\(\int x^{14} (a+b x^3+c x^6)^{3/2} \, dx\) [201]

Optimal result

Integrand size = 20, antiderivative size = 293 \[ \int x^{14} \left (a+b x^3+c x^6\right )^{3/2} \, dx=-\frac {\left (b^2-4 a c\right ) \left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{16384 c^6}+\frac {\left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{6144 c^5}-\frac {11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac {x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c}-\frac {\left (3 b \left (77 b^2-124 a c\right )-10 c \left (33 b^2-28 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}+\frac {\left (b^2-4 a c\right )^2 \left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{32768 c^{13/2}} \]

[Out]

1/6144*(16*a^2*c^2-72*a*b^2*c+33*b^4)*(2*c*x^3+b)*(c*x^6+b*x^3+a)^(3/2)/c^5-11/336*b*x^6*(c*x^6+b*x^3+a)^(5/2)
/c^2+1/24*x^9*(c*x^6+b*x^3+a)^(5/2)/c-1/13440*(3*b*(-124*a*c+77*b^2)-10*c*(-28*a*c+33*b^2)*x^3)*(c*x^6+b*x^3+a
)^(5/2)/c^4+1/32768*(-4*a*c+b^2)^2*(16*a^2*c^2-72*a*b^2*c+33*b^4)*arctanh(1/2*(2*c*x^3+b)/c^(1/2)/(c*x^6+b*x^3
+a)^(1/2))/c^(13/2)-1/16384*(-4*a*c+b^2)*(16*a^2*c^2-72*a*b^2*c+33*b^4)*(2*c*x^3+b)*(c*x^6+b*x^3+a)^(1/2)/c^6

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1371, 756, 846, 793, 626, 635, 212} \[ \int x^{14} \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {\left (b^2-4 a c\right )^2 \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{32768 c^{13/2}}-\frac {\left (b^2-4 a c\right ) \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{16384 c^6}+\frac {\left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{6144 c^5}-\frac {\left (3 b \left (77 b^2-124 a c\right )-10 c x^3 \left (33 b^2-28 a c\right )\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}-\frac {11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac {x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c} \]

[In]

Int[x^14*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

-1/16384*((b^2 - 4*a*c)*(33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/c^6 + ((33*b
^4 - 72*a*b^2*c + 16*a^2*c^2)*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(6144*c^5) - (11*b*x^6*(a + b*x^3 + c*x
^6)^(5/2))/(336*c^2) + (x^9*(a + b*x^3 + c*x^6)^(5/2))/(24*c) - ((3*b*(77*b^2 - 124*a*c) - 10*c*(33*b^2 - 28*a
*c)*x^3)*(a + b*x^3 + c*x^6)^(5/2))/(13440*c^4) + ((b^2 - 4*a*c)^2*(33*b^4 - 72*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])])/(32768*c^(13/2))

