3.3008 \(\int \frac {x}{a+b (c x^n)^{\frac {1}{n}}} \, dx\)

Optimal. Leaf size=53 \[ \frac {x^2 \left (c x^n\right )^{-1/n}}{b}-\frac {a x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {368, 43} \[ \frac {x^2 \left (c x^n\right )^{-1/n}}{b}-\frac {a x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*(c*x^n)^n^(-1)),x]

[Out]

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

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

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Mathematica [A]  time = 0.02, size = 49, normalized size = 0.92 \[ x^2 \left (c x^n\right )^{-2/n} \left (\frac {\left (c x^n\right )^{\frac {1}{n}}}{b}-\frac {a \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*(c*x^n)^n^(-1)),x]

[Out]

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

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fricas [A]  time = 0.78, size = 36, normalized size = 0.68 \[ \frac {b c^{\left (\frac {1}{n}\right )} x - a \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{b^{2} c^{\frac {2}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/((c*x^n)^(1/n)*b + a), x)

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maple [C]  time = 0.12, size = 233, normalized size = 4.40 \[ -\frac {a \,x^{2} \left (c^{-\frac {1}{n}}\right )^{2} \left (\left (x^{n}\right )^{-\frac {1}{n}}\right )^{2} {\mathrm e}^{-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{n}} \ln \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )}{b^{2}}+\frac {x^{2} c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/((c*x^n)^(1/n)*b + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x}{a+b\,{\left (c\,x^n\right )}^{1/n}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*(c*x^n)^(1/n)),x)

[Out]

int(x/(a + b*(c*x^n)^(1/n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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