3.1.31 \(\int \frac {(b x^2)^{5/2}}{x^7} \, dx\)

Optimal. Leaf size=17 \[ -\frac {b^2 \sqrt {b x^2}}{x^2} \]

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Rubi [A]  time = 0.00, antiderivative size = 17, 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} \begin {gather*} -\frac {b^2 \sqrt {b x^2}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2)^(5/2)/x^7,x]

[Out]

-((b^2*Sqrt[b*x^2])/x^2)

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 {\left (b x^2\right )^{5/2}}{x^7} \, dx &=\frac {\left (b^2 \sqrt {b x^2}\right ) \int \frac {1}{x^2} \, dx}{x}\\ &=-\frac {b^2 \sqrt {b x^2}}{x^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 0.82 \begin {gather*} -\frac {\left (b x^2\right )^{5/2}}{x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2)^(5/2)/x^7,x]

[Out]

-((b*x^2)^(5/2)/x^6)

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IntegrateAlgebraic [A]  time = 0.02, size = 17, normalized size = 1.00 \begin {gather*} -\frac {b^2 \sqrt {b x^2}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b*x^2)^(5/2)/x^7,x]

[Out]

-((b^2*Sqrt[b*x^2])/x^2)

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fricas [A]  time = 0.83, size = 15, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {b x^{2}} b^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(5/2)/x^7,x, algorithm="fricas")

[Out]

-sqrt(b*x^2)*b^2/x^2

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giac [A]  time = 0.15, size = 10, normalized size = 0.59 \begin {gather*} -\frac {b^{\frac {5}{2}} \mathrm {sgn}\relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(5/2)/x^7,x, algorithm="giac")

[Out]

-b^(5/2)*sgn(x)/x

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maple [A]  time = 0.00, size = 13, normalized size = 0.76 \begin {gather*} -\frac {\left (b \,x^{2}\right )^{\frac {5}{2}}}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(5/2)/x^7,x)

[Out]

-(b*x^2)^(5/2)/x^6

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(5/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 0.93, size = 10, normalized size = 0.59 \begin {gather*} -\frac {b^{5/2}}{\sqrt {x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(5/2)/x^7,x)

[Out]

-b^(5/2)/(x^2)^(1/2)

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sympy [A]  time = 1.87, size = 15, normalized size = 0.88 \begin {gather*} - \frac {b^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2)**(5/2)/x**7,x)

[Out]

-b**(5/2)*(x**2)**(5/2)/x**6

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