3.105 \(\int x (a+b x)^7 \, dx\)

Optimal. Leaf size=30 \[ \frac{(a+b x)^9}{9 b^2}-\frac{a (a+b x)^8}{8 b^2} \]

[Out]

-(a*(a + b*x)^8)/(8*b^2) + (a + b*x)^9/(9*b^2)

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Rubi [A]  time = 0.0081933, 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} \[ \frac{(a+b x)^9}{9 b^2}-\frac{a (a+b x)^8}{8 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^7,x]

[Out]

-(a*(a + b*x)^8)/(8*b^2) + (a + b*x)^9/(9*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)^7 \, dx &=\int \left (-\frac{a (a+b x)^7}{b}+\frac{(a+b x)^8}{b}\right ) \, dx\\ &=-\frac{a (a+b x)^8}{8 b^2}+\frac{(a+b x)^9}{9 b^2}\\ \end{align*}

Mathematica [B]  time = 0.0025274, size = 91, normalized size = 3.03 \[ 3 a^2 b^5 x^7+\frac{35}{6} a^3 b^4 x^6+7 a^4 b^3 x^5+\frac{21}{4} a^5 b^2 x^4+\frac{7}{3} a^6 b x^3+\frac{a^7 x^2}{2}+\frac{7}{8} a b^6 x^8+\frac{b^7 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^7,x]

[Out]

(a^7*x^2)/2 + (7*a^6*b*x^3)/3 + (21*a^5*b^2*x^4)/4 + 7*a^4*b^3*x^5 + (35*a^3*b^4*x^6)/6 + 3*a^2*b^5*x^7 + (7*a
*b^6*x^8)/8 + (b^7*x^9)/9

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Maple [B]  time = 0.002, size = 80, normalized size = 2.7 \begin{align*}{\frac{{b}^{7}{x}^{9}}{9}}+{\frac{7\,a{b}^{6}{x}^{8}}{8}}+3\,{a}^{2}{b}^{5}{x}^{7}+{\frac{35\,{a}^{3}{b}^{4}{x}^{6}}{6}}+7\,{a}^{4}{b}^{3}{x}^{5}+{\frac{21\,{a}^{5}{b}^{2}{x}^{4}}{4}}+{\frac{7\,{a}^{6}b{x}^{3}}{3}}+{\frac{{a}^{7}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x+a)^7,x)

[Out]

1/9*b^7*x^9+7/8*a*b^6*x^8+3*a^2*b^5*x^7+35/6*a^3*b^4*x^6+7*a^4*b^3*x^5+21/4*a^5*b^2*x^4+7/3*a^6*b*x^3+1/2*a^7*
x^2

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Maxima [B]  time = 1.01952, size = 107, normalized size = 3.57 \begin{align*} \frac{1}{9} \, b^{7} x^{9} + \frac{7}{8} \, a b^{6} x^{8} + 3 \, a^{2} b^{5} x^{7} + \frac{35}{6} \, a^{3} b^{4} x^{6} + 7 \, a^{4} b^{3} x^{5} + \frac{21}{4} \, a^{5} b^{2} x^{4} + \frac{7}{3} \, a^{6} b x^{3} + \frac{1}{2} \, a^{7} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^7,x, algorithm="maxima")

[Out]

1/9*b^7*x^9 + 7/8*a*b^6*x^8 + 3*a^2*b^5*x^7 + 35/6*a^3*b^4*x^6 + 7*a^4*b^3*x^5 + 21/4*a^5*b^2*x^4 + 7/3*a^6*b*
x^3 + 1/2*a^7*x^2

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Fricas [B]  time = 1.38148, size = 174, normalized size = 5.8 \begin{align*} \frac{1}{9} x^{9} b^{7} + \frac{7}{8} x^{8} b^{6} a + 3 x^{7} b^{5} a^{2} + \frac{35}{6} x^{6} b^{4} a^{3} + 7 x^{5} b^{3} a^{4} + \frac{21}{4} x^{4} b^{2} a^{5} + \frac{7}{3} x^{3} b a^{6} + \frac{1}{2} x^{2} a^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^7,x, algorithm="fricas")

[Out]

1/9*x^9*b^7 + 7/8*x^8*b^6*a + 3*x^7*b^5*a^2 + 35/6*x^6*b^4*a^3 + 7*x^5*b^3*a^4 + 21/4*x^4*b^2*a^5 + 7/3*x^3*b*
a^6 + 1/2*x^2*a^7

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Sympy [B]  time = 0.0818, size = 90, normalized size = 3. \begin{align*} \frac{a^{7} x^{2}}{2} + \frac{7 a^{6} b x^{3}}{3} + \frac{21 a^{5} b^{2} x^{4}}{4} + 7 a^{4} b^{3} x^{5} + \frac{35 a^{3} b^{4} x^{6}}{6} + 3 a^{2} b^{5} x^{7} + \frac{7 a b^{6} x^{8}}{8} + \frac{b^{7} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)**7,x)

[Out]

a**7*x**2/2 + 7*a**6*b*x**3/3 + 21*a**5*b**2*x**4/4 + 7*a**4*b**3*x**5 + 35*a**3*b**4*x**6/6 + 3*a**2*b**5*x**
7 + 7*a*b**6*x**8/8 + b**7*x**9/9

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Giac [B]  time = 1.21016, size = 107, normalized size = 3.57 \begin{align*} \frac{1}{9} \, b^{7} x^{9} + \frac{7}{8} \, a b^{6} x^{8} + 3 \, a^{2} b^{5} x^{7} + \frac{35}{6} \, a^{3} b^{4} x^{6} + 7 \, a^{4} b^{3} x^{5} + \frac{21}{4} \, a^{5} b^{2} x^{4} + \frac{7}{3} \, a^{6} b x^{3} + \frac{1}{2} \, a^{7} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^7,x, algorithm="giac")

[Out]

1/9*b^7*x^9 + 7/8*a*b^6*x^8 + 3*a^2*b^5*x^7 + 35/6*a^3*b^4*x^6 + 7*a^4*b^3*x^5 + 21/4*a^5*b^2*x^4 + 7/3*a^6*b*
x^3 + 1/2*a^7*x^2