3.3076 \(\int \frac {(c x^n)^{\frac {1}{n}}}{(a+b (c x^n)^{\frac {1}{n}})^3} \, dx\)

Optimal. Leaf size=32 \[ \frac {x \left (c x^n\right )^{\frac {1}{n}}}{2 a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {15, 368, 37} \[ \frac {x \left (c x^n\right )^{\frac {1}{n}}}{2 a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 37

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] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 368

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 32, normalized size = 1.00 \[ \frac {x \left (c x^n\right )^{\frac {1}{n}}}{2 a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 56, normalized size = 1.75 \[ -\frac {2 \, b c^{\left (\frac {1}{n}\right )} x + a}{2 \, {\left (b^{4} c^{\frac {3}{n}} x^{2} + 2 \, a b^{3} c^{\frac {2}{n}} x + a^{2} b^{2} c^{\left (\frac {1}{n}\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.07, size = 137, normalized size = 4.28 \[ \frac {x \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}}{2 \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )^{2} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*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/(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)^2

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maxima [A]  time = 0.67, size = 60, normalized size = 1.88 \[ \frac {c^{\left (\frac {1}{n}\right )} x {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}}{2 \, {\left (a b^{2} c^{\frac {2}{n}} {\left (x^{n}\right )}^{\frac {2}{n}} + 2 \, a^{2} b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.18, size = 30, normalized size = 0.94 \[ \frac {x\,{\left (c\,x^n\right )}^{1/n}}{2\,a\,{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 178.51, size = 160, normalized size = 5.00 \[ \begin {cases} \tilde {\infty } c^{- \frac {2}{n}} x \left (x^{n}\right )^{- \frac {2}{n}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {x}{b^{3} \left (2 \cdot 0^{n} \tilde {\infty }^{n} \left (0^{n}\right )^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} - \left (0^{n}\right )^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}}\right )} & \text {for}\: a = 0 \wedge c = 0^{n} \\- \frac {c^{- \frac {2}{n}} x \left (x^{n}\right )^{- \frac {2}{n}}}{b^{3}} & \text {for}\: a = 0 \\\tilde {\infty } c^{\frac {1}{n}} x \left (x^{n}\right )^{\frac {1}{n}} & \text {for}\: a = - b c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} \\\frac {c^{\frac {1}{n}} x \left (x^{n}\right )^{\frac {1}{n}}}{2 a^{3} + 4 a^{2} b c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} + 2 a b^{2} c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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