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

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

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^10,x]

[Out]

IntegrateAlgebraic[(a + b*x)^10, x]

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fricas [B]  time = 1.26, size = 108, normalized size = 7.71 \begin {gather*} \frac {1}{11} x^{11} b^{10} + x^{10} b^{9} a + 5 x^{9} b^{8} a^{2} + 15 x^{8} b^{7} a^{3} + 30 x^{7} b^{6} a^{4} + 42 x^{6} b^{5} a^{5} + 42 x^{5} b^{4} a^{6} + 30 x^{4} b^{3} a^{7} + 15 x^{3} b^{2} a^{8} + 5 x^{2} b a^{9} + x a^{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.16, 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="giac")

[Out]

1/11*(b*x + a)^11/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.33, 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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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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sympy [B]  time = 0.11, 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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