3.217 \(\int \frac{1}{(a+b x)^7} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{6 b (a+b x)^6} \]

[Out]

-1/(6*b*(a + b*x)^6)

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Rubi [A]  time = 0.00687739, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{1}{6 b (a+b x)^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(-7),x]

[Out]

-1/(6*b*(a + b*x)^6)

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Rubi in Sympy [A]  time = 1.27233, size = 12, normalized size = 0.86 \[ - \frac{1}{6 b \left (a + b x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**7,x)

[Out]

-1/(6*b*(a + b*x)**6)

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Mathematica [A]  time = 0.00437641, size = 14, normalized size = 1. \[ -\frac{1}{6 b (a+b x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(-7),x]

[Out]

-1/(6*b*(a + b*x)^6)

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Maple [A]  time = 0.001, size = 13, normalized size = 0.9 \[ -{\frac{1}{6\,b \left ( bx+a \right ) ^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^7,x)

[Out]

-1/6/b/(b*x+a)^6

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Maxima [A]  time = 1.34606, size = 16, normalized size = 1.14 \[ -\frac{1}{6 \,{\left (b x + a\right )}^{6} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(-7),x, algorithm="maxima")

[Out]

-1/6/((b*x + a)^6*b)

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Fricas [A]  time = 0.201074, size = 92, normalized size = 6.57 \[ -\frac{1}{6 \,{\left (b^{7} x^{6} + 6 \, a b^{6} x^{5} + 15 \, a^{2} b^{5} x^{4} + 20 \, a^{3} b^{4} x^{3} + 15 \, a^{4} b^{3} x^{2} + 6 \, a^{5} b^{2} x + a^{6} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(-7),x, algorithm="fricas")

[Out]

-1/6/(b^7*x^6 + 6*a*b^6*x^5 + 15*a^2*b^5*x^4 + 20*a^3*b^4*x^3 + 15*a^4*b^3*x^2 +
6*a^5*b^2*x + a^6*b)

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Sympy [A]  time = 2.12242, size = 73, normalized size = 5.21 \[ - \frac{1}{6 a^{6} b + 36 a^{5} b^{2} x + 90 a^{4} b^{3} x^{2} + 120 a^{3} b^{4} x^{3} + 90 a^{2} b^{5} x^{4} + 36 a b^{6} x^{5} + 6 b^{7} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**7,x)

[Out]

-1/(6*a**6*b + 36*a**5*b**2*x + 90*a**4*b**3*x**2 + 120*a**3*b**4*x**3 + 90*a**2
*b**5*x**4 + 36*a*b**6*x**5 + 6*b**7*x**6)

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GIAC/XCAS [A]  time = 0.217755, size = 16, normalized size = 1.14 \[ -\frac{1}{6 \,{\left (b x + a\right )}^{6} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(-7),x, algorithm="giac")

[Out]

-1/6/((b*x + a)^6*b)