^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5
/2))))/(8*c)))/(10*c) - (2*a*(((8*c*(a + c*x^6) - 3*b^2 + 2*b*c*x^3)*(a + b*x^3 + c*x^6)^(1/2))/(24*c^2) + (lo
g(2*(a + b*x^3 + c*x^6)^(1/2) + (b + 2*c*x^3)/c^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2))))/(5*c)))/(4*c) + (a*((a*
((b/(4*c) + x^3/2)*(a + b*x^3 + c*x^6)^(1/2) + (log((a + b*x^3 + c*x^6)^(1/2) + (b/2 + c*x^3)/c^(1/2))*(a*c -
b^2/4))/(2*c^(3/2))))/(4*c) - (x^3*(a + b*x^3 + c*x^6)^(3/2))/(4*c) + (5*b*(((8*c*(a + c*x^6) - 3*b^2 + 2*b*c*
x^3)*(a + b*x^3 + c*x^6)^(1/2))/(24*c^2) + (log(2*(a + b*x^3 + c*x^6)^(1/2) + (b + 2*c*x^3)/c^(1/2))*(b^3 - 4*
a*b*c))/(16*c^(5/2))))/(8*c)))/(6*c)

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