3.1.93 \(\int (b+2 c x) (a+b x+c x^2)^{13} \, dx\) [93]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {643} \begin {gather*} \frac {1}{14} \left (a+b x+c x^2\right )^{14} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x + c*x^2)^14/14

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]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(201\) vs. \(2(16)=32\).
time = 0.12, size = 201, normalized size = 12.56 \begin {gather*} \frac {1}{14} x (b+c x) \left (14 a^{13}+91 a^{12} x (b+c x)+364 a^{11} x^2 (b+c x)^2+1001 a^{10} x^3 (b+c x)^3+2002 a^9 x^4 (b+c x)^4+3003 a^8 x^5 (b+c x)^5+3432 a^7 x^6 (b+c x)^6+3003 a^6 x^7 (b+c x)^7+2002 a^5 x^8 (b+c x)^8+1001 a^4 x^9 (b+c x)^9+364 a^3 x^{10} (b+c x)^{10}+91 a^2 x^{11} (b+c x)^{11}+14 a x^{12} (b+c x)^{12}+x^{13} (b+c x)^{13}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b + c*x)*(14*a^13 + 91*a^12*x*(b + c*x) + 364*a^11*x^2*(b + c*x)^2 + 1001*a^10*x^3*(b + c*x)^3 + 2002*a^9*
x^4*(b + c*x)^4 + 3003*a^8*x^5*(b + c*x)^5 + 3432*a^7*x^6*(b + c*x)^6 + 3003*a^6*x^7*(b + c*x)^7 + 2002*a^5*x^
8*(b + c*x)^8 + 1001*a^4*x^9*(b + c*x)^9 + 364*a^3*x^10*(b + c*x)^10 + 91*a^2*x^11*(b + c*x)^11 + 14*a*x^12*(b
+ c*x)^12 + x^13*(b + c*x)^13))/14

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Maple [A]
time = 0.31, size = 15, normalized size = 0.94 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 14, normalized size = 0.88 \begin {gather*} \frac {1}{14} \, {\left (c x^{2} + b x + a\right )}^{14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*(c*x^2 + b*x + a)^14

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1234 vs. \(2 (14) = 28\).
time = 0.36, size = 1234, normalized size = 77.12 \begin {gather*} \frac {1}{14} \, c^{14} x^{28} + b c^{13} x^{27} + \frac {1}{2} \, {\left (13 \, b^{2} c^{12} + 2 \, a c^{13}\right )} x^{26} + 13 \, {\left (2 \, b^{3} c^{11} + a b c^{12}\right )} x^{25} + \frac {13}{2} \, {\left (11 \, b^{4} c^{10} + 12 \, a b^{2} c^{11} + a^{2} c^{12}\right )} x^{24} + 13 \, {\left (11 \, b^{5} c^{9} + 22 \, a b^{3} c^{10} + 6 \, a^{2} b c^{11}\right )} x^{23} + \frac {13}{2} \, {\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} + \frac {143}{7} \, {\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} + \frac {143}{2} \, {\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} + 143 \, {\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} + \frac {143}{2} \, {\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} + 13 \, {\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} + \frac {13}{2} \, {\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} + {\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} + a^{13} b x + \frac {1}{14} \, {\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} + {\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} + \frac {13}{2} \, {\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} + 13 \, {\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} + \frac {143}{2} \, {\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} + 143 \, {\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} + \frac {143}{2} \, {\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} + \frac {143}{7} \, {\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} + \frac {13}{2} \, {\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} + 13 \, {\left (11 \, a^{9} b^{5} + 22 \, a^{10} b^{3} c + 6 \, a^{11} b c^{2}\right )} x^{5} + \frac {13}{2} \, {\left (11 \, a^{10} b^{4} + 12 \, a^{11} b^{2} c + a^{12} c^{2}\right )} x^{4} + 13 \, {\left (2 \, a^{11} b^{3} + a^{12} b c\right )} x^{3} + \frac {1}{2} \, {\left (13 \, a^{12} b^{2} + 2 \, a^{13} c\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*c^14*x^28 + b*c^13*x^27 + 1/2*(13*b^2*c^12 + 2*a*c^13)*x^26 + 13*(2*b^3*c^11 + a*b*c^12)*x^25 + 13/2*(11*
b^4*c^10 + 12*a*b^2*c^11 + a^2*c^12)*x^24 + 13*(11*b^5*c^9 + 22*a*b^3*c^10 + 6*a^2*b*c^11)*x^23 + 13/2*(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 + 143/7*(12*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 + 143/2*(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 + 1
43*(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 + 143/2*(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 + 13*(2*b^11*c^3 + 55*a*b^9*c^4 + 39
6*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 + 13/2*(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 + (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 + a^13
*b*x + 1/14*(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 + (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 + 13/2*(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)*x^12 + 13*(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)*x^11 + 143/2*(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 + 143*(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 + 143/2*(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 + 143/7*(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 + 13/2*(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 + 13*(11*a^9*b^5 + 22*a^10*b^3*c + 6*a^11*b*c^2)*x^5 + 13/2*
(11*a^10*b^4 + 12*a^11*b^2*c + a^12*c^2)*x^4 + 13*(2*a^11*b^3 + a^12*b*c)*x^3 + 1/2*(13*a^12*b^2 + 2*a^13*c)*x
^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1326 vs. \(2 (12) = 24\).
