3.77 \(\int x^6 (a+b x)^5 \, dx\)

Optimal. Leaf size=66 \[ a^2 b^3 x^{10}+\frac{10}{9} a^3 b^2 x^9+\frac{5}{8} a^4 b x^8+\frac{a^5 x^7}{7}+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{12}}{12} \]

[Out]

(a^5*x^7)/7 + (5*a^4*b*x^8)/8 + (10*a^3*b^2*x^9)/9 + a^2*b^3*x^10 + (5*a*b^4*x^11)/11 + (b^5*x^12)/12

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Rubi [A]  time = 0.031675, antiderivative size = 66, 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} \[ a^2 b^3 x^{10}+\frac{10}{9} a^3 b^2 x^9+\frac{5}{8} a^4 b x^8+\frac{a^5 x^7}{7}+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(a + b*x)^5,x]

[Out]

(a^5*x^7)/7 + (5*a^4*b*x^8)/8 + (10*a^3*b^2*x^9)/9 + a^2*b^3*x^10 + (5*a*b^4*x^11)/11 + (b^5*x^12)/12

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^6 (a+b x)^5 \, dx &=\int \left (a^5 x^6+5 a^4 b x^7+10 a^3 b^2 x^8+10 a^2 b^3 x^9+5 a b^4 x^{10}+b^5 x^{11}\right ) \, dx\\ &=\frac{a^5 x^7}{7}+\frac{5}{8} a^4 b x^8+\frac{10}{9} a^3 b^2 x^9+a^2 b^3 x^{10}+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{12}}{12}\\ \end{align*}

Mathematica [A]  time = 0.0028657, size = 66, normalized size = 1. \[ a^2 b^3 x^{10}+\frac{10}{9} a^3 b^2 x^9+\frac{5}{8} a^4 b x^8+\frac{a^5 x^7}{7}+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*x)^5,x]

[Out]

(a^5*x^7)/7 + (5*a^4*b*x^8)/8 + (10*a^3*b^2*x^9)/9 + a^2*b^3*x^10 + (5*a*b^4*x^11)/11 + (b^5*x^12)/12

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Maple [A]  time = 0.001, size = 57, normalized size = 0.9 \begin{align*}{\frac{{a}^{5}{x}^{7}}{7}}+{\frac{5\,{a}^{4}b{x}^{8}}{8}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{9}}{9}}+{a}^{2}{b}^{3}{x}^{10}+{\frac{5\,a{b}^{4}{x}^{11}}{11}}+{\frac{{b}^{5}{x}^{12}}{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x+a)^5,x)

[Out]

1/7*a^5*x^7+5/8*a^4*b*x^8+10/9*a^3*b^2*x^9+a^2*b^3*x^10+5/11*a*b^4*x^11+1/12*b^5*x^12

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Maxima [A]  time = 1.01192, size = 76, normalized size = 1.15 \begin{align*} \frac{1}{12} \, b^{5} x^{12} + \frac{5}{11} \, a b^{4} x^{11} + a^{2} b^{3} x^{10} + \frac{10}{9} \, a^{3} b^{2} x^{9} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{7} \, a^{5} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^5,x, algorithm="maxima")

[Out]

1/12*b^5*x^12 + 5/11*a*b^4*x^11 + a^2*b^3*x^10 + 10/9*a^3*b^2*x^9 + 5/8*a^4*b*x^8 + 1/7*a^5*x^7

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Fricas [A]  time = 1.29006, size = 131, normalized size = 1.98 \begin{align*} \frac{1}{12} x^{12} b^{5} + \frac{5}{11} x^{11} b^{4} a + x^{10} b^{3} a^{2} + \frac{10}{9} x^{9} b^{2} a^{3} + \frac{5}{8} x^{8} b a^{4} + \frac{1}{7} x^{7} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^5,x, algorithm="fricas")

[Out]

1/12*x^12*b^5 + 5/11*x^11*b^4*a + x^10*b^3*a^2 + 10/9*x^9*b^2*a^3 + 5/8*x^8*b*a^4 + 1/7*x^7*a^5

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Sympy [A]  time = 0.087194, size = 63, normalized size = 0.95 \begin{align*} \frac{a^{5} x^{7}}{7} + \frac{5 a^{4} b x^{8}}{8} + \frac{10 a^{3} b^{2} x^{9}}{9} + a^{2} b^{3} x^{10} + \frac{5 a b^{4} x^{11}}{11} + \frac{b^{5} x^{12}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b*x+a)**5,x)

[Out]

a**5*x**7/7 + 5*a**4*b*x**8/8 + 10*a**3*b**2*x**9/9 + a**2*b**3*x**10 + 5*a*b**4*x**11/11 + b**5*x**12/12

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Giac [A]  time = 1.18553, size = 76, normalized size = 1.15 \begin{align*} \frac{1}{12} \, b^{5} x^{12} + \frac{5}{11} \, a b^{4} x^{11} + a^{2} b^{3} x^{10} + \frac{10}{9} \, a^{3} b^{2} x^{9} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{7} \, a^{5} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^5,x, algorithm="giac")

[Out]

1/12*b^5*x^12 + 5/11*a*b^4*x^11 + a^2*b^3*x^10 + 10/9*a^3*b^2*x^9 + 5/8*a^4*b*x^8 + 1/7*a^5*x^7