∫ x−1+n(b + 2cxn)(−a + bxn + cx2n)13 dx [100]

3.1.100 \(\int x^{-1+n} (b+2 c x^n) (-a+b x^n+c x^{2 n})^{13} \, dx\) [100]

Optimal. Leaf size=25 \[ \frac {\left (a-b x^n-c x^{2 n}\right )^{14}}{14 n} \]

[Out]

1/14*(a-b*x^n-c*x^(2*n))^14/n

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Rubi [A]
time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1482, 643} \begin {gather*} \frac {\left (a-b x^n-c x^{2 n}\right )^{14}}{14 n} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a - b*x^n - c*x^(2*n))^14/(14*n)

Rule 643

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*((a + b*x + c*x^2)^(p +

1)/(b*(p + 1))), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1482

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>

Dist[1/n, Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x]

&& EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(260\) vs. \(2(25)=50\).
time = 0.40, size = 260, normalized size = 10.40 \begin {gather*} \frac {x^n \left (b+c x^n\right ) \left (-14 a^{13}+91 a^{12} x^n \left (b+c x^n\right )-364 a^{11} x^{2 n} \left (b+c x^n\right )^2+1001 a^{10} x^{3 n} \left (b+c x^n\right )^3-2002 a^9 x^{4 n} \left (b+c x^n\right )^4+3003 a^8 x^{5 n} \left (b+c x^n\right )^5-3432 a^7 x^{6 n} \left (b+c x^n\right )^6+3003 a^6 x^{7 n} \left (b+c x^n\right )^7-2002 a^5 x^{8 n} \left (b+c x^n\right )^8+1001 a^4 x^{9 n} \left (b+c x^n\right )^9-364 a^3 x^{10 n} \left (b+c x^n\right )^{10}+91 a^2 x^{11 n} \left (b+c x^n\right )^{11}-14 a x^{12 n} \left (b+c x^n\right )^{12}+x^{13 n} \left (b+c x^n\right )^{13}\right )}{14 n} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x^n*(b + c*x^n)*(-14*a^13 + 91*a^12*x^n*(b + c*x^n) - 364*a^11*x^(2*n)*(b + c*x^n)^2 + 1001*a^10*x^(3*n)*(b +

c*x^n)^3 - 2002*a^9*x^(4*n)*(b + c*x^n)^4 + 3003*a^8*x^(5*n)*(b + c*x^n)^5 - 3432*a^7*x^(6*n)*(b + c*x^n)^6 +

3003*a^6*x^(7*n)*(b + c*x^n)^7 - 2002*a^5*x^(8*n)*(b + c*x^n)^8 + 1001*a^4*x^(9*n)*(b + c*x^n)^9 - 364*a^3*x^

(10*n)*(b + c*x^n)^10 + 91*a^2*x^(11*n)*(b + c*x^n)^11 - 14*a*x^(12*n)*(b + c*x^n)^12 + x^(13*n)*(b + c*x^n)^1

3))/(14*n)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2045\) vs. \(2(23)=46\).
time = 0.05, size = 2046, normalized size = 81.84


method	result	size

risch	\(\text {Expression too large to display}\)	\(2046\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+n)*(b+2*c*x^n)*(-a+b*x^n+c*x^(2*n))^13,x,method=_RETURNVERBOSE)

[Out]

26*b^3*c^11/n*(x^n)^25+b^13*c/n*(x^n)^15-1716/7*a^7*b^7/n*(x^n)^7+143/2*c^10/n*(x^n)^20*a^4+429/2*c^6/n*(x^n)^

20*b^8-a^13/n*(x^n)^2*c+13/2*a^12/n*(x^n)^2*b^2-26*c^11/n*(x^n)^22*a^3+429/2*c^8/n*(x^n)^22*b^6+b*c^13/n*(x^n)

^27-b*a^13/n*x^n+715*a^4*b^9/n*(x^n)^11*c+13*b*a^12/n*(x^n)^3*c-13*b*c^12/n*(x^n)^25*a+1716*b*c^7/n*(x^n)^15*a

^6-12012*b^3*c^6/n*(x^n)^15*a^5+18018*b^5*c^5/n*(x^n)^15*a^4-8580*b^7*c^4/n*(x^n)^15*a^3+1430*b^9*c^3/n*(x^n)^

15*a^2-78*b^11*c^2/n*(x^n)^15*a+26*b^11*c^3/n*(x^n)^17-143*a^5*b^9/n*(x^n)^9+143*b^5*c^9/n*(x^n)^23-26*a^3*b^1

1/n*(x^n)^11-26*b^3*a^11/n*(x^n)^3+286*a^10*b/n*(x^n)^7*c^3-1430*a^9*b^3/n*(x^n)^7*c^2+1287*a^8*b^5/n*(x^n)^7*

c+429*c^10/n*(x^n)^22*a^2*b^2-715*c^9/n*(x^n)^22*a*b^4-1430*c^9/n*(x^n)^20*a^3*b^2+6435/2*c^8/n*(x^n)^20*a^2*b

^4-1716*c^7/n*(x^n)^20*a*b^6+143/2*a^10/n*(x^n)^8*c^4+429/2*a^6/n*(x^n)^8*b^8-26*a^11/n*(x^n)^6*c^3-1287*b*c^8

/n*(x^n)^17*a^5+8580*b^3*c^7/n*(x^n)^17*a^4-12012*b^5*c^6/n*(x^n)^17*a^3+5148*b^7*c^5/n*(x^n)^17*a^2-715*b^9*c

^4/n*(x^n)^17*a-715*a^9*b/n*(x^n)^9*c^4+4290*a^8*b^3/n*(x^n)^9*c^3-5148*a^7*b^5/n*(x^n)^9*c^2+1716*a^6*b^7/n*(

x^n)^9*c+78*b*c^11/n*(x^n)^23*a^2-286*b^3*c^10/n*(x^n)^23*a+1287*a^8*b/n*(x^n)^11*c^5-8580*a^7*b^3/n*(x^n)^11*

c^4+12012*a^6*b^5/n*(x^n)^11*c^3-5148*a^5*b^7/n*(x^n)^11*c^2+429/2*a^8/n*(x^n)^6*b^6+429/2*a^8/n*(x^n)^12*c^6+

13/2*a^2/n*(x^n)^12*b^12-1716/7/n*(x^n)^14*a^7*c^7-c^13/n*(x^n)^26*a+13/2*c^12/n*(x^n)^26*b^2-143*c^9/n*(x^n)^

18*a^5+143/2*c^4/n*(x^n)^18*b^10+429/2*c^8/n*(x^n)^16*a^6+13/2*c^2/n*(x^n)^16*b^12+1/14/n*(x^n)^14*b^14+13/2*a

