3.4.52 \(\int x^{24} (a+b x^{25})^{12} \, dx\) [352]

Optimal. Leaf size=16 \[ \frac {\left (a+b x^{25}\right )^{13}}{325 b} \]

[Out]

1/325*(b*x^25+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 = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \begin {gather*} \frac {\left (a+b x^{25}\right )^{13}}{325 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^24*(a + b*x^25)^12,x]

[Out]

(a + b*x^25)^13/(325*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]

Rubi steps

\begin {align*} \int x^{24} \left (a+b x^{25}\right )^{12} \, dx &=\frac {\left (a+b x^{25}\right )^{13}}{325 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^{25}}{25}+\frac {6}{25} a^{11} b x^{50}+\frac {22}{25} a^{10} b^2 x^{75}+\frac {11}{5} a^9 b^3 x^{100}+\frac {99}{25} a^8 b^4 x^{125}+\frac {132}{25} a^7 b^5 x^{150}+\frac {132}{25} a^6 b^6 x^{175}+\frac {99}{25} a^5 b^7 x^{200}+\frac {11}{5} a^4 b^8 x^{225}+\frac {22}{25} a^3 b^9 x^{250}+\frac {6}{25} a^2 b^{10} x^{275}+\frac {1}{25} a b^{11} x^{300}+\frac {b^{12} x^{325}}{325} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^24*(a + b*x^25)^12,x]

[Out]

(a^12*x^25)/25 + (6*a^11*b*x^50)/25 + (22*a^10*b^2*x^75)/25 + (11*a^9*b^3*x^100)/5 + (99*a^8*b^4*x^125)/25 + (
132*a^7*b^5*x^150)/25 + (132*a^6*b^6*x^175)/25 + (99*a^5*b^7*x^200)/25 + (11*a^4*b^8*x^225)/5 + (22*a^3*b^9*x^
250)/25 + (6*a^2*b^10*x^275)/25 + (a*b^11*x^300)/25 + (b^12*x^325)/325

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Maple [A]
time = 0.53, size = 15, normalized size = 0.94

method	result	size
default	\(\frac {\left (b \,x^{25}+a \right )^{13}}{325 b}\)	\(15\)
gosper	\(\frac {132}{25} a^{6} b^{6} x^{175}+\frac {11}{5} b^{3} a^{9} x^{100}+\frac {22}{25} a^{10} b^{2} x^{75}+\frac {6}{25} b \,a^{11} x^{50}+\frac {1}{25} a^{12} x^{25}+\frac {132}{25} b^{5} a^{7} x^{150}+\frac {99}{25} a^{8} b^{4} x^{125}+\frac {1}{325} b^{12} x^{325}+\frac {1}{25} a \,b^{11} x^{300}+\frac {11}{5} a^{4} b^{8} x^{225}+\frac {6}{25} a^{2} b^{10} x^{275}+\frac {22}{25} a^{3} b^{9} x^{250}+\frac {99}{25} a^{5} b^{7} x^{200}\)	\(135\)
risch	\(\frac {b^{12} x^{325}}{325}+\frac {a \,b^{11} x^{300}}{25}+\frac {6 a^{2} b^{10} x^{275}}{25}+\frac {22 a^{3} b^{9} x^{250}}{25}+\frac {11 a^{4} b^{8} x^{225}}{5}+\frac {99 a^{5} b^{7} x^{200}}{25}+\frac {132 a^{6} b^{6} x^{175}}{25}+\frac {132 b^{5} a^{7} x^{150}}{25}+\frac {99 a^{8} b^{4} x^{125}}{25}+\frac {11 b^{3} a^{9} x^{100}}{5}+\frac {22 a^{10} b^{2} x^{75}}{25}+\frac {6 b \,a^{11} x^{50}}{25}+\frac {a^{12} x^{25}}{25}+\frac {a^{13}}{325 b}\)	\(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^24*(b*x^25+a)^12,x,method=_RETURNVERBOSE)

[Out]

1/325*(b*x^25+a)^13/b

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Maxima [A]
time = 0.29, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{25} + a\right )}^{13}}{325 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^24*(b*x^25+a)^12,x, algorithm="maxima")

[Out]

1/325*(b*x^25 + a)^13/b

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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 = 1.62, size = 134, normalized size = 8.38 \begin {gather*} \frac {1}{325} \, b^{12} x^{325} + \frac {1}{25} \, a b^{11} x^{300} + \frac {6}{25} \, a^{2} b^{10} x^{275} + \frac {22}{25} \, a^{3} b^{9} x^{250} + \frac {11}{5} \, a^{4} b^{8} x^{225} + \frac {99}{25} \, a^{5} b^{7} x^{200} + \frac {132}{25} \, a^{6} b^{6} x^{175} + \frac {132}{25} \, a^{7} b^{5} x^{150} + \frac {99}{25} \, a^{8} b^{4} x^{125} + \frac {11}{5} \, a^{9} b^{3} x^{100} + \frac {22}{25} \, a^{10} b^{2} x^{75} + \frac {6}{25} \, a^{11} b x^{50} + \frac {1}{25} \, a^{12} x^{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^24*(b*x^25+a)^12,x, algorithm="fricas")

[Out]

1/325*b^12*x^325 + 1/25*a*b^11*x^300 + 6/25*a^2*b^10*x^275 + 22/25*a^3*b^9*x^250 + 11/5*a^4*b^8*x^225 + 99/25*
a^5*b^7*x^200 + 132/25*a^6*b^6*x^175 + 132/25*a^7*b^5*x^150 + 99/25*a^8*b^4*x^125 + 11/5*a^9*b^3*x^100 + 22/25
*a^10*b^2*x^75 + 6/25*a^11*b*x^50 + 1/25*a^12*x^25

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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^{25}}{25} + \frac {6 a^{11} b x^{50}}{25} + \frac {22 a^{10} b^{2} x^{75}}{25} + \frac {11 a^{9} b^{3} x^{100}}{5} + \frac {99 a^{8} b^{4} x^{125}}{25} + \frac {132 a^{7} b^{5} x^{150}}{25} + \frac {132 a^{6} b^{6} x^{175}}{25} + \frac {99 a^{5} b^{7} x^{200}}{25} + \frac {11 a^{4} b^{8} x^{225}}{5} + \frac {22 a^{3} b^{9} x^{250}}{25} + \frac {6 a^{2} b^{10} x^{275}}{25} + \frac {a b^{11} x^{300}}{25} + \frac {b^{12} x^{325}}{325} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**24*(b*x**25+a)**12,x)

[Out]

a**12*x**25/25 + 6*a**11*b*x**50/25 + 22*a**10*b**2*x**75/25 + 11*a**9*b**3*x**100/5 + 99*a**8*b**4*x**125/25
+ 132*a**7*b**5*x**150/25 + 132*a**6*b**6*x**175/25 + 99*a**5*b**7*x**200/25 + 11*a**4*b**8*x**225/5 + 22*a**3
*b**9*x**250/25 + 6*a**2*b**10*x**275/25 + a*b**11*x**300/25 + b**12*x**325/325

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Giac [A]
time = 0.58, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{25} + a\right )}^{13}}{325 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^24*(b*x^25+a)^12,x, algorithm="giac")

[Out]

1/325*(b*x^25 + a)^13/b

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Mupad [B]
time = 5.20, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b\,x^{25}+a\right )}^{13}}{325\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^24*(a + b*x^25)^12,x)

[Out]

(a + b*x^25)^13/(325*b)

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