\(\int x^2 (b+2 c x^3) (b x^3+c x^6)^{13} \, dx\) [103]

Optimal result

Integrand size = 25, antiderivative size = 16 \[ \int x^2 \left (b+2 c x^3\right ) \left (b x^3+c x^6\right )^{13} \, dx=\frac {1}{42} x^{42} \left (b+c x^3\right )^{14} \]

[Out]

1/42*x^42*(c*x^3+b)^14

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1598, 457, 75} \[ \int x^2 \left (b+2 c x^3\right ) \left (b x^3+c x^6\right )^{13} \, dx=\frac {1}{42} x^{42} \left (b+c x^3\right )^{14} \]

[In]

Int[x^2*(b + 2*c*x^3)*(b*x^3 + c*x^6)^13,x]

[Out]

(x^42*(b + c*x^3)^14)/42

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps \begin{align*} \text {integral}& = \int x^{41} \left (b+c x^3\right )^{13} \left (b+2 c x^3\right ) \, dx \\ & = \frac {1}{3} \text {Subst}\left (\int x^{13} (b+c x)^{13} (b+2 c x) \, dx,x,x^3\right ) \\ & = \frac {1}{42} x^{42} \left (b+c x^3\right )^{14} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(186\) vs. \(2(16)=32\).

Time = 0.00 (sec) , antiderivative size = 186, normalized size of antiderivative = 11.62 \[ \int x^2 \left (b+2 c x^3\right ) \left (b x^3+c x^6\right )^{13} \, dx=\frac {b^{14} x^{42}}{42}+\frac {1}{3} b^{13} c x^{45}+\frac {13}{6} b^{12} c^2 x^{48}+\frac {26}{3} b^{11} c^3 x^{51}+\frac {143}{6} b^{10} c^4 x^{54}+\frac {143}{3} b^9 c^5 x^{57}+\frac {143}{2} b^8 c^6 x^{60}+\frac {572}{7} b^7 c^7 x^{63}+\frac {143}{2} b^6 c^8 x^{66}+\frac {143}{3} b^5 c^9 x^{69}+\frac {143}{6} b^4 c^{10} x^{72}+\frac {26}{3} b^3 c^{11} x^{75}+\frac {13}{6} b^2 c^{12} x^{78}+\frac {1}{3} b c^{13} x^{81}+\frac {c^{14} x^{84}}{42} \]

[In]

Integrate[x^2*(b + 2*c*x^3)*(b*x^3 + c*x^6)^13,x]

[Out]

(b^14*x^42)/42 + (b^13*c*x^45)/3 + (13*b^12*c^2*x^48)/6 + (26*b^11*c^3*x^51)/3 + (143*b^10*c^4*x^54)/6 + (143*
b^9*c^5*x^57)/3 + (143*b^8*c^6*x^60)/2 + (572*b^7*c^7*x^63)/7 + (143*b^6*c^8*x^66)/2 + (143*b^5*c^9*x^69)/3 +
(143*b^4*c^10*x^72)/6 + (26*b^3*c^11*x^75)/3 + (13*b^2*c^12*x^78)/6 + (b*c^13*x^81)/3 + (c^14*x^84)/42

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
gosper	\(\frac {x^{42} \left (c \,x^{3}+b \right )^{14}}{42}\)	\(15\)
default	\(\frac {\left (b^{2}-\left (2 c \,x^{3}+b \right )^{2}\right )^{14}}{11274289152 c^{14}}\)	\(24\)
risch	\(\frac {1}{42} c^{14} x^{84}+\frac {13}{6} x^{78} b^{2} c^{12}+\frac {1}{3} b \,c^{13} x^{81}+\frac {143}{3} x^{69} b^{5} c^{9}+\frac {143}{6} x^{72} b^{4} c^{10}+\frac {26}{3} x^{75} b^{3} c^{11}+\frac {572}{7} x^{63} b^{7} c^{7}+\frac {143}{2} x^{66} b^{6} c^{8}+\frac {143}{6} x^{54} b^{10} c^{4}+\frac {143}{3} x^{57} b^{9} c^{5}+\frac {143}{2} x^{60} b^{8} c^{6}+\frac {13}{6} x^{48} b^{12} c^{2}+\frac {26}{3} x^{51} b^{11} c^{3}+\frac {1}{42} x^{42} b^{14}+\frac {1}{3} x^{45} b^{13} c\)	\(157\)
parallelrisch	\(\frac {1}{42} c^{14} x^{84}+\frac {13}{6} x^{78} b^{2} c^{12}+\frac {1}{3} b \,c^{13} x^{81}+\frac {143}{3} x^{69} b^{5} c^{9}+\frac {143}{6} x^{72} b^{4} c^{10}+\frac {26}{3} x^{75} b^{3} c^{11}+\frac {572}{7} x^{63} b^{7} c^{7}+\frac {143}{2} x^{66} b^{6} c^{8}+\frac {143}{6} x^{54} b^{10} c^{4}+\frac {143}{3} x^{57} b^{9} c^{5}+\frac {143}{2} x^{60} b^{8} c^{6}+\frac {13}{6} x^{48} b^{12} c^{2}+\frac {26}{3} x^{51} b^{11} c^{3}+\frac {1}{42} x^{42} b^{14}+\frac {1}{3} x^{45} b^{13} c\)	\(157\)