^12/n*(x^n)^4*c^2+143/2*a^10/n*(x^n)^4*b^4+13/2*c^12/n*(x^n)^24*a^2+143/2*c^10/n*(x^n)^24*b^4+6435/2*c^8/n*(x^

n)^18*a^4*b^2-8580*c^7/n*(x^n)^18*a^3*b^4+6006*c^6/n*(x^n)^18*a^2*b^6-1287*c^5/n*(x^n)^18*a*b^8-5148*c^7/n*(x^

n)^16*a^5*b^2+15015*c^6/n*(x^n)^16*a^4*b^4-12012*c^5/n*(x^n)^16*a^3*b^6+6435/2*c^4/n*(x^n)^16*a^2*b^8-286*c^3/

n*(x^n)^16*a*b^10-78*a^11/n*(x^n)^4*b^2*c-78*c^11/n*(x^n)^24*a*b^2-a*b^13/n*(x^n)^13-143*a^9*b^5/n*(x^n)^5+171

6/7*b^7*c^7/n*(x^n)^21+143*b^9*c^5/n*(x^n)^19-143*a^9/n*(x^n)^10*c^5+143/2*a^4/n*(x^n)^10*b^10+1/14*c^14/n*(x^

n)^28+1430*b^3*c^9/n*(x^n)^21*a^2-1287*b^5*c^8/n*(x^n)^21*a+715*b*c^9/n*(x^n)^19*a^4-4290*b^3*c^8/n*(x^n)^19*a

^3+5148*b^5*c^7/n*(x^n)^19*a^2-1716*b^7*c^6/n*(x^n)^19*a+6435/2*a^8/n*(x^n)^10*b^2*c^4-8580*a^7/n*(x^n)^10*b^4

*c^3+6006*a^6/n*(x^n)^10*b^6*c^2-1287*a^5/n*(x^n)^10*b^8*c-1716*a^7*b/n*(x^n)^13*c^6+12012*a^6*b^3/n*(x^n)^13*

c^5-18018*a^5*b^5/n*(x^n)^13*c^4+8580*a^4*b^7/n*(x^n)^13*c^3-1430*a^3*b^9/n*(x^n)^13*c^2+78*a^2*b^11/n*(x^n)^1

3*c-78*a^11*b/n*(x^n)^5*c^2+286*a^10*b^3/n*(x^n)^5*c-286*b*c^10/n*(x^n)^21*a^3-1430*a^9/n*(x^n)^8*b^2*c^3+6435

/2*a^8/n*(x^n)^8*b^4*c^2-1716*a^7/n*(x^n)^8*b^6*c+429*a^10/n*(x^n)^6*b^2*c^2-715*a^9/n*(x^n)^6*b^4*c-5148*a^7/

n*(x^n)^12*b^2*c^5+15015*a^6/n*(x^n)^12*b^4*c^4-12012*a^5/n*(x^n)^12*b^6*c^3+6435/2*a^4/n*(x^n)^12*b^8*c^2-286

*a^3/n*(x^n)^12*b^10*c+6006/n*(x^n)^14*a^6*b^2*c^6-18018/n*(x^n)^14*a^5*b^4*c^5+15015/n*(x^n)^14*a^4*b^6*c^4-4

290/n*(x^n)^14*a^3*b^8*c^3+429/n*(x^n)^14*a^2*b^10*c^2-13/n*(x^n)^14*a*b^12*c

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2045 vs. \(2 (23) = 46\).
time = 0.35, size = 2045, normalized size = 81.80 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*c^14*x^(28*n)/n + b*c^13*x^(27*n)/n + 13/2*b^2*c^12*x^(26*n)/n - a*c^13*x^(26*n)/n + 26*b^3*c^11*x^(25*n)

/n - 13*a*b*c^12*x^(25*n)/n + 143/2*b^4*c^10*x^(24*n)/n - 78*a*b^2*c^11*x^(24*n)/n + 13/2*a^2*c^12*x^(24*n)/n

+ 143*b^5*c^9*x^(23*n)/n - 286*a*b^3*c^10*x^(23*n)/n + 78*a^2*b*c^11*x^(23*n)/n + 429/2*b^6*c^8*x^(22*n)/n - 7

15*a*b^4*c^9*x^(22*n)/n + 429*a^2*b^2*c^10*x^(22*n)/n - 26*a^3*c^11*x^(22*n)/n + 1716/7*b^7*c^7*x^(21*n)/n - 1

287*a*b^5*c^8*x^(21*n)/n + 1430*a^2*b^3*c^9*x^(21*n)/n - 286*a^3*b*c^10*x^(21*n)/n + 429/2*b^8*c^6*x^(20*n)/n

- 1716*a*b^6*c^7*x^(20*n)/n + 6435/2*a^2*b^4*c^8*x^(20*n)/n - 1430*a^3*b^2*c^9*x^(20*n)/n + 143/2*a^4*c^10*x^(

20*n)/n + 143*b^9*c^5*x^(19*n)/n - 1716*a*b^7*c^6*x^(19*n)/n + 5148*a^2*b^5*c^7*x^(19*n)/n - 4290*a^3*b^3*c^8*

x^(19*n)/n + 715*a^4*b*c^9*x^(19*n)/n + 143/2*b^10*c^4*x^(18*n)/n - 1287*a*b^8*c^5*x^(18*n)/n + 6006*a^2*b^6*c

^6*x^(18*n)/n - 8580*a^3*b^4*c^7*x^(18*n)/n + 6435/2*a^4*b^2*c^8*x^(18*n)/n - 143*a^5*c^9*x^(18*n)/n + 26*b^11

*c^3*x^(17*n)/n - 715*a*b^9*c^4*x^(17*n)/n + 5148*a^2*b^7*c^5*x^(17*n)/n - 12012*a^3*b^5*c^6*x^(17*n)/n + 8580

*a^4*b^3*c^7*x^(17*n)/n - 1287*a^5*b*c^8*x^(17*n)/n + 13/2*b^12*c^2*x^(16*n)/n - 286*a*b^10*c^3*x^(16*n)/n + 6

435/2*a^2*b^8*c^4*x^(16*n)/n - 12012*a^3*b^6*c^5*x^(16*n)/n + 15015*a^4*b^4*c^6*x^(16*n)/n - 5148*a^5*b^2*c^7*

x^(16*n)/n + 429/2*a^6*c^8*x^(16*n)/n + b^13*c*x^(15*n)/n - 78*a*b^11*c^2*x^(15*n)/n + 1430*a^2*b^9*c^3*x^(15*

n)/n - 8580*a^3*b^7*c^4*x^(15*n)/n + 18018*a^4*b^5*c^5*x^(15*n)/n - 12012*a^5*b^3*c^6*x^(15*n)/n + 1716*a^6*b*

c^7*x^(15*n)/n + 1/14*b^14*x^(14*n)/n - 13*a*b^12*c*x^(14*n)/n + 429*a^2*b^10*c^2*x^(14*n)/n - 4290*a^3*b^8*c^

3*x^(14*n)/n + 15015*a^4*b^6*c^4*x^(14*n)/n - 18018*a^5*b^4*c^5*x^(14*n)/n + 6006*a^6*b^2*c^6*x^(14*n)/n - 171

