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

Optimal. Leaf size=87 \[ \frac {b^2 \log (x) \left (c x^n\right )^{2/n}}{a^3 x^2}-\frac {b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^3 x^2}+\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a^2 x^2}-\frac {1}{2 a x^2} \]

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Rubi [A]  time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {368, 44} \begin {gather*} \frac {b^2 \log (x) \left (c x^n\right )^{2/n}}{a^3 x^2}-\frac {b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^3 x^2}+\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a^2 x^2}-\frac {1}{2 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*(c*x^n)^n^(-1))),x]

[Out]

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

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[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 {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx &=\frac {\left (c x^n\right )^{2/n} \operatorname {Subst}\left (\int \frac {1}{x^3 (a+b x)} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{x^2}\\ &=\frac {\left (c x^n\right )^{2/n} \operatorname {Subst}\left (\int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{x^2}\\ &=-\frac {1}{2 a x^2}+\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a^2 x^2}+\frac {b^2 \left (c x^n\right )^{2/n} \log (x)}{a^3 x^2}-\frac {b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^3 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 75, normalized size = 0.86 \begin {gather*} -\frac {a^2+2 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )-2 a b \left (c x^n\right )^{\frac {1}{n}}-2 b^2 \log (x) \left (c x^n\right )^{2/n}}{2 a^3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.99, size = 65, normalized size = 0.75 \begin {gather*} -\frac {2 \, b^{2} c^{\frac {2}{n}} x^{2} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right ) - 2 \, b^{2} c^{\frac {2}{n}} x^{2} \log \relax (x) - 2 \, a b c^{\left (\frac {1}{n}\right )} x + a^{2}}{2 \, a^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.18, size = 302, normalized size = 3.47 \begin {gather*} \frac {b^{2} c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{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}} \ln \relax (x )}{a^{3} x^{2}}-\frac {b^{2} c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{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}} \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 )}{a^{3} x^{2}}+\frac {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^{2} x^{2}}-\frac {1}{2 a \,x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^3\,\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*(c*x^n)^(1/n))),x)

[Out]

int(1/(x^3*(a + b*(c*x^n)^(1/n))), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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