[In]

int(x^2*(2*c*x^3+b)*(c*x^6+b*x^3)^13,x,method=_RETURNVERBOSE)

[Out]

1/42*x^42*(c*x^3+b)^14

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (14) = 28\).

Time = 0.25 (sec) , antiderivative size = 156, normalized size of antiderivative = 9.75 \[ \int x^2 \left (b+2 c x^3\right ) \left (b x^3+c x^6\right )^{13} \, dx=\frac {1}{42} \, c^{14} x^{84} + \frac {1}{3} \, b c^{13} x^{81} + \frac {13}{6} \, b^{2} c^{12} x^{78} + \frac {26}{3} \, b^{3} c^{11} x^{75} + \frac {143}{6} \, b^{4} c^{10} x^{72} + \frac {143}{3} \, b^{5} c^{9} x^{69} + \frac {143}{2} \, b^{6} c^{8} x^{66} + \frac {572}{7} \, b^{7} c^{7} x^{63} + \frac {143}{2} \, b^{8} c^{6} x^{60} + \frac {143}{3} \, b^{9} c^{5} x^{57} + \frac {143}{6} \, b^{10} c^{4} x^{54} + \frac {26}{3} \, b^{11} c^{3} x^{51} + \frac {13}{6} \, b^{12} c^{2} x^{48} + \frac {1}{3} \, b^{13} c x^{45} + \frac {1}{42} \, b^{14} x^{42} \]

[In]

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

[Out]

1/42*c^14*x^84 + 1/3*b*c^13*x^81 + 13/6*b^2*c^12*x^78 + 26/3*b^3*c^11*x^75 + 143/6*b^4*c^10*x^72 + 143/3*b^5*c
^9*x^69 + 143/2*b^6*c^8*x^66 + 572/7*b^7*c^7*x^63 + 143/2*b^8*c^6*x^60 + 143/3*b^9*c^5*x^57 + 143/6*b^10*c^4*x
^54 + 26/3*b^11*c^3*x^51 + 13/6*b^12*c^2*x^48 + 1/3*b^13*c*x^45 + 1/42*b^14*x^42

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (12) = 24\).

Time = 0.05 (sec) , antiderivative size = 185, normalized size of antiderivative = 11.56 \[ \int x^2 \left (b+2 c x^3\right ) \left (b x^3+c x^6\right )^{13} \, dx=\frac {b^{14} x^{42}}{42} + \frac {b^{13} c x^{45}}{3} + \frac {13 b^{12} c^{2} x^{48}}{6} + \frac {26 b^{11} c^{3} x^{51}}{3} + \frac {143 b^{10} c^{4} x^{54}}{6} + \frac {143 b^{9} c^{5} x^{57}}{3} + \frac {143 b^{8} c^{6} x^{60}}{2} + \frac {572 b^{7} c^{7} x^{63}}{7} + \frac {143 b^{6} c^{8} x^{66}}{2} + \frac {143 b^{5} c^{9} x^{69}}{3} + \frac {143 b^{4} c^{10} x^{72}}{6} + \frac {26 b^{3} c^{11} x^{75}}{3} + \frac {13 b^{2} c^{12} x^{78}}{6} + \frac {b c^{13} x^{81}}{3} + \frac {c^{14} x^{84}}{42} \]

[In]

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

[Out]

b**14*x**42/42 + b**13*c*x**45/3 + 13*b**12*c**2*x**48/6 + 26*b**11*c**3*x**51/3 + 143*b**10*c**4*x**54/6 + 14
3*b**9*c**5*x**57/3 + 143*b**8*c**6*x**60/2 + 572*b**7*c**7*x**63/7 + 143*b**6*c**8*x**66/2 + 143*b**5*c**9*x*
*69/3 + 143*b**4*c**10*x**72/6 + 26*b**3*c**11*x**75/3 + 13*b**2*c**12*x**78/6 + b*c**13*x**81/3 + c**14*x**84
/42

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (14) = 28\).

