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

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

[Out]

(a + b*x)^11/(11*b)

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Rubi [A]  time = 0.0015093, antiderivative size = 14, normalized size of antiderivative = 1., 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} \[ \frac{(a+b x)^{11}}{11 b} \]

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*}

Mathematica [A]  time = 0.0015456, size = 14, normalized size = 1. \[ \frac{(a+b x)^{11}}{11 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10,x]

[Out]

(a + b*x)^11/(11*b)

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Maple [A]  time = 0., size = 13, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ) ^{11}}{11\,b}} \end{align*}

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 = 1.01827, size = 16, normalized size = 1.14 \begin{align*} \frac{{\left (b x + a\right )}^{11}}{11 \, b} \end{align*}

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]  time = 1.64011, size = 230, normalized size = 16.43 \begin{align*} \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{align*}

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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Sympy [B]  time = 0.093853, size = 114, normalized size = 8.14 \begin{align*} 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{align*}

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 = 1.19818, size = 16, normalized size = 1.14 \begin{align*} \frac{{\left (b x + a\right )}^{11}}{11 \, b} \end{align*}

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