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 = 0.0290225, 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 \[ \frac{(a+b x)^9}{9 b^2}-\frac{a (a+b x)^8}{8 b^2} \]

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)

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Rubi in Sympy [A]  time = 8.87201, size = 24, normalized size = 0.8 \[ - \frac{a \left (a + b x\right )^{8}}{8 b^{2}} + \frac{\left (a + b x\right )^{9}}{9 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x+a)**7,x)

[Out]

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

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Mathematica [B]  time = 0.00281233, size = 91, normalized size = 3.03 \[ \frac{a^7 x^2}{2}+\frac{7}{3} a^6 b x^3+\frac{21}{4} a^5 b^2 x^4+7 a^4 b^3 x^5+\frac{35}{6} a^3 b^4 x^6+3 a^2 b^5 x^7+\frac{7}{8} a b^6 x^8+\frac{b^7 x^9}{9} \]

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 = 0.001, size = 80, normalized size = 2.7 \[{\frac{{b}^{7}{x}^{9}}{9}}+{\frac{7\,a{b}^{6}{x}^{8}}{8}}+3\,{a}^{2}{b}^{5}{x}^{7}+{\frac{35\,{a}^{3}{b}^{4}{x}^{6}}{6}}+7\,{a}^{4}{b}^{3}{x}^{5}+{\frac{21\,{a}^{5}{b}^{2}{x}^{4}}{4}}+{\frac{7\,{a}^{6}b{x}^{3}}{3}}+{\frac{{a}^{7}{x}^{2}}{2}} \]

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 [A]  time = 1.34543, size = 107, normalized size = 3.57 \[ \frac{1}{9} \, b^{7} x^{9} + \frac{7}{8} \, a b^{6} x^{8} + 3 \, a^{2} b^{5} x^{7} + \frac{35}{6} \, a^{3} b^{4} x^{6} + 7 \, a^{4} b^{3} x^{5} + \frac{21}{4} \, a^{5} b^{2} x^{4} + \frac{7}{3} \, a^{6} b x^{3} + \frac{1}{2} \, a^{7} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7*x,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 [A]  time = 0.18109, size = 1, normalized size = 0.03 \[ \frac{1}{9} x^{9} b^{7} + \frac{7}{8} x^{8} b^{6} a + 3 x^{7} b^{5} a^{2} + \frac{35}{6} x^{6} b^{4} a^{3} + 7 x^{5} b^{3} a^{4} + \frac{21}{4} x^{4} b^{2} a^{5} + \frac{7}{3} x^{3} b a^{6} + \frac{1}{2} x^{2} a^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7*x,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 [A]  time = 0.150287, size = 90, normalized size = 3. \[ \frac{a^{7} x^{2}}{2} + \frac{7 a^{6} b x^{3}}{3} + \frac{21 a^{5} b^{2} x^{4}}{4} + 7 a^{4} b^{3} x^{5} + \frac{35 a^{3} b^{4} x^{6}}{6} + 3 a^{2} b^{5} x^{7} + \frac{7 a b^{6} x^{8}}{8} + \frac{b^{7} x^{9}}{9} \]

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/XCAS [A]  time = 0.210255, size = 107, normalized size = 3.57 \[ \frac{1}{9} \, b^{7} x^{9} + \frac{7}{8} \, a b^{6} x^{8} + 3 \, a^{2} b^{5} x^{7} + \frac{35}{6} \, a^{3} b^{4} x^{6} + 7 \, a^{4} b^{3} x^{5} + \frac{21}{4} \, a^{5} b^{2} x^{4} + \frac{7}{3} \, a^{6} b x^{3} + \frac{1}{2} \, a^{7} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7*x,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