3.2825 \(\int (\frac{c}{(a+b x)^2})^{5/2} \, dx\)

Optimal. Leaf size=30 \[ -\frac{c^2 \sqrt{\frac{c}{(a+b x)^2}}}{4 b (a+b x)^3} \]

[Out]

-(c^2*Sqrt[c/(a + b*x)^2])/(4*b*(a + b*x)^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0087175, 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} \[ -\frac{c^2 \sqrt{\frac{c}{(a+b x)^2}}}{4 b (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(c/(a + b*x)^2)^(5/2),x]

[Out]

-(c^2*Sqrt[c/(a + b*x)^2])/(4*b*(a + b*x)^3)

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)^2}\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{c}{x^2}\right )^{5/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c^2 \sqrt{\frac{c}{(a+b x)^2}} (a+b x)\right ) \operatorname{Subst}\left (\int \frac{1}{x^5} \, dx,x,a+b x\right )}{b}\\ &=-\frac{c^2 \sqrt{\frac{c}{(a+b x)^2}}}{4 b (a+b x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0104442, size = 25, normalized size = 0.83 \[ -\frac{(a+b x) \left (\frac{c}{(a+b x)^2}\right )^{5/2}}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c/(a + b*x)^2)^(5/2),x]

[Out]

-((c/(a + b*x)^2)^(5/2)*(a + b*x))/(4*b)

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 22, normalized size = 0.7 \begin{align*} -{\frac{bx+a}{4\,b} \left ({\frac{c}{ \left ( bx+a \right ) ^{2}}} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(b*x+a)/b*(c/(b*x+a)^2)^(5/2)

________________________________________________________________________________________

Maxima [A]  time = 1.19904, size = 66, normalized size = 2.2 \begin{align*} -\frac{c^{\frac{5}{2}}}{4 \,{\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*c^(5/2)/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b)

________________________________________________________________________________________

Fricas [B]  time = 1.31098, size = 122, normalized size = 4.07 \begin{align*} -\frac{c^{2} \sqrt{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{4 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*c^2*sqrt(c/(b^2*x^2 + 2*a*b*x + a^2))/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\frac{c}{\left (a + b x\right )^{2}}\right )^{\frac{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c/(a + b*x)**2)**(5/2), x)

________________________________________________________________________________________

Giac [A]  time = 1.11585, size = 1, normalized size = 0.03 \begin{align*} +\infty \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

+Infinity