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

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

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {1}{6 b (a+b x)^6} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{6 b (a+b x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-7),x]

[Out]

-1/6*1/(b*(a + b*x)^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(a+b x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^(-7),x]

[Out]

IntegrateAlgebraic[(a + b*x)^(-7), x]

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fricas [B]  time = 1.20, size = 68, normalized size = 4.86 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 1.09, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{6 \, {\left (b x + a\right )}^{6} b} \end {gather*}

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)

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maple [A]  time = 0.00, size = 13, normalized size = 0.93 \begin {gather*} -\frac {1}{6 \left (b x +a \right )^{6} b} \end {gather*}

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 = 1.41, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{6 \, {\left (b x + a\right )}^{6} b} \end {gather*}

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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mupad [B]  time = 0.06, size = 70, normalized size = 5.00 \begin {gather*} -\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x)^7,x)

[Out]

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

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sympy [B]  time = 0.46, size = 73, normalized size = 5.21 \begin {gather*} - \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 {gather*}

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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