3.4.32 \(\int (a x+b x^{14})^{12} \, dx\) [332]

Optimal. Leaf size=16 \[ \frac {\left (a+b x^{13}\right )^{13}}{169 b} \]

[Out]

1/169*(b*x^13+a)^13/b

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1607, 267} \begin {gather*} \frac {\left (a+b x^{13}\right )^{13}}{169 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^14)^12,x]

[Out]

(a + b*x^13)^13/(169*b)

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \left (a x+b x^{14}\right )^{12} \, dx &=\int x^{12} \left (a+b x^{13}\right )^{12} \, dx\\ &=\frac {\left (a+b x^{13}\right )^{13}}{169 b}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(160\) vs. \(2(16)=32\).
time = 0.00, size = 160, normalized size = 10.00 \begin {gather*} \frac {a^{12} x^{13}}{13}+\frac {6}{13} a^{11} b x^{26}+\frac {22}{13} a^{10} b^2 x^{39}+\frac {55}{13} a^9 b^3 x^{52}+\frac {99}{13} a^8 b^4 x^{65}+\frac {132}{13} a^7 b^5 x^{78}+\frac {132}{13} a^6 b^6 x^{91}+\frac {99}{13} a^5 b^7 x^{104}+\frac {55}{13} a^4 b^8 x^{117}+\frac {22}{13} a^3 b^9 x^{130}+\frac {6}{13} a^2 b^{10} x^{143}+\frac {1}{13} a b^{11} x^{156}+\frac {b^{12} x^{169}}{169} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x^14)^12,x]

[Out]

(a^12*x^13)/13 + (6*a^11*b*x^26)/13 + (22*a^10*b^2*x^39)/13 + (55*a^9*b^3*x^52)/13 + (99*a^8*b^4*x^65)/13 + (1
32*a^7*b^5*x^78)/13 + (132*a^6*b^6*x^91)/13 + (99*a^5*b^7*x^104)/13 + (55*a^4*b^8*x^117)/13 + (22*a^3*b^9*x^13
0)/13 + (6*a^2*b^10*x^143)/13 + (a*b^11*x^156)/13 + (b^12*x^169)/169

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(134\) vs. \(2(14)=28\).
time = 0.34, size = 135, normalized size = 8.44

method	result	size
default	\(\frac {1}{169} b^{12} x^{169}+\frac {1}{13} a \,b^{11} x^{156}+\frac {6}{13} a^{2} b^{10} x^{143}+\frac {22}{13} a^{3} b^{9} x^{130}+\frac {55}{13} a^{4} b^{8} x^{117}+\frac {99}{13} a^{5} b^{7} x^{104}+\frac {132}{13} a^{6} b^{6} x^{91}+\frac {132}{13} b^{5} a^{7} x^{78}+\frac {99}{13} a^{8} b^{4} x^{65}+\frac {55}{13} b^{3} a^{9} x^{52}+\frac {22}{13} a^{10} b^{2} x^{39}+\frac {6}{13} b \,a^{11} x^{26}+\frac {1}{13} a^{12} x^{13}\)	\(135\)
gosper	\(\frac {x^{13} \left (b^{12} x^{156}+13 a \,b^{11} x^{143}+78 a^{2} b^{10} x^{130}+286 a^{3} b^{9} x^{117}+715 a^{4} b^{8} x^{104}+1287 a^{5} b^{7} x^{91}+1716 a^{6} b^{6} x^{78}+1716 b^{5} a^{7} x^{65}+1287 a^{8} b^{4} x^{52}+715 b^{3} a^{9} x^{39}+286 a^{10} b^{2} x^{26}+78 b \,a^{11} x^{13}+13 a^{12}\right )}{169}\)	\(136\)
risch	\(\frac {b^{12} x^{169}}{169}+\frac {a \,b^{11} x^{156}}{13}+\frac {6 a^{2} b^{10} x^{143}}{13}+\frac {22 a^{3} b^{9} x^{130}}{13}+\frac {55 a^{4} b^{8} x^{117}}{13}+\frac {99 a^{5} b^{7} x^{104}}{13}+\frac {132 a^{6} b^{6} x^{91}}{13}+\frac {132 b^{5} a^{7} x^{78}}{13}+\frac {99 a^{8} b^{4} x^{65}}{13}+\frac {55 b^{3} a^{9} x^{52}}{13}+\frac {22 a^{10} b^{2} x^{39}}{13}+\frac {6 b \,a^{11} x^{26}}{13}+\frac {a^{12} x^{13}}{13}+\frac {a^{13}}{169 b}\)	\(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^14+a*x)^12,x,method=_RETURNVERBOSE)

