3.13.5 \(\int (c+d x)^{10} \, dx\)

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

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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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (c+d x)^{10} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^10,x]

[Out]

IntegrateAlgebraic[(c + d*x)^10, x]

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fricas [B]  time = 0.98, size = 108, normalized size = 7.71 \begin {gather*} \frac {1}{11} x^{11} d^{10} + x^{10} d^{9} c + 5 x^{9} d^{8} c^{2} + 15 x^{8} d^{7} c^{3} + 30 x^{7} d^{6} c^{4} + 42 x^{6} d^{5} c^{5} + 42 x^{5} d^{4} c^{6} + 30 x^{4} d^{3} c^{7} + 15 x^{3} d^{2} c^{8} + 5 x^{2} d c^{9} + x c^{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.28, 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="giac")

[Out]

1/11*(d*x + c)^11/d

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maple [A]  time = 0.00, size = 13, normalized size = 0.93 \begin {gather*} \frac {\left (d x +c \right )^{11}}{11 d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.37, 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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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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sympy [B]  time = 0.09, 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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