6/7*a^7*c^7*x^(14*n)/n - a*b^13*x^(13*n)/n + 78*a^2*b^11*c*x^(13*n)/n - 1430*a^3*b^9*c^2*x^(13*n)/n + 8580*a^4

*b^7*c^3*x^(13*n)/n - 18018*a^5*b^5*c^4*x^(13*n)/n + 12012*a^6*b^3*c^5*x^(13*n)/n - 1716*a^7*b*c^6*x^(13*n)/n

+ 13/2*a^2*b^12*x^(12*n)/n - 286*a^3*b^10*c*x^(12*n)/n + 6435/2*a^4*b^8*c^2*x^(12*n)/n - 12012*a^5*b^6*c^3*x^(

12*n)/n + 15015*a^6*b^4*c^4*x^(12*n)/n - 5148*a^7*b^2*c^5*x^(12*n)/n + 429/2*a^8*c^6*x^(12*n)/n - 26*a^3*b^11*

x^(11*n)/n + 715*a^4*b^9*c*x^(11*n)/n - 5148*a^5*b^7*c^2*x^(11*n)/n + 12012*a^6*b^5*c^3*x^(11*n)/n - 8580*a^7*

b^3*c^4*x^(11*n)/n + 1287*a^8*b*c^5*x^(11*n)/n + 143/2*a^4*b^10*x^(10*n)/n - 1287*a^5*b^8*c*x^(10*n)/n + 6006*

a^6*b^6*c^2*x^(10*n)/n - 8580*a^7*b^4*c^3*x^(10*n)/n + 6435/2*a^8*b^2*c^4*x^(10*n)/n - 143*a^9*c^5*x^(10*n)/n

- 143*a^5*b^9*x^(9*n)/n + 1716*a^6*b^7*c*x^(9*n)/n - 5148*a^7*b^5*c^2*x^(9*n)/n + 4290*a^8*b^3*c^3*x^(9*n)/n -

715*a^9*b*c^4*x^(9*n)/n + 429/2*a^6*b^8*x^(8*n)/n - 1716*a^7*b^6*c*x^(8*n)/n + 6435/2*a^8*b^4*c^2*x^(8*n)/n -

1430*a^9*b^2*c^3*x^(8*n)/n + 143/2*a^10*c^4*x^(8*n)/n - 1716/7*a^7*b^7*x^(7*n)/n + 1287*a^8*b^5*c*x^(7*n)/n -

1430*a^9*b^3*c^2*x^(7*n)/n + 286*a^10*b*c^3*x^(7*n)/n + 429/2*a^8*b^6*x^(6*n)/n - 715*a^9*b^4*c*x^(6*n)/n + 4

29*a^10*b^2*c^2*x^(6*n)/n - 26*a^11*c^3*x^(6*n)/n - 143*a^9*b^5*x^(5*n)/n + 286*a^10*b^3*c*x^(5*n)/n - 78*a^11

*b*c^2*x^(5*n)/n + 143/2*a^10*b^4*x^(4*n)/n - 78*a^11*b^2*c*x^(4*n)/n + 13/2*a^12*c^2*x^(4*n)/n - 26*a^11*b^3*

x^(3*n)/n + 13*a^12*b*c*x^(3*n)/n + 13/2*a^12*b^2*x^(2*n)/n - a^13*c*x^(2*n)/n - a^13*b*x^n/n

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1299 vs. \(2 (23) = 46\).
time = 0.40, size = 1299, normalized size = 51.96 \begin {gather*} \frac {c^{14} x^{28 \, n} + 14 \, b c^{13} x^{27 \, n} - 14 \, a^{13} b x^{n} + 7 \, {\left (13 \, b^{2} c^{12} - 2 \, a c^{13}\right )} x^{26 \, n} + 182 \, {\left (2 \, b^{3} c^{11} - a b c^{12}\right )} x^{25 \, n} + 91 \, {\left (11 \, b^{4} c^{10} - 12 \, a b^{2} c^{11} + a^{2} c^{12}\right )} x^{24 \, n} + 182 \, {\left (11 \, b^{5} c^{9} - 22 \, a b^{3} c^{10} + 6 \, a^{2} b c^{11}\right )} x^{23 \, n} + 91 \, {\left (33 \, b^{6} c^{8} - 110 \, a b^{4} c^{9} + 66 \, a^{2} b^{2} c^{10} - 4 \, a^{3} c^{11}\right )} x^{22 \, n} + 286 \, {\left (12 \, b^{7} c^{7} - 63 \, a b^{5} c^{8} + 70 \, a^{2} b^{3} c^{9} - 14 \, a^{3} b c^{10}\right )} x^{21 \, n} + 1001 \, {\left (3 \, b^{8} c^{6} - 24 \, a b^{6} c^{7} + 45 \, a^{2} b^{4} c^{8} - 20 \, a^{3} b^{2} c^{9} + a^{4} c^{10}\right )} x^{20 \, n} + 2002 \, {\left (b^{9} c^{5} - 12 \, a b^{7} c^{6} + 36 \, a^{2} b^{5} c^{7} - 30 \, a^{3} b^{3} c^{8} + 5 \, a^{4} b c^{9}\right )} x^{19 \, n} + 1001 \, {\left (b^{10} c^{4} - 18 \, a b^{8} c^{5} + 84 \, a^{2} b^{6} c^{6} - 120 \, a^{3} b^{4} c^{7} + 45 \, a^{4} b^{2} c^{8} - 2 \, a^{5} c^{9}\right )} x^{18 \, n} + 182 \, {\left (2 \, b^{11} c^{3} - 55 \, a b^{9} c^{4} + 396 \, a^{2} b^{7} c^{5} - 924 \, a^{3} b^{5} c^{6} + 660 \, a^{4} b^{3} c^{7} - 99 \, a^{5} b c^{8}\right )} x^{17 \, n} + 91 \, {\left (b^{12} c^{2} - 44 \, a b^{10} c^{3} + 495 \, a^{2} b^{8} c^{4} - 1848 \, a^{3} b^{6} c^{5} + 2310 \, a^{4} b^{4} c^{6} - 792 \, a^{5} b^{2} c^{7} + 33 \, a^{6} c^{8}\right )} x^{16 \, n} + 14 \, {\left (b^{13} c - 78 \, a b^{11} c^{2} + 1430 \, a^{2} b^{9} c^{3} - 8580 \, a^{3} b^{7} c^{4} + 18018 \, a^{4} b^{5} c^{5} - 12012 \, a^{5} b^{3} c^{6} + 1716 \, a^{6} b c^{7}\right )} x^{15 \, n} + {\left (b^{14} - 182 \, a b^{12} c + 6006 \, a^{2} b^{10} c^{2} - 60060 \, a^{3} b^{8} c^{3} + 210210 \, a^{4} b^{6} c^{4} - 252252 \, a^{5} b^{4} c^{5} + 84084 \, a^{6} b^{2} c^{6} - 3432 \, a^{7} c^{7}\right )} x^{14 \, n} - 14 \, {\left (a b^{13} - 78 \, a^{2} b^{11} c + 1430 \, a^{3} b^{9} c^{2} - 8580 \, a^{4} b^{7} c^{3} + 18018 \, a^{5} b^{5} c^{4} - 12012 \, a^{6} b^{3} c^{5} + 1716 \, a^{7} b c^{6}\right )} x^{13 \, n} + 91 \, {\left (a^{2} b^{12} - 44 \, a^{3} b^{10} c + 495 \, a^{4} b^{8} c^{2} - 1848 \, a^{5} b^{6} c^{3} + 2310 \, a^{6} b^{4} c^{4} - 792 \, a^{7} b^{2} c^{5} + 33 \, a^{8} c^{6}\right )} x^{12 \, n} - 182 \, {\left (2 \, a^{3} b^{11} - 55 \, a^{4} b^{9} c + 396 \, a^{5} b^{7} c^{2} - 924 \, a^{6} b^{5} c^{3} + 660 \, a^{7} b^{3} c^{4} - 99 \, a^{8} b c^{5}\right )} x^{11 \, n} + 1001 \, {\left (a^{4} b^{10} - 18 \, a^{5} b^{8} c + 84 \, a^{6} b^{6} c^{2} - 120 \, a^{7} b^{4} c^{3} + 45 \, a^{8} b^{2} c^{4} - 2 \, a^{9} c^{5}\right )} x^{10 \, n} - 2002 \, {\left (a^{5} b^{9} - 12 \, a^{6} b^{7} c + 36 \, a^{7} b^{5} c^{2} - 30 \, a^{8} b^{3} c^{3} + 5 \, a^{9} b c^{4}\right )} x^{9 \, n} + 1001 \, {\left (3 \, a^{6} b^{8} - 24 \, a^{7} b^{6} c + 45 \, a^{8} b^{4} c^{2} - 20 \, a^{9} b^{2} c^{3} + a^{10} c^{4}\right )} x^{8 \, n} - 286 \, {\left (12 \, a^{7} b^{7} - 63 \, a^{8} b^{5} c + 70 \, a^{9} b^{3} c^{2} - 14 \, a^{10} b c^{3}\right )} x^{7 \, n} + 91 \, {\left (33 \, a^{8} b^{6} - 110 \, a^{9} b^{4} c + 66 \, a^{10} b^{2} c^{2} - 4 \, a^{11} c^{3}\right )} x^{6 \, n} - 182 \, {\left (11 \, a^{9} b^{5} - 22 \, a^{10} b^{3} c + 6 \, a^{11} b c^{2}\right )} x^{5 \, n} + 91 \, {\left (11 \, a^{10} b^{4} - 12 \, a^{11} b^{2} c + a^{12} c^{2}\right )} x^{4 \, n} - 182 \, {\left (2 \, a^{11} b^{3} - a^{12} b c\right )} x^{3 \, n} + 7 \, {\left (13 \, a^{12} b^{2} - 2 \, a^{13} c\right )} x^{2 \, n}}{14 \, n} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(b+2*c*x^n)*(-a+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) - 14*a^13*b*x^n + 7*(13*b^2*c^12 - 2*a*c^13)*x^(26*n) + 182*(2*b^3*c^