time = 0.14, size = 1326, normalized size = 82.88 \begin {gather*} a^{13} b x + b c^{13} x^{27} + \frac {c^{14} x^{28}}{14} + x^{26} \left (a c^{13} + \frac {13 b^{2} c^{12}}{2}\right ) + x^{25} \cdot \left (13 a b c^{12} + 26 b^{3} c^{11}\right ) + x^{24} \cdot \left (\frac {13 a^{2} c^{12}}{2} + 78 a b^{2} c^{11} + \frac {143 b^{4} c^{10}}{2}\right ) + x^{23} \cdot \left (78 a^{2} b c^{11} + 286 a b^{3} c^{10} + 143 b^{5} c^{9}\right ) + x^{22} \cdot \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 ) + x^{21} \cdot \left (286 a^{3} b c^{10} + 1430 a^{2} b^{3} c^{9} + 1287 a b^{5} c^{8} + \frac {1716 b^{7} c^{7}}{7}\right ) + x^{20} \cdot \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 ) + x^{19} \cdot \left (715 a^{4} b c^{9} + 4290 a^{3} b^{3} c^{8} + 5148 a^{2} b^{5} c^{7} + 1716 a b^{7} c^{6} + 143 b^{9} c^{5}\right ) + x^{18} \cdot \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 ) + x^{17} \cdot \left (1287 a^{5} b c^{8} + 8580 a^{4} b^{3} c^{7} + 12012 a^{3} b^{5} c^{6} + 5148 a^{2} b^{7} c^{5} + 715 a b^{9} c^{4} + 26 b^{11} c^{3}\right ) + x^{16} \cdot \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 ) + x^{15} \cdot \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 ) + x^{14} \cdot \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 ) + x^{13} \cdot \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 ) + x^{12} \cdot \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 ) + x^{11} \cdot \left (1287 a^{8} b c^{5} + 8580 a^{7} b^{3} c^{4} + 12012 a^{6} b^{5} c^{3} + 5148 a^{5} b^{7} c^{2} + 715 a^{4} b^{9} c + 26 a^{3} b^{11}\right ) + x^{10} \cdot \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 ) + x^{9} \cdot \left (715 a^{9} b c^{4} + 4290 a^{8} b^{3} c^{3} + 5148 a^{7} b^{5} c^{2} + 1716 a^{6} b^{7} c + 143 a^{5} b^{9}\right ) + x^{8} \cdot \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 ) + x^{7} \cdot \left (286 a^{10} b c^{3} + 1430 a^{9} b^{3} c^{2} + 1287 a^{8} b^{5} c + \frac {1716 a^{7} b^{7}}{7}\right ) + x^{6} \cdot \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 ) + x^{5} \cdot \left (78 a^{11} b c^{2} + 286 a^{10} b^{3} c + 143 a^{9} b^{5}\right ) + x^{4} \cdot \left (\frac {13 a^{12} c^{2}}{2} + 78 a^{11} b^{2} c + \frac {143 a^{10} b^{4}}{2}\right ) + x^{3} \cdot \left (13 a^{12} b c + 26 a^{11} b^{3}\right ) + x^{2} \left (a^{13} c + \frac {13 a^{12} b^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**13*b*x + b*c**13*x**27 + c**14*x**28/14 + x**26*(a*c**13 + 13*b**2*c**12/2) + x**25*(13*a*b*c**12 + 26*b**3
*c**11) + x**24*(13*a**2*c**12/2 + 78*a*b**2*c**11 + 143*b**4*c**10/2) + x**23*(78*a**2*b*c**11 + 286*a*b**3*c
**10 + 143*b**5*c**9) + x**22*(26*a**3*c**11 + 429*a**2*b**2*c**10 + 715*a*b**4*c**9 + 429*b**6*c**8/2) + x**2
1*(286*a**3*b*c**10 + 1430*a**2*b**3*c**9 + 1287*a*b**5*c**8 + 1716*b**7*c**7/7) + x**20*(143*a**4*c**10/2 + 1
430*a**3*b**2*c**9 + 6435*a**2*b**4*c**8/2 + 1716*a*b**6*c**7 + 429*b**8*c**6/2) + x**19*(715*a**4*b*c**9 + 42
90*a**3*b**3*c**8 + 5148*a**2*b**5*c**7 + 1716*a*b**7*c**6 + 143*b**9*c**5) + x**18*(143*a**5*c**9 + 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 + 143*b**10*c**4/2) + x**17*(1287*
