\(\int (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5)^2 \, dx\) [64]

Optimal result

Integrand size = 51, antiderivative size = 14 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^2 \, dx=\frac {(a+b x)^{11}}{11 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {2084, 32} \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^2 \, dx=\frac {(a+b x)^{11}}{11 b} \]

[In]

Int[(a^5 + 5*a^4*b*x + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a*b^4*x^4 + b^5*x^5)^2,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]

Rule 2084

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[u^p, x] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x]
&& IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^{10} \, dx \\ & = \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 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^2 \, dx=\frac {(a+b x)^{11}}{11 b} \]

[In]

Integrate[(a^5 + 5*a^4*b*x + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a*b^4*x^4 + b^5*x^5)^2,x]

[Out]

(a + b*x)^11/(11*b)

Maple [B] (verified)

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

Time = 0.04 (sec) , antiderivative size = 109, normalized size of antiderivative = 7.79

method	result	size
default	\(\frac {1}{11} b^{10} x^{11}+a \,b^{9} x^{10}+5 a^{2} b^{8} x^{9}+15 a^{3} b^{7} x^{8}+30 a^{4} b^{6} x^{7}+42 a^{5} b^{5} x^{6}+42 a^{6} b^{4} x^{5}+30 a^{7} b^{3} x^{4}+15 a^{8} b^{2} x^{3}+5 a^{9} b \,x^{2}+a^{10} x\)	\(109\)
norman	\(\frac {1}{11} b^{10} x^{11}+a \,b^{9} x^{10}+5 a^{2} b^{8} x^{9}+15 a^{3} b^{7} x^{8}+30 a^{4} b^{6} x^{7}+42 a^{5} b^{5} x^{6}+42 a^{6} b^{4} x^{5}+30 a^{7} b^{3} x^{4}+15 a^{8} b^{2} x^{3}+5 a^{9} b \,x^{2}+a^{10} x\)	\(109\)
risch	\(\frac {1}{11} b^{10} x^{11}+a \,b^{9} x^{10}+5 a^{2} b^{8} x^{9}+15 a^{3} b^{7} x^{8}+30 a^{4} b^{6} x^{7}+42 a^{5} b^{5} x^{6}+42 a^{6} b^{4} x^{5}+30 a^{7} b^{3} x^{4}+15 a^{8} b^{2} x^{3}+5 a^{9} b \,x^{2}+a^{10} x\)	\(109\)
parallelrisch	\(\frac {1}{11} b^{10} x^{11}+a \,b^{9} x^{10}+5 a^{2} b^{8} x^{9}+15 a^{3} b^{7} x^{8}+30 a^{4} b^{6} x^{7}+42 a^{5} b^{5} x^{6}+42 a^{6} b^{4} x^{5}+30 a^{7} b^{3} x^{4}+15 a^{8} b^{2} x^{3}+5 a^{9} b \,x^{2}+a^{10} x\)	\(109\)
gosper	\(\frac {x \left (b^{10} x^{10}+11 a \,b^{9} x^{9}+55 a^{2} b^{8} x^{8}+165 a^{3} b^{7} x^{7}+330 a^{4} b^{6} x^{6}+462 a^{5} b^{5} x^{5}+462 a^{6} b^{4} x^{4}+330 a^{7} b^{3} x^{3}+165 a^{8} b^{2} x^{2}+55 a^{9} b x +11 a^{10}\right )}{11}\)	\(110\)

[In]

int((b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5)^2,x,method=_RETURNVERBOSE)

[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

Fricas [B] (verification not implemented)

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

Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 7.71 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^2 \, dx=\frac {1}{11} \, b^{10} x^{11} + a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{9} + 15 \, a^{3} b^{7} x^{8} + 30 \, a^{4} b^{6} x^{7} + 42 \, a^{5} b^{5} x^{6} + 42 \, a^{6} b^{4} x^{5} + 30 \, a^{7} b^{3} x^{4} + 15 \, a^{8} b^{2} x^{3} + 5 \, a^{9} b x^{2} + a^{10} x \]

[In]

integrate((b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5)^2,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 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^2 \, dx=a^{10} x + 5 a^{9} b x^{2} + 15 a^{8} b^{2} x^{3} + 30 a^{7} b^{3} x^{4} + 42 a^{6} b^{4} x^{5} + 42 a^{5} b^{5} x^{6} + 30 a^{4} b^{6} x^{7} + 15 a^{3} b^{7} x^{8} + 5 a^{2} b^{8} x^{9} + a b^{9} x^{10} + \frac {b^{10} x^{11}}{11} \]

[In]

integrate((b**5*x**5+5*a*b**4*x**4+10*a**2*b**3*x**3+10*a**3*b**2*x**2+5*a**4*b*x+a**5)**2,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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (12) = 24\).

