3.201 \(\int \frac{1}{(a+b x)^4} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{3 b (a+b x)^3} \]

[Out]

-1/(3*b*(a + b*x)^3)

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Rubi [A]  time = 0.0015438, 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} \[ -\frac{1}{3 b (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-4),x]

[Out]

-1/(3*b*(a + b*x)^3)

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)^4} \, dx &=-\frac{1}{3 b (a+b x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0030585, size = 14, normalized size = 1. \[ -\frac{1}{3 b (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-4),x]

[Out]

-1/(3*b*(a + b*x)^3)

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Maple [A]  time = 0.001, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,b \left ( bx+a \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^4,x)

[Out]

-1/3/b/(b*x+a)^3

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Maxima [A]  time = 1.07077, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{3 \,{\left (b x + a\right )}^{3} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^4,x, algorithm="maxima")

[Out]

-1/3/((b*x + a)^3*b)

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Fricas [B]  time = 1.38961, size = 70, normalized size = 5. \begin{align*} -\frac{1}{3 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^4,x, algorithm="fricas")

[Out]

-1/3/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b)

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Sympy [B]  time = 0.449615, size = 37, normalized size = 2.64 \begin{align*} - \frac{1}{3 a^{3} b + 9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**4,x)

[Out]

-1/(3*a**3*b + 9*a**2*b**2*x + 9*a*b**3*x**2 + 3*b**4*x**3)

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Giac [A]  time = 1.16565, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{3 \,{\left (b x + a\right )}^{3} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^4,x, algorithm="giac")

[Out]

-1/3/((b*x + a)^3*b)