3.14.11 \(\int (c+d x)^{10} \, dx\) [1311]

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

[Out]

1/11*(d*x+c)^11/d

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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 {(c+d x)^{11}}{11 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10,x]

[Out]

(c + d*x)^11/(11*d)

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 (c+d x)^{10} \, dx &=\frac {(c+d x)^{11}}{11 d}\\ \end {align*}

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

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10,x]

[Out]

(c + d*x)^11/(11*d)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(14)=28\).
time = 2.01, size = 106, normalized size = 7.57 \begin {gather*} x \left (c^{10}+5 c^9 d x+15 c^8 d^2 x^2+30 c^7 d^3 x^3+42 c^6 d^4 x^4+42 c^5 d^5 x^5+30 c^4 d^6 x^6+15 c^3 d^7 x^7+5 c^2 d^8 x^8+c d^9 x^9+\frac {d^{10} x^{10}}{11}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.11, size = 13, normalized size = 0.93

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10,x,method=_RETURNVERBOSE)

[Out]

1/11*(d*x+c)^11/d

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Maxima [A]
time = 0.27, size = 12, normalized size = 0.86 \begin {gather*} \frac {{\left (d x + c\right )}^{11}}{11 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10,x, algorithm="maxima")

[Out]

1/11*(d*x + c)^11/d

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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 = 0.29, size = 108, normalized size = 7.71 \begin {gather*} \frac {1}{11} \, d^{10} x^{11} + c d^{9} x^{10} + 5 \, c^{2} d^{8} x^{9} + 15 \, c^{3} d^{7} x^{8} + 30 \, c^{4} d^{6} x^{7} + 42 \, c^{5} d^{5} x^{6} + 42 \, c^{6} d^{4} x^{5} + 30 \, c^{7} d^{3} x^{4} + 15 \, c^{8} d^{2} x^{3} + 5 \, c^{9} d x^{2} + c^{10} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10,x, algorithm="fricas")

[Out]

1/11*d^10*x^11 + c*d^9*x^10 + 5*c^2*d^8*x^9 + 15*c^3*d^7*x^8 + 30*c^4*d^6*x^7 + 42*c^5*d^5*x^6 + 42*c^6*d^4*x^
5 + 30*c^7*d^3*x^4 + 15*c^8*d^2*x^3 + 5*c^9*d*x^2 + c^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*} c^{10} x + 5 c^{9} d x^{2} + 15 c^{8} d^{2} x^{3} + 30 c^{7} d^{3} x^{4} + 42 c^{6} d^{4} x^{5} + 42 c^{5} d^{5} x^{6} + 30 c^{4} d^{6} x^{7} + 15 c^{3} d^{7} x^{8} + 5 c^{2} d^{8} x^{9} + c d^{9} x^{10} + \frac {d^{10} x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**10,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10,x)

[Out]

1/11*(d*x + c)^11/d

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Mupad [B]
time = 0.08, size = 108, normalized size = 7.71 \begin {gather*} c^{10}\,x+5\,c^9\,d\,x^2+15\,c^8\,d^2\,x^3+30\,c^7\,d^3\,x^4+42\,c^6\,d^4\,x^5+42\,c^5\,d^5\,x^6+30\,c^4\,d^6\,x^7+15\,c^3\,d^7\,x^8+5\,c^2\,d^8\,x^9+c\,d^9\,x^{10}+\frac {d^{10}\,x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10,x)

[Out]

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

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