3.669 \(\int \frac{1}{x (-a+b (c x)^n)^{5/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac{2}{a^2 n \sqrt{b (c x)^n-a}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{a^{5/2} n}-\frac{2}{3 a n \left (b (c x)^n-a\right )^{3/2}} \]

[Out]

-2/(3*a*n*(-a + b*(c*x)^n)^(3/2)) + 2/(a^2*n*Sqrt[-a + b*(c*x)^n]) + (2*ArcTan[Sqrt[-a + b*(c*x)^n]/Sqrt[a]])/
(a^(5/2)*n)

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Rubi [A]  time = 0.0609469, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {367, 12, 266, 51, 63, 205} \[ \frac{2}{a^2 n \sqrt{b (c x)^n-a}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{a^{5/2} n}-\frac{2}{3 a n \left (b (c x)^n-a\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(-a + b*(c*x)^n)^(5/2)),x]

[Out]

-2/(3*a*n*(-a + b*(c*x)^n)^(3/2)) + 2/(a^2*n*Sqrt[-a + b*(c*x)^n]) + (2*ArcTan[Sqrt[-a + b*(c*x)^n]/Sqrt[a]])/
(a^(5/2)*n)

Rule 367

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_)*(x_))^(n_))^(p_.), x_Symbol] :> Dist[1/c, Subst[Int[((d*x)/c)^m*(a
+ b*x^n)^p, x], x, c*x], x] /; FreeQ[{a, b, c, d, m, n, p}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [C]  time = 0.0382383, size = 46, normalized size = 0.57 \[ -\frac{2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};1-\frac{b (c x)^n}{a}\right )}{3 a n \left (b (c x)^n-a\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(-a + b*(c*x)^n)^(5/2)),x]

[Out]

(-2*Hypergeometric2F1[-3/2, 1, -1/2, 1 - (b*(c*x)^n)/a])/(3*a*n*(-a + b*(c*x)^n)^(3/2))

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Maple [A]  time = 0.006, size = 70, normalized size = 0.9 \begin{align*} -{\frac{2}{3\,an} \left ( -a+b \left ( cx \right ) ^{n} \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{1}{{a}^{5/2}n}\arctan \left ({\frac{\sqrt{-a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) }+2\,{\frac{1}{{a}^{2}n\sqrt{-a+b \left ( cx \right ) ^{n}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(-a+b*(c*x)^n)^(5/2),x)

[Out]

-2/3/a/n/(-a+b*(c*x)^n)^(3/2)+2*arctan((-a+b*(c*x)^n)^(1/2)/a^(1/2))/a^(5/2)/n+2/a^2/n/(-a+b*(c*x)^n)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (\left (c x\right )^{n} b - a\right )}^{\frac{5}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-a+b*(c*x)^n)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/(((c*x)^n*b - a)^(5/2)*x), x)

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Fricas [A]  time = 1.55115, size = 602, normalized size = 7.43 \begin{align*} \left [-\frac{3 \,{\left (2 \, \left (c x\right )^{n} \sqrt{-a} a b - \left (c x\right )^{2 \, n} \sqrt{-a} b^{2} - \sqrt{-a} a^{2}\right )} \log \left (\frac{\left (c x\right )^{n} b - 2 \, \sqrt{\left (c x\right )^{n} b - a} \sqrt{-a} - 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \,{\left (3 \, \left (c x\right )^{n} a b - 4 \, a^{2}\right )} \sqrt{\left (c x\right )^{n} b - a}}{3 \,{\left (2 \, \left (c x\right )^{n} a^{4} b n - \left (c x\right )^{2 \, n} a^{3} b^{2} n - a^{5} n\right )}}, \frac{2 \,{\left (3 \,{\left (2 \, \left (c x\right )^{n} a^{\frac{3}{2}} b - \left (c x\right )^{2 \, n} \sqrt{a} b^{2} - a^{\frac{5}{2}}\right )} \arctan \left (\frac{\sqrt{\left (c x\right )^{n} b - a}}{\sqrt{a}}\right ) -{\left (3 \, \left (c x\right )^{n} a b - 4 \, a^{2}\right )} \sqrt{\left (c x\right )^{n} b - a}\right )}}{3 \,{\left (2 \, \left (c x\right )^{n} a^{4} b n - \left (c x\right )^{2 \, n} a^{3} b^{2} n - a^{5} n\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-a+b*(c*x)^n)^(5/2),x, algorithm="fricas")

[Out]

[-1/3*(3*(2*(c*x)^n*sqrt(-a)*a*b - (c*x)^(2*n)*sqrt(-a)*b^2 - sqrt(-a)*a^2)*log(((c*x)^n*b - 2*sqrt((c*x)^n*b
- a)*sqrt(-a) - 2*a)/(c*x)^n) + 2*(3*(c*x)^n*a*b - 4*a^2)*sqrt((c*x)^n*b - a))/(2*(c*x)^n*a^4*b*n - (c*x)^(2*n
)*a^3*b^2*n - a^5*n), 2/3*(3*(2*(c*x)^n*a^(3/2)*b - (c*x)^(2*n)*sqrt(a)*b^2 - a^(5/2))*arctan(sqrt((c*x)^n*b -
a)/sqrt(a)) - (3*(c*x)^n*a*b - 4*a^2)*sqrt((c*x)^n*b - a))/(2*(c*x)^n*a^4*b*n - (c*x)^(2*n)*a^3*b^2*n - a^5*n
)]

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Sympy [A]  time = 10.965, size = 63, normalized size = 0.78 \begin{align*} - \frac{2}{3 a n \left (- a + b \left (c x\right )^{n}\right )^{\frac{3}{2}}} + \frac{2}{a^{2} n \sqrt{- a + b \left (c x\right )^{n}}} + \frac{2 \operatorname{atan}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-a+b*(c*x)**n)**(5/2),x)

[Out]

-2/(3*a*n*(-a + b*(c*x)**n)**(3/2)) + 2/(a**2*n*sqrt(-a + b*(c*x)**n)) + 2*atan(sqrt(-a + b*(c*x)**n)/sqrt(a))
/(a**(5/2)*n)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (\left (c x\right )^{n} b - a\right )}^{\frac{5}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-a+b*(c*x)^n)^(5/2),x, algorithm="giac")

[Out]

integrate(1/(((c*x)^n*b - a)^(5/2)*x), x)