3.2.87 \(\int \frac {1}{(a+b x)^3} \, dx\) [187]

Optimal. Leaf size=14 \[ -\frac {1}{2 b (a+b x)^2} \]

[Out]

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

________________________________________________________________________________________

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}{2 b (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-3),x]

[Out]

-1/2*1/(b*(a + b*x)^2)

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)^3} \, dx &=-\frac {1}{2 b (a+b x)^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{2 b (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-3),x]

[Out]

-1/2*1/(b*(a + b*x)^2)

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 13, normalized size = 0.93

method	result	size
gosper	\(-\frac {1}{2 b \left (b x +a \right )^{2}}\)	\(13\)
default	\(-\frac {1}{2 b \left (b x +a \right )^{2}}\)	\(13\)
norman	\(-\frac {1}{2 b \left (b x +a \right )^{2}}\)	\(13\)
risch	\(-\frac {1}{2 b \left (b x +a \right )^{2}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{2 \, {\left (b x + a\right )}^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2/((b*x + a)^2*b)

________________________________________________________________________________________

Fricas [A]
time = 0.91, size = 24, normalized size = 1.71 \begin {gather*} -\frac {1}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/2/(b^3*x^2 + 2*a*b^2*x + a^2*b)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).
time = 0.07, size = 26, normalized size = 1.86 \begin {gather*} - \frac {1}{2 a^{2} b + 4 a b^{2} x + 2 b^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.58, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{2 \, {\left (b x + a\right )}^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^3,x, algorithm="giac")

[Out]

-1/2/((b*x + a)^2*b)

________________________________________________________________________________________

Mupad [B]
time = 0.07, size = 26, normalized size = 1.86 \begin {gather*} -\frac {1}{2\,a^2\,b+4\,a\,b^2\,x+2\,b^3\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x)^3,x)

[Out]

-1/(2*a^2*b + 2*b^3*x^2 + 4*a*b^2*x)

________________________________________________________________________________________