### 3.103 $$\int x^2 (b+2 c x^3) (b x^3+c x^6)^{13} \, dx$$

Optimal. Leaf size=16 $\frac{1}{42} x^{42} \left (b+c x^3\right )^{14}$

[Out]

(x^42*(b + c*x^3)^14)/42

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Rubi [A]  time = 0.0557805, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.12, Rules used = {1584, 446, 74} $\frac{1}{42} x^{42} \left (b+c x^3\right )^{14}$

Antiderivative was successfully veriﬁed.

[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 1584

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]

Rule 446

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 74

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]

Rubi steps

\begin{align*} \int x^2 \left (b+2 c x^3\right ) \left (b x^3+c x^6\right )^{13} \, dx &=\int x^{41} \left (b+c x^3\right )^{13} \left (b+2 c x^3\right ) \, dx\\ &=\frac{1}{3} \operatorname{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]  time = 0.006226, size = 186, normalized size = 11.62 $\frac{13}{6} b^2 c^{12} x^{78}+\frac{26}{3} b^3 c^{11} x^{75}+\frac{143}{6} b^4 c^{10} x^{72}+\frac{143}{3} b^5 c^9 x^{69}+\frac{143}{2} b^6 c^8 x^{66}+\frac{572}{7} b^7 c^7 x^{63}+\frac{143}{2} b^8 c^6 x^{60}+\frac{143}{3} b^9 c^5 x^{57}+\frac{143}{6} b^{10} c^4 x^{54}+\frac{26}{3} b^{11} c^3 x^{51}+\frac{13}{6} b^{12} c^2 x^{48}+\frac{1}{3} b^{13} c x^{45}+\frac{b^{14} x^{42}}{42}+\frac{1}{3} b c^{13} x^{81}+\frac{c^{14} x^{84}}{42}$

Antiderivative was successfully veriﬁed.

[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

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Maple [B]  time = 0.004, size = 157, normalized size = 9.8 \begin{align*}{\frac{{c}^{14}{x}^{84}}{42}}+{\frac{b{c}^{13}{x}^{81}}{3}}+{\frac{13\,{b}^{2}{c}^{12}{x}^{78}}{6}}+{\frac{26\,{b}^{3}{c}^{11}{x}^{75}}{3}}+{\frac{143\,{b}^{4}{c}^{10}{x}^{72}}{6}}+{\frac{143\,{b}^{5}{c}^{9}{x}^{69}}{3}}+{\frac{143\,{b}^{6}{c}^{8}{x}^{66}}{2}}+{\frac{572\,{b}^{7}{c}^{7}{x}^{63}}{7}}+{\frac{143\,{b}^{8}{c}^{6}{x}^{60}}{2}}+{\frac{143\,{b}^{9}{c}^{5}{x}^{57}}{3}}+{\frac{143\,{b}^{10}{c}^{4}{x}^{54}}{6}}+{\frac{26\,{b}^{11}{c}^{3}{x}^{51}}{3}}+{\frac{13\,{b}^{12}{c}^{2}{x}^{48}}{6}}+{\frac{{b}^{13}c{x}^{45}}{3}}+{\frac{{b}^{14}{x}^{42}}{42}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[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+14
3/2*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

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Maxima [B]  time = 1.17024, size = 211, normalized size = 13.19 \begin{align*} \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} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 0.818319, size = 408, normalized size = 25.5 \begin{align*} \frac{1}{42} x^{84} c^{14} + \frac{1}{3} x^{81} c^{13} b + \frac{13}{6} x^{78} c^{12} b^{2} + \frac{26}{3} x^{75} c^{11} b^{3} + \frac{143}{6} x^{72} c^{10} b^{4} + \frac{143}{3} x^{69} c^{9} b^{5} + \frac{143}{2} x^{66} c^{8} b^{6} + \frac{572}{7} x^{63} c^{7} b^{7} + \frac{143}{2} x^{60} c^{6} b^{8} + \frac{143}{3} x^{57} c^{5} b^{9} + \frac{143}{6} x^{54} c^{4} b^{10} + \frac{26}{3} x^{51} c^{3} b^{11} + \frac{13}{6} x^{48} c^{2} b^{12} + \frac{1}{3} x^{45} c b^{13} + \frac{1}{42} x^{42} b^{14} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.120845, size = 185, normalized size = 11.56 \begin{align*} \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} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

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Giac [B]  time = 1.0941, size = 211, normalized size = 13.19 \begin{align*} \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} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[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