3.402 \(\int \frac{1}{c x (a+b x^n)^{3/2}} \, dx\)

Optimal. Leaf size=54 \[ \frac{2}{a c n \sqrt{a+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} c n} \]

[Out]

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

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Rubi [A]  time = 0.0332688, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {12, 266, 51, 63, 208} \[ \frac{2}{a c n \sqrt{a+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} c n} \]

Antiderivative was successfully verified.

[In]

Int[1/(c*x*(a + b*x^n)^(3/2)),x]

[Out]

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

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 208

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

Rubi steps

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

Mathematica [C]  time = 0.0140132, size = 40, normalized size = 0.74 \[ \frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b x^n}{a}+1\right )}{a c n \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 42, normalized size = 0.8 \begin{align*}{\frac{1}{cn} \left ( -2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{a+b{x}^{n}}}{\sqrt{a}}} \right ) }+2\,{\frac{1}{a\sqrt{a+b{x}^{n}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/c/x/(a+b*x^n)^(3/2),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.846647, size = 335, normalized size = 6.2 \begin{align*} \left [\frac{{\left (\sqrt{a} b x^{n} + a^{\frac{3}{2}}\right )} \log \left (\frac{b x^{n} - 2 \, \sqrt{b x^{n} + a} \sqrt{a} + 2 \, a}{x^{n}}\right ) + 2 \, \sqrt{b x^{n} + a} a}{a^{2} b c n x^{n} + a^{3} c n}, \frac{2 \,{\left ({\left (\sqrt{-a} b x^{n} + \sqrt{-a} a\right )} \arctan \left (\frac{\sqrt{b x^{n} + a} \sqrt{-a}}{a}\right ) + \sqrt{b x^{n} + a} a\right )}}{a^{2} b c n x^{n} + a^{3} c n}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 3.49617, size = 185, normalized size = 3.43 \begin{align*} \frac{\frac{2 a^{3} \sqrt{1 + \frac{b x^{n}}{a}}}{a^{\frac{9}{2}} n + a^{\frac{7}{2}} b n x^{n}} + \frac{a^{3} \log{\left (\frac{b x^{n}}{a} \right )}}{a^{\frac{9}{2}} n + a^{\frac{7}{2}} b n x^{n}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{n}}{a}} + 1 \right )}}{a^{\frac{9}{2}} n + a^{\frac{7}{2}} b n x^{n}} + \frac{a^{2} b x^{n} \log{\left (\frac{b x^{n}}{a} \right )}}{a^{\frac{9}{2}} n + a^{\frac{7}{2}} b n x^{n}} - \frac{2 a^{2} b x^{n} \log{\left (\sqrt{1 + \frac{b x^{n}}{a}} + 1 \right )}}{a^{\frac{9}{2}} n + a^{\frac{7}{2}} b n x^{n}}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a**3*sqrt(1 + b*x**n/a)/(a**(9/2)*n + a**(7/2)*b*n*x**n) + a**3*log(b*x**n/a)/(a**(9/2)*n + a**(7/2)*b*n*x*
*n) - 2*a**3*log(sqrt(1 + b*x**n/a) + 1)/(a**(9/2)*n + a**(7/2)*b*n*x**n) + a**2*b*x**n*log(b*x**n/a)/(a**(9/2
)*n + a**(7/2)*b*n*x**n) - 2*a**2*b*x**n*log(sqrt(1 + b*x**n/a) + 1)/(a**(9/2)*n + a**(7/2)*b*n*x**n))/c

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{3}{2}} c x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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