a**5*b*c**8 + 8580*a**4*b**3*c**7 + 12012*a**3*b**5*c**6 + 5148*a**2*b**7*c**5 + 715*a*b**9*c**4 + 26*b**11*c*
*3) + x**16*(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 + 6435*a**2*b
**8*c**4/2 + 286*a*b**10*c**3 + 13*b**12*c**2/2) + x**15*(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) + x**14*(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 + b**14/14) + x**13*(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) + x**12*(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 + 6435*a**4*b**8*c**2/2 + 286*a**3*b**10*c + 13*a**2*b**12/2) +
x**11*(1287*a**8*b*c**5 + 8580*a**7*b**3*c**4 + 12012*a**6*b**5*c**3 + 5148*a**5*b**7*c**2 + 715*a**4*b**9*c
+ 26*a**3*b**11) + x**10*(143*a**9*c**5 + 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 + 143*a**4*b**10/2) + x**9*(715*a**9*b*c**4 + 4290*a**8*b**3*c**3 + 5148*a**7*b**5*c**2 + 171
6*a**6*b**7*c + 143*a**5*b**9) + x**8*(143*a**10*c**4/2 + 1430*a**9*b**2*c**3 + 6435*a**8*b**4*c**2/2 + 1716*a
**7*b**6*c + 429*a**6*b**8/2) + x**7*(286*a**10*b*c**3 + 1430*a**9*b**3*c**2 + 1287*a**8*b**5*c + 1716*a**7*b*
*7/7) + x**6*(26*a**11*c**3 + 429*a**10*b**2*c**2 + 715*a**9*b**4*c + 429*a**8*b**6/2) + x**5*(78*a**11*b*c**2
+ 286*a**10*b**3*c + 143*a**9*b**5) + x**4*(13*a**12*c**2/2 + 78*a**11*b**2*c + 143*a**10*b**4/2) + x**3*(13*
a**12*b*c + 26*a**11*b**3) + x**2*(a**13*c + 13*a**12*b**2/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (14) = 28\).
time = 3.20, size = 216, normalized size = 13.50 \begin {gather*} \frac {1}{14} \, {\left (c x^{2} + b x\right )}^{14} + {\left (c x^{2} + b x\right )}^{13} a + \frac {13}{2} \, {\left (c x^{2} + b x\right )}^{12} a^{2} + 26 \, {\left (c x^{2} + b x\right )}^{11} a^{3} + \frac {143}{2} \, {\left (c x^{2} + b x\right )}^{10} a^{4} + 143 \, {\left (c x^{2} + b x\right )}^{9} a^{5} + \frac {429}{2} \, {\left (c x^{2} + b x\right )}^{8} a^{6} + \frac {1716}{7} \, {\left (c x^{2} + b x\right )}^{7} a^{7} + \frac {429}{2} \, {\left (c x^{2} + b x\right )}^{6} a^{8} + 143 \, {\left (c x^{2} + b x\right )}^{5} a^{9} + \frac {143}{2} \, {\left (c x^{2} + b x\right )}^{4} a^{10} + 26 \, {\left (c x^{2} + b x\right )}^{3} a^{11} + \frac {13}{2} \, {\left (c x^{2} + b x\right )}^{2} a^{12} + {\left (c x^{2} + b x\right )} a^{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*(c*x^2 + b*x)^14 + (c*x^2 + b*x)^13*a + 13/2*(c*x^2 + b*x)^12*a^2 + 26*(c*x^2 + b*x)^11*a^3 + 143/2*(c*x^
2 + b*x)^10*a^4 + 143*(c*x^2 + b*x)^9*a^5 + 429/2*(c*x^2 + b*x)^8*a^6 + 1716/7*(c*x^2 + b*x)^7*a^7 + 429/2*(c*
x^2 + b*x)^6*a^8 + 143*(c*x^2 + b*x)^5*a^9 + 143/2*(c*x^2 + b*x)^4*a^10 + 26*(c*x^2 + b*x)^3*a^11 + 13/2*(c*x^
2 + b*x)^2*a^12 + (c*x^2 + b*x)*a^13

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Mupad [B]
