3.3016 \(\int x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p \, dx\)

Optimal. Leaf size=126 \[ \frac{a^2 x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b^3 (p+1)}-\frac{2 a x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+2}}{b^3 (p+2)}+\frac{x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+3}}{b^3 (p+3)} \]

[Out]

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

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Rubi [A]  time = 0.104915, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{a^2 x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b^3 (p+1)}-\frac{2 a x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+2}}{b^3 (p+2)}+\frac{x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+3}}{b^3 (p+3)} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*(c*x^n)^n^(-1))^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 18.5395, size = 107, normalized size = 0.85 \[ \frac{a^{2} x^{3} \left (c x^{n}\right )^{- \frac{3}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 1}}{b^{3} \left (p + 1\right )} - \frac{2 a x^{3} \left (c x^{n}\right )^{- \frac{3}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 2}}{b^{3} \left (p + 2\right )} + \frac{x^{3} \left (c x^{n}\right )^{- \frac{3}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 3}}{b^{3} \left (p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.355206, size = 207, normalized size = 1.64 \[ \frac{x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p \left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^{-p} \left (2 a^3 \left (\left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^p-1\right )-2 a^2 b p \left (c x^n\right )^{\frac{1}{n}} \left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^p+b^3 \left (p^2+3 p+2\right ) \left (c x^n\right )^{3/n} \left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^p+a b^2 p (p+1) \left (c x^n\right )^{2/n} \left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^p\right )}{b^3 (p+1) (p+2) (p+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*(c*x^n)^n^(-1))^p,x]

[Out]

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

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Maple [C]  time = 140.953, size = 2258, normalized size = 17.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a+b*(c*x^n)^(1/n))^p,x)

[Out]

1/(1+p)*x^3*(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+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*x^2*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)-2/(1+p
)^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)^(1+p)/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^2+4/(1+p)^2/b/(c^(1/n))/(3+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)*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)^(1+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)+4/(1+p)^2/b^2/(c^(1/n))^2*a/(2+p)/(3+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)*c
sgn(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)^(1+p)*e
xp(-(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)+4/(1+
p)^2/b^2/(c^(1/n))^2*a/(2+p)/(3+p)*x*(b*exp(1/2*(-I*Pi*csgn(I*x^n)*csgn(I*c)*csg
n(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)^(1+p)*p*exp(-(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)-4/(1+p)^2/b^3/(2+p)/(c^
(1/n))^3*a^2/(3+p)*(b*exp(1/2*(-I*Pi*csgn(I*x^n)*csgn(I*c)*csgn(I*c*x^n)+I*Pi*cs
gn(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)^(1+p)*exp(-3/2*(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)-2/(c^(1/n))/b/(1+p)^2*a*x^3/((x^n)^(1/n
))*(b*c^(1/n)*(x^n)^(1/n)*exp(1/2*I*Pi*csgn(I*c*x^n)*(csgn(I*c*x^n)-csgn(I*c))*(
-csgn(I*c*x^n)+csgn(I*x^n))/n)+a)^p*exp(-1/2*I*Pi*csgn(I*c*x^n)*(csgn(I*c*x^n)-c
sgn(I*c))*(-csgn(I*c*x^n)+csgn(I*x^n))/n)-2/(c^(1/n))^2/b^2/(1+p)^2*a^2*x^3/((x^
n)^(1/n))^2*(b*c^(1/n)*(x^n)^(1/n)*exp(1/2*I*Pi*csgn(I*c*x^n)*(csgn(I*c*x^n)-csg
n(I*c))*(-csgn(I*c*x^n)+csgn(I*x^n))/n)+a)^p*exp(-I*Pi*csgn(I*c*x^n)*(csgn(I*c*x
^n)-csgn(I*c))*(-csgn(I*c*x^n)+csgn(I*x^n))/n)+2/(c^(1/n))/b/(1+p)^2*a*x^3/((x^n
)^(1/n))/(2+p)*(b*c^(1/n)*(x^n)^(1/n)*exp(1/2*I*Pi*csgn(I*c*x^n)*(csgn(I*c*x^n)-
csgn(I*c))*(-csgn(I*c*x^n)+csgn(I*x^n))/n)+a)^p*exp(-1/2*I*Pi*csgn(I*c*x^n)*(csg
n(I*c*x^n)-csgn(I*c))*(-csgn(I*c*x^n)+csgn(I*x^n))/n)+4/(c^(1/n))^2/b^2/(1+p)^2*
a^2*x^3/((x^n)^(1/n))^2/(2+p)*(b*c^(1/n)*(x^n)^(1/n)*exp(1/2*I*Pi*csgn(I*c*x^n)*
(csgn(I*c*x^n)-csgn(I*c))*(-csgn(I*c*x^n)+csgn(I*x^n))/n)+a)^p*exp(-I*Pi*csgn(I*
c*x^n)*(csgn(I*c*x^n)-csgn(I*c))*(-csgn(I*c*x^n)+csgn(I*x^n))/n)+2/(c^(1/n))^3/b
^3/(1+p)^2*a^3*x^3/((x^n)^(1/n))^3/(2+p)*(b*c^(1/n)*(x^n)^(1/n)*exp(1/2*I*Pi*csg
n(I*c*x^n)*(csgn(I*c*x^n)-csgn(I*c))*(-csgn(I*c*x^n)+csgn(I*x^n))/n)+a)^p*exp(-3
/2*I*Pi*csgn(I*c*x^n)*(csgn(I*c*x^n)-csgn(I*c))*(-csgn(I*c*x^n)+csgn(I*x^n))/n)