11 - a*b*c^12)*x^(25*n) + 91*(11*b^4*c^10 - 12*a*b^2*c^11 + a^2*c^12)*x^(24*n) + 182*(11*b^5*c^9 - 22*a*b^3*c^

10 + 6*a^2*b*c^11)*x^(23*n) + 91*(33*b^6*c^8 - 110*a*b^4*c^9 + 66*a^2*b^2*c^10 - 4*a^3*c^11)*x^(22*n) + 286*(1

2*b^7*c^7 - 63*a*b^5*c^8 + 70*a^2*b^3*c^9 - 14*a^3*b*c^10)*x^(21*n) + 1001*(3*b^8*c^6 - 24*a*b^6*c^7 + 45*a^2*

b^4*c^8 - 20*a^3*b^2*c^9 + a^4*c^10)*x^(20*n) + 2002*(b^9*c^5 - 12*a*b^7*c^6 + 36*a^2*b^5*c^7 - 30*a^3*b^3*c^8

+ 5*a^4*b*c^9)*x^(19*n) + 1001*(b^10*c^4 - 18*a*b^8*c^5 + 84*a^2*b^6*c^6 - 120*a^3*b^4*c^7 + 45*a^4*b^2*c^8 -

2*a^5*c^9)*x^(18*n) + 182*(2*b^11*c^3 - 55*a*b^9*c^4 + 396*a^2*b^7*c^5 - 924*a^3*b^5*c^6 + 660*a^4*b^3*c^7 -

99*a^5*b*c^8)*x^(17*n) + 91*(b^12*c^2 - 44*a*b^10*c^3 + 495*a^2*b^8*c^4 - 1848*a^3*b^6*c^5 + 2310*a^4*b^4*c^6

- 792*a^5*b^2*c^7 + 33*a^6*c^8)*x^(16*n) + 14*(b^13*c - 78*a*b^11*c^2 + 1430*a^2*b^9*c^3 - 8580*a^3*b^7*c^4 +

18018*a^4*b^5*c^5 - 12012*a^5*b^3*c^6 + 1716*a^6*b*c^7)*x^(15*n) + (b^14 - 182*a*b^12*c + 6006*a^2*b^10*c^2 -

60060*a^3*b^8*c^3 + 210210*a^4*b^6*c^4 - 252252*a^5*b^4*c^5 + 84084*a^6*b^2*c^6 - 3432*a^7*c^7)*x^(14*n) - 14*

(a*b^13 - 78*a^2*b^11*c + 1430*a^3*b^9*c^2 - 8580*a^4*b^7*c^3 + 18018*a^5*b^5*c^4 - 12012*a^6*b^3*c^5 + 1716*a

^7*b*c^6)*x^(13*n) + 91*(a^2*b^12 - 44*a^3*b^10*c + 495*a^4*b^8*c^2 - 1848*a^5*b^6*c^3 + 2310*a^6*b^4*c^4 - 79

2*a^7*b^2*c^5 + 33*a^8*c^6)*x^(12*n) - 182*(2*a^3*b^11 - 55*a^4*b^9*c + 396*a^5*b^7*c^2 - 924*a^6*b^5*c^3 + 66

0*a^7*b^3*c^4 - 99*a^8*b*c^5)*x^(11*n) + 1001*(a^4*b^10 - 18*a^5*b^8*c + 84*a^6*b^6*c^2 - 120*a^7*b^4*c^3 + 45

