\(\int \frac {1}{(a+b x)^{4/3}} \, dx\) [416]

Optimal result

Integrand size = 9, antiderivative size = 14 \[ \int \frac {1}{(a+b x)^{4/3}} \, dx=-\frac {3}{b \sqrt [3]{a+b x}} \]

[Out]

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

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {32} \[ \int \frac {1}{(a+b x)^{4/3}} \, dx=-\frac {3}{b \sqrt [3]{a+b x}} \]

[In]

Int[(a + b*x)^(-4/3),x]

[Out]

-3/(b*(a + b*x)^(1/3))

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 {3}{b \sqrt [3]{a+b x}} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[(a + b*x)^(-4/3),x]

[Out]

-3/(b*(a + b*x)^(1/3))

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
gosper	\(-\frac {3}{b \left (b x +a \right )^{\frac {1}{3}}}\)	\(13\)
derivativedivides	\(-\frac {3}{b \left (b x +a \right )^{\frac {1}{3}}}\)	\(13\)
default	\(-\frac {3}{b \left (b x +a \right )^{\frac {1}{3}}}\)	\(13\)
trager	\(-\frac {3}{b \left (b x +a \right )^{\frac {1}{3}}}\)	\(13\)
pseudoelliptic	\(-\frac {3}{b \left (b x +a \right )^{\frac {1}{3}}}\)	\(13\)

[In]

int(1/(b*x+a)^(4/3),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {1}{(a+b x)^{4/3}} \, dx=-\frac {3 \, {\left (b x + a\right )}^{\frac {2}{3}}}{b^{2} x + a b} \]

[In]

integrate(1/(b*x+a)^(4/3),x, algorithm="fricas")

[Out]

-3*(b*x + a)^(2/3)/(b^2*x + a*b)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^{4/3}} \, dx=- \frac {3}{b \sqrt [3]{a + b x}} \]

[In]

integrate(1/(b*x+a)**(4/3),x)

[Out]

-3/(b*(a + b*x)**(1/3))

Maxima [A] (verification not implemented)

none

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

[In]

integrate(1/(b*x+a)^(4/3),x, algorithm="maxima")

[Out]

-3/((b*x + a)^(1/3)*b)

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^{4/3}} \, dx=-\frac {3}{{\left (b x + a\right )}^{\frac {1}{3}} b} \]

[In]

integrate(1/(b*x+a)^(4/3),x, algorithm="giac")

[Out]

-3/((b*x + a)^(1/3)*b)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^{4/3}} \, dx=-\frac {3}{b\,{\left (a+b\,x\right )}^{1/3}} \]

[In]

int(1/(a + b*x)^(4/3),x)

[Out]

-3/(b*(a + b*x)^(1/3))