time = 3.34, size = 1203, normalized size = 75.19 \begin {gather*} x^{12}\,\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 )+x^{16}\,\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 )+x^{13}\,\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 )+x^{15}\,\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 )+x^6\,\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 )+x^{22}\,\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 )+x^{10}\,\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 )+x^{18}\,\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 )+x^{14}\,\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 )+x^8\,\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 )+x^{20}\,\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 )+\frac {c^{14}\,x^{28}}{14}+x^2\,\left (c\,a^{13}+\frac {13\,a^{12}\,b^2}{2}\right )+\frac {13\,a^{10}\,x^4\,\left (a^2\,c^2+12\,a\,b^2\,c+11\,b^4\right )}{2}+\frac {13\,c^{10}\,x^{24}\,\left (a^2\,c^2+12\,a\,b^2\,c+11\,b^4\right )}{2}+b\,c^{13}\,x^{27}+\frac {c^{12}\,x^{26}\,\left (13\,b^2+2\,a\,c\right )}{2}+a^{13}\,b\,x+\frac {143\,a^7\,b\,x^7\,\left (14\,a^3\,c^3+70\,a^2\,b^2\,c^2+63\,a\,b^4\,c+12\,b^6\right )}{7}+\frac {143\,b\,c^7\,x^{21}\,\left (14\,a^3\,c^3+70\,a^2\,b^2\,c^2+63\,a\,b^4\,c+12\,b^6\right )}{7}+143\,a^5\,b\,x^9\,\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 )+143\,b\,c^5\,x^{19}\,\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 )+13\,a^3\,b\,x^{11}\,\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 )+13\,b\,c^3\,x^{17}\,\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 )+13\,a^9\,b\,x^5\,\left (6\,a^2\,c^2+22\,a\,b^2\,c+11\,b^4\right )+13\,b\,c^9\,x^{23}\,\left (6\,a^2\,c^2+22\,a\,b^2\,c+11\,b^4\right )+13\,a^{11}\,b\,x^3\,\left (2\,b^2+a\,c\right )+13\,b\,c^{11}\,x^{25}\,\left (2\,b^2+a\,c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^12*((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) + x^16*((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) + x^13*(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) + x^15*(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) + x^6*((429*a^
8*b^6)/2 + 26*a^11*c^3 + 715*a^9*b^4*c + 429*a^10*b^2*c^2) + x^22*(26*a^3*c^11 + (429*b^6*c^8)/2 + 715*a*b^4*c
^9 + 429*a^2*b^2*c^10) + x^10*((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) + x^18*(143*a^5*c^9 + (143*b^10*c^4)/2 + 1287*a*b^8*c^5 + 6006*a^2*b^6*c^6 + 85
80*a^3*b^4*c^7 + (6435*a^4*b^2*c^8)/2) + x^14*(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) + x^8*((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) + x^20*((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) + (c^14*x^28)/14 + x^2*(a^13*c + (13*a^12*b^2)/2)
+ (13*a^10*x^4*(11*b^4 + a^2*c^2 + 12*a*b^2*c))/2 + (13*c^10*x^24*(11*b^4 + a^2*c^2 + 12*a*b^2*c))/2 + b*c^13
*x^27 + (c^12*x^26*(2*a*c + 13*b^2))/2 + a^13*b*x + (143*a^7*b*x^7*(12*b^6 + 14*a^3*c^3 + 70*a^2*b^2*c^2 + 63*
a*b^4*c))/7 + (143*b*c^7*x^21*(12*b^6 + 14*a^3*c^3 + 70*a^2*b^2*c^2 + 63*a*b^4*c))/7 + 143*a^5*b*x^9*(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) + 143*b*c^5*x^19*(b^8 + 5*a^4*c^4 + 36*a^2*b^4*c^2 + 3
0*a^3*b^2*c^3 + 12*a*b^6*c) + 13*a^3*b*x^11*(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) + 13*b*c^3*x^17*(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) + 13*a^9*b*x^5*(11*b^4 + 6*a^2*c^2 + 22*a*b^2*c) + 13*b*c^9*x^23*(11*b^4 + 6*a^2*c^2 + 22*a*
b^2*c) + 13*a^11*b*x^3*(a*c + 2*b^2) + 13*b*c^11*x^25*(a*c + 2*b^2)

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