3.102 $$\int x (b+2 c x^2) (b x^2+c x^4)^{13} \, dx$$

Optimal. Leaf size=16 $\frac{1}{28} x^{28} \left (b+c x^2\right )^{14}$

[Out]

(x^28*(b + c*x^2)^14)/28

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Rubi [A]  time = 0.0541807, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.13, Rules used = {1584, 446, 74} $\frac{1}{28} x^{28} \left (b+c x^2\right )^{14}$

Antiderivative was successfully veriﬁed.

[In]

Int[x*(b + 2*c*x^2)*(b*x^2 + c*x^4)^13,x]

[Out]

(x^28*(b + c*x^2)^14)/28

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 \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{13} \, dx &=\int x^{27} \left (b+c x^2\right )^{13} \left (b+2 c x^2\right ) \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int x^{13} (b+c x)^{13} (b+2 c x) \, dx,x,x^2\right )\\ &=\frac{1}{28} x^{28} \left (b+c x^2\right )^{14}\\ \end{align*}

Mathematica [B]  time = 0.0058507, size = 182, normalized size = 11.38 $\frac{13}{4} b^2 c^{12} x^{52}+13 b^3 c^{11} x^{50}+\frac{143}{4} b^4 c^{10} x^{48}+\frac{143}{2} b^5 c^9 x^{46}+\frac{429}{4} b^6 c^8 x^{44}+\frac{858}{7} b^7 c^7 x^{42}+\frac{429}{4} b^8 c^6 x^{40}+\frac{143}{2} b^9 c^5 x^{38}+\frac{143}{4} b^{10} c^4 x^{36}+13 b^{11} c^3 x^{34}+\frac{13}{4} b^{12} c^2 x^{32}+\frac{1}{2} b^{13} c x^{30}+\frac{b^{14} x^{28}}{28}+\frac{1}{2} b c^{13} x^{54}+\frac{c^{14} x^{56}}{28}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*(b + 2*c*x^2)*(b*x^2 + c*x^4)^13,x]

[Out]

(b^14*x^28)/28 + (b^13*c*x^30)/2 + (13*b^12*c^2*x^32)/4 + 13*b^11*c^3*x^34 + (143*b^10*c^4*x^36)/4 + (143*b^9*
c^5*x^38)/2 + (429*b^8*c^6*x^40)/4 + (858*b^7*c^7*x^42)/7 + (429*b^6*c^8*x^44)/4 + (143*b^5*c^9*x^46)/2 + (143
*b^4*c^10*x^48)/4 + 13*b^3*c^11*x^50 + (13*b^2*c^12*x^52)/4 + (b*c^13*x^54)/2 + (c^14*x^56)/28

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Maple [B]  time = 0.004, size = 157, normalized size = 9.8 \begin{align*}{\frac{{c}^{14}{x}^{56}}{28}}+{\frac{b{c}^{13}{x}^{54}}{2}}+{\frac{13\,{b}^{2}{c}^{12}{x}^{52}}{4}}+13\,{b}^{3}{c}^{11}{x}^{50}+{\frac{143\,{b}^{4}{c}^{10}{x}^{48}}{4}}+{\frac{143\,{b}^{5}{c}^{9}{x}^{46}}{2}}+{\frac{429\,{b}^{6}{c}^{8}{x}^{44}}{4}}+{\frac{858\,{b}^{7}{c}^{7}{x}^{42}}{7}}+{\frac{429\,{b}^{8}{c}^{6}{x}^{40}}{4}}+{\frac{143\,{b}^{9}{c}^{5}{x}^{38}}{2}}+{\frac{143\,{b}^{10}{c}^{4}{x}^{36}}{4}}+13\,{b}^{11}{c}^{3}{x}^{34}+{\frac{13\,{b}^{12}{c}^{2}{x}^{32}}{4}}+{\frac{{b}^{13}c{x}^{30}}{2}}+{\frac{{b}^{14}{x}^{28}}{28}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*(2*c*x^2+b)*(c*x^4+b*x^2)^13,x)

[Out]

1/28*c^14*x^56+1/2*b*c^13*x^54+13/4*b^2*c^12*x^52+13*b^3*c^11*x^50+143/4*b^4*c^10*x^48+143/2*b^5*c^9*x^46+429/
4*b^6*c^8*x^44+858/7*b^7*c^7*x^42+429/4*b^8*c^6*x^40+143/2*b^9*c^5*x^38+143/4*b^10*c^4*x^36+13*b^11*c^3*x^34+1
3/4*b^12*c^2*x^32+1/2*b^13*c*x^30+1/28*b^14*x^28

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Maxima [B]  time = 1.13387, size = 211, normalized size = 13.19 \begin{align*} \frac{1}{28} \, c^{14} x^{56} + \frac{1}{2} \, b c^{13} x^{54} + \frac{13}{4} \, b^{2} c^{12} x^{52} + 13 \, b^{3} c^{11} x^{50} + \frac{143}{4} \, b^{4} c^{10} x^{48} + \frac{143}{2} \, b^{5} c^{9} x^{46} + \frac{429}{4} \, b^{6} c^{8} x^{44} + \frac{858}{7} \, b^{7} c^{7} x^{42} + \frac{429}{4} \, b^{8} c^{6} x^{40} + \frac{143}{2} \, b^{9} c^{5} x^{38} + \frac{143}{4} \, b^{10} c^{4} x^{36} + 13 \, b^{11} c^{3} x^{34} + \frac{13}{4} \, b^{12} c^{2} x^{32} + \frac{1}{2} \, b^{13} c x^{30} + \frac{1}{28} \, b^{14} x^{28} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*c*x^2+b)*(c*x^4+b*x^2)^13,x, algorithm="maxima")

[Out]