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*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int x^4 \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right ) \\ & = \frac {x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c}+\frac {\text {Subst}\left (\int x^2 \left (-3 a-\frac {11 b x}{2}\right ) \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{24 c} \\ & = -\frac {11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac {x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c}+\frac {\text {Subst}\left (\int x \left (11 a b+\frac {3}{4} \left (33 b^2-28 a c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{168 c^2} \\ & = -\frac {11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac {x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c}-\frac {\left (3 b \left (77 b^2-124 a c\right )-10 c \left (33 b^2-28 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}+\frac {\left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \text {Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{768 c^4} \\ & = \frac {\left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{6144 c^5}-\frac {11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac {x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c}-\frac {\left (3 b \left (77 b^2-124 a c\right )-10 c \left (33 b^2-28 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}-\frac {\left (\left (b^2-4 a c\right ) \left (33 b^4-72 a b^2 c+16 a^2 c^2\right )\right ) \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^3\right )}{4096 c^5} \\ & = -\frac {\left (b^2-4 a c\right ) \left (33 b^4-72 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}}{16384 c^6}+\frac {\left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{6144 c^5}-\frac {11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac {x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c}-\frac {\left (3 b \left (77 b^2-124 a c\right )-10 c \left (33 b^2-28 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}+\frac {\left (\left (b^2-4 a c\right )^2 \left (33 b^4-72 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 )}{32768 c^6} \\ & = -\frac {\left (b^2-4 a c\right ) \left (33 b^4-72 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}}{16384 c^6}+\frac {\left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{6144 c^5}-\frac {11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac {x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c}-\frac {\left (3 b \left (77 b^2-124 a c\right )-10 c \left (33 b^2-28 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}+\frac {\left (\left (b^2-4 a c\right )^2 \left (33 b^4-72 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 )}{16384 c^6} \\ & = -\frac {\left (b^2-4 a c\right ) \left (33 b^4-72 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}}{16384 c^6}+\frac {\left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{6144 c^5}-\frac {11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac {x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c}-\frac {\left (3 b \left (77 b^2-124 a c\right )-10 c \left (33 b^2-28 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}+\frac {\left (b^2-4 a c\right )^2 \left (33 b^4-72 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 )}{32768 c^{13/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.99 \[ \int x^{14} \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {2 \sqrt {c} \sqrt {a+b x^3+c x^6} \left (-3465 b^7+2310 b^6 c x^3+84 b^5 c \left (365 a-22 c x^6\right )+24 b^4 c^2 x^3 \left (-749 a+66 c x^6\right )+32 b^2 c^3 x^3 \left (1181 a^2-284 a c x^6+40 c^2 x^{12}\right )-16 b^3 c^2 \left (5103 a^2-780 a c x^6+88 c^2 x^{12}\right )+4480 c^4 x^3 \left (-3 a^3+2 a^2 c x^6+24 a c^2 x^{12}+16 c^3 x^{18}\right )+64 b c^3 \left (919 a^3-302 a^2 c x^6+104 a c^2 x^{12}+1360 c^3 x^{18}\right )\right )-105 \left (b^2-4 a c\right )^2 \left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \log \left (b+2 c x^3-2 \sqrt {c} \sqrt {a+b x^3+c x^6}\right )}{3440640 c^{13/2}} \]

[In]

Integrate[x^14*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]*(-3465*b^7 + 2310*b^6*c*x^3 + 84*b^5*c*(365*a - 22*c*x^6) + 24*b^4*c^2*x^3*
(-749*a + 66*c*x^6) + 32*b^2*c^3*x^3*(1181*a^2 - 284*a*c*x^6 + 40*c^2*x^12) - 16*b^3*c^2*(5103*a^2 - 780*a*c*x
^6 + 88*c^2*x^12) + 4480*c^4*x^3*(-3*a^3 + 2*a^2*c*x^6 + 24*a*c^2*x^12 + 16*c^3*x^18) + 64*b*c^3*(919*a^3 - 30
2*a^2*c*x^6 + 104*a*c^2*x^12 + 1360*c^3*x^18)) - 105*(b^2 - 4*a*c)^2*(33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*Log[b
+ 2*c*x^3 - 2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]])/(3440640*c^(13/2))

Maple [F]

