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

Optimal. Leaf size=30 \[ \frac{(a+b x)^7}{7 b^2}-\frac{a (a+b x)^6}{6 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0079793, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ \frac{(a+b x)^7}{7 b^2}-\frac{a (a+b x)^6}{6 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^5,x]

[Out]

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

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

Mathematica [B]  time = 0.0020969, size = 67, normalized size = 2.23 \[ 2 a^2 b^3 x^5+\frac{5}{2} a^3 b^2 x^4+\frac{5}{3} a^4 b x^3+\frac{a^5 x^2}{2}+\frac{5}{6} a b^4 x^6+\frac{b^5 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^5,x]

[Out]

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

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Maple [B]  time = 0., size = 58, normalized size = 1.9 \begin{align*}{\frac{{b}^{5}{x}^{7}}{7}}+{\frac{5\,a{b}^{4}{x}^{6}}{6}}+2\,{a}^{2}{b}^{3}{x}^{5}+{\frac{5\,{a}^{3}{b}^{2}{x}^{4}}{2}}+{\frac{5\,{a}^{4}b{x}^{3}}{3}}+{\frac{{a}^{5}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.03267, size = 77, normalized size = 2.57 \begin{align*} \frac{1}{7} \, b^{5} x^{7} + \frac{5}{6} \, a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{5} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{5}{3} \, a^{4} b x^{3} + \frac{1}{2} \, a^{5} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.35533, size = 126, normalized size = 4.2 \begin{align*} \frac{1}{7} x^{7} b^{5} + \frac{5}{6} x^{6} b^{4} a + 2 x^{5} b^{3} a^{2} + \frac{5}{2} x^{4} b^{2} a^{3} + \frac{5}{3} x^{3} b a^{4} + \frac{1}{2} x^{2} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.076103, size = 65, normalized size = 2.17 \begin{align*} \frac{a^{5} x^{2}}{2} + \frac{5 a^{4} b x^{3}}{3} + \frac{5 a^{3} b^{2} x^{4}}{2} + 2 a^{2} b^{3} x^{5} + \frac{5 a b^{4} x^{6}}{6} + \frac{b^{5} x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.16592, size = 77, normalized size = 2.57 \begin{align*} \frac{1}{7} \, b^{5} x^{7} + \frac{5}{6} \, a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{5} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{5}{3} \, a^{4} b x^{3} + \frac{1}{2} \, a^{5} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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