\(\int (\frac {c}{a+b x})^{3/2} \, dx\) [2820]

Optimal result

Integrand size = 13, antiderivative size = 19 \[ \int \left (\frac {c}{a+b x}\right )^{3/2} \, dx=-\frac {2 c \sqrt {\frac {c}{a+b x}}}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {253, 15, 30} \[ \int \left (\frac {c}{a+b x}\right )^{3/2} \, dx=-\frac {2 c \sqrt {\frac {c}{a+b x}}}{b} \]

[In]

Int[(c/(a + b*x))^(3/2),x]

[Out]

(-2*c*Sqrt[c/(a + b*x)])/b

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 253

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \left (\frac {c}{a+b x}\right )^{3/2} \, dx=-\frac {2 c \sqrt {\frac {c}{a+b x}}}{b} \]

[In]

Integrate[(c/(a + b*x))^(3/2),x]

[Out]

(-2*c*Sqrt[c/(a + b*x)])/b

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

method	result	size
trager	\(-\frac {2 c \sqrt {\frac {c}{b x +a}}}{b}\)	\(18\)
gosper	\(-\frac {2 \left (b x +a \right ) \left (\frac {c}{b x +a}\right )^{\frac {3}{2}}}{b}\)	\(22\)
default	\(-\frac {2 \left (b x +a \right ) \left (\frac {c}{b x +a}\right )^{\frac {3}{2}}}{b}\)	\(22\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \left (\frac {c}{a+b x}\right )^{3/2} \, dx=-\frac {2 \, c \sqrt {\frac {c}{b x + a}}}{b} \]

[In]

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

[Out]

-2*c*sqrt(c/(b*x + a))/b

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).

Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.95 \[ \int \left (\frac {c}{a+b x}\right )^{3/2} \, dx=\begin {cases} - \frac {2 a \left (\frac {c}{a + b x}\right )^{\frac {3}{2}}}{b} - 2 x \left (\frac {c}{a + b x}\right )^{\frac {3}{2}} & \text {for}\: b \neq 0 \\x \left (\frac {c}{a}\right )^{\frac {3}{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2*a*(c/(a + b*x))**(3/2)/b - 2*x*(c/(a + b*x))**(3/2), Ne(b, 0)), (x*(c/a)**(3/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \left (\frac {c}{a+b x}\right )^{3/2} \, dx=-\frac {2 \, c \sqrt {\frac {c}{b x + a}}}{b} \]

[In]

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

[Out]

-2*c*sqrt(c/(b*x + a))/b

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \left (\frac {c}{a+b x}\right )^{3/2} \, dx=-\frac {2 \, c^{2} \mathrm {sgn}\left (b x + a\right )}{\sqrt {b c x + a c} b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.96 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \left (\frac {c}{a+b x}\right )^{3/2} \, dx=-\frac {2\,c\,\sqrt {\frac {c}{a+b\,x}}}{b} \]

[In]

int((c/(a + b*x))^(3/2),x)

[Out]

-(2*c*(c/(a + b*x))^(1/2))/b