3.2819 \(\int (\frac{c}{a+b x})^{5/2} \, dx\)

Optimal. Leaf size=30 \[ -\frac{2 c^2 \sqrt{\frac{c}{a+b x}}}{3 b (a+b x)} \]

[Out]

(-2*c^2*Sqrt[c/(a + b*x)])/(3*b*(a + b*x))

________________________________________________________________________________________

Rubi [A]  time = 0.0079889, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ -\frac{2 c^2 \sqrt{\frac{c}{a+b x}}}{3 b (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(c/(a + b*x))^(5/2),x]

[Out]

(-2*c^2*Sqrt[c/(a + b*x)])/(3*b*(a + b*x))

Rule 247

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]

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]

Rubi steps

\begin{align*} \int \left (\frac{c}{a+b x}\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{c}{x}\right )^{5/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c^2 \sqrt{\frac{c}{a+b x}} \sqrt{a+b x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{5/2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{2 c^2 \sqrt{\frac{c}{a+b x}}}{3 b (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0115844, size = 21, normalized size = 0.7 \[ -\frac{2 c \left (\frac{c}{a+b x}\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c/(a + b*x))^(5/2),x]

[Out]

(-2*c*(c/(a + b*x))^(3/2))/(3*b)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 22, normalized size = 0.7 \begin{align*} -{\frac{2\,bx+2\,a}{3\,b} \left ({\frac{c}{bx+a}} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c/(b*x+a))^(5/2),x)

[Out]

-2/3*(b*x+a)*(c/(b*x+a))^(5/2)/b

________________________________________________________________________________________

Maxima [A]  time = 0.970897, size = 23, normalized size = 0.77 \begin{align*} -\frac{2 \, c \left (\frac{c}{b x + a}\right )^{\frac{3}{2}}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c/(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

-2/3*c*(c/(b*x + a))^(3/2)/b

________________________________________________________________________________________

Fricas [A]  time = 1.3314, size = 57, normalized size = 1.9 \begin{align*} -\frac{2 \, c^{2} \sqrt{\frac{c}{b x + a}}}{3 \,{\left (b^{2} x + a b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c/(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

-2/3*c^2*sqrt(c/(b*x + a))/(b^2*x + a*b)

________________________________________________________________________________________

Sympy [A]  time = 2.70138, size = 51, normalized size = 1.7 \begin{align*} \begin{cases} - \frac{2 a c^{\frac{5}{2}} \left (\frac{1}{a + b x}\right )^{\frac{5}{2}}}{3 b} - \frac{2 c^{\frac{5}{2}} x \left (\frac{1}{a + b x}\right )^{\frac{5}{2}}}{3} & \text{for}\: b \neq 0 \\x \left (\frac{c}{a}\right )^{\frac{5}{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c/(b*x+a))**(5/2),x)

[Out]

Piecewise((-2*a*c**(5/2)*(1/(a + b*x))**(5/2)/(3*b) - 2*c**(5/2)*x*(1/(a + b*x))**(5/2)/3, Ne(b, 0)), (x*(c/a)
**(5/2), True))

________________________________________________________________________________________

Giac [A]  time = 1.15108, size = 32, normalized size = 1.07 \begin{align*} -\frac{2 \, c^{4} \mathrm{sgn}\left (b x + a\right )}{3 \,{\left (b c x + a c\right )}^{\frac{3}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c/(b*x+a))^(5/2),x, algorithm="giac")

[Out]

-2/3*c^4*sgn(b*x + a)/((b*c*x + a*c)^(3/2)*b)