Time = 0.21 (sec) , antiderivative size = 228, normalized size of antiderivative = 16.29 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^2 \, dx=\frac {1}{11} \, b^{10} x^{11} + a b^{9} x^{10} + \frac {25}{9} \, a^{2} b^{8} x^{9} + \frac {100}{7} \, a^{4} b^{6} x^{7} + 20 \, a^{6} b^{4} x^{5} + \frac {25}{3} \, a^{8} b^{2} x^{3} + a^{10} x + \frac {1}{3} \, {\left (b^{5} x^{6} + 6 \, a b^{4} x^{5} + 15 \, a^{2} b^{3} x^{4} + 20 \, a^{3} b^{2} x^{3} + 15 \, a^{4} b x^{2}\right )} a^{5} + \frac {5}{21} \, {\left (6 \, b^{5} x^{7} + 35 \, a b^{4} x^{6} + 84 \, a^{2} b^{3} x^{5} + 105 \, a^{3} b^{2} x^{4}\right )} a^{4} b + \frac {5}{42} \, {\left (21 \, b^{5} x^{8} + 120 \, a b^{4} x^{7} + 280 \, a^{2} b^{3} x^{6}\right )} a^{3} b^{2} + \frac {5}{18} \, {\left (8 \, b^{5} x^{9} + 45 \, a b^{4} x^{8}\right )} a^{2} b^{3} \]

[In]

integrate((b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5)^2,x, algorithm="maxima")

[Out]

1/11*b^10*x^11 + a*b^9*x^10 + 25/9*a^2*b^8*x^9 + 100/7*a^4*b^6*x^7 + 20*a^6*b^4*x^5 + 25/3*a^8*b^2*x^3 + a^10*
x + 1/3*(b^5*x^6 + 6*a*b^4*x^5 + 15*a^2*b^3*x^4 + 20*a^3*b^2*x^3 + 15*a^4*b*x^2)*a^5 + 5/21*(6*b^5*x^7 + 35*a*
b^4*x^6 + 84*a^2*b^3*x^5 + 105*a^3*b^2*x^4)*a^4*b + 5/42*(21*b^5*x^8 + 120*a*b^4*x^7 + 280*a^2*b^3*x^6)*a^3*b^
2 + 5/18*(8*b^5*x^9 + 45*a*b^4*x^8)*a^2*b^3

Giac [B] (verification not implemented)

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

Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 7.71 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^2 \, dx=\frac {1}{11} \, b^{10} x^{11} + a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{9} + 15 \, a^{3} b^{7} x^{8} + 30 \, a^{4} b^{6} x^{7} + 42 \, a^{5} b^{5} x^{6} + 42 \, a^{6} b^{4} x^{5} + 30 \, a^{7} b^{3} x^{4} + 15 \, a^{8} b^{2} x^{3} + 5 \, a^{9} b x^{2} + a^{10} x \]

[In]

integrate((b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5)^2,x, algorithm="giac")

[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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 108, normalized size of antiderivative = 7.71 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^2 \, dx=a^{10}\,x+5\,a^9\,b\,x^2+15\,a^8\,b^2\,x^3+30\,a^7\,b^3\,x^4+42\,a^6\,b^4\,x^5+42\,a^5\,b^5\,x^6+30\,a^4\,b^6\,x^7+15\,a^3\,b^7\,x^8+5\,a^2\,b^8\,x^9+a\,b^9\,x^{10}+\frac {b^{10}\,x^{11}}{11} \]

[In]

int((a^5 + b^5*x^5 + 5*a*b^4*x^4 + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a^4*b*x)^2,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