*a^8*b^2*c^4 - 2*a^9*c^5)*x^(10*n) - 2002*(a^5*b^9 - 12*a^6*b^7*c + 36*a^7*b^5*c^2 - 30*a^8*b^3*c^3 + 5*a^9*b*

c^4)*x^(9*n) + 1001*(3*a^6*b^8 - 24*a^7*b^6*c + 45*a^8*b^4*c^2 - 20*a^9*b^2*c^3 + a^10*c^4)*x^(8*n) - 286*(12*

a^7*b^7 - 63*a^8*b^5*c + 70*a^9*b^3*c^2 - 14*a^10*b*c^3)*x^(7*n) + 91*(33*a^8*b^6 - 110*a^9*b^4*c + 66*a^10*b^

2*c^2 - 4*a^11*c^3)*x^(6*n) - 182*(11*a^9*b^5 - 22*a^10*b^3*c + 6*a^11*b*c^2)*x^(5*n) + 91*(11*a^10*b^4 - 12*a

^11*b^2*c + a^12*c^2)*x^(4*n) - 182*(2*a^11*b^3 - a^12*b*c)*x^(3*n) + 7*(13*a^12*b^2 - 2*a^13*c)*x^(2*n))/n

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 7316 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1693 vs. \(2 (23) = 46\).
time = 3.77, size = 1693, normalized size = 67.72 \begin {gather*} \frac {c^{14} x^{28 \, n} + 14 \, b c^{13} x^{27 \, n} + 91 \, b^{2} c^{12} x^{26 \, n} - 14 \, a c^{13} x^{26 \, n} + 364 \, b^{3} c^{11} x^{25 \, n} - 182 \, a b c^{12} x^{25 \, n} + 1001 \, b^{4} c^{10} x^{24 \, n} - 1092 \, a b^{2} c^{11} x^{24 \, n} + 91 \, a^{2} c^{12} x^{24 \, n} + 2002 \, b^{5} c^{9} x^{23 \, n} - 4004 \, a b^{3} c^{10} x^{23 \, n} + 1092 \, a^{2} b c^{11} x^{23 \, n} + 3003 \, b^{6} c^{8} x^{22 \, n} - 10010 \, a b^{4} c^{9} x^{22 \, n} + 6006 \, a^{2} b^{2} c^{10} x^{22 \, n} - 364 \, a^{3} c^{11} x^{22 \, n} + 3432 \, b^{7} c^{7} x^{21 \, n} - 18018 \, a b^{5} c^{8} x^{21 \, n} + 20020 \, a^{2} b^{3} c^{9} x^{21 \, n} - 4004 \, a^{3} b c^{10} x^{21 \, n} + 3003 \, b^{8} c^{6} x^{20 \, n} - 24024 \, a b^{6} c^{7} x^{20 \, n} + 45045 \, a^{2} b^{4} c^{8} x^{20 \, n} - 20020 \, a^{3} b^{2} c^{9} x^{20 \, n} + 1001 \, a^{4} c^{10} x^{20 \, n} + 2002 \, b^{9} c^{5} x^{19 \, n} - 24024 \, a b^{7} c^{6} x^{19 \, n} + 72072 \, a^{2} b^{5} c^{7} x^{19 \, n} - 60060 \, a^{3} b^{3} c^{8} x^{19 \, n} + 10010 \, a^{4} b c^{9} x^{19 \, n} + 1001 \, b^{10} c^{4} x^{18 \, n} - 18018 \, a b^{8} c^{5} x^{18 \, n} + 84084 \, a^{2} b^{6} c^{6} x^{18 \, n} - 120120 \, a^{3} b^{4} c^{7} x^{18 \, n} + 45045 \, a^{4} b^{2} c^{8} x^{18 \, n} - 2002 \, a^{5} c^{9} x^{18 \, n} + 364 \, b^{11} c^{3} x^{17 \, n} - 10010 \, a b^{9} c^{4} x^{17 \, n} + 72072 \, a^{2} b^{7} c^{5} x^{17 \, n} - 168168 \, a^{3} b^{5} c^{6} x^{17 \, n} + 120120 \, a^{4} b^{3} c^{7} x^{17 \, n} - 18018 \, a^{5} b c^{8} x^{17 \, n} + 91 \, b^{12} c^{2} x^{16 \, n} - 4004 \, a b^{10} c^{3} x^{16 \, n} + 45045 \, a^{2} b^{8} c^{4} x^{16 \, n} - 168168 \, a^{3} b^{6} c^{5} x^{16 \, n} + 210210 \, a^{4} b^{4} c^{6} x^{16 \, n} - 72072 \, a^{5} b^{2} c^{7} x^{16 \, n} + 3003 \, a^{6} c^{8} x^{16 \, n} + 14 \, b^{13} c x^{15 \, n} - 1092 \, a b^{11} c^{2} x^{15 \, n} + 20020 \, a^{2} b^{9} c^{3} x^{15 \, n} - 120120 \, a^{3} b^{7} c^{4} x^{15 \, n} + 252252 \, a^{4} b^{5} c^{5} x^{15 \, n} - 168168 \, a^{5} b^{3} c^{6} x^{15 \, n} + 24024 \, a^{6} b c^{7} x^{15 \, n} + b^{14} x^{14 \, n} - 182 \, a b^{12} c x^{14 \, n} + 6006 \, a^{2} b^{10} c^{2} x^{14 \, n} - 60060 \, a^{3} b^{8} c^{3} x^{14 \, n} + 210210 \, a^{4} b^{6} c^{4} x^{14 \, n} - 252252 \, a^{5} b^{4} c^{5} x^{14 \, n} + 84084 \, a^{6} b^{2} c^{6} x^{14 \, n} - 3432 \, a^{7} c^{7} x^{14 \, n} - 14 \, a b^{13} x^{13 \, n} + 1092 \, a^{2} b^{11} c x^{13 \, n} - 20020 \, a^{3} b^{9} c^{2} x^{13 \, n} + 120120 \, a^{4} b^{7} c^{3} x^{13 \, n} - 252252 \, a^{5} b^{5} c^{4} x^{13 \, n} + 168168 \, a^{6} b^{3} c^{5} x^{13 \, n} - 24024 \, a^{7} b c^{6} x^{13 \, n} + 91 \, a^{2} b^{12} x^{12 \, n} - 4004 \, a^{3} b^{10} c x^{12 \, n} + 45045 \, a^{4} b^{8} c^{2} x^{12 \, n} - 168168 \, a^{5} b^{6} c^{3} x^{12 \, n} + 210210 \, a^{6} b^{4} c^{4} x^{12 \, n} - 72072 \, a^{7} b^{2} c^{5} x^{12 \, n} + 3003 \, a^{8} c^{6} x^{12 \, n} - 364 \, a^{3} b^{11} x^{11 \, n} + 10010 \, a^{4} b^{9} c x^{11 \, n} - 72072 \, a^{5} b^{7} c^{2} x^{11 \, n} + 168168 \, a^{6} b^{5} c^{3} x^{11 \, n} - 120120 \, a^{7} b^{3} c^{4} x^{11 \, n} + 18018 \, a^{8} b c^{5} x^{11 \, n} + 1001 \, a^{4} b^{10} x^{10 \, n} - 18018 \, a^{5} b^{8} c x^{10 \, n} + 84084 \, a^{6} b^{6} c^{2} x^{10 \, n} - 120120 \, a^{7} b^{4} c^{3} x^{10 \, n} + 45045 \, a^{8} b^{2} c^{4} x^{10 \, n} - 2002 \, a^{9} c^{5} x^{10 \, n} - 2002 \, a^{5} b^{9} x^{9 \, n} + 24024 \, a^{6} b^{7} c x^{9 \, n} - 72072 \, a^{7} b^{5} c^{2} x^{9 \, n} + 60060 \, a^{8} b^{3} c^{3} x^{9 \, n} - 10010 \, a^{9} b c^{4} x^{9 \, n} + 3003 \, a^{6} b^{8} x^{8 \, n} - 24024 \, a^{7} b^{6} c x^{8 \, n} + 45045 \, a^{8} b^{4} c^{2} x^{8 \, n} - 20020 \, a^{9} b^{2} c^{3} x^{8 \, n} + 1001 \, a^{10} c^{4} x^{8 \, n} - 3432 \, a^{7} b^{7} x^{7 \, n} + 18018 \, a^{8} b^{5} c x^{7 \, n} - 20020 \, a^{9} b^{3} c^{2} x^{7 \, n} + 4004 \, a^{10} b c^{3} x^{7 \, n} + 3003 \, a^{8} b^{6} x^{6 \, n} - 10010 \, a^{9} b^{4} c x^{6 \, n} + 6006 \, a^{10} b^{2} c^{2} x^{6 \, n} - 364 \, a^{11} c^{3} x^{6 \, n} - 2002 \, a^{9} b^{5} x^{5 \, n} + 4004 \, a^{10} b^{3} c x^{5 \, n} - 1092 \, a^{11} b c^{2} x^{5 \, n} + 1001 \, a^{10} b^{4} x^{4 \, n} - 1092 \, a^{11} b^{2} c x^{4 \, n} + 91 \, a^{12} c^{2} x^{4 \, n} - 364 \, a^{11} b^{3} x^{3 \, n} + 182 \, a^{12} b c x^{3 \, n} + 91 \, a^{12} b^{2} x^{2 \, n} - 14 \, a^{13} c x^{2 \, n} - 14 \, a^{13} b x^{n}}{14 \, n} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(b+2*c*x^n)*(-a+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) - 14*a*c^13*x^(26*n) + 364*b^3*c^11*x^(25*n) -