\[\int x^{14} \left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 641, normalized size of antiderivative = 2.19 \[ \int x^{14} \left (a+b x^3+c x^6\right )^{3/2} \, dx=\left [\frac {105 \, {\left (33 \, b^{8} - 336 \, a b^{6} c + 1120 \, a^{2} b^{4} c^{2} - 1280 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\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 (71680 \, c^{8} x^{21} + 87040 \, b c^{7} x^{18} + 1280 \, {\left (b^{2} c^{6} + 84 \, a c^{7}\right )} x^{15} - 128 \, {\left (11 \, b^{3} c^{5} - 52 \, a b c^{6}\right )} x^{12} + 16 \, {\left (99 \, b^{4} c^{4} - 568 \, a b^{2} c^{5} + 560 \, a^{2} c^{6}\right )} x^{9} - 3465 \, b^{7} c + 30660 \, a b^{5} c^{2} - 81648 \, a^{2} b^{3} c^{3} + 58816 \, a^{3} b c^{4} - 8 \, {\left (231 \, b^{5} c^{3} - 1560 \, a b^{3} c^{4} + 2416 \, a^{2} b c^{5}\right )} x^{6} + 2 \, {\left (1155 \, b^{6} c^{2} - 8988 \, a b^{4} c^{3} + 18896 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{6881280 \, c^{7}}, -\frac {105 \, {\left (33 \, b^{8} - 336 \, a b^{6} c + 1120 \, a^{2} b^{4} c^{2} - 1280 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\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 (71680 \, c^{8} x^{21} + 87040 \, b c^{7} x^{18} + 1280 \, {\left (b^{2} c^{6} + 84 \, a c^{7}\right )} x^{15} - 128 \, {\left (11 \, b^{3} c^{5} - 52 \, a b c^{6}\right )} x^{12} + 16 \, {\left (99 \, b^{4} c^{4} - 568 \, a b^{2} c^{5} + 560 \, a^{2} c^{6}\right )} x^{9} - 3465 \, b^{7} c + 30660 \, a b^{5} c^{2} - 81648 \, a^{2} b^{3} c^{3} + 58816 \, a^{3} b c^{4} - 8 \, {\left (231 \, b^{5} c^{3} - 1560 \, a b^{3} c^{4} + 2416 \, a^{2} b c^{5}\right )} x^{6} + 2 \, {\left (1155 \, b^{6} c^{2} - 8988 \, a b^{4} c^{3} + 18896 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{3440640 \, c^{7}}\right ] \]

[In]

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

[Out]

[1/6881280*(105*(33*b^8 - 336*a*b^6*c + 1120*a^2*b^4*c^2 - 1280*a^3*b^2*c^3 + 256*a^4*c^4)*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*(71680*c^8*x^21 + 87040*b
*c^7*x^18 + 1280*(b^2*c^6 + 84*a*c^7)*x^15 - 128*(11*b^3*c^5 - 52*a*b*c^6)*x^12 + 16*(99*b^4*c^4 - 568*a*b^2*c
^5 + 560*a^2*c^6)*x^9 - 3465*b^7*c + 30660*a*b^5*c^2 - 81648*a^2*b^3*c^3 + 58816*a^3*b*c^4 - 8*(231*b^5*c^3 -
1560*a*b^3*c^4 + 2416*a^2*b*c^5)*x^6 + 2*(1155*b^6*c^2 - 8988*a*b^4*c^3 + 18896*a^2*b^2*c^4 - 6720*a^3*c^5)*x^
3)*sqrt(c*x^6 + b*x^3 + a))/c^7, -1/3440640*(105*(33*b^8 - 336*a*b^6*c + 1120*a^2*b^4*c^2 - 1280*a^3*b^2*c^3 +
256*a^4*c^4)*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*(71680*c^8*x^21 + 87040*b*c^7*x^18 + 1280*(b^2*c^6 + 84*a*c^7)*x^15 - 128*(11*b^3*c^5 - 52*a*b*c^6)*x^12 + 1
6*(99*b^4*c^4 - 568*a*b^2*c^5 + 560*a^2*c^6)*x^9 - 3465*b^7*c + 30660*a*b^5*c^2 - 81648*a^2*b^3*c^3 + 58816*a^
3*b*c^4 - 8*(231*b^5*c^3 - 1560*a*b^3*c^4 + 2416*a^2*b*c^5)*x^6 + 2*(1155*b^6*c^2 - 8988*a*b^4*c^3 + 18896*a^2
*b^2*c^4 - 6720*a^3*c^5)*x^3)*sqrt(c*x^6 + b*x^3 + a))/c^7]

Sympy [F]

\[ \int x^{14} \left (a+b x^3+c x^6\right )^{3/2} \, dx=\int x^{14} \left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int x^{14} \left (a+b x^3+c x^6\right )^{3/2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

\[ \int x^{14} \left (a+b x^3+c x^6\right )^{3/2} \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}} x^{14} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^{14} \left (a+b x^3+c x^6\right )^{3/2} \, dx=\int x^{14}\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2} \,d x \]

[In]

int(x^14*(a + b*x^3 + c*x^6)^(3/2),x)

[Out]

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