\(\int \frac {1}{(a+b x)^{3/2}} \, dx\) [347]

Optimal result

Integrand size = 9, antiderivative size = 14 \[ \int \frac {1}{(a+b x)^{3/2}} \, dx=-\frac {2}{b \sqrt {a+b x}} \]

[Out]

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

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} \[ \int \frac {1}{(a+b x)^{3/2}} \, dx=-\frac {2}{b \sqrt {a+b x}} \]

[In]

Int[(a + b*x)^(-3/2),x]

[Out]

-2/(b*Sqrt[a + b*x])

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 {2}{b \sqrt {a+b x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b x)^{3/2}} \, dx=-\frac {2}{b \sqrt {a+b x}} \]

[In]

Integrate[(a + b*x)^(-3/2),x]

[Out]

-2/(b*Sqrt[a + b*x])

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
gosper	\(-\frac {2}{b \sqrt {b x +a}}\)	\(13\)
derivativedivides	\(-\frac {2}{b \sqrt {b x +a}}\)	\(13\)
default	\(-\frac {2}{b \sqrt {b x +a}}\)	\(13\)
trager	\(-\frac {2}{b \sqrt {b x +a}}\)	\(13\)
pseudoelliptic	\(-\frac {2}{b \sqrt {b x +a}}\)	\(13\)

[In]

int(1/(b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

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

[In]

integrate(1/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2*sqrt(b*x + a)/(b^2*x + a*b)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^{3/2}} \, dx=- \frac {2}{b \sqrt {a + b x}} \]

[In]

integrate(1/(b*x+a)**(3/2),x)

[Out]

-2/(b*sqrt(a + b*x))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^{3/2}} \, dx=-\frac {2}{\sqrt {b x + a} b} \]

[In]

integrate(1/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

-2/(sqrt(b*x + a)*b)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^{3/2}} \, dx=-\frac {2}{\sqrt {b x + a} b} \]

[In]

integrate(1/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

-2/(sqrt(b*x + a)*b)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^{3/2}} \, dx=-\frac {2}{b\,\sqrt {a+b\,x}} \]

[In]

int(1/(a + b*x)^(3/2),x)

[Out]

-2/(b*(a + b*x)^(1/2))