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

Optimal. Leaf size=20 \[ \frac {x}{a^2+a b \left (c x^n\right )^{\frac {1}{n}}} \]

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.60, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {254, 32} \begin {gather*} -\frac {x \left (c x^n\right )^{-1/n}}{b \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x^n)^n^(-1))^(-2),x]

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 254

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_.), x_Symbol] :> Dist[x/(c*x^q)^(1/q), Subst[Int[(a + b*x^(n*q))
^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 33, normalized size = 1.65 \begin {gather*} -\frac {x \left (c x^n\right )^{-1/n}}{a b+b^2 \left (c x^n\right )^{\frac {1}{n}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[IntegrateAlgebraic][(a + b*(c*x^n)^n^(-1))^(-2), x]

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fricas [A]  time = 0.94, size = 25, normalized size = 1.25 \begin {gather*} -\frac {1}{b^{2} c^{\frac {2}{n}} x + a b c^{\left (\frac {1}{n}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b^2*c^(2/n)*x + a*b*c^(1/n))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.03, size = 74, normalized size = 3.70 \begin {gather*} \frac {x}{\left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right ) a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*x/(b*c^(1/n)*(x^n)^(1/n)*exp(1/2*I*Pi*(csgn(I*c)-csgn(I*c*x^n))*(-csgn(I*x^n)+csgn(I*c*x^n))/n*csgn(I*c*x^
n))+a)

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maxima [A]  time = 0.54, size = 23, normalized size = 1.15 \begin {gather*} \frac {x}{a b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(a*b*c^(1/n)*(x^n)^(1/n) + a^2)

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mupad [B]  time = 1.19, size = 20, normalized size = 1.00 \begin {gather*} \frac {x}{a\,\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*(c*x^n)^(1/n))^2,x)

[Out]

x/(a*(a + b*(c*x^n)^(1/n)))

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sympy [A]  time = 169.22, size = 80, normalized size = 4.00 \begin {gather*} \begin {cases} \tilde {\infty } c^{- \frac {2}{n}} x \left (x^{n}\right )^{- \frac {2}{n}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {c^{- \frac {2}{n}} x \left (x^{n}\right )^{- \frac {2}{n}}}{b^{2}} & \text {for}\: a = 0 \\\tilde {\infty } c^{\frac {2}{n}} x \left (x^{n}\right )^{\frac {2}{n}} & \text {for}\: b = - a c^{- \frac {1}{n}} \left (x^{n}\right )^{- \frac {1}{n}} \\\frac {x}{a^{2} + a b c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*c**(-2/n)*x*(x**n)**(-2/n), Eq(a, 0) & Eq(b, 0)), (-c**(-2/n)*x*(x**n)**(-2/n)/b**2, Eq(a, 0)),
(zoo*c**(2/n)*x*(x**n)**(2/n), Eq(b, -a*c**(-1/n)*(x**n)**(-1/n))), (x/(a**2 + a*b*c**(1/n)*(x**n)**(1/n)), T
rue))

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