3.31.20 \(\int \frac {1}{(a+b (c x^n)^{\frac {1}{n}})^3} \, dx\) [3020]

Optimal. Leaf size=34 \[ -\frac {x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 = {260, 32} \begin {gather*} -\frac {x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x^n)^n^(-1))^(-3),x]

[Out]

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

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 260

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

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Mathematica [A]
time = 0.06, size = 34, normalized size = 1.00 \begin {gather*} -\frac {x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.27, size = 143, normalized size = 4.21

method	result	size
risch	\(\frac {x \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (\mathrm {csgn}\left (i c \,x^{n}\right )-\mathrm {csgn}\left (i x^{n}\right )\right )}{2 n}}+2 a \right )}{2 a^{2} \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (\mathrm {csgn}\left (i c \,x^{n}\right )-\mathrm {csgn}\left (i x^{n}\right )\right )}{2 n}}+a \right )^{2}}\)	\(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*(c*x^n)^(1/n))^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (32) = 64\).
time = 0.35, size = 69, normalized size = 2.03 \begin {gather*} \frac {b c^{\left (\frac {1}{n}\right )} x {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + 2 \, a x}{2 \, {\left (a^{2} b^{2} c^{\frac {2}{n}} {\left (x^{n}\right )}^{\frac {2}{n}} + 2 \, a^{3} b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.38, size = 43, normalized size = 1.26 \begin {gather*} -\frac {1}{2 \, {\left (b^{3} c^{\frac {3}{n}} x^{2} + 2 \, a b^{2} c^{\frac {2}{n}} x + a^{2} b c^{\left (\frac {1}{n}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (27) = 54\).
time = 4.52, size = 107, normalized size = 3.15 \begin {gather*} \begin {cases} \frac {2 a x}{2 a^{4} + 4 a^{3} b \left (c x^{n}\right )^{\frac {1}{n}} + 2 a^{2} b^{2} \left (c x^{n}\right )^{\frac {2}{n}}} + \frac {b x \left (c x^{n}\right )^{\frac {1}{n}}}{2 a^{4} + 4 a^{3} b \left (c x^{n}\right )^{\frac {1}{n}} + 2 a^{2} b^{2} \left (c x^{n}\right )^{\frac {2}{n}}} & \text {for}\: a \neq 0 \\- \frac {x \left (c x^{n}\right )^{- \frac {3}{n}}}{2 b^{3}} & \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))**3,x)

[Out]

Piecewise((2*a*x/(2*a**4 + 4*a**3*b*(c*x**n)**(1/n) + 2*a**2*b**2*(c*x**n)**(2/n)) + b*x*(c*x**n)**(1/n)/(2*a*
*4 + 4*a**3*b*(c*x**n)**(1/n) + 2*a**2*b**2*(c*x**n)**(2/n)), Ne(a, 0)), (-x/(2*b**3*(c*x**n)**(3/n)), True))

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

[Out]

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

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Mupad [B]
time = 1.20, size = 36, normalized size = 1.06 \begin {gather*} \frac {x\,\left (2\,a+b\,{\left (c\,x^n\right )}^{1/n}\right )}{2\,a^2\,{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*(c*x^n)^(1/n))^3,x)

[Out]

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

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