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Maxima [A]  time = 7.56487, size = 134, normalized size = 1.06 \[ \frac{{\left ({\left (p^{2} + 3 \, p + 2\right )} b^{3} c^{\frac{3}{n}} x^{3} +{\left (p^{2} + p\right )} a b^{2} c^{\frac{2}{n}} x^{2} - 2 \, a^{2} b c^{\left (\frac{1}{n}\right )} p x + 2 \, a^{3}\right )}{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p} c^{-\frac{3}{n}}}{{\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(1/n)*b + a)^p*x^2,x, algorithm="maxima")

[Out]

((p^2 + 3*p + 2)*b^3*c^(3/n)*x^3 + (p^2 + p)*a*b^2*c^(2/n)*x^2 - 2*a^2*b*c^(1/n)
*p*x + 2*a^3)*(b*c^(1/n)*x + a)^p*c^(-3/n)/((p^3 + 6*p^2 + 11*p + 6)*b^3)

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Fricas [A]  time = 0.239885, size = 174, normalized size = 1.38 \[ -\frac{{\left (2 \, a^{2} b c^{\left (\frac{1}{n}\right )} p x -{\left (b^{3} p^{2} + 3 \, b^{3} p + 2 \, b^{3}\right )} c^{\frac{3}{n}} x^{3} -{\left (a b^{2} p^{2} + a b^{2} p\right )} c^{\frac{2}{n}} x^{2} - 2 \, a^{3}\right )}{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p}}{{\left (b^{3} p^{3} + 6 \, b^{3} p^{2} + 11 \, b^{3} p + 6 \, b^{3}\right )} c^{\frac{3}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(1/n)*b + a)^p*x^2,x, algorithm="fricas")

[Out]

-(2*a^2*b*c^(1/n)*p*x - (b^3*p^2 + 3*b^3*p + 2*b^3)*c^(3/n)*x^3 - (a*b^2*p^2 + a
*b^2*p)*c^(2/n)*x^2 - 2*a^3)*(b*c^(1/n)*x + a)^p/((b^3*p^3 + 6*b^3*p^2 + 11*b^3*
p + 6*b^3)*c^(3/n))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(a+b*(c*x**n)**(1/n))**p,x)

[Out]

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

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GIAC/XCAS [A]  time = 23.5303, size = 381, normalized size = 3.02 \[ \frac{b^{3} p^{2} x^{3} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + 3 \, b^{3} p x^{3} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + a b^{2} p^{2} x^{2} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{2 \,{\rm ln}\left (c\right )}{n}\right )} + 2 \, b^{3} x^{3} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + a b^{2} p x^{2} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{2 \,{\rm ln}\left (c\right )}{n}\right )} - 2 \, a^{2} b p x e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{{\rm ln}\left (c\right )}{n}\right )} + 2 \, a^{3} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right )\right )}}{b^{3} p^{3} e^{\left (\frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + 6 \, b^{3} p^{2} e^{\left (\frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + 11 \, b^{3} p e^{\left (\frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + 6 \, b^{3} e^{\left (\frac{3 \,{\rm ln}\left (c\right )}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(1/n)*b + a)^p*x^2,x, algorithm="giac")

[Out]

(b^3*p^2*x^3*e^(p*ln(b*x*e^(ln(c)/n) + a) + 3*ln(c)/n) + 3*b^3*p*x^3*e^(p*ln(b*x
*e^(ln(c)/n) + a) + 3*ln(c)/n) + a*b^2*p^2*x^2*e^(p*ln(b*x*e^(ln(c)/n) + a) + 2*
ln(c)/n) + 2*b^3*x^3*e^(p*ln(b*x*e^(ln(c)/n) + a) + 3*ln(c)/n) + a*b^2*p*x^2*e^(
p*ln(b*x*e^(ln(c)/n) + a) + 2*ln(c)/n) - 2*a^2*b*p*x*e^(p*ln(b*x*e^(ln(c)/n) + a
) + ln(c)/n) + 2*a^3*e^(p*ln(b*x*e^(ln(c)/n) + a)))/(b^3*p^3*e^(3*ln(c)/n) + 6*b
^3*p^2*e^(3*ln(c)/n) + 11*b^3*p*e^(3*ln(c)/n) + 6*b^3*e^(3*ln(c)/n))