3.3015 \(\int x^3 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p \, dx\)

Optimal. Leaf size=171 \[ -\frac{a^3 x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b^4 (p+1)}+\frac{3 a^2 x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+2}}{b^4 (p+2)}-\frac{3 a x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+3}}{b^4 (p+3)}+\frac{x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+4}}{b^4 (p+4)} \]

[Out]

-((a^3*x^4*(a + b*(c*x^n)^n^(-1))^(1 + p))/(b^4*(1 + p)*(c*x^n)^(4/n))) + (3*a^2
*x^4*(a + b*(c*x^n)^n^(-1))^(2 + p))/(b^4*(2 + p)*(c*x^n)^(4/n)) - (3*a*x^4*(a +
b*(c*x^n)^n^(-1))^(3 + p))/(b^4*(3 + p)*(c*x^n)^(4/n)) + (x^4*(a + b*(c*x^n)^n^
(-1))^(4 + p))/(b^4*(4 + p)*(c*x^n)^(4/n))

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Rubi [A]  time = 0.146232, antiderivative size = 171, 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 \[ -\frac{a^3 x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b^4 (p+1)}+\frac{3 a^2 x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+2}}{b^4 (p+2)}-\frac{3 a x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+3}}{b^4 (p+3)}+\frac{x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+4}}{b^4 (p+4)} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*(c*x^n)^n^(-1))^p,x]

[Out]

-((a^3*x^4*(a + b*(c*x^n)^n^(-1))^(1 + p))/(b^4*(1 + p)*(c*x^n)^(4/n))) + (3*a^2
*x^4*(a + b*(c*x^n)^n^(-1))^(2 + p))/(b^4*(2 + p)*(c*x^n)^(4/n)) - (3*a*x^4*(a +
b*(c*x^n)^n^(-1))^(3 + p))/(b^4*(3 + p)*(c*x^n)^(4/n)) + (x^4*(a + b*(c*x^n)^n^
(-1))^(4 + p))/(b^4*(4 + p)*(c*x^n)^(4/n))

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Rubi in Sympy [A]  time = 25.4345, size = 146, normalized size = 0.85 \[ - \frac{a^{3} x^{4} \left (c x^{n}\right )^{- \frac{4}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 1}}{b^{4} \left (p + 1\right )} + \frac{3 a^{2} x^{4} \left (c x^{n}\right )^{- \frac{4}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 2}}{b^{4} \left (p + 2\right )} - \frac{3 a x^{4} \left (c x^{n}\right )^{- \frac{4}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 3}}{b^{4} \left (p + 3\right )} + \frac{x^{4} \left (c x^{n}\right )^{- \frac{4}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 4}}{b^{4} \left (p + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(a+b*(c*x**n)**(1/n))**p,x)

[Out]

-a**3*x**4*(c*x**n)**(-4/n)*(a + b*(c*x**n)**(1/n))**(p + 1)/(b**4*(p + 1)) + 3*
a**2*x**4*(c*x**n)**(-4/n)*(a + b*(c*x**n)**(1/n))**(p + 2)/(b**4*(p + 2)) - 3*a
*x**4*(c*x**n)**(-4/n)*(a + b*(c*x**n)**(1/n))**(p + 3)/(b**4*(p + 3)) + x**4*(c
*x**n)**(-4/n)*(a + b*(c*x**n)**(1/n))**(p + 4)/(b**4*(p + 4))

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Mathematica [A]  time = 0.442841, size = 263, normalized size = 1.54 \[ \frac{x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p \left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^{-p} \left (-6 a^4 \left (\left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^p-1\right )+6 a^3 b p \left (c x^n\right )^{\frac{1}{n}} \left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^p-3 a^2 b^2 p (p+1) \left (c x^n\right )^{2/n} \left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^p+b^4 \left (p^3+6 p^2+11 p+6\right ) \left (c x^n\right )^{4/n} \left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^p+a b^3 p \left (p^2+3 p+2\right ) \left (c x^n\right )^{3/n} \left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^p\right )}{b^4 (p+1) (p+2) (p+3) (p+4)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*(c*x^n)^n^(-1))^p,x]

