3.154 \(\int \frac {1}{x^3 (b x^n)^{3/2}} \, dx\)

Optimal. Leaf size=28 \[ -\frac {2 x^{-n-2}}{b (3 n+4) \sqrt {b x^n}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \[ -\frac {2 x^{-n-2}}{b (3 n+4) \sqrt {b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(b*x^n)^(3/2)),x]

[Out]

(-2*x^(-2 - n))/(b*(4 + 3*n)*Sqrt[b*x^n])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{x^3 \left (b x^n\right )^{3/2}} \, dx &=\frac {x^{n/2} \int x^{-3-\frac {3 n}{2}} \, dx}{b \sqrt {b x^n}}\\ &=-\frac {2 x^{-2-n}}{b (4+3 n) \sqrt {b x^n}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 0.79 \[ \frac {1}{\left (-\frac {3 n}{2}-2\right ) x^2 \left (b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(b*x^n)^(3/2)),x]

[Out]

1/((-2 - (3*n)/2)*x^2*(b*x^n)^(3/2))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^n)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x^{n}\right )^{\frac {3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^n)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((b*x^n)^(3/2)*x^3), x)

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maple [A]  time = 0.00, size = 20, normalized size = 0.71 \[ -\frac {2}{\left (3 n +4\right ) \left (b \,x^{n}\right )^{\frac {3}{2}} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b*x^n)^(3/2),x)

[Out]

-2/x^2/(3*n+4)/(b*x^n)^(3/2)

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maxima [A]  time = 1.33, size = 19, normalized size = 0.68 \[ -\frac {2}{\left (b x^{n}\right )^{\frac {3}{2}} {\left (3 \, n + 4\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^n)^(3/2),x, algorithm="maxima")

[Out]

-2/((b*x^n)^(3/2)*(3*n + 4)*x^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{x^3\,{\left (b\,x^n\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(b*x^n)^(3/2)),x)

[Out]

int(1/(x^3*(b*x^n)^(3/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} - \frac {2}{3 b^{\frac {3}{2}} n x^{2} \left (x^{n}\right )^{\frac {3}{2}} + 4 b^{\frac {3}{2}} x^{2} \left (x^{n}\right )^{\frac {3}{2}}} & \text {for}\: n \neq - \frac {4}{3} \\\int \frac {1}{x^{3} \left (\frac {b}{x^{\frac {4}{3}}}\right )^{\frac {3}{2}}}\, dx & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(b*x**n)**(3/2),x)

[Out]

Piecewise((-2/(3*b**(3/2)*n*x**2*(x**n)**(3/2) + 4*b**(3/2)*x**2*(x**n)**(3/2)), Ne(n, -4/3)), (Integral(1/(x*
*3*(b/x**(4/3))**(3/2)), x), True))

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