3.2470 \(\int \frac{(a+b x^n)^3}{x^2} \, dx\)

Optimal. Leaf size=66 \[ -\frac{3 a^2 b x^{n-1}}{1-n}-\frac{a^3}{x}-\frac{3 a b^2 x^{2 n-1}}{1-2 n}-\frac{b^3 x^{3 n-1}}{1-3 n} \]

[Out]

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

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Rubi [A]  time = 0.0319257, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ -\frac{3 a^2 b x^{n-1}}{1-n}-\frac{a^3}{x}-\frac{3 a b^2 x^{2 n-1}}{1-2 n}-\frac{b^3 x^{3 n-1}}{1-3 n} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^3/x^2,x]

[Out]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^n\right )^3}{x^2} \, dx &=\int \left (\frac{a^3}{x^2}+3 a^2 b x^{-2+n}+3 a b^2 x^{2 (-1+n)}+b^3 x^{-2+3 n}\right ) \, dx\\ &=-\frac{a^3}{x}-\frac{3 a^2 b x^{-1+n}}{1-n}-\frac{3 a b^2 x^{-1+2 n}}{1-2 n}-\frac{b^3 x^{-1+3 n}}{1-3 n}\\ \end{align*}

Mathematica [A]  time = 0.050773, size = 58, normalized size = 0.88 \[ \frac{\frac{3 a^2 b x^n}{n-1}-a^3+\frac{3 a b^2 x^{2 n}}{2 n-1}+\frac{b^3 x^{3 n}}{3 n-1}}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^n)^3/x^2,x]

[Out]

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

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Maple [A]  time = 0.012, size = 65, normalized size = 1. \begin{align*}{\frac{1}{x} \left ({\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{-1+3\,n}}-{a}^{3}+3\,{\frac{b{a}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{-1+n}}+3\,{\frac{{b}^{2}a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{-1+2\,n}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^n)^3/x^2,x)

[Out]

(b^3/(-1+3*n)*exp(n*ln(x))^3-a^3+3*b*a^2/(-1+n)*exp(n*ln(x))+3*b^2*a/(-1+2*n)*exp(n*ln(x))^2)/x

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^3/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.04034, size = 270, normalized size = 4.09 \begin{align*} -\frac{6 \, a^{3} n^{3} - 11 \, a^{3} n^{2} + 6 \, a^{3} n - a^{3} -{\left (2 \, b^{3} n^{2} - 3 \, b^{3} n + b^{3}\right )} x^{3 \, n} - 3 \,{\left (3 \, a b^{2} n^{2} - 4 \, a b^{2} n + a b^{2}\right )} x^{2 \, n} - 3 \,{\left (6 \, a^{2} b n^{2} - 5 \, a^{2} b n + a^{2} b\right )} x^{n}}{{\left (6 \, n^{3} - 11 \, n^{2} + 6 \, n - 1\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^3/x^2,x, algorithm="fricas")

[Out]

-(6*a^3*n^3 - 11*a^3*n^2 + 6*a^3*n - a^3 - (2*b^3*n^2 - 3*b^3*n + b^3)*x^(3*n) - 3*(3*a*b^2*n^2 - 4*a*b^2*n +
a*b^2)*x^(2*n) - 3*(6*a^2*b*n^2 - 5*a^2*b*n + a^2*b)*x^n)/((6*n^3 - 11*n^2 + 6*n - 1)*x)

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Sympy [A]  time = 1.57028, size = 508, normalized size = 7.7 \begin{align*} \begin{cases} - \frac{a^{3}}{x} - \frac{9 a^{2} b}{2 x^{\frac{2}{3}}} - \frac{9 a b^{2}}{\sqrt [3]{x}} + b^{3} \log{\left (x \right )} & \text{for}\: n = \frac{1}{3} \\- \frac{a^{3}}{x} - \frac{6 a^{2} b}{\sqrt{x}} + 3 a b^{2} \log{\left (x \right )} + 2 b^{3} \sqrt{x} & \text{for}\: n = \frac{1}{2} \\- \frac{a^{3}}{x} + 3 a^{2} b \log{\left (x \right )} + 3 a b^{2} x + \frac{b^{3} x^{2}}{2} & \text{for}\: n = 1 \\- \frac{6 a^{3} n^{3}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac{11 a^{3} n^{2}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} - \frac{6 a^{3} n}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac{a^{3}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac{18 a^{2} b n^{2} x^{n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} - \frac{15 a^{2} b n x^{n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac{3 a^{2} b x^{n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac{9 a b^{2} n^{2} x^{2 n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} - \frac{12 a b^{2} n x^{2 n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac{3 a b^{2} x^{2 n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac{2 b^{3} n^{2} x^{3 n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} - \frac{3 b^{3} n x^{3 n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac{b^{3} x^{3 n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**n)**3/x**2,x)

[Out]

Piecewise((-a**3/x - 9*a**2*b/(2*x**(2/3)) - 9*a*b**2/x**(1/3) + b**3*log(x), Eq(n, 1/3)), (-a**3/x - 6*a**2*b
/sqrt(x) + 3*a*b**2*log(x) + 2*b**3*sqrt(x), Eq(n, 1/2)), (-a**3/x + 3*a**2*b*log(x) + 3*a*b**2*x + b**3*x**2/
2, Eq(n, 1)), (-6*a**3*n**3/(6*n**3*x - 11*n**2*x + 6*n*x - x) + 11*a**3*n**2/(6*n**3*x - 11*n**2*x + 6*n*x -
x) - 6*a**3*n/(6*n**3*x - 11*n**2*x + 6*n*x - x) + a**3/(6*n**3*x - 11*n**2*x + 6*n*x - x) + 18*a**2*b*n**2*x*
*n/(6*n**3*x - 11*n**2*x + 6*n*x - x) - 15*a**2*b*n*x**n/(6*n**3*x - 11*n**2*x + 6*n*x - x) + 3*a**2*b*x**n/(6
*n**3*x - 11*n**2*x + 6*n*x - x) + 9*a*b**2*n**2*x**(2*n)/(6*n**3*x - 11*n**2*x + 6*n*x - x) - 12*a*b**2*n*x**
(2*n)/(6*n**3*x - 11*n**2*x + 6*n*x - x) + 3*a*b**2*x**(2*n)/(6*n**3*x - 11*n**2*x + 6*n*x - x) + 2*b**3*n**2*
x**(3*n)/(6*n**3*x - 11*n**2*x + 6*n*x - x) - 3*b**3*n*x**(3*n)/(6*n**3*x - 11*n**2*x + 6*n*x - x) + b**3*x**(
3*n)/(6*n**3*x - 11*n**2*x + 6*n*x - x), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{n} + a\right )}^{3}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^3/x^2,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^3/x^2, x)