3.3001 \(\int \frac{x}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx\)

Optimal. Leaf size=53 \[ \frac{x^2 \left (c x^n\right )^{-1/n}}{b}-\frac{a x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0493574, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x^2 \left (c x^n\right )^{-1/n}}{b}-\frac{a x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*(c*x^n)^n^(-1)),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 4.64948, size = 0, normalized size = 0. \[ \int \frac{x}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [C]  time = 0.084, size = 325, normalized size = 6.1 \[{\frac{x}{\sqrt [n]{c}b}{{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ) \pi \,{\it csgn} \left ( ic \right ) -i \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\pi +i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,n\ln \left ( x \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}}-{\frac{a}{ \left ( \sqrt [n]{c} \right ) ^{2}{b}^{2}}\ln \left ( b{{\rm e}^{{\frac{-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) +i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\pi +2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) -2\,n\ln \left ( x \right ) }{2\,n}}}}x+a \right ){{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ) \pi \,{\it csgn} \left ( ic \right ) -i \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\pi +i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,n\ln \left ( x \right ) +2\,\ln \left ({x}^{n} \right ) }{n}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*(c*x^n)^(1/n)),x)

[Out]

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

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Maxima [A]  time = 22.6675, size = 50, normalized size = 0.94 \[ \frac{c^{-\frac{1}{n}} x}{b} - \frac{a c^{-\frac{2}{n}} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.253974, size = 49, normalized size = 0.92 \[ \frac{b c^{\left (\frac{1}{n}\right )} x - a \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{2} c^{\frac{2}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((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^(2/n))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/((c*x^n)^(1/n)*b + a), x)