3.3033 \(\int \left (a+b \left (c x^n\right )^{3/n}\right )^3 \, dx\)

Optimal. Leaf size=65 \[ a^3 x+\frac{3}{4} a^2 b x \left (c x^n\right )^{3/n}+\frac{3}{7} a b^2 x \left (c x^n\right )^{6/n}+\frac{1}{10} b^3 x \left (c x^n\right )^{9/n} \]

[Out]

a^3*x + (3*a^2*b*x*(c*x^n)^(3/n))/4 + (3*a*b^2*x*(c*x^n)^(6/n))/7 + (b^3*x*(c*x^
n)^(9/n))/10

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Rubi [A]  time = 0.0517521, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ a^3 x+\frac{3}{4} a^2 b x \left (c x^n\right )^{3/n}+\frac{3}{7} a b^2 x \left (c x^n\right )^{6/n}+\frac{1}{10} b^3 x \left (c x^n\right )^{9/n} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c*x^n)^(3/n))^3,x]

[Out]

a^3*x + (3*a^2*b*x*(c*x^n)^(3/n))/4 + (3*a*b^2*x*(c*x^n)^(6/n))/7 + (b^3*x*(c*x^
n)^(9/n))/10

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a^{2} b x \left (c x^{n}\right )^{\frac{3}{n}}}{4} + \frac{3 a b^{2} x \left (c x^{n}\right )^{\frac{6}{n}}}{7} + \frac{b^{3} x \left (c x^{n}\right )^{\frac{9}{n}}}{10} + x \left (c x^{n}\right )^{- \frac{1}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} a^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**2*b*x*(c*x**n)**(3/n)/4 + 3*a*b**2*x*(c*x**n)**(6/n)/7 + b**3*x*(c*x**n)**(
9/n)/10 + x*(c*x**n)**(-1/n)*Integral(a**3, (x, (c*x**n)**(1/n)))

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Mathematica [A]  time = 0.210039, size = 65, normalized size = 1. \[ a^3 x+\frac{3}{4} a^2 b x \left (c x^n\right )^{3/n}+\frac{3}{7} a b^2 x \left (c x^n\right )^{6/n}+\frac{1}{10} b^3 x \left (c x^n\right )^{9/n} \]

Antiderivative was successfully verified.

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

[Out]

a^3*x + (3*a^2*b*x*(c*x^n)^(3/n))/4 + (3*a*b^2*x*(c*x^n)^(6/n))/7 + (b^3*x*(c*x^
n)^(9/n))/10

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int \left ( a+b \left ( c{x}^{n} \right ) ^{3\,{n}^{-1}} \right ) ^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^n)^(3/n))^3,x)

[Out]

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

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Maxima [A]  time = 1.50232, size = 72, normalized size = 1.11 \[ \frac{1}{10} \, b^{3} c^{\frac{9}{n}} x^{10} + \frac{3}{7} \, a b^{2} c^{\frac{6}{n}} x^{7} + \frac{3}{4} \, a^{2} b c^{\frac{3}{n}} x^{4} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*b^3*c^(9/n)*x^10 + 3/7*a*b^2*c^(6/n)*x^7 + 3/4*a^2*b*c^(3/n)*x^4 + a^3*x

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Fricas [A]  time = 0.249848, size = 72, normalized size = 1.11 \[ \frac{1}{10} \, b^{3} c^{\frac{9}{n}} x^{10} + \frac{3}{7} \, a b^{2} c^{\frac{6}{n}} x^{7} + \frac{3}{4} \, a^{2} b c^{\frac{3}{n}} x^{4} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*b^3*c^(9/n)*x^10 + 3/7*a*b^2*c^(6/n)*x^7 + 3/4*a^2*b*c^(3/n)*x^4 + a^3*x

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Sympy [A]  time = 2.34324, size = 66, normalized size = 1.02 \[ a^{3} x + \frac{3 a^{2} b c^{\frac{3}{n}} x \left (x^{n}\right )^{\frac{3}{n}}}{4} + \frac{3 a b^{2} c^{\frac{6}{n}} x \left (x^{n}\right )^{\frac{6}{n}}}{7} + \frac{b^{3} c^{\frac{9}{n}} x \left (x^{n}\right )^{\frac{9}{n}}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x + 3*a**2*b*c**(3/n)*x*(x**n)**(3/n)/4 + 3*a*b**2*c**(6/n)*x*(x**n)**(6/n)
/7 + b**3*c**(9/n)*x*(x**n)**(9/n)/10

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GIAC/XCAS [A]  time = 0.224421, size = 76, normalized size = 1.17 \[ \frac{1}{10} \, b^{3} x^{10} e^{\left (\frac{9 \,{\rm ln}\left (c\right )}{n}\right )} + \frac{3}{7} \, a b^{2} x^{7} e^{\left (\frac{6 \,{\rm ln}\left (c\right )}{n}\right )} + \frac{3}{4} \, a^{2} b x^{4} e^{\left (\frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*b^3*x^10*e^(9*ln(c)/n) + 3/7*a*b^2*x^7*e^(6*ln(c)/n) + 3/4*a^2*b*x^4*e^(3*l
n(c)/n) + a^3*x