182*a*b*c^12*x^(25*n) + 1001*b^4*c^10*x^(24*n) - 1092*a*b^2*c^11*x^(24*n) + 91*a^2*c^12*x^(24*n) + 2002*b^5*c

^9*x^(23*n) - 4004*a*b^3*c^10*x^(23*n) + 1092*a^2*b*c^11*x^(23*n) + 3003*b^6*c^8*x^(22*n) - 10010*a*b^4*c^9*x^

(22*n) + 6006*a^2*b^2*c^10*x^(22*n) - 364*a^3*c^11*x^(22*n) + 3432*b^7*c^7*x^(21*n) - 18018*a*b^5*c^8*x^(21*n)

+ 20020*a^2*b^3*c^9*x^(21*n) - 4004*a^3*b*c^10*x^(21*n) + 3003*b^8*c^6*x^(20*n) - 24024*a*b^6*c^7*x^(20*n) +

45045*a^2*b^4*c^8*x^(20*n) - 20020*a^3*b^2*c^9*x^(20*n) + 1001*a^4*c^10*x^(20*n) + 2002*b^9*c^5*x^(19*n) - 240

24*a*b^7*c^6*x^(19*n) + 72072*a^2*b^5*c^7*x^(19*n) - 60060*a^3*b^3*c^8*x^(19*n) + 10010*a^4*b*c^9*x^(19*n) + 1

001*b^10*c^4*x^(18*n) - 18018*a*b^8*c^5*x^(18*n) + 84084*a^2*b^6*c^6*x^(18*n) - 120120*a^3*b^4*c^7*x^(18*n) +

45045*a^4*b^2*c^8*x^(18*n) - 2002*a^5*c^9*x^(18*n) + 364*b^11*c^3*x^(17*n) - 10010*a*b^9*c^4*x^(17*n) + 72072*

a^2*b^7*c^5*x^(17*n) - 168168*a^3*b^5*c^6*x^(17*n) + 120120*a^4*b^3*c^7*x^(17*n) - 18018*a^5*b*c^8*x^(17*n) +

91*b^12*c^2*x^(16*n) - 4004*a*b^10*c^3*x^(16*n) + 45045*a^2*b^8*c^4*x^(16*n) - 168168*a^3*b^6*c^5*x^(16*n) + 2

10210*a^4*b^4*c^6*x^(16*n) - 72072*a^5*b^2*c^7*x^(16*n) + 3003*a^6*c^8*x^(16*n) + 14*b^13*c*x^(15*n) - 1092*a*

b^11*c^2*x^(15*n) + 20020*a^2*b^9*c^3*x^(15*n) - 120120*a^3*b^7*c^4*x^(15*n) + 252252*a^4*b^5*c^5*x^(15*n) - 1

68168*a^5*b^3*c^6*x^(15*n) + 24024*a^6*b*c^7*x^(15*n) + b^14*x^(14*n) - 182*a*b^12*c*x^(14*n) + 6006*a^2*b^10*

c^2*x^(14*n) - 60060*a^3*b^8*c^3*x^(14*n) + 210210*a^4*b^6*c^4*x^(14*n) - 252252*a^5*b^4*c^5*x^(14*n) + 84084*

a^6*b^2*c^6*x^(14*n) - 3432*a^7*c^7*x^(14*n) - 14*a*b^13*x^(13*n) + 1092*a^2*b^11*c*x^(13*n) - 20020*a^3*b^9*c

^2*x^(13*n) + 120120*a^4*b^7*c^3*x^(13*n) - 252252*a^5*b^5*c^4*x^(13*n) + 168168*a^6*b^3*c^5*x^(13*n) - 24024*

a^7*b*c^6*x^(13*n) + 91*a^2*b^12*x^(12*n) - 4004*a^3*b^10*c*x^(12*n) + 45045*a^4*b^8*c^2*x^(12*n) - 168168*a^5

*b^6*c^3*x^(12*n) + 210210*a^6*b^4*c^4*x^(12*n) - 72072*a^7*b^2*c^5*x^(12*n) + 3003*a^8*c^6*x^(12*n) - 364*a^3

*b^11*x^(11*n) + 10010*a^4*b^9*c*x^(11*n) - 72072*a^5*b^7*c^2*x^(11*n) + 168168*a^6*b^5*c^3*x^(11*n) - 120120*

a^7*b^3*c^4*x^(11*n) + 18018*a^8*b*c^5*x^(11*n) + 1001*a^4*b^10*x^(10*n) - 18018*a^5*b^8*c*x^(10*n) + 84084*a^

