3.5.94 \(\int \frac {x^{-1+3 n}}{b x^n+c x^{2 n}} \, dx\) [494]

Optimal. Leaf size=28 \[ \frac {x^n}{c n}-\frac {b \log \left (b+c x^n\right )}{c^2 n} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1598, 272, 45} \begin {gather*} \frac {x^n}{c n}-\frac {b \log \left (b+c x^n\right )}{c^2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 45

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {x^{-1+3 n}}{b x^n+c x^{2 n}} \, dx &=\int \frac {x^{-1+2 n}}{b+c x^n} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {x}{b+c x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{c}-\frac {b}{c (b+c x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {x^n}{c n}-\frac {b \log \left (b+c x^n\right )}{c^2 n}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 27, normalized size = 0.96 \begin {gather*} \frac {c x^n-b \log \left (c n \left (b+c x^n\right )\right )}{c^2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.22, size = 31, normalized size = 1.11

method	result	size
risch	\(\frac {x^{n}}{c n}-\frac {b \ln \left (x^{n}+\frac {b}{c}\right )}{c^{2} n}\)	\(31\)
norman	\(\frac {{\mathrm e}^{n \ln \left (x \right )}}{c n}-\frac {b \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}+b \right )}{c^{2} n}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+3*n)/(b*x^n+c*x^(2*n)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 32, normalized size = 1.14 \begin {gather*} \frac {x^{n}}{c n} - \frac {b \log \left (\frac {c x^{n} + b}{c}\right )}{c^{2} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+3*n)/(b*x^n+c*x^(2*n)),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.45, size = 24, normalized size = 0.86 \begin {gather*} \frac {c x^{n} - b \log \left (c x^{n} + b\right )}{c^{2} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+3*n)/(b*x^n+c*x^(2*n)),x, algorithm="fricas")

[Out]

(c*x^n - b*log(c*x^n + b))/(c^2*n)

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Sympy [A]
time = 6.82, size = 26, normalized size = 0.93 \begin {gather*} - \frac {b \left (\begin {cases} \frac {x^{n}}{b} & \text {for}\: c = 0 \\\frac {\log {\left (b + c x^{n} \right )}}{c} & \text {otherwise} \end {cases}\right )}{c n} + \frac {x^{n}}{c n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+3*n)/(b*x**n+c*x**(2*n)),x)

[Out]

-b*Piecewise((x**n/b, Eq(c, 0)), (log(b + c*x**n)/c, True))/(c*n) + x**n/(c*n)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+3*n)/(b*x^n+c*x^(2*n)),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^{3\,n-1}}{b\,x^n+c\,x^{2\,n}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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