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

Optimal result

Integrand size = 7, antiderivative size = 14 \[ \int (a+b x)^{10} \, dx=\frac {(a+b x)^{11}}{11 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \[ \int (a+b x)^{10} \, dx=\frac {(a+b x)^{11}}{11 b} \]

[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*} \text {integral}& = \frac {(a+b x)^{11}}{11 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int (a+b x)^{10} \, dx=\frac {(a+b x)^{11}}{11 b} \]

[In]

Integrate[(a + b*x)^10,x]

[Out]

(a + b*x)^11/(11*b)

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 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	\(\frac {1}{11} b^{10} x^{11}+a \,b^{9} x^{10}+5 a^{2} b^{8} x^{9}+15 a^{3} b^{7} x^{8}+30 a^{4} b^{6} x^{7}+42 a^{5} b^{5} x^{6}+42 a^{6} b^{4} x^{5}+30 a^{7} b^{3} x^{4}+15 a^{8} b^{2} x^{3}+5 a^{9} b \,x^{2}+a^{10} x\)	\(109\)
parallelrisch	\(\frac {1}{11} b^{10} x^{11}+a \,b^{9} x^{10}+5 a^{2} b^{8} x^{9}+15 a^{3} b^{7} x^{8}+30 a^{4} b^{6} x^{7}+42 a^{5} b^{5} x^{6}+42 a^{6} b^{4} x^{5}+30 a^{7} b^{3} x^{4}+15 a^{8} b^{2} x^{3}+5 a^{9} b \,x^{2}+a^{10} x\)	\(109\)
risch	\(\frac {b^{10} x^{11}}{11}+a \,b^{9} x^{10}+5 a^{2} b^{8} x^{9}+15 a^{3} b^{7} x^{8}+30 a^{4} b^{6} x^{7}+42 a^{5} b^{5} x^{6}+42 a^{6} b^{4} x^{5}+30 a^{7} b^{3} x^{4}+15 a^{8} b^{2} x^{3}+5 a^{9} b \,x^{2}+a^{10} x +\frac {a^{11}}{11 b}\)	\(117\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (12) = 24\).

Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 7.71 \[ \int (a+b x)^{10} \, dx=\frac {1}{11} \, b^{10} x^{11} + a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{9} + 15 \, a^{3} b^{7} x^{8} + 30 \, a^{4} b^{6} x^{7} + 42 \, a^{5} b^{5} x^{6} + 42 \, a^{6} b^{4} x^{5} + 30 \, a^{7} b^{3} x^{4} + 15 \, a^{8} b^{2} x^{3} + 5 \, a^{9} b x^{2} + a^{10} x \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (8) = 16\).

Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 8.14 \[ \int (a+b x)^{10} \, dx=a^{10} x + 5 a^{9} b x^{2} + 15 a^{8} b^{2} x^{3} + 30 a^{7} b^{3} x^{4} + 42 a^{6} b^{4} x^{5} + 42 a^{5} b^{5} x^{6} + 30 a^{4} b^{6} x^{7} + 15 a^{3} b^{7} x^{8} + 5 a^{2} b^{8} x^{9} + a b^{9} x^{10} + \frac {b^{10} x^{11}}{11} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int (a+b x)^{10} \, dx=\frac {{\left (b x + a\right )}^{11}}{11 \, b} \]

[In]

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

[Out]

1/11*(b*x + a)^11/b

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int (a+b x)^{10} \, dx=\frac {{\left (b x + a\right )}^{11}}{11 \, b} \]

[In]

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

[Out]

1/11*(b*x + a)^11/b

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 7.71 \[ \int (a+b x)^{10} \, dx=a^{10}\,x+5\,a^9\,b\,x^2+15\,a^8\,b^2\,x^3+30\,a^7\,b^3\,x^4+42\,a^6\,b^4\,x^5+42\,a^5\,b^5\,x^6+30\,a^4\,b^6\,x^7+15\,a^3\,b^7\,x^8+5\,a^2\,b^8\,x^9+a\,b^9\,x^{10}+\frac {b^{10}\,x^{11}}{11} \]

[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