[Out]

1/169*b^12*x^169+1/13*a*b^11*x^156+6/13*a^2*b^10*x^143+22/13*a^3*b^9*x^130+55/13*a^4*b^8*x^117+99/13*a^5*b^7*x
^104+132/13*a^6*b^6*x^91+132/13*b^5*a^7*x^78+99/13*a^8*b^4*x^65+55/13*b^3*a^9*x^52+22/13*a^10*b^2*x^39+6/13*b*
a^11*x^26+1/13*a^12*x^13

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (14) = 28\).
time = 0.28, size = 134, normalized size = 8.38 \begin {gather*} \frac {1}{169} \, b^{12} x^{169} + \frac {1}{13} \, a b^{11} x^{156} + \frac {6}{13} \, a^{2} b^{10} x^{143} + \frac {22}{13} \, a^{3} b^{9} x^{130} + \frac {55}{13} \, a^{4} b^{8} x^{117} + \frac {99}{13} \, a^{5} b^{7} x^{104} + \frac {132}{13} \, a^{6} b^{6} x^{91} + \frac {132}{13} \, a^{7} b^{5} x^{78} + \frac {99}{13} \, a^{8} b^{4} x^{65} + \frac {55}{13} \, a^{9} b^{3} x^{52} + \frac {22}{13} \, a^{10} b^{2} x^{39} + \frac {6}{13} \, a^{11} b x^{26} + \frac {1}{13} \, a^{12} x^{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^14+a*x)^12,x, algorithm="maxima")

[Out]

1/169*b^12*x^169 + 1/13*a*b^11*x^156 + 6/13*a^2*b^10*x^143 + 22/13*a^3*b^9*x^130 + 55/13*a^4*b^8*x^117 + 99/13
*a^5*b^7*x^104 + 132/13*a^6*b^6*x^91 + 132/13*a^7*b^5*x^78 + 99/13*a^8*b^4*x^65 + 55/13*a^9*b^3*x^52 + 22/13*a
^10*b^2*x^39 + 6/13*a^11*b*x^26 + 1/13*a^12*x^13

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (14) = 28\).
time = 2.03, size = 134, normalized size = 8.38 \begin {gather*} \frac {1}{169} \, b^{12} x^{169} + \frac {1}{13} \, a b^{11} x^{156} + \frac {6}{13} \, a^{2} b^{10} x^{143} + \frac {22}{13} \, a^{3} b^{9} x^{130} + \frac {55}{13} \, a^{4} b^{8} x^{117} + \frac {99}{13} \, a^{5} b^{7} x^{104} + \frac {132}{13} \, a^{6} b^{6} x^{91} + \frac {132}{13} \, a^{7} b^{5} x^{78} + \frac {99}{13} \, a^{8} b^{4} x^{65} + \frac {55}{13} \, a^{9} b^{3} x^{52} + \frac {22}{13} \, a^{10} b^{2} x^{39} + \frac {6}{13} \, a^{11} b x^{26} + \frac {1}{13} \, a^{12} x^{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^14+a*x)^12,x, algorithm="fricas")

[Out]

1/169*b^12*x^169 + 1/13*a*b^11*x^156 + 6/13*a^2*b^10*x^143 + 22/13*a^3*b^9*x^130 + 55/13*a^4*b^8*x^117 + 99/13
*a^5*b^7*x^104 + 132/13*a^6*b^6*x^91 + 132/13*a^7*b^5*x^78 + 99/13*a^8*b^4*x^65 + 55/13*a^9*b^3*x^52 + 22/13*a
^10*b^2*x^39 + 6/13*a^11*b*x^26 + 1/13*a^12*x^13

