3.3018 \(\int \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0278667, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c*x^n)^n^(-1))^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.14208, size = 29, normalized size = 0.76 \[ \frac{x \left (c x^{n}\right )^{- \frac{1}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 1}}{b \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(c*x**n)**(-1/n)*(a + b*(c*x**n)**(1/n))**(p + 1)/(b*(p + 1))

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Mathematica [A]  time = 0.192924, size = 64, normalized size = 1.68 \[ \frac{x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p \left (\frac{a \left (c x^n\right )^{-1/n} \left (1-\left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^{-p}\right )}{b}+1\right )}{p+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.192, size = 336, normalized size = 8.8 \[{\frac{x}{1+p} \left ( b{{\rm e}^{{\frac{-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) +i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) -2\,n\ln \left ( x \right ) }{2\,n}}}}x+a \right ) ^{p}}+{\frac{a}{\sqrt [n]{c}b \left ( 1+p \right ) } \left ( b{{\rm e}^{{\frac{-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) +i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) -2\,n\ln \left ( x \right ) }{2\,n}}}}x+a \right ) ^{p}{{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,n\ln \left ( x \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^n)^(1/n))^p,x)

[Out]

1/(1+p)*x*(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*l
n(x^n)-2*n*ln(x))/n)*x+a)^p+1/(c^(1/n))/b/(1+p)*a*(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)^p*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*csg
n(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 = 1.84947, size = 51, normalized size = 1.34 \[ \frac{{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p} c^{-\frac{1}{n}}}{b{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.263128, size = 50, normalized size = 1.32 \[ \frac{{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p}}{{\left (b p + b\right )} c^{\left (\frac{1}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 11.9381, size = 92, normalized size = 2.42 \[ \frac{b x e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{{\rm ln}\left (c\right )}{n}\right )} + a e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right )\right )}}{b p e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + b e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x*e^(p*ln(b*x*e^(ln(c)/n) + a) + ln(c)/n) + a*e^(p*ln(b*x*e^(ln(c)/n) + a)))/
(b*p*e^(ln(c)/n) + b*e^(ln(c)/n))