3.3064 \(\int \left (c x^n\right )^{\frac{1}{n}} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3 \, dx\)

Optimal. Leaf size=70 \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^5}{5 b^2}-\frac{a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}{4 b^2} \]

[Out]

-(a*x*(a + b*(c*x^n)^n^(-1))^4)/(4*b^2*(c*x^n)^n^(-1)) + (x*(a + b*(c*x^n)^n^(-1
))^5)/(5*b^2*(c*x^n)^n^(-1))

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Rubi [A]  time = 0.0689983, antiderivative size = 70, 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 \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^5}{5 b^2}-\frac{a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a*x*(a + b*(c*x^n)^n^(-1))^4)/(4*b^2*(c*x^n)^n^(-1)) + (x*(a + b*(c*x^n)^n^(-1
))^5)/(5*b^2*(c*x^n)^n^(-1))

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Rubi in Sympy [A]  time = 18.4884, size = 58, normalized size = 0.83 \[ - \frac{a x \left (c x^{n}\right )^{- \frac{1}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{4}}{4 b^{2}} + \frac{x \left (c x^{n}\right )^{- \frac{1}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{5}}{5 b^{2}} \]

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))**3,x)

[Out]

-a*x*(c*x**n)**(-1/n)*(a + b*(c*x**n)**(1/n))**4/(4*b**2) + x*(c*x**n)**(-1/n)*(
a + b*(c*x**n)**(1/n))**5/(5*b**2)

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Mathematica [A]  time = 0.209742, size = 68, normalized size = 0.97 \[ \frac{1}{20} x \left (c x^n\right )^{\frac{1}{n}} \left (10 a^3+20 a^2 b \left (c x^n\right )^{\frac{1}{n}}+15 a b^2 \left (c x^n\right )^{2/n}+4 b^3 \left (c x^n\right )^{3/n}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(c*x^n)^n^(-1)*(10*a^3 + 20*a^2*b*(c*x^n)^n^(-1) + 15*a*b^2*(c*x^n)^(2/n) + 4
*b^3*(c*x^n)^(3/n)))/20

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int \sqrt [n]{c{x}^{n}} \left ( a+b\sqrt [n]{c{x}^{n}} \right ) ^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^3,x)

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Maxima [A]  time = 1.51231, size = 81, normalized size = 1.16 \[ \frac{1}{5} \, b^{3} c^{\frac{4}{n}} x^{5} + \frac{3}{4} \, a b^{2} c^{\frac{3}{n}} x^{4} + a^{2} b c^{\frac{2}{n}} x^{3} + \frac{1}{2} \, a^{3} c^{\left (\frac{1}{n}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^3*c^(4/n)*x^5 + 3/4*a*b^2*c^(3/n)*x^4 + a^2*b*c^(2/n)*x^3 + 1/2*a^3*c^(1/n
)*x^2

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Fricas [A]  time = 0.217041, size = 81, normalized size = 1.16 \[ \frac{1}{5} \, b^{3} c^{\frac{4}{n}} x^{5} + \frac{3}{4} \, a b^{2} c^{\frac{3}{n}} x^{4} + a^{2} b c^{\frac{2}{n}} x^{3} + \frac{1}{2} \, a^{3} c^{\left (\frac{1}{n}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^3*c^(4/n)*x^5 + 3/4*a*b^2*c^(3/n)*x^4 + a^2*b*c^(2/n)*x^3 + 1/2*a^3*c^(1/n
)*x^2

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Sympy [A]  time = 4.4127, size = 76, normalized size = 1.09 \[ \frac{a^{3} c^{\frac{1}{n}} x \left (x^{n}\right )^{\frac{1}{n}}}{2} + a^{2} b c^{\frac{2}{n}} x \left (x^{n}\right )^{\frac{2}{n}} + \frac{3 a b^{2} c^{\frac{3}{n}} x \left (x^{n}\right )^{\frac{3}{n}}}{4} + \frac{b^{3} c^{\frac{4}{n}} x \left (x^{n}\right )^{\frac{4}{n}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*c**(1/n)*x*(x**n)**(1/n)/2 + a**2*b*c**(2/n)*x*(x**n)**(2/n) + 3*a*b**2*c**
(3/n)*x*(x**n)**(3/n)/4 + b**3*c**(4/n)*x*(x**n)**(4/n)/5

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GIAC/XCAS [A]  time = 0.245762, size = 88, normalized size = 1.26 \[ \frac{1}{5} \, b^{3} x^{5} e^{\left (\frac{4 \,{\rm ln}\left (c\right )}{n}\right )} + \frac{3}{4} \, a b^{2} x^{4} e^{\left (\frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + a^{2} b x^{3} e^{\left (\frac{2 \,{\rm ln}\left (c\right )}{n}\right )} + \frac{1}{2} \, a^{3} x^{2} e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^3*x^5*e^(4*ln(c)/n) + 3/4*a*b^2*x^4*e^(3*ln(c)/n) + a^2*b*x^3*e^(2*ln(c)/n
) + 1/2*a^3*x^2*e^(ln(c)/n)