[Out]

(x^4*(a + b*(c*x^n)^n^(-1))^p*(6*a^3*b*p*(c*x^n)^n^(-1)*(1 + (b*(c*x^n)^n^(-1))/
a)^p - 3*a^2*b^2*p*(1 + p)*(c*x^n)^(2/n)*(1 + (b*(c*x^n)^n^(-1))/a)^p + a*b^3*p*
(2 + 3*p + p^2)*(c*x^n)^(3/n)*(1 + (b*(c*x^n)^n^(-1))/a)^p + b^4*(6 + 11*p + 6*p
^2 + p^3)*(c*x^n)^(4/n)*(1 + (b*(c*x^n)^n^(-1))/a)^p - 6*a^4*(-1 + (1 + (b*(c*x^
n)^n^(-1))/a)^p)))/(b^4*(1 + p)*(2 + p)*(3 + p)*(4 + p)*(c*x^n)^(4/n)*(1 + (b*(c
*x^n)^n^(-1))/a)^p)

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Maple [C]  time = 0.593, size = 2923, normalized size = 17.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(a+b*(c*x^n)^(1/n))^p,x)

[Out]

x^4/(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*l
n(x^n)-2*n*ln(x))/n)*x+a)^p+x^3/(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*c
sgn(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)-3/(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^3+9/(1+p)^2/b/(c^(1/n))/(4+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)^(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*l
n(x^n))/n)+9/(1+p)^2/b^2/(c^(1/n))^2*a/(3+p)/(4+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)*csg
n(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
(-(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)+9/(1+p)
^2/b^2/(c^(1/n))^2*a/(3+p)/(4+p)*x^2*(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)+18/(1+p)^2/b^4/(3+p)/(c
^(1/n))^4/(2+p)*a^3/(4+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+a)^(1+p)*exp(-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)-18/(1+p)^2/b^3/(c^(1/n))^3*a^2/(3+
p)/(2+p)/(4+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*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)*c
sgn(I*c*x^n)^2-2*n*ln(x)+2*ln(x^n))/n)-18/(1+p)^2/b^3/(c^(1/n))^3*a^2/(3+p)/(2+p
)/(4+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)^(1+p)*p*exp(-3/2*(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*P
i*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)-3/(c^(1/n))/b/(1+p)*a/(3+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*cs
gn(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)-3/(c^(1/n))^2/b^2/(1+p)*a^2*p/(2+p)/(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)^p*exp(-(I*Pi*c
sgn(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)-6/(c^(1/n))^4/b^
4/(1+p)^2*a^4/(2+p)/(3+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+a)^p*exp(-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)+6/(1+p)^2/b^3/(c^(1/n))^3*a^3/(2+p)/(3
+p)*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)*c
sgn(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(-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)

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Maxima [A]  time = 1.80074, size = 188, normalized size = 1.1 \[ \frac{{\left ({\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{4} c^{\frac{4}{n}} x^{4} +{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} a b^{3} c^{\frac{3}{n}} x^{3} - 3 \,{\left (p^{2} + p\right )} a^{2} b^{2} c^{\frac{2}{n}} x^{2} + 6 \, a^{3} b c^{\left (\frac{1}{n}\right )} p x - 6 \, a^{4}\right )}{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p} c^{-\frac{4}{n}}}{{\left (p^{4} + 10 \, p^{3} + 35 \, p^{2} + 50 \, p + 24\right )} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(1/n)*b + a)^p*x^3,x, algorithm="maxima")

[Out]

((p^3 + 6*p^2 + 11*p + 6)*b^4*c^(4/n)*x^4 + (p^3 + 3*p^2 + 2*p)*a*b^3*c^(3/n)*x^
3 - 3*(p^2 + p)*a^2*b^2*c^(2/n)*x^2 + 6*a^3*b*c^(1/n)*p*x - 6*a^4)*(b*c^(1/n)*x
+ a)^p*c^(-4/n)/((p^4 + 10*p^3 + 35*p^2 + 50*p + 24)*b^4)

