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

Optimal result

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

[Out]

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

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} \[ \int (c+d x)^{10} \, dx=\frac {(c+d x)^{11}}{11 d} \]

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

Mathematica [A] (verified)

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

[In]

Integrate[(c + d*x)^10,x]

[Out]

(c + d*x)^11/(11*d)

Maple [A] (verified)

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

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

Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 7.71 \[ \int (c+d x)^{10} \, dx=\frac {1}{11} \, d^{10} x^{11} + c d^{9} x^{10} + 5 \, c^{2} d^{8} x^{9} + 15 \, c^{3} d^{7} x^{8} + 30 \, c^{4} d^{6} x^{7} + 42 \, c^{5} d^{5} x^{6} + 42 \, c^{6} d^{4} x^{5} + 30 \, c^{7} d^{3} x^{4} + 15 \, c^{8} d^{2} x^{3} + 5 \, c^{9} d x^{2} + c^{10} x \]

[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

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 \[ \int (c+d x)^{10} \, dx=c^{10} x + 5 c^{9} d x^{2} + 15 c^{8} d^{2} x^{3} + 30 c^{7} d^{3} x^{4} + 42 c^{6} d^{4} x^{5} + 42 c^{5} d^{5} x^{6} + 30 c^{4} d^{6} x^{7} + 15 c^{3} d^{7} x^{8} + 5 c^{2} d^{8} x^{9} + c d^{9} x^{10} + \frac {d^{10} x^{11}}{11} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int (c+d x)^{10} \, dx=\frac {{\left (d x + c\right )}^{11}}{11 \, d} \]

[In]

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

[Out]

1/11*(d*x + c)^11/d

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int (c+d x)^{10} \, dx=\frac {{\left (d x + c\right )}^{11}}{11 \, d} \]

[In]

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

[Out]

1/11*(d*x + c)^11/d

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 7.71 \[ \int (c+d x)^{10} \, dx=c^{10}\,x+5\,c^9\,d\,x^2+15\,c^8\,d^2\,x^3+30\,c^7\,d^3\,x^4+42\,c^6\,d^4\,x^5+42\,c^5\,d^5\,x^6+30\,c^4\,d^6\,x^7+15\,c^3\,d^7\,x^8+5\,c^2\,d^8\,x^9+c\,d^9\,x^{10}+\frac {d^{10}\,x^{11}}{11} \]

[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