3.3078 \(\int \frac{(c x^n)^{\frac{1}{n}}}{(a+b (c x^n)^{\frac{1}{n}})^5} \, dx\)

Optimal. Leaf size=70 \[ \frac{a x \left (c x^n\right )^{-1/n}}{4 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}-\frac{x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.034735, 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} \[ \frac{a x \left (c x^n\right )^{-1/n}}{4 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}-\frac{x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(c*x^n)^n^(-1)/(a + b*(c*x^n)^n^(-1))^5,x]

[Out]

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

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 )^5} \, 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 )^5} \, dx}{x}\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x}{(a+b x)^5} \, 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)^5}+\frac{1}{b (a+b x)^4}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{a x \left (c x^n\right )^{-1/n}}{4 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}-\frac{x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3}\\ \end{align*}

Mathematica [A]  time = 0.0248681, size = 48, normalized size = 0.69 \[ -\frac{x \left (c x^n\right )^{-1/n} \left (a+4 b \left (c x^n\right )^{\frac{1}{n}}\right )}{12 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x^n)^n^(-1)/(a + b*(c*x^n)^n^(-1))^5,x]

[Out]

-(x*(a + 4*b*(c*x^n)^n^(-1)))/(12*b^2*(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1))^4)

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Maple [C]  time = 0.059, size = 316, normalized size = 4.5 \begin{align*}{\frac{x}{12\,{a}^{3}} \left ({b}^{2} \left ( \sqrt [n]{{x}^{n}} \right ) ^{3} \left ( \sqrt [n]{c} \right ) ^{3}{{\rm e}^{{\frac{-{\frac{3\,i}{2}}\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \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}}}}+4\,ba \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}}}}+6\,{a}^{2}\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 ) ^{-4}} \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))^5,x)

[Out]

1/12*x/a^3/(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))^4*(b^2*((x^n)^(1/n))^3*(c^(1/n))^3*exp(-3/2*I
*Pi*csgn(I*c*x^n)*(csgn(I*c)-csgn(I*c*x^n))*(csgn(I*x^n)-csgn(I*c*x^n))/n)+4*b*a*((x^n)^(1/n))^2*(c^(1/n))^2*e
xp(-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)+6*a^2*(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 [B]  time = 1.08316, size = 213, normalized size = 3.04 \begin{align*} \frac{b^{2} c^{\frac{3}{n}} x{\left (x^{n}\right )}^{\frac{3}{n}} + 4 \, a b c^{\frac{2}{n}} x{\left (x^{n}\right )}^{\frac{2}{n}} + 6 \, a^{2} c^{\left (\frac{1}{n}\right )} x{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}}{12 \,{\left (a^{3} b^{4} c^{\frac{4}{n}}{\left (x^{n}\right )}^{\frac{4}{n}} + 4 \, a^{4} b^{3} c^{\frac{3}{n}}{\left (x^{n}\right )}^{\frac{3}{n}} + 6 \, a^{5} b^{2} c^{\frac{2}{n}}{\left (x^{n}\right )}^{\frac{2}{n}} + 4 \, a^{6} b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{7}\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))^5,x, algorithm="maxima")

[Out]

1/12*(b^2*c^(3/n)*x*(x^n)^(3/n) + 4*a*b*c^(2/n)*x*(x^n)^(2/n) + 6*a^2*c^(1/n)*x*(x^n)^(1/n))/(a^3*b^4*c^(4/n)*
(x^n)^(4/n) + 4*a^4*b^3*c^(3/n)*(x^n)^(3/n) + 6*a^5*b^2*c^(2/n)*(x^n)^(2/n) + 4*a^6*b*c^(1/n)*(x^n)^(1/n) + a^
7)

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Fricas [A]  time = 1.66837, size = 177, normalized size = 2.53 \begin{align*} -\frac{4 \, b c^{\left (\frac{1}{n}\right )} x + a}{12 \,{\left (b^{6} c^{\frac{5}{n}} x^{4} + 4 \, a b^{5} c^{\frac{4}{n}} x^{3} + 6 \, a^{2} b^{4} c^{\frac{3}{n}} x^{2} + 4 \, a^{3} b^{3} c^{\frac{2}{n}} x + a^{4} 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))^5,x, algorithm="fricas")

[Out]

-1/12*(4*b*c^(1/n)*x + a)/(b^6*c^(5/n)*x^4 + 4*a*b^5*c^(4/n)*x^3 + 6*a^2*b^4*c^(3/n)*x^2 + 4*a^3*b^3*c^(2/n)*x
+ a^4*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))**5,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 )}^{5}}\,{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))^5,x, algorithm="giac")

[Out]

integrate((c*x^n)^(1/n)/((c*x^n)^(1/n)*b + a)^5, x)