### 3.104 $$\int x^{-1+n} (b+2 c x^n) (b x^n+c x^{2 n})^{13} \, dx$$

Optimal. Leaf size=21 $\frac{x^{14 n} \left (b+c x^n\right )^{14}}{14 n}$

[Out]

(x^(14*n)*(b + c*x^n)^14)/(14*n)

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Rubi [A]  time = 0.0327483, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.103, Rules used = {1584, 446, 74} $\frac{x^{14 n} \left (b+c x^n\right )^{14}}{14 n}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^(-1 + n)*(b + 2*c*x^n)*(b*x^n + c*x^(2*n))^13,x]

[Out]

(x^(14*n)*(b + c*x^n)^14)/(14*n)

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

Mathematica [A]  time = 0.118071, size = 21, normalized size = 1. $\frac{x^{14 n} \left (b+c x^n\right )^{14}}{14 n}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^(-1 + n)*(b + 2*c*x^n)*(b*x^n + c*x^(2*n))^13,x]

[Out]

(x^(14*n)*(b + c*x^n)^14)/(14*n)

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Maple [B]  time = 0.033, size = 230, normalized size = 11. \begin{align*}{\frac{{c}^{14} \left ({x}^{n} \right ) ^{28}}{14\,n}}+{\frac{b{c}^{13} \left ({x}^{n} \right ) ^{27}}{n}}+{\frac{13\,{c}^{12} \left ({x}^{n} \right ) ^{26}{b}^{2}}{2\,n}}+26\,{\frac{{b}^{3}{c}^{11} \left ({x}^{n} \right ) ^{25}}{n}}+{\frac{143\,{c}^{10} \left ({x}^{n} \right ) ^{24}{b}^{4}}{2\,n}}+143\,{\frac{{b}^{5}{c}^{9} \left ({x}^{n} \right ) ^{23}}{n}}+{\frac{429\,{c}^{8} \left ({x}^{n} \right ) ^{22}{b}^{6}}{2\,n}}+{\frac{1716\,{b}^{7}{c}^{7} \left ({x}^{n} \right ) ^{21}}{7\,n}}+{\frac{429\,{c}^{6} \left ({x}^{n} \right ) ^{20}{b}^{8}}{2\,n}}+143\,{\frac{{b}^{9}{c}^{5} \left ({x}^{n} \right ) ^{19}}{n}}+{\frac{143\,{c}^{4} \left ({x}^{n} \right ) ^{18}{b}^{10}}{2\,n}}+26\,{\frac{{b}^{11}{c}^{3} \left ({x}^{n} \right ) ^{17}}{n}}+{\frac{13\,{c}^{2} \left ({x}^{n} \right ) ^{16}{b}^{12}}{2\,n}}+{\frac{{b}^{13}c \left ({x}^{n} \right ) ^{15}}{n}}+{\frac{ \left ({x}^{n} \right ) ^{14}{b}^{14}}{14\,n}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+n)*(b+2*c*x^n)*(b*x^n+c*x^(2*n))^13,x)

[Out]

1/14*c^14/n*(x^n)^28+b*c^13/n*(x^n)^27+13/2*c^12/n*(x^n)^26*b^2+26*b^3*c^11/n*(x^n)^25+143/2*c^10/n*(x^n)^24*b
^4+143*b^5*c^9/n*(x^n)^23+429/2*c^8/n*(x^n)^22*b^6+1716/7*b^7*c^7/n*(x^n)^21+429/2*c^6/n*(x^n)^20*b^8+143*b^9*
c^5/n*(x^n)^19+143/2*c^4/n*(x^n)^18*b^10+26*b^11*c^3/n*(x^n)^17+13/2*c^2/n*(x^n)^16*b^12+b^13*c/n*(x^n)^15+1/1
4/n*(x^n)^14*b^14

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(b+2*c*x^n)*(b*x^n+c*x^(2*n))^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.10325, size = 467, normalized size = 22.24 \begin{align*} \frac{c^{14} x^{28 \, n} + 14 \, b c^{13} x^{27 \, n} + 91 \, b^{2} c^{12} x^{26 \, n} + 364 \, b^{3} c^{11} x^{25 \, n} + 1001 \, b^{4} c^{10} x^{24 \, n} + 2002 \, b^{5} c^{9} x^{23 \, n} + 3003 \, b^{6} c^{8} x^{22 \, n} + 3432 \, b^{7} c^{7} x^{21 \, n} + 3003 \, b^{8} c^{6} x^{20 \, n} + 2002 \, b^{9} c^{5} x^{19 \, n} + 1001 \, b^{10} c^{4} x^{18 \, n} + 364 \, b^{11} c^{3} x^{17 \, n} + 91 \, b^{12} c^{2} x^{16 \, n} + 14 \, b^{13} c x^{15 \, n} + b^{14} x^{14 \, n}}{14 \, n} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(b+2*c*x^n)*(b*x^n+c*x^(2*n))^13,x, algorithm="fricas")

[Out]

1/14*(c^14*x^(28*n) + 14*b*c^13*x^(27*n) + 91*b^2*c^12*x^(26*n) + 364*b^3*c^11*x^(25*n) + 1001*b^4*c^10*x^(24*
n) + 2002*b^5*c^9*x^(23*n) + 3003*b^6*c^8*x^(22*n) + 3432*b^7*c^7*x^(21*n) + 3003*b^8*c^6*x^(20*n) + 2002*b^9*
c^5*x^(19*n) + 1001*b^10*c^4*x^(18*n) + 364*b^11*c^3*x^(17*n) + 91*b^12*c^2*x^(16*n) + 14*b^13*c*x^(15*n) + b^
14*x^(14*n))/n

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+n)*(b+2*c*x**n)*(b*x**n+c*x**(2*n))**13,x)

[Out]

Timed out

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Giac [B]  time = 1.13235, size = 255, normalized size = 12.14 \begin{align*} \frac{c^{14} x^{28 \, n} + 14 \, b c^{13} x^{27 \, n} + 91 \, b^{2} c^{12} x^{26 \, n} + 364 \, b^{3} c^{11} x^{25 \, n} + 1001 \, b^{4} c^{10} x^{24 \, n} + 2002 \, b^{5} c^{9} x^{23 \, n} + 3003 \, b^{6} c^{8} x^{22 \, n} + 3432 \, b^{7} c^{7} x^{21 \, n} + 3003 \, b^{8} c^{6} x^{20 \, n} + 2002 \, b^{9} c^{5} x^{19 \, n} + 1001 \, b^{10} c^{4} x^{18 \, n} + 364 \, b^{11} c^{3} x^{17 \, n} + 91 \, b^{12} c^{2} x^{16 \, n} + 14 \, b^{13} c x^{15 \, n} + b^{14} x^{14 \, n}}{14 \, n} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(b+2*c*x^n)*(b*x^n+c*x^(2*n))^13,x, algorithm="giac")

[Out]

1/14*(c^14*x^(28*n) + 14*b*c^13*x^(27*n) + 91*b^2*c^12*x^(26*n) + 364*b^3*c^11*x^(25*n) + 1001*b^4*c^10*x^(24*
n) + 2002*b^5*c^9*x^(23*n) + 3003*b^6*c^8*x^(22*n) + 3432*b^7*c^7*x^(21*n) + 3003*b^8*c^6*x^(20*n) + 2002*b^9*
c^5*x^(19*n) + 1001*b^10*c^4*x^(18*n) + 364*b^11*c^3*x^(17*n) + 91*b^12*c^2*x^(16*n) + 14*b^13*c*x^(15*n) + b^
14*x^(14*n))/n