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 = 0.0017102, 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, Rules used = {32} \[ -\frac{1}{6 b (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-7),x]

[Out]

-1/(6*b*(a + b*x)^6)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{(a+b x)^7} \, dx &=-\frac{1}{6 b (a+b x)^6}\\ \end{align*}

Mathematica [A]  time = 0.0044064, 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 \begin{align*} -{\frac{1}{6\,b \left ( bx+a \right ) ^{6}}} \end{align*}

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 = 1.06624, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{6 \,{\left (b x + a\right )}^{6} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^7,x, algorithm="maxima")

[Out]

-1/6/((b*x + a)^6*b)

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Fricas [B]  time = 1.54331, size = 139, normalized size = 9.93 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 [B]  time = 0.806741, size = 73, normalized size = 5.21 \begin{align*} - \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}} \end{align*}

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 [A]  time = 1.19839, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{6 \,{\left (b x + a\right )}^{6} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^7,x, algorithm="giac")

[Out]

-1/6/((b*x + a)^6*b)