3.2.34 \(\int (a+b x)^{10} \, dx\) [134]

Optimal. Leaf size=14 \[ \frac {(a+b x)^{11}}{11 b} \]

[Out]

1/11*(b*x+a)^11/b

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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \begin {gather*} \frac {(a+b x)^{11}}{11 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10,x]

[Out]

(a + b*x)^11/(11*b)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {align*} \int (a+b x)^{10} \, dx &=\frac {(a+b x)^{11}}{11 b}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {(a+b x)^{11}}{11 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10,x]

[Out]

(a + b*x)^11/(11*b)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(14)=28\).
time = 2.10, size = 106, normalized size = 7.57 \begin {gather*} x \left (a^{10}+5 a^9 b x+15 a^8 b^2 x^2+30 a^7 b^3 x^3+42 a^6 b^4 x^4+42 a^5 b^5 x^5+30 a^4 b^6 x^6+15 a^3 b^7 x^7+5 a^2 b^8 x^8+a b^9 x^9+\frac {b^{10} x^{10}}{11}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^0*(a + b*x)^10,x]')

[Out]

x (a ^ 10 + 5 a ^ 9 b x + 15 a ^ 8 b ^ 2 x ^ 2 + 30 a ^ 7 b ^ 3 x ^ 3 + 42 a ^ 6 b ^ 4 x ^ 4 + 42 a ^ 5 b ^ 5
x ^ 5 + 30 a ^ 4 b ^ 6 x ^ 6 + 15 a ^ 3 b ^ 7 x ^ 7 + 5 a ^ 2 b ^ 8 x ^ 8 + a b ^ 9 x ^ 9 + b ^ 10 x ^ 10 / 11
)

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Maple [A]
time = 0.08, size = 13, normalized size = 0.93

method	result	size
default	\(\frac {\left (b x +a \right )^{11}}{11 b}\)	\(13\)
gosper	\(\frac {1}{11} b^{10} x^{11}+a \,b^{9} x^{10}+5 a^{2} b^{8} x^{9}+15 a^{3} b^{7} x^{8}+30 a^{4} b^{6} x^{7}+42 a^{5} b^{5} x^{6}+42 a^{6} b^{4} x^{5}+30 a^{7} b^{3} x^{4}+15 a^{8} b^{2} x^{3}+5 a^{9} b \,x^{2}+a^{10} x\)	\(109\)
norman	\(\frac {1}{11} b^{10} x^{11}+a \,b^{9} x^{10}+5 a^{2} b^{8} x^{9}+15 a^{3} b^{7} x^{8}+30 a^{4} b^{6} x^{7}+42 a^{5} b^{5} x^{6}+42 a^{6} b^{4} x^{5}+30 a^{7} b^{3} x^{4}+15 a^{8} b^{2} x^{3}+5 a^{9} b \,x^{2}+a^{10} x\)	\(109\)
risch	\(\frac {b^{10} x^{11}}{11}+a \,b^{9} x^{10}+5 a^{2} b^{8} x^{9}+15 a^{3} b^{7} x^{8}+30 a^{4} b^{6} x^{7}+42 a^{5} b^{5} x^{6}+42 a^{6} b^{4} x^{5}+30 a^{7} b^{3} x^{4}+15 a^{8} b^{2} x^{3}+5 a^{9} b \,x^{2}+a^{10} x +\frac {a^{11}}{11 b}\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10,x,method=_RETURNVERBOSE)

[Out]

1/11*(b*x+a)^11/b

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Maxima [A]
time = 0.25, size = 12, normalized size = 0.86 \begin {gather*} \frac {{\left (b x + a\right )}^{11}}{11 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10,x, algorithm="maxima")

[Out]

1/11*(b*x + a)^11/b

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (12) = 24\).
time = 0.30, size = 108, normalized size = 7.71 \begin {gather*} \frac {1}{11} \, b^{10} x^{11} + a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{9} + 15 \, a^{3} b^{7} x^{8} + 30 \, a^{4} b^{6} x^{7} + 42 \, a^{5} b^{5} x^{6} + 42 \, a^{6} b^{4} x^{5} + 30 \, a^{7} b^{3} x^{4} + 15 \, a^{8} b^{2} x^{3} + 5 \, a^{9} b x^{2} + a^{10} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10,x, algorithm="fricas")

[Out]

1/11*b^10*x^11 + a*b^9*x^10 + 5*a^2*b^8*x^9 + 15*a^3*b^7*x^8 + 30*a^4*b^6*x^7 + 42*a^5*b^5*x^6 + 42*a^6*b^4*x^
5 + 30*a^7*b^3*x^4 + 15*a^8*b^2*x^3 + 5*a^9*b*x^2 + a^10*x

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (8) = 16\).
time = 0.04, size = 114, normalized size = 8.14 \begin {gather*} a^{10} x + 5 a^{9} b x^{2} + 15 a^{8} b^{2} x^{3} + 30 a^{7} b^{3} x^{4} + 42 a^{6} b^{4} x^{5} + 42 a^{5} b^{5} x^{6} + 30 a^{4} b^{6} x^{7} + 15 a^{3} b^{7} x^{8} + 5 a^{2} b^{8} x^{9} + a b^{9} x^{10} + \frac {b^{10} x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10,x)

[Out]

a**10*x + 5*a**9*b*x**2 + 15*a**8*b**2*x**3 + 30*a**7*b**3*x**4 + 42*a**6*b**4*x**5 + 42*a**5*b**5*x**6 + 30*a
**4*b**6*x**7 + 15*a**3*b**7*x**8 + 5*a**2*b**8*x**9 + a*b**9*x**10 + b**10*x**11/11

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Giac [A]
time = 0.00, size = 12, normalized size = 0.86 \begin {gather*} \frac {\left (b x+a\right )^{11}}{11 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10,x)

[Out]

1/11*(b*x + a)^11/b

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Mupad [B]
time = 0.11, size = 108, normalized size = 7.71 \begin {gather*} a^{10}\,x+5\,a^9\,b\,x^2+15\,a^8\,b^2\,x^3+30\,a^7\,b^3\,x^4+42\,a^6\,b^4\,x^5+42\,a^5\,b^5\,x^6+30\,a^4\,b^6\,x^7+15\,a^3\,b^7\,x^8+5\,a^2\,b^8\,x^9+a\,b^9\,x^{10}+\frac {b^{10}\,x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^10,x)

[Out]

a^10*x + (b^10*x^11)/11 + 5*a^9*b*x^2 + a*b^9*x^10 + 15*a^8*b^2*x^3 + 30*a^7*b^3*x^4 + 42*a^6*b^4*x^5 + 42*a^5
*b^5*x^6 + 30*a^4*b^6*x^7 + 15*a^3*b^7*x^8 + 5*a^2*b^8*x^9

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