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]

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

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Rubi [A]  time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, 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 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])

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 )^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*}

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Mathematica [A]  time = 0.03, 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]

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

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fricas [A]  time = 0.42, size = 92, normalized size = 1.31 \[ -\frac {4 \, b c^{\left (\frac {1}{n}\right )} x + a}{12 \, {\left (b^{6} c^{\frac {5}{n}} x^{4} + 4 \, a b^{5} c^{\frac {4}{n}} x^{3} + 6 \, a^{2} b^{4} c^{\frac {3}{n}} x^{2} + 4 \, a^{3} b^{3} c^{\frac {2}{n}} x + a^{4} b^{2} c^{\left (\frac {1}{n}\right )}\right )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x^{n}\right )^{\left (\frac {1}{n}\right )}}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^n)^(1/n)/(a+b*(c*x^n)^(1/n))^5,x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.18, size = 279, normalized size = 3.99 \[ \frac {\left (4 a b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{n}}+b^{2} c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {3 i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+6 a^{2} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}\right ) x \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}}}{12 \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )^{4} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x*(x^n)^(1/n)*c^(1/n)/a^3/(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)+csg
n(I*c*x^n))/n*csgn(I*c*x^n))+a)^4*(((x^n)^(1/n))^2*(c^(1/n))^2*b^2*exp(3/2*I*Pi*(csgn(I*c)-csgn(I*c*x^n))*(-cs
gn(I*x^n)+csgn(I*c*x^n))/n*csgn(I*c*x^n))+4*c^(1/n)*(x^n)^(1/n)*a*b*exp(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))+6*a^2*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)))

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maxima [B]  time = 0.65, size = 158, normalized size = 2.26 \[ \frac {b^{2} c^{\frac {3}{n}} x {\left (x^{n}\right )}^{\frac {3}{n}} + 4 \, a b c^{\frac {2}{n}} x {\left (x^{n}\right )}^{\frac {2}{n}} + 6 \, a^{2} c^{\left (\frac {1}{n}\right )} x {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}}{12 \, {\left (a^{3} b^{4} c^{\frac {4}{n}} {\left (x^{n}\right )}^{\frac {4}{n}} + 4 \, a^{4} b^{3} c^{\frac {3}{n}} {\left (x^{n}\right )}^{\frac {3}{n}} + 6 \, a^{5} b^{2} c^{\frac {2}{n}} {\left (x^{n}\right )}^{\frac {2}{n}} + 4 \, a^{6} b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{7}\right )}} \]

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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mupad [B]  time = 1.20, size = 208, normalized size = 2.97 \[ \frac {x}{12\,a\,b\,\left (b^3\,{\left (c\,x^n\right )}^{3/n}+a^3+3\,a\,b^2\,{\left (c\,x^n\right )}^{2/n}+3\,a^2\,b\,{\left (c\,x^n\right )}^{1/n}\right )}-\frac {x}{4\,b\,\left (b^4\,{\left (c\,x^n\right )}^{4/n}+a^4+4\,a\,b^3\,{\left (c\,x^n\right )}^{3/n}+6\,a^2\,b^2\,{\left (c\,x^n\right )}^{2/n}+4\,a^3\,b\,{\left (c\,x^n\right )}^{1/n}\right )}+\frac {x}{12\,a^3\,b\,\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}+\frac {x}{12\,a^2\,b\,\left (b^2\,{\left (c\,x^n\right )}^{2/n}+a^2+2\,a\,b\,{\left (c\,x^n\right )}^{1/n}\right )} \]

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]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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