3.