3.3063 \(\int \left (c x^n\right )^{\frac{1}{n}} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p \, dx\)

Optimal. Leaf size=79 \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+2}}{b^2 (p+2)}-\frac{a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b^2 (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0875262, antiderivative size = 79, 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 \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+2}}{b^2 (p+2)}-\frac{a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b^2 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.9106, size = 65, normalized size = 0.82 \[ - \frac{a x \left (c x^{n}\right )^{- \frac{1}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 1}}{b^{2} \left (p + 1\right )} + \frac{x \left (c x^{n}\right )^{- \frac{1}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 2}}{b^{2} \left (p + 2\right )} \]

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))**p,x)

[Out]

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

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Mathematica [A]  time = 0.263018, size = 88, normalized size = 1.11 \[ \frac{x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p \left (a^2 \left (c x^n\right )^{-1/n} \left (\left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^{-p}-1\right )+a b p+b^2 (p+1) \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2 (p+1) (p+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.421, size = 571, normalized size = 7.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*p/b/(2+p)/(1+p)*x*(b*exp(1/2*(-I*Pi*csgn(I*x^n)*csgn(I*c)*csgn(I*c*x^n)+I*Pi*c
sgn(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+c^(1/n)/(2+p)*x^2*(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)*cs
gn(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*P
i*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))/b^2/(1+p)/(2+p)*a^2*(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*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 = 1.77188, size = 89, normalized size = 1.13 \[ \frac{{\left (b^{2} c^{\frac{2}{n}}{\left (p + 1\right )} x^{2} + a b c^{\left (\frac{1}{n}\right )} p x - a^{2}\right )}{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p} c^{-\frac{1}{n}}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^2*c^(2/n)*(p + 1)*x^2 + a*b*c^(1/n)*p*x - a^2)*(b*c^(1/n)*x + a)^p*c^(-1/n)/(
(p^2 + 3*p + 2)*b^2)

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Fricas [A]  time = 0.226544, size = 104, normalized size = 1.32 \[ \frac{{\left (a b c^{\left (\frac{1}{n}\right )} p x +{\left (b^{2} p + b^{2}\right )} c^{\frac{2}{n}} x^{2} - a^{2}\right )}{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p}}{{\left (b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}\right )} c^{\left (\frac{1}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*b*c^(1/n)*p*x + (b^2*p + b^2)*c^(2/n)*x^2 - a^2)*(b*c^(1/n)*x + a)^p/((b^2*p^
2 + 3*b^2*p + 2*b^2)*c^(1/n))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 23.3487, size = 211, normalized size = 2.67 \[ \frac{b^{2} p x^{2} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{2 \,{\rm ln}\left (c\right )}{n}\right )} + b^{2} x^{2} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{2 \,{\rm ln}\left (c\right )}{n}\right )} + a b p x e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{{\rm ln}\left (c\right )}{n}\right )} - a^{2} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right )\right )}}{b^{2} p^{2} e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + 3 \, b^{2} p e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + 2 \, b^{2} e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^2*p*x^2*e^(p*ln(b*x*e^(ln(c)/n) + a) + 2*ln(c)/n) + b^2*x^2*e^(p*ln(b*x*e^(ln
(c)/n) + a) + 2*ln(c)/n) + a*b*p*x*e^(p*ln(b*x*e^(ln(c)/n) + a) + ln(c)/n) - a^2
*e^(p*ln(b*x*e^(ln(c)/n) + a)))/(b^2*p^2*e^(ln(c)/n) + 3*b^2*p*e^(ln(c)/n) + 2*b
^2*e^(ln(c)/n))