\(\int \frac {1}{(a+b x)^4} \, dx\) [201]

Optimal result

Integrand size = 7, antiderivative size = 14 \[ \int \frac {1}{(a+b x)^4} \, dx=-\frac {1}{3 b (a+b x)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \[ \int \frac {1}{(a+b x)^4} \, dx=-\frac {1}{3 b (a+b x)^3} \]

[In]

Int[(a + b*x)^(-4),x]

[Out]

-1/3*1/(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*} \text {integral}& = -\frac {1}{3 b (a+b x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b x)^4} \, dx=-\frac {1}{3 b (a+b x)^3} \]

[In]

Integrate[(a + b*x)^(-4),x]

[Out]

-1/3*1/(b*(a + b*x)^3)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
gosper	\(-\frac {1}{3 b \left (b x +a \right )^{3}}\)	\(13\)
default	\(-\frac {1}{3 b \left (b x +a \right )^{3}}\)	\(13\)
norman	\(-\frac {1}{3 b \left (b x +a \right )^{3}}\)	\(13\)
risch	\(-\frac {1}{3 b \left (b x +a \right )^{3}}\)	\(13\)
parallelrisch	\(-\frac {1}{3 b \left (b x +a \right )^{3}}\)	\(13\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (12) = 24\).

Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.50 \[ \int \frac {1}{(a+b x)^4} \, dx=-\frac {1}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (12) = 24\).

Time = 0.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.64 \[ \int \frac {1}{(a+b x)^4} \, dx=- \frac {1}{3 a^{3} b + 9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{4} x^{3}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^4} \, dx=-\frac {1}{3 \, {\left (b x + a\right )}^{3} b} \]

[In]

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

[Out]

-1/3/((b*x + a)^3*b)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^4} \, dx=-\frac {1}{3 \, {\left (b x + a\right )}^{3} b} \]

[In]

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

[Out]

-1/3/((b*x + a)^3*b)

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.64 \[ \int \frac {1}{(a+b x)^4} \, dx=-\frac {1}{3\,a^3\,b+9\,a^2\,b^2\,x+9\,a\,b^3\,x^2+3\,b^4\,x^3} \]

[In]

int(1/(a + b*x)^4,x)

[Out]

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