3.3067 \(\int \frac{\left (c x^n\right )^{\frac{1}{n}}}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx\)

Optimal. Leaf size=38 \[ \frac{x}{b}-\frac{a x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0555695, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{x}{b}-\frac{a x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a x \left (c x^{n}\right )^{- \frac{1}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{b^{2}} + x \left (c x^{n}\right )^{- \frac{1}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} \frac{1}{b}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*x*(c*x**n)**(-1/n)*log(a + b*(c*x**n)**(1/n))/b**2 + x*(c*x**n)**(-1/n)*Integ
ral(1/b, (x, (c*x**n)**(1/n)))

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Mathematica [A]  time = 4.18747, size = 0, normalized size = 0. \[ \int \frac{\left (c x^n\right )^{\frac{1}{n}}}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [C]  time = 0.058, size = 222, normalized size = 5.8 \[{\frac{x}{b}}-{\frac{a}{{b}^{2}\sqrt [n]{c}}\ln \left ( b{{\rm e}^{{\frac{-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) +i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) -2\,n\ln \left ( x \right ) }{2\,n}}}}x+a \right ){{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,n\ln \left ( x \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^n)^(1/n)/(a+b*(c*x^n)^(1/n)),x)

[Out]

x/b-a/b^2/(c^(1/n))*ln(b*exp(1/2*(-I*Pi*csgn(I*x^n)*csgn(I*c)*csgn(I*c*x^n)+I*Pi
*csgn(I*x^n)*csgn(I*c*x^n)^2+I*Pi*csgn(I*c)*csgn(I*c*x^n)^2-I*Pi*csgn(I*c*x^n)^3
+2*ln(c)+2*ln(x^n)-2*n*ln(x))/n)*x+a)*exp(-1/2*(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2
-I*Pi*csgn(I*x^n)*csgn(I*c)*csgn(I*c*x^n)-I*Pi*csgn(I*c*x^n)^3+I*Pi*csgn(I*c)*cs
gn(I*c*x^n)^2-2*n*ln(x)+2*ln(x^n))/n)

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Maxima [A]  time = 22.8551, size = 46, normalized size = 1.21 \[ -\frac{a c^{-\frac{1}{n}} \log \left (b^{2} c^{\left (\frac{1}{n}\right )} x + a b\right )}{b^{2}} + \frac{x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.219818, size = 46, normalized size = 1.21 \[ \frac{b c^{\left (\frac{1}{n}\right )} x - a \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{2} c^{\left (\frac{1}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{n}\right )^{\frac{1}{n}}}{a + b \left (c x^{n}\right )^{\frac{1}{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**n)**(1/n)/(a+b*(c*x**n)**(1/n)),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )}}{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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