Time = 0.19 (sec) , antiderivative size = 156, normalized size of antiderivative = 9.75 \[ \int x^2 \left (b+2 c x^3\right ) \left (b x^3+c x^6\right )^{13} \, dx=\frac {1}{42} \, c^{14} x^{84} + \frac {1}{3} \, b c^{13} x^{81} + \frac {13}{6} \, b^{2} c^{12} x^{78} + \frac {26}{3} \, b^{3} c^{11} x^{75} + \frac {143}{6} \, b^{4} c^{10} x^{72} + \frac {143}{3} \, b^{5} c^{9} x^{69} + \frac {143}{2} \, b^{6} c^{8} x^{66} + \frac {572}{7} \, b^{7} c^{7} x^{63} + \frac {143}{2} \, b^{8} c^{6} x^{60} + \frac {143}{3} \, b^{9} c^{5} x^{57} + \frac {143}{6} \, b^{10} c^{4} x^{54} + \frac {26}{3} \, b^{11} c^{3} x^{51} + \frac {13}{6} \, b^{12} c^{2} x^{48} + \frac {1}{3} \, b^{13} c x^{45} + \frac {1}{42} \, b^{14} x^{42} \]

[In]

integrate(x^2*(2*c*x^3+b)*(c*x^6+b*x^3)^13,x, algorithm="maxima")

[Out]

1/42*c^14*x^84 + 1/3*b*c^13*x^81 + 13/6*b^2*c^12*x^78 + 26/3*b^3*c^11*x^75 + 143/6*b^4*c^10*x^72 + 143/3*b^5*c
^9*x^69 + 143/2*b^6*c^8*x^66 + 572/7*b^7*c^7*x^63 + 143/2*b^8*c^6*x^60 + 143/3*b^9*c^5*x^57 + 143/6*b^10*c^4*x
^54 + 26/3*b^11*c^3*x^51 + 13/6*b^12*c^2*x^48 + 1/3*b^13*c*x^45 + 1/42*b^14*x^42

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int x^2 \left (b+2 c x^3\right ) \left (b x^3+c x^6\right )^{13} \, dx=\frac {1}{42} \, {\left (c x^{6} + b x^{3}\right )}^{14} \]

[In]

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

[Out]

1/42*(c*x^6 + b*x^3)^14

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 156, normalized size of antiderivative = 9.75 \[ \int x^2 \left (b+2 c x^3\right ) \left (b x^3+c x^6\right )^{13} \, dx=\frac {b^{14}\,x^{42}}{42}+\frac {b^{13}\,c\,x^{45}}{3}+\frac {13\,b^{12}\,c^2\,x^{48}}{6}+\frac {26\,b^{11}\,c^3\,x^{51}}{3}+\frac {143\,b^{10}\,c^4\,x^{54}}{6}+\frac {143\,b^9\,c^5\,x^{57}}{3}+\frac {143\,b^8\,c^6\,x^{60}}{2}+\frac {572\,b^7\,c^7\,x^{63}}{7}+\frac {143\,b^6\,c^8\,x^{66}}{2}+\frac {143\,b^5\,c^9\,x^{69}}{3}+\frac {143\,b^4\,c^{10}\,x^{72}}{6}+\frac {26\,b^3\,c^{11}\,x^{75}}{3}+\frac {13\,b^2\,c^{12}\,x^{78}}{6}+\frac {b\,c^{13}\,x^{81}}{3}+\frac {c^{14}\,x^{84}}{42} \]

[In]

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

[Out]

(b^14*x^42)/42 + (c^14*x^84)/42 + (b^13*c*x^45)/3 + (b*c^13*x^81)/3 + (13*b^12*c^2*x^48)/6 + (26*b^11*c^3*x^51
)/3 + (143*b^10*c^4*x^54)/6 + (143*b^9*c^5*x^57)/3 + (143*b^8*c^6*x^60)/2 + (572*b^7*c^7*x^63)/7 + (143*b^6*c^
8*x^66)/2 + (143*b^5*c^9*x^69)/3 + (143*b^4*c^10*x^72)/6 + (26*b^3*c^11*x^75)/3 + (13*b^2*c^12*x^78)/6