3.78 \(\int x^5 (a+b x)^5 \, dx\)

Optimal. Leaf size=69 \[ \frac{10}{9} a^2 b^3 x^9+\frac{5}{4} a^3 b^2 x^8+\frac{5}{7} a^4 b x^7+\frac{a^5 x^6}{6}+\frac{1}{2} a b^4 x^{10}+\frac{b^5 x^{11}}{11} \]

[Out]

(a^5*x^6)/6 + (5*a^4*b*x^7)/7 + (5*a^3*b^2*x^8)/4 + (10*a^2*b^3*x^9)/9 + (a*b^4*x^10)/2 + (b^5*x^11)/11

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Rubi [A]  time = 0.0276943, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{10}{9} a^2 b^3 x^9+\frac{5}{4} a^3 b^2 x^8+\frac{5}{7} a^4 b x^7+\frac{a^5 x^6}{6}+\frac{1}{2} a b^4 x^{10}+\frac{b^5 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x)^5,x]

[Out]

(a^5*x^6)/6 + (5*a^4*b*x^7)/7 + (5*a^3*b^2*x^8)/4 + (10*a^2*b^3*x^9)/9 + (a*b^4*x^10)/2 + (b^5*x^11)/11

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^5 (a+b x)^5 \, dx &=\int \left (a^5 x^5+5 a^4 b x^6+10 a^3 b^2 x^7+10 a^2 b^3 x^8+5 a b^4 x^9+b^5 x^{10}\right ) \, dx\\ &=\frac{a^5 x^6}{6}+\frac{5}{7} a^4 b x^7+\frac{5}{4} a^3 b^2 x^8+\frac{10}{9} a^2 b^3 x^9+\frac{1}{2} a b^4 x^{10}+\frac{b^5 x^{11}}{11}\\ \end{align*}

Mathematica [A]  time = 0.002325, size = 69, normalized size = 1. \[ \frac{10}{9} a^2 b^3 x^9+\frac{5}{4} a^3 b^2 x^8+\frac{5}{7} a^4 b x^7+\frac{a^5 x^6}{6}+\frac{1}{2} a b^4 x^{10}+\frac{b^5 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x)^5,x]

[Out]

(a^5*x^6)/6 + (5*a^4*b*x^7)/7 + (5*a^3*b^2*x^8)/4 + (10*a^2*b^3*x^9)/9 + (a*b^4*x^10)/2 + (b^5*x^11)/11

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Maple [A]  time = 0., size = 58, normalized size = 0.8 \begin{align*}{\frac{{a}^{5}{x}^{6}}{6}}+{\frac{5\,{a}^{4}b{x}^{7}}{7}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{8}}{4}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{9}}{9}}+{\frac{a{b}^{4}{x}^{10}}{2}}+{\frac{{b}^{5}{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x+a)^5,x)

[Out]

1/6*a^5*x^6+5/7*a^4*b*x^7+5/4*a^3*b^2*x^8+10/9*a^2*b^3*x^9+1/2*a*b^4*x^10+1/11*b^5*x^11

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Maxima [A]  time = 1.02887, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{11} \, b^{5} x^{11} + \frac{1}{2} \, a b^{4} x^{10} + \frac{10}{9} \, a^{2} b^{3} x^{9} + \frac{5}{4} \, a^{3} b^{2} x^{8} + \frac{5}{7} \, a^{4} b x^{7} + \frac{1}{6} \, a^{5} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^5,x, algorithm="maxima")

[Out]

1/11*b^5*x^11 + 1/2*a*b^4*x^10 + 10/9*a^2*b^3*x^9 + 5/4*a^3*b^2*x^8 + 5/7*a^4*b*x^7 + 1/6*a^5*x^6

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Fricas [A]  time = 1.32468, size = 134, normalized size = 1.94 \begin{align*} \frac{1}{11} x^{11} b^{5} + \frac{1}{2} x^{10} b^{4} a + \frac{10}{9} x^{9} b^{3} a^{2} + \frac{5}{4} x^{8} b^{2} a^{3} + \frac{5}{7} x^{7} b a^{4} + \frac{1}{6} x^{6} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^5,x, algorithm="fricas")

[Out]

1/11*x^11*b^5 + 1/2*x^10*b^4*a + 10/9*x^9*b^3*a^2 + 5/4*x^8*b^2*a^3 + 5/7*x^7*b*a^4 + 1/6*x^6*a^5

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Sympy [A]  time = 0.084904, size = 65, normalized size = 0.94 \begin{align*} \frac{a^{5} x^{6}}{6} + \frac{5 a^{4} b x^{7}}{7} + \frac{5 a^{3} b^{2} x^{8}}{4} + \frac{10 a^{2} b^{3} x^{9}}{9} + \frac{a b^{4} x^{10}}{2} + \frac{b^{5} x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x+a)**5,x)

[Out]

a**5*x**6/6 + 5*a**4*b*x**7/7 + 5*a**3*b**2*x**8/4 + 10*a**2*b**3*x**9/9 + a*b**4*x**10/2 + b**5*x**11/11

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Giac [A]  time = 1.14324, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{11} \, b^{5} x^{11} + \frac{1}{2} \, a b^{4} x^{10} + \frac{10}{9} \, a^{2} b^{3} x^{9} + \frac{5}{4} \, a^{3} b^{2} x^{8} + \frac{5}{7} \, a^{4} b x^{7} + \frac{1}{6} \, a^{5} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^5,x, algorithm="giac")

[Out]

1/11*b^5*x^11 + 1/2*a*b^4*x^10 + 10/9*a^2*b^3*x^9 + 5/4*a^3*b^2*x^8 + 5/7*a^4*b*x^7 + 1/6*a^5*x^6