3.81 \(\int x^2 (a+b x)^5 \, dx\)

Optimal. Leaf size=47 \[ \frac{a^2 (a+b x)^6}{6 b^3}+\frac{(a+b x)^8}{8 b^3}-\frac{2 a (a+b x)^7}{7 b^3} \]

[Out]

(a^2*(a + b*x)^6)/(6*b^3) - (2*a*(a + b*x)^7)/(7*b^3) + (a + b*x)^8/(8*b^3)

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Rubi [A]  time = 0.0205076, antiderivative size = 47, 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} \[ \frac{a^2 (a+b x)^6}{6 b^3}+\frac{(a+b x)^8}{8 b^3}-\frac{2 a (a+b x)^7}{7 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^5,x]

[Out]

(a^2*(a + b*x)^6)/(6*b^3) - (2*a*(a + b*x)^7)/(7*b^3) + (a + b*x)^8/(8*b^3)

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

Mathematica [A]  time = 0.0048094, size = 67, normalized size = 1.43 \[ \frac{5}{3} a^2 b^3 x^6+2 a^3 b^2 x^5+\frac{5}{4} a^4 b x^4+\frac{a^5 x^3}{3}+\frac{5}{7} a b^4 x^7+\frac{b^5 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^5,x]

[Out]

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

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Maple [A]  time = 0.001, size = 58, normalized size = 1.2 \begin{align*}{\frac{{b}^{5}{x}^{8}}{8}}+{\frac{5\,a{b}^{4}{x}^{7}}{7}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{6}}{3}}+2\,{a}^{3}{b}^{2}{x}^{5}+{\frac{5\,{a}^{4}b{x}^{4}}{4}}+{\frac{{a}^{5}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0072, size = 77, normalized size = 1.64 \begin{align*} \frac{1}{8} \, b^{5} x^{8} + \frac{5}{7} \, a b^{4} x^{7} + \frac{5}{3} \, a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{5} + \frac{5}{4} \, a^{4} b x^{4} + \frac{1}{3} \, a^{5} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.39859, size = 126, normalized size = 2.68 \begin{align*} \frac{1}{8} x^{8} b^{5} + \frac{5}{7} x^{7} b^{4} a + \frac{5}{3} x^{6} b^{3} a^{2} + 2 x^{5} b^{2} a^{3} + \frac{5}{4} x^{4} b a^{4} + \frac{1}{3} x^{3} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.081626, size = 65, normalized size = 1.38 \begin{align*} \frac{a^{5} x^{3}}{3} + \frac{5 a^{4} b x^{4}}{4} + 2 a^{3} b^{2} x^{5} + \frac{5 a^{2} b^{3} x^{6}}{3} + \frac{5 a b^{4} x^{7}}{7} + \frac{b^{5} x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x+a)**5,x)

[Out]

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

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Giac [A]  time = 1.17811, size = 77, normalized size = 1.64 \begin{align*} \frac{1}{8} \, b^{5} x^{8} + \frac{5}{7} \, a b^{4} x^{7} + \frac{5}{3} \, a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{5} + \frac{5}{4} \, a^{4} b x^{4} + \frac{1}{3} \, a^{5} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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