3.564 \(\int \frac{x^{-1+n}}{a c+b c x^n+d \sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=27 \[ \frac{2 \log \left (c \sqrt{a+b x^n}+d\right )}{b c n} \]

[Out]

(2*Log[d + c*Sqrt[a + b*x^n]])/(b*c*n)

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Rubi [A]  time = 0.10991, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {2155, 31} \[ \frac{2 \log \left (c \sqrt{a+b x^n}+d\right )}{b c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Log[d + c*Sqrt[a + b*x^n]])/(b*c*n)

Rule 2155

Int[(x_)^(m_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[1/n, Subst[Int
[x^((m + 1)/n - 1)/(c + d*x + e*Sqrt[a + b*x]), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[b*c
- a*d, 0] && IntegerQ[(m + 1)/n]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0609164, size = 27, normalized size = 1. \[ \frac{2 \log \left (c \sqrt{a+b x^n}+d\right )}{b c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Log[d + c*Sqrt[a + b*x^n]])/(b*c*n)

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Maple [F]  time = 0.013, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1+n} \left ( ac+bc{x}^{n}+d\sqrt{a+b{x}^{n}} \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.32554, size = 82, normalized size = 3.04 \begin{align*} -\frac{\log \left (\frac{b x^{n} + a}{b}\right )}{b c n} + \frac{2 \, \log \left (\frac{b c x^{n} + a c + \sqrt{b x^{n} + a} d}{d}\right )}{b c n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.86726, size = 51, normalized size = 1.89 \begin{align*} \frac{2 \, \log \left (\sqrt{b x^{n} + a} c + d\right )}{b c n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(sqrt(b*x^n + a)*c + d)/(b*c*n)

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Sympy [A]  time = 18.4952, size = 32, normalized size = 1.19 \begin{align*} \frac{2 \left (\begin{cases} \frac{\sqrt{a + b x^{n}}}{d} & \text{for}\: c = 0 \\\frac{\log{\left (c \sqrt{a + b x^{n}} + d \right )}}{c} & \text{otherwise} \end{cases}\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*Piecewise((sqrt(a + b*x**n)/d, Eq(c, 0)), (log(c*sqrt(a + b*x**n) + d)/c, True))/(b*n)

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Giac [A]  time = 1.12408, size = 55, normalized size = 2.04 \begin{align*} \frac{2 \, \log \left ({\left | \sqrt{b x^{n} + a} c + d \right |}\right )}{b c n} - \frac{2 \, \log \left ({\left | d \right |}\right )}{b c n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(abs(sqrt(b*x^n + a)*c + d))/(b*c*n) - 2*log(abs(d))/(b*c*n)