6*b^6*c^2*x^(10*n) - 120120*a^7*b^4*c^3*x^(10*n) + 45045*a^8*b^2*c^4*x^(10*n) - 2002*a^9*c^5*x^(10*n) - 2002*a

^5*b^9*x^(9*n) + 24024*a^6*b^7*c*x^(9*n) - 72072*a^7*b^5*c^2*x^(9*n) + 60060*a^8*b^3*c^3*x^(9*n) - 10010*a^9*b

*c^4*x^(9*n) + 3003*a^6*b^8*x^(8*n) - 24024*a^7*b^6*c*x^(8*n) + 45045*a^8*b^4*c^2*x^(8*n) - 20020*a^9*b^2*c^3*

x^(8*n) + 1001*a^10*c^4*x^(8*n) - 3432*a^7*b^7*x^(7*n) + 18018*a^8*b^5*c*x^(7*n) - 20020*a^9*b^3*c^2*x^(7*n) +

4004*a^10*b*c^3*x^(7*n) + 3003*a^8*b^6*x^(6*n) - 10010*a^9*b^4*c*x^(6*n) + 6006*a^10*b^2*c^2*x^(6*n) - 364*a^

11*c^3*x^(6*n) - 2002*a^9*b^5*x^(5*n) + 4004*a^10*b^3*c*x^(5*n) - 1092*a^11*b*c^2*x^(5*n) + 1001*a^10*b^4*x^(4

*n) - 1092*a^11*b^2*c*x^(4*n) + 91*a^12*c^2*x^(4*n) - 364*a^11*b^3*x^(3*n) + 182*a^12*b*c*x^(3*n) + 91*a^12*b^

2*x^(2*n) - 14*a^13*c*x^(2*n) - 14*a^13*b*x^n)/n

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Mupad [B]
time = 5.78, size = 1401, normalized size = 56.04 \begin {gather*} x^{n-1}\,\left (\frac {x^{11\,n+1}\,\left (\frac {429\,a^8\,c^6}{2}-5148\,a^7\,b^2\,c^5+15015\,a^6\,b^4\,c^4-12012\,a^5\,b^6\,c^3+\frac {6435\,a^4\,b^8\,c^2}{2}-286\,a^3\,b^{10}\,c+\frac {13\,a^2\,b^{12}}{2}\right )}{n}+\frac {x^{15\,n+1}\,\left (\frac {429\,a^6\,c^8}{2}-5148\,a^5\,b^2\,c^7+15015\,a^4\,b^4\,c^6-12012\,a^3\,b^6\,c^5+\frac {6435\,a^2\,b^8\,c^4}{2}-286\,a\,b^{10}\,c^3+\frac {13\,b^{12}\,c^2}{2}\right )}{n}-\frac {x^{12\,n+1}\,\left (1716\,a^7\,b\,c^6-12012\,a^6\,b^3\,c^5+18018\,a^5\,b^5\,c^4-8580\,a^4\,b^7\,c^3+1430\,a^3\,b^9\,c^2-78\,a^2\,b^{11}\,c+a\,b^{13}\right )}{n}+\frac {x^{14\,n+1}\,\left (1716\,a^6\,b\,c^7-12012\,a^5\,b^3\,c^6+18018\,a^4\,b^5\,c^5-8580\,a^3\,b^7\,c^4+1430\,a^2\,b^9\,c^3-78\,a\,b^{11}\,c^2+b^{13}\,c\right )}{n}+\frac {x^{5\,n+1}\,\left (-26\,a^{11}\,c^3+429\,a^{10}\,b^2\,c^2-715\,a^9\,b^4\,c+\frac {429\,a^8\,b^6}{2}\right )}{n}-\frac {x^{21\,n+1}\,\left (26\,a^3\,c^{11}-429\,a^2\,b^2\,c^{10}+715\,a\,b^4\,c^9-\frac {429\,b^6\,c^8}{2}\right )}{n}+\frac {x^{9\,n+1}\,\left (-143\,a^9\,c^5+\frac {6435\,a^8\,b^2\,c^4}{2}-8580\,a^7\,b^4\,c^3+6006\,a^6\,b^6\,c^2-1287\,a^5\,b^8\,c+\frac {143\,a^4\,b^{10}}{2}\right )}{n}-\frac {x^{17\,n+1}\,\left (143\,a^5\,c^9-\frac {6435\,a^4\,b^2\,c^8}{2}+8580\,a^3\,b^4\,c^7-6006\,a^2\,b^6\,c^6+1287\,a\,b^8\,c^5-\frac {143\,b^{10}\,c^4}{2}\right )}{n}+\frac {x^{13\,n+1}\,\left (-\frac {1716\,a^7\,c^7}{7}+6006\,a^6\,b^2\,c^6-18018\,a^5\,b^4\,c^5+15015\,a^4\,b^6\,c^4-4290\,a^3\,b^8\,c^3+429\,a^2\,b^{10}\,c^2-13\,a\,b^{12}\,c+\frac {b^{14}}{14}\right )}{n}+\frac {x^{7\,n+1}\,\left (\frac {143\,a^{10}\,c^4}{2}-1430\,a^9\,b^2\,c^3+\frac {6435\,a^8\,b^4\,c^2}{2}-1716\,a^7\,b^6\,c+\frac {429\,a^6\,b^8}{2}\right )}{n}+\frac {x^{19\,n+1}\,\left (\frac {143\,a^4\,c^{10}}{2}-1430\,a^3\,b^2\,c^9+\frac {6435\,a^2\,b^4\,c^8}{2}-1716\,a\,b^6\,c^7+\frac {429\,b^8\,c^6}{2}\right )}{n}+\frac {c^{14}\,x^{27\,n+1}}{14\,n}-\frac {a^{12}\,x^{n+1}\,\left (a\,c-\frac {13\,b^2}{2}\right )}{n}+\frac {a^{10}\,x^{3\,n+1}\,\left (\frac {13\,a^2\,c^2}{2}-78\,a\,b^2\,c+\frac {143\,b^4}{2}\right )}{n}+\frac {c^{10}\,x^{23\,n+1}\,\left (\frac {13\,a^2\,c^2}{2}-78\,a\,b^2\,c+\frac {143\,b^4}{2}\right )}{n}+\frac {b\,c^{13}\,x^{26\,n+1}}{n}-\frac {c^{12}\,x^{25\,n+1}\,\left (a\,c-\frac {13\,b^2}{2}\right )}{n}-\frac {a^{13}\,b\,x}{n}-\frac {143\,a^7\,b\,x^{6\,n+1}\,\left (-14\,a^3\,c^3+70\,a^2\,b^2\,c^2-63\,a\,b^4\,c+12\,b^6\right )}{7\,n}+\frac {143\,b\,c^7\,x^{20\,n+1}\,\left (-14\,a^3\,c^3+70\,a^2\,b^2\,c^2-63\,a\,b^4\,c+12\,b^6\right )}{7\,n}-\frac {143\,a^5\,b\,x^{8\,n+1}\,\left (5\,a^4\,c^4-30\,a^3\,b^2\,c^3+36\,a^2\,b^4\,c^2-12\,a\,b^6\,c+b^8\right )}{n}+\frac {143\,b\,c^5\,x^{18\,n+1}\,\left (5\,a^4\,c^4-30\,a^3\,b^2\,c^3+36\,a^2\,b^4\,c^2-12\,a\,b^6\,c+b^8\right )}{n}-\frac {13\,a^3\,b\,x^{10\,n+1}\,\left (-99\,a^5\,c^5+660\,a^4\,b^2\,c^4-924\,a^3\,b^4\,c^3+396\,a^2\,b^6\,c^2-55\,a\,b^8\,c+2\,b^{10}\right )}{n}+\frac {13\,b\,c^3\,x^{16\,n+1}\,\left (-99\,a^5\,c^5+660\,a^4\,b^2\,c^4-924\,a^3\,b^4\,c^3+396\,a^2\,b^6\,c^2-55\,a\,b^8\,c+2\,b^{10}\right )}{n}-\frac {13\,a^9\,b\,x^{4\,n+1}\,\left (6\,a^2\,c^2-22\,a\,b^2\,c+11\,b^4\right )}{n}+\frac {13\,b\,c^9\,x^{22\,n+1}\,\left (6\,a^2\,c^2-22\,a\,b^2\,c+11\,b^4\right )}{n}+\frac {13\,a^{11}\,b\,x^{2\,n+1}\,\left (a\,c-2\,b^2\right )}{n}-\frac {13\,b\,c^{11}\,x^{24\,n+1}\,\left (a\,c-2\,b^2\right )}{n}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(n - 1)*((x^(11*n + 1)*((13*a^2*b^12)/2 + (429*a^8*c^6)/2 - 286*a^3*b^10*c + (6435*a^4*b^8*c^2)/2 - 12012*a^