1/28*c^14*x^56 + 1/2*b*c^13*x^54 + 13/4*b^2*c^12*x^52 + 13*b^3*c^11*x^50 + 143/4*b^4*c^10*x^48 + 143/2*b^5*c^9
*x^46 + 429/4*b^6*c^8*x^44 + 858/7*b^7*c^7*x^42 + 429/4*b^8*c^6*x^40 + 143/2*b^9*c^5*x^38 + 143/4*b^10*c^4*x^3
6 + 13*b^11*c^3*x^34 + 13/4*b^12*c^2*x^32 + 1/2*b^13*c*x^30 + 1/28*b^14*x^28

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Fricas [B]  time = 0.897156, size = 402, normalized size = 25.12 \begin{align*} \frac{1}{28} x^{56} c^{14} + \frac{1}{2} x^{54} c^{13} b + \frac{13}{4} x^{52} c^{12} b^{2} + 13 x^{50} c^{11} b^{3} + \frac{143}{4} x^{48} c^{10} b^{4} + \frac{143}{2} x^{46} c^{9} b^{5} + \frac{429}{4} x^{44} c^{8} b^{6} + \frac{858}{7} x^{42} c^{7} b^{7} + \frac{429}{4} x^{40} c^{6} b^{8} + \frac{143}{2} x^{38} c^{5} b^{9} + \frac{143}{4} x^{36} c^{4} b^{10} + 13 x^{34} c^{3} b^{11} + \frac{13}{4} x^{32} c^{2} b^{12} + \frac{1}{2} x^{30} c b^{13} + \frac{1}{28} x^{28} b^{14} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*c*x^2+b)*(c*x^4+b*x^2)^13,x, algorithm="fricas")

[Out]

1/28*x^56*c^14 + 1/2*x^54*c^13*b + 13/4*x^52*c^12*b^2 + 13*x^50*c^11*b^3 + 143/4*x^48*c^10*b^4 + 143/2*x^46*c^
9*b^5 + 429/4*x^44*c^8*b^6 + 858/7*x^42*c^7*b^7 + 429/4*x^40*c^6*b^8 + 143/2*x^38*c^5*b^9 + 143/4*x^36*c^4*b^1
0 + 13*x^34*c^3*b^11 + 13/4*x^32*c^2*b^12 + 1/2*x^30*c*b^13 + 1/28*x^28*b^14

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Sympy [B]  time = 0.156746, size = 182, normalized size = 11.38 \begin{align*} \frac{b^{14} x^{28}}{28} + \frac{b^{13} c x^{30}}{2} + \frac{13 b^{12} c^{2} x^{32}}{4} + 13 b^{11} c^{3} x^{34} + \frac{143 b^{10} c^{4} x^{36}}{4} + \frac{143 b^{9} c^{5} x^{38}}{2} + \frac{429 b^{8} c^{6} x^{40}}{4} + \frac{858 b^{7} c^{7} x^{42}}{7} + \frac{429 b^{6} c^{8} x^{44}}{4} + \frac{143 b^{5} c^{9} x^{46}}{2} + \frac{143 b^{4} c^{10} x^{48}}{4} + 13 b^{3} c^{11} x^{50} + \frac{13 b^{2} c^{12} x^{52}}{4} + \frac{b c^{13} x^{54}}{2} + \frac{c^{14} x^{56}}{28} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*c*x**2+b)*(c*x**4+b*x**2)**13,x)

[Out]

b**14*x**28/28 + b**13*c*x**30/2 + 13*b**12*c**2*x**32/4 + 13*b**11*c**3*x**34 + 143*b**10*c**4*x**36/4 + 143*
b**9*c**5*x**38/2 + 429*b**8*c**6*x**40/4 + 858*b**7*c**7*x**42/7 + 429*b**6*c**8*x**44/4 + 143*b**5*c**9*x**4
6/2 + 143*b**4*c**10*x**48/4 + 13*b**3*c**11*x**50 + 13*b**2*c**12*x**52/4 + b*c**13*x**54/2 + c**14*x**56/28

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Giac [B]  time = 1.08891, size = 211, normalized size = 13.19 \begin{align*} \frac{1}{28} \, c^{14} x^{56} + \frac{1}{2} \, b c^{13} x^{54} + \frac{13}{4} \, b^{2} c^{12} x^{52} + 13 \, b^{3} c^{11} x^{50} + \frac{143}{4} \, b^{4} c^{10} x^{48} + \frac{143}{2} \, b^{5} c^{9} x^{46} + \frac{429}{4} \, b^{6} c^{8} x^{44} + \frac{858}{7} \, b^{7} c^{7} x^{42} + \frac{429}{4} \, b^{8} c^{6} x^{40} + \frac{143}{2} \, b^{9} c^{5} x^{38} + \frac{143}{4} \, b^{10} c^{4} x^{36} + 13 \, b^{11} c^{3} x^{34} + \frac{13}{4} \, b^{12} c^{2} x^{32} + \frac{1}{2} \, b^{13} c x^{30} + \frac{1}{28} \, b^{14} x^{28} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*c*x^2+b)*(c*x^4+b*x^2)^13,x, algorithm="giac")

[Out]

1/28*c^14*x^56 + 1/2*b*c^13*x^54 + 13/4*b^2*c^12*x^52 + 13*b^3*c^11*x^50 + 143/4*b^4*c^10*x^48 + 143/2*b^5*c^9
*x^46 + 429/4*b^6*c^8*x^44 + 858/7*b^7*c^7*x^42 + 429/4*b^8*c^6*x^40 + 143/2*b^9*c^5*x^38 + 143/4*b^10*c^4*x^3
6 + 13*b^11*c^3*x^34 + 13/4*b^12*c^2*x^32 + 1/2*b^13*c*x^30 + 1/28*b^14*x^28