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

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 = 0.017942, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 51, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.039, Rules used = {2059, 32} \[ \frac{(a+b x)^{11}}{11 b} \]

Antiderivative was successfully verified.

[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 2059

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

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 \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 &=\int (a+b x)^{10} \, dx\\ &=\frac{(a+b x)^{11}}{11 b}\\ \end{align*}

Mathematica [A]  time = 0.0014514, size = 14, normalized size = 1. \[ \frac{(a+b x)^{11}}{11 b} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.001, size = 109, normalized size = 7.8 \begin{align*}{\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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

[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

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Maxima [B]  time = 1.15947, size = 308, normalized size = 22. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.53508, 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^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*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 = 0.086512, 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**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

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Giac [B]  time = 1.10229, size = 146, normalized size = 10.43 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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