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

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Rubi [A]  time = 0.11, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 31

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

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]

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*}

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Mathematica [A]  time = 0.06, size = 27, normalized size = 1.00 \[ \frac {2 \log \left (c \sqrt {a+b x^n}+d\right )}{b c n} \]

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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fricas [A]  time = 0.45, size = 25, normalized size = 0.93 \[ \frac {2 \, \log \left (\sqrt {b x^{n} + a} c + d\right )}{b c n} \]

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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giac [A]  time = 0.37, size = 26, normalized size = 0.96 \[ \frac {2 \, \log \left ({\left | \sqrt {b x^{n} + a} c + d \right |}\right )}{b c n} \]

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)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {x^{n -1}}{b c \,x^{n}+a c +\sqrt {b \,x^{n}+a}\, d}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.65, size = 61, normalized size = 2.26 \[ -\frac {\log \left (\frac {b x^{n} + a}{b}\right )}{b c n} + \frac {2 \, \log \left (\frac {b c x^{n} + a c + \sqrt {b x^{n} + a} d}{d}\right )}{b c n} \]

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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mupad [B]  time = 3.35, size = 25, normalized size = 0.93 \[ \frac {2\,\ln \left (d+c\,\sqrt {a+b\,x^n}\right )}{b\,c\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 33.63, size = 32, normalized size = 1.19 \[ \frac {2 \left (\begin {cases} \frac {\sqrt {a + b x^{n}}}{d} & \text {for}\: c = 0 \\\frac {\log {\left (c \sqrt {a + b x^{n}} + d \right )}}{c} & \text {otherwise} \end {cases}\right )}{b n} \]

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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