\(\int \frac {1}{(c+d (a+b x))^{3/2}} \, dx\) [41]

Optimal result

Integrand size = 13, antiderivative size = 21 \[ \int \frac {1}{(c+d (a+b x))^{3/2}} \, dx=-\frac {2}{b d \sqrt {c+d (a+b x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {33, 32} \[ \int \frac {1}{(c+d (a+b x))^{3/2}} \, dx=-\frac {2}{b d \sqrt {d (a+b x)+c}} \]

[In]

Int[(c + d*(a + b*x))^(-3/2),x]

[Out]

-2/(b*d*Sqrt[c + d*(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]

Rule 33

Int[((a_.) + (b_.)*(u_))^(m_), x_Symbol] :> Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x)^m, x], x, u], x]
/; FreeQ[{a, b, m}, x] && LinearQ[u, x] && NeQ[u, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{(c+d x)^{3/2}} \, dx,x,a+b x\right )}{b} \\ & = -\frac {2}{b d \sqrt {c+d (a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d (a+b x))^{3/2}} \, dx=-\frac {2}{b d \sqrt {c+a d+b d x}} \]

[In]

Integrate[(c + d*(a + b*x))^(-3/2),x]

[Out]

-2/(b*d*Sqrt[c + a*d + b*d*x])

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95

method	result	size
gosper	\(-\frac {2}{\sqrt {b d x +a d +c}\, d b}\)	\(20\)
derivativedivides	\(-\frac {2}{\sqrt {b d x +a d +c}\, d b}\)	\(20\)
default	\(-\frac {2}{\sqrt {b d x +a d +c}\, d b}\)	\(20\)
trager	\(-\frac {2}{\sqrt {b d x +a d +c}\, d b}\)	\(20\)
pseudoelliptic	\(-\frac {2}{b d \sqrt {c +d \left (b x +a \right )}}\)	\(20\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int \frac {1}{(c+d (a+b x))^{3/2}} \, dx=-\frac {2 \, \sqrt {b d x + a d + c}}{b^{2} d^{2} x + a b d^{2} + b c d} \]

[In]

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

[Out]

-2*sqrt(b*d*x + a*d + c)/(b^2*d^2*x + a*b*d^2 + b*c*d)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (17) = 34\).

Time = 0.33 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.76 \[ \int \frac {1}{(c+d (a+b x))^{3/2}} \, dx=\begin {cases} \frac {x}{c^{\frac {3}{2}}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x}{\left (a d + c\right )^{\frac {3}{2}}} & \text {for}\: b = 0 \\\frac {x}{c^{\frac {3}{2}}} & \text {for}\: d = 0 \\- \frac {2 \sqrt {a d + b d x + c}}{a b d^{2} + b^{2} d^{2} x + b c d} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x/c**(3/2), Eq(b, 0) & Eq(d, 0)), (x/(a*d + c)**(3/2), Eq(b, 0)), (x/c**(3/2), Eq(d, 0)), (-2*sqrt(
a*d + b*d*x + c)/(a*b*d**2 + b**2*d**2*x + b*c*d), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(c+d (a+b x))^{3/2}} \, dx=-\frac {2}{\sqrt {{\left (b x + a\right )} d + c} b d} \]

[In]

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

[Out]

-2/(sqrt((b*x + a)*d + c)*b*d)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(c+d (a+b x))^{3/2}} \, dx=-\frac {2}{\sqrt {b d x + a d + c} b d} \]

[In]

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

[Out]

-2/(sqrt(b*d*x + a*d + c)*b*d)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(c+d (a+b x))^{3/2}} \, dx=-\frac {2}{b\,d\,\sqrt {c+d\,\left (a+b\,x\right )}} \]

[In]

int(1/(c + d*(a + b*x))^(3/2),x)

[Out]

-2/(b*d*(c + d*(a + b*x))^(1/2))