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

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 = 0.188998, 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 \[ \frac{2 \log \left (c \sqrt{a+b x^n}+d\right )}{b c n} \]

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)

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Rubi in Sympy [A]  time = 10.5515, size = 20, normalized size = 0.74 \[ \frac{2 \log{\left (c \sqrt{a + b x^{n}} + d \right )}}{b c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+n)/(a*c+b*c*x**n+d*(a+b*x**n)**(1/2)),x)

[Out]

2*log(c*sqrt(a + b*x**n) + d)/(b*c*n)

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Mathematica [A]  time = 0.0308307, 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 = 0.021, size = 0, normalized size = 0. \[ \int{{x}^{-1+n} \left ( ac+bc{x}^{n}+d\sqrt{a+b{x}^{n}} \right ) ^{-1}}\, dx \]

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 [A]  time = 0.770031, size = 82, normalized size = 3.04 \[ -\frac{\log \left (\frac{b x^{n} + a}{b}\right )}{b c n} + \frac{2 \, \log \left (\frac{b c x^{n} + a c + \sqrt{b x^{n} + a} d}{d}\right )}{b c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(n - 1)/(b*c*x^n + a*c + sqrt(b*x^n + a)*d),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 = 0.285102, size = 34, normalized size = 1.26 \[ \frac{2 \, \log \left (\sqrt{b x^{n} + a} c + d\right )}{b c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(sqrt(b*x^n + a)*c + d)/(b*c*n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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GIAC/XCAS [A]  time = 0.384767, size = 55, normalized size = 2.04 \[ \frac{2 \,{\rm ln}\left ({\left | \sqrt{b x^{n} + a} c + d \right |}\right )}{b c n} - \frac{2 \,{\rm ln}\left ({\left | d \right |}\right )}{b c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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