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Fricas [A]  time = 0.239772, size = 247, normalized size = 1.44 \[ \frac{{\left (6 \, a^{3} b c^{\left (\frac{1}{n}\right )} p x +{\left (b^{4} p^{3} + 6 \, b^{4} p^{2} + 11 \, b^{4} p + 6 \, b^{4}\right )} c^{\frac{4}{n}} x^{4} +{\left (a b^{3} p^{3} + 3 \, a b^{3} p^{2} + 2 \, a b^{3} p\right )} c^{\frac{3}{n}} x^{3} - 6 \, a^{4} - 3 \,{\left (a^{2} b^{2} p^{2} + a^{2} b^{2} p\right )} c^{\frac{2}{n}} x^{2}\right )}{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p}}{{\left (b^{4} p^{4} + 10 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 50 \, b^{4} p + 24 \, b^{4}\right )} c^{\frac{4}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(1/n)*b + a)^p*x^3,x, algorithm="fricas")

[Out]

(6*a^3*b*c^(1/n)*p*x + (b^4*p^3 + 6*b^4*p^2 + 11*b^4*p + 6*b^4)*c^(4/n)*x^4 + (a
*b^3*p^3 + 3*a*b^3*p^2 + 2*a*b^3*p)*c^(3/n)*x^3 - 6*a^4 - 3*(a^2*b^2*p^2 + a^2*b
^2*p)*c^(2/n)*x^2)*(b*c^(1/n)*x + a)^p/((b^4*p^4 + 10*b^4*p^3 + 35*b^4*p^2 + 50*
b^4*p + 24*b^4)*c^(4/n))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{3} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(a+b*(c*x**n)**(1/n))**p,x)

[Out]

Integral(x**3*(a + b*(c*x**n)**(1/n))**p, x)

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GIAC/XCAS [A]  time = 23.7683, size = 599, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(1/n)*b + a)^p*x^3,x, algorithm="giac")

[Out]

(b^4*p^3*x^4*e^(p*ln(b*x*e^(ln(c)/n) + a) + 4*ln(c)/n) + 6*b^4*p^2*x^4*e^(p*ln(b
*x*e^(ln(c)/n) + a) + 4*ln(c)/n) + a*b^3*p^3*x^3*e^(p*ln(b*x*e^(ln(c)/n) + a) +
3*ln(c)/n) + 11*b^4*p*x^4*e^(p*ln(b*x*e^(ln(c)/n) + a) + 4*ln(c)/n) + 3*a*b^3*p^
2*x^3*e^(p*ln(b*x*e^(ln(c)/n) + a) + 3*ln(c)/n) + 6*b^4*x^4*e^(p*ln(b*x*e^(ln(c)
/n) + a) + 4*ln(c)/n) + 2*a*b^3*p*x^3*e^(p*ln(b*x*e^(ln(c)/n) + a) + 3*ln(c)/n)
- 3*a^2*b^2*p^2*x^2*e^(p*ln(b*x*e^(ln(c)/n) + a) + 2*ln(c)/n) - 3*a^2*b^2*p*x^2*
e^(p*ln(b*x*e^(ln(c)/n) + a) + 2*ln(c)/n) + 6*a^3*b*p*x*e^(p*ln(b*x*e^(ln(c)/n)
+ a) + ln(c)/n) - 6*a^4*e^(p*ln(b*x*e^(ln(c)/n) + a)))/(b^4*p^4*e^(4*ln(c)/n) +
10*b^4*p^3*e^(4*ln(c)/n) + 35*b^4*p^2*e^(4*ln(c)/n) + 50*b^4*p*e^(4*ln(c)/n) + 2
4*b^4*e^(4*ln(c)/n))