5*b^6*c^3 + 15015*a^6*b^4*c^4 - 5148*a^7*b^2*c^5))/n + (x^(15*n + 1)*((429*a^6*c^8)/2 + (13*b^12*c^2)/2 - 286*

a*b^10*c^3 + (6435*a^2*b^8*c^4)/2 - 12012*a^3*b^6*c^5 + 15015*a^4*b^4*c^6 - 5148*a^5*b^2*c^7))/n - (x^(12*n +

1)*(a*b^13 - 78*a^2*b^11*c + 1716*a^7*b*c^6 + 1430*a^3*b^9*c^2 - 8580*a^4*b^7*c^3 + 18018*a^5*b^5*c^4 - 12012*

a^6*b^3*c^5))/n + (x^(14*n + 1)*(b^13*c - 78*a*b^11*c^2 + 1716*a^6*b*c^7 + 1430*a^2*b^9*c^3 - 8580*a^3*b^7*c^4

+ 18018*a^4*b^5*c^5 - 12012*a^5*b^3*c^6))/n + (x^(5*n + 1)*((429*a^8*b^6)/2 - 26*a^11*c^3 - 715*a^9*b^4*c + 4

29*a^10*b^2*c^2))/n - (x^(21*n + 1)*(26*a^3*c^11 - (429*b^6*c^8)/2 + 715*a*b^4*c^9 - 429*a^2*b^2*c^10))/n + (x

^(9*n + 1)*((143*a^4*b^10)/2 - 143*a^9*c^5 - 1287*a^5*b^8*c + 6006*a^6*b^6*c^2 - 8580*a^7*b^4*c^3 + (6435*a^8*

b^2*c^4)/2))/n - (x^(17*n + 1)*(143*a^5*c^9 - (143*b^10*c^4)/2 + 1287*a*b^8*c^5 - 6006*a^2*b^6*c^6 + 8580*a^3*

b^4*c^7 - (6435*a^4*b^2*c^8)/2))/n + (x^(13*n + 1)*(b^14/14 - (1716*a^7*c^7)/7 + 429*a^2*b^10*c^2 - 4290*a^3*b

^8*c^3 + 15015*a^4*b^6*c^4 - 18018*a^5*b^4*c^5 + 6006*a^6*b^2*c^6 - 13*a*b^12*c))/n + (x^(7*n + 1)*((429*a^6*b

^8)/2 + (143*a^10*c^4)/2 - 1716*a^7*b^6*c + (6435*a^8*b^4*c^2)/2 - 1430*a^9*b^2*c^3))/n + (x^(19*n + 1)*((143*

a^4*c^10)/2 + (429*b^8*c^6)/2 - 1716*a*b^6*c^7 + (6435*a^2*b^4*c^8)/2 - 1430*a^3*b^2*c^9))/n + (c^14*x^(27*n +

1))/(14*n) - (a^12*x^(n + 1)*(a*c - (13*b^2)/2))/n + (a^10*x^(3*n + 1)*((143*b^4)/2 + (13*a^2*c^2)/2 - 78*a*b

^2*c))/n + (c^10*x^(23*n + 1)*((143*b^4)/2 + (13*a^2*c^2)/2 - 78*a*b^2*c))/n + (b*c^13*x^(26*n + 1))/n - (c^12

*x^(25*n + 1)*(a*c - (13*b^2)/2))/n - (a^13*b*x)/n - (143*a^7*b*x^(6*n + 1)*(12*b^6 - 14*a^3*c^3 + 70*a^2*b^2*

c^2 - 63*a*b^4*c))/(7*n) + (143*b*c^7*x^(20*n + 1)*(12*b^6 - 14*a^3*c^3 + 70*a^2*b^2*c^2 - 63*a*b^4*c))/(7*n)

- (143*a^5*b*x^(8*n + 1)*(b^8 + 5*a^4*c^4 + 36*a^2*b^4*c^2 - 30*a^3*b^2*c^3 - 12*a*b^6*c))/n + (143*b*c^5*x^(1

8*n + 1)*(b^8 + 5*a^4*c^4 + 36*a^2*b^4*c^2 - 30*a^3*b^2*c^3 - 12*a*b^6*c))/n - (13*a^3*b*x^(10*n + 1)*(2*b^10

- 99*a^5*c^5 + 396*a^2*b^6*c^2 - 924*a^3*b^4*c^3 + 660*a^4*b^2*c^4 - 55*a*b^8*c))/n + (13*b*c^3*x^(16*n + 1)*(

2*b^10 - 99*a^5*c^5 + 396*a^2*b^6*c^2 - 924*a^3*b^4*c^3 + 660*a^4*b^2*c^4 - 55*a*b^8*c))/n - (13*a^9*b*x^(4*n

+ 1)*(11*b^4 + 6*a^2*c^2 - 22*a*b^2*c))/n + (13*b*c^9*x^(22*n + 1)*(11*b^4 + 6*a^2*c^2 - 22*a*b^2*c))/n + (13*

a^11*b*x^(2*n + 1)*(a*c - 2*b^2))/n - (13*b*c^11*x^(24*n + 1)*(a*c - 2*b^2))/n)

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