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (10) = 20\).
time = 0.03, size = 160, normalized size = 10.00 \begin {gather*} \frac {a^{12} x^{13}}{13} + \frac {6 a^{11} b x^{26}}{13} + \frac {22 a^{10} b^{2} x^{39}}{13} + \frac {55 a^{9} b^{3} x^{52}}{13} + \frac {99 a^{8} b^{4} x^{65}}{13} + \frac {132 a^{7} b^{5} x^{78}}{13} + \frac {132 a^{6} b^{6} x^{91}}{13} + \frac {99 a^{5} b^{7} x^{104}}{13} + \frac {55 a^{4} b^{8} x^{117}}{13} + \frac {22 a^{3} b^{9} x^{130}}{13} + \frac {6 a^{2} b^{10} x^{143}}{13} + \frac {a b^{11} x^{156}}{13} + \frac {b^{12} x^{169}}{169} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**14+a*x)**12,x)

[Out]

a**12*x**13/13 + 6*a**11*b*x**26/13 + 22*a**10*b**2*x**39/13 + 55*a**9*b**3*x**52/13 + 99*a**8*b**4*x**65/13 +
132*a**7*b**5*x**78/13 + 132*a**6*b**6*x**91/13 + 99*a**5*b**7*x**104/13 + 55*a**4*b**8*x**117/13 + 22*a**3*b
**9*x**130/13 + 6*a**2*b**10*x**143/13 + a*b**11*x**156/13 + b**12*x**169/169

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (14) = 28\).
time = 0.48, size = 134, normalized size = 8.38 \begin {gather*} \frac {1}{169} \, b^{12} x^{169} + \frac {1}{13} \, a b^{11} x^{156} + \frac {6}{13} \, a^{2} b^{10} x^{143} + \frac {22}{13} \, a^{3} b^{9} x^{130} + \frac {55}{13} \, a^{4} b^{8} x^{117} + \frac {99}{13} \, a^{5} b^{7} x^{104} + \frac {132}{13} \, a^{6} b^{6} x^{91} + \frac {132}{13} \, a^{7} b^{5} x^{78} + \frac {99}{13} \, a^{8} b^{4} x^{65} + \frac {55}{13} \, a^{9} b^{3} x^{52} + \frac {22}{13} \, a^{10} b^{2} x^{39} + \frac {6}{13} \, a^{11} b x^{26} + \frac {1}{13} \, a^{12} x^{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^14+a*x)^12,x, algorithm="giac")

[Out]

1/169*b^12*x^169 + 1/13*a*b^11*x^156 + 6/13*a^2*b^10*x^143 + 22/13*a^3*b^9*x^130 + 55/13*a^4*b^8*x^117 + 99/13
*a^5*b^7*x^104 + 132/13*a^6*b^6*x^91 + 132/13*a^7*b^5*x^78 + 99/13*a^8*b^4*x^65 + 55/13*a^9*b^3*x^52 + 22/13*a
^10*b^2*x^39 + 6/13*a^11*b*x^26 + 1/13*a^12*x^13

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Mupad [B]
time = 0.00, size = 134, normalized size = 8.38 \begin {gather*} \frac {a^{12}\,x^{13}}{13}+\frac {6\,a^{11}\,b\,x^{26}}{13}+\frac {22\,a^{10}\,b^2\,x^{39}}{13}+\frac {55\,a^9\,b^3\,x^{52}}{13}+\frac {99\,a^8\,b^4\,x^{65}}{13}+\frac {132\,a^7\,b^5\,x^{78}}{13}+\frac {132\,a^6\,b^6\,x^{91}}{13}+\frac {99\,a^5\,b^7\,x^{104}}{13}+\frac {55\,a^4\,b^8\,x^{117}}{13}+\frac {22\,a^3\,b^9\,x^{130}}{13}+\frac {6\,a^2\,b^{10}\,x^{143}}{13}+\frac {a\,b^{11}\,x^{156}}{13}+\frac {b^{12}\,x^{169}}{169} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + b*x^14)^12,x)

[Out]

(a^12*x^13)/13 + (b^12*x^169)/169 + (6*a^11*b*x^26)/13 + (a*b^11*x^156)/13 + (22*a^10*b^2*x^39)/13 + (55*a^9*b
^3*x^52)/13 + (99*a^8*b^4*x^65)/13 + (132*a^7*b^5*x^78)/13 + (132*a^6*b^6*x^91)/13 + (99*a^5*b^7*x^104)/13 + (
55*a^4*b^8*x^117)/13 + (22*a^3*b^9*x^130)/13 + (6*a^2*b^10*x^143)/13

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