3.3077 \(\int \frac{(c x^n)^{\frac{1}{n}}}{(a+b (c x^n)^{\frac{1}{n}})^4} \, dx\)

Optimal. Leaf size=70 \[ \frac{a x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3}-\frac{x \left (c x^n\right )^{-1/n}}{2 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0347402, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {15, 368, 43} \[ \frac{a x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3}-\frac{x \left (c x^n\right )^{-1/n}}{2 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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

Mathematica [A]  time = 0.0236064, size = 48, normalized size = 0.69 \[ -\frac{x \left (c x^n\right )^{-1/n} \left (a+3 b \left (c x^n\right )^{\frac{1}{n}}\right )}{6 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*(a + 3*b*(c*x^n)^n^(-1)))/(6*b^2*(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1))^3)

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Maple [C]  time = 0.079, size = 242, normalized size = 3.5 \begin{align*}{\frac{x}{6\,{a}^{2}} \left ( b \left ( \sqrt [n]{{x}^{n}} \right ) ^{2} \left ( \sqrt [n]{c} \right ) ^{2}{{\rm e}^{{\frac{-i{\it csgn} \left ( ic{x}^{n} \right ) \pi \, \left ({\it csgn} \left ( ic \right ) -{\it csgn} \left ( ic{x}^{n} \right ) \right ) \left ({\it csgn} \left ( i{x}^{n} \right ) -{\it csgn} \left ( ic{x}^{n} \right ) \right ) }{n}}}}+3\,a\sqrt [n]{{x}^{n}}\sqrt [n]{c}{{\rm e}^{{\frac{-i/2{\it csgn} \left ( ic{x}^{n} \right ) \pi \, \left ({\it csgn} \left ( ic \right ) -{\it csgn} \left ( ic{x}^{n} \right ) \right ) \left ({\it csgn} \left ( i{x}^{n} \right ) -{\it csgn} \left ( ic{x}^{n} \right ) \right ) }{n}}}} \right ) \left ( a+b{{\rm e}^{-{\frac{i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) +i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}} \right ) ^{-3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x/(a+b*exp(-1/2*(I*Pi*csgn(I*c*x^n)^3-I*Pi*csgn(I*c*x^n)^2*csgn(I*c)-I*Pi*csgn(I*c*x^n)^2*csgn(I*x^n)+I*Pi
*csgn(I*c*x^n)*csgn(I*c)*csgn(I*x^n)-2*ln(c)-2*ln(x^n))/n))^3/a^2*(b*((x^n)^(1/n))^2*(c^(1/n))^2*exp(-I*csgn(I
*c*x^n)*Pi*(csgn(I*c)-csgn(I*c*x^n))*(csgn(I*x^n)-csgn(I*c*x^n))/n)+3*a*(x^n)^(1/n)*c^(1/n)*exp(-1/2*I*csgn(I*
c*x^n)*Pi*(csgn(I*c)-csgn(I*c*x^n))*(csgn(I*x^n)-csgn(I*c*x^n))/n))

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Maxima [A]  time = 1.0759, size = 147, normalized size = 2.1 \begin{align*} \frac{b c^{\frac{2}{n}} x{\left (x^{n}\right )}^{\frac{2}{n}} + 3 \, a c^{\left (\frac{1}{n}\right )} x{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}}{6 \,{\left (a^{2} b^{3} c^{\frac{3}{n}}{\left (x^{n}\right )}^{\frac{3}{n}} + 3 \, a^{3} b^{2} c^{\frac{2}{n}}{\left (x^{n}\right )}^{\frac{2}{n}} + 3 \, a^{4} b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b*c^(2/n)*x*(x^n)^(2/n) + 3*a*c^(1/n)*x*(x^n)^(1/n))/(a^2*b^3*c^(3/n)*(x^n)^(3/n) + 3*a^3*b^2*c^(2/n)*(x^
n)^(2/n) + 3*a^4*b*c^(1/n)*(x^n)^(1/n) + a^5)

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Fricas [A]  time = 1.65935, size = 143, normalized size = 2.04 \begin{align*} -\frac{3 \, b c^{\left (\frac{1}{n}\right )} x + a}{6 \,{\left (b^{5} c^{\frac{4}{n}} x^{3} + 3 \, a b^{4} c^{\frac{3}{n}} x^{2} + 3 \, a^{2} b^{3} c^{\frac{2}{n}} x + a^{3} b^{2} c^{\left (\frac{1}{n}\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*b*c^(1/n)*x + a)/(b^5*c^(4/n)*x^3 + 3*a*b^4*c^(3/n)*x^2 + 3*a^2*b^3*c^(2/n)*x + a^3*b^2*c^(1/n))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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