3.5.4 \(\int \frac {1}{c^4 x^4 (\frac {a}{x^2}+b x^n)^{3/2}} \, dx\) [404]

Optimal. Leaf size=72 \[ \frac {2}{a c^4 (2+n) x \sqrt {\frac {a}{x^2}+b x^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{a^{3/2} c^4 (2+n)} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 2055, 2054, 212} \begin {gather*} \frac {2}{a c^4 (n+2) x \sqrt {\frac {a}{x^2}+b x^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{a^{3/2} c^4 (n+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(c^4*x^4*(a/x^2 + b*x^n)^(3/2)),x]

[Out]

2/(a*c^4*(2 + n)*x*Sqrt[a/x^2 + b*x^n]) - (2*ArcTanh[Sqrt[a]/(x*Sqrt[a/x^2 + b*x^n])])/(a^(3/2)*c^4*(2 + 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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2054

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rule 2055

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-c^(j - 1))*(c*x)^(m - j
+ 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j)*(p + 1))), x] + Dist[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1)))
, Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] && ILtQ[p + 1/2, 0] && NeQ
[n, j] && EqQ[Simplify[m + j*p + 1], 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 74, normalized size = 1.03 \begin {gather*} \frac {2 \left (\sqrt {a}-\sqrt {a+b x^{2+n}} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{2+n}}}{\sqrt {a}}\right )\right )}{a^{3/2} c^4 (2+n) x \sqrt {\frac {a}{x^2}+b x^n}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(c^4*x^4*(a/x^2 + b*x^n)^(3/2)),x]

[Out]

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

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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {1}{c^{4} x^{4} \left (\frac {a}{x^{2}}+b \,x^{n}\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/c^4/x^4/(a/x^2+b*x^n)^(3/2),x)

[Out]

int(1/c^4/x^4/(a/x^2+b*x^n)^(3/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/c^4/x^4/(a/x^2+b*x^n)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^n + a/x^2)^(3/2)*x^4), x)/c^4

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/c^4/x^4/(a/x^2+b*x^n)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{a x^{2} \sqrt {\frac {a}{x^{2}} + b x^{n}} + b x^{4} x^{n} \sqrt {\frac {a}{x^{2}} + b x^{n}}}\, dx}{c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/c**4/x**4/(a/x**2+b*x**n)**(3/2),x)

[Out]

Integral(1/(a*x**2*sqrt(a/x**2 + b*x**n) + b*x**4*x**n*sqrt(a/x**2 + b*x**n)), x)/c**4

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/c^4/x^4/(a/x^2+b*x^n)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((b*x^n + a/x^2)^(3/2)*c^4*x^4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{c^4\,x^4\,{\left (b\,x^n+\frac {a}{x^2}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c^4*x^4*(b*x^n + a/x^2)^(3/2)),x)

[Out]

int(1/(c^4*x^4*(b*x^n + a/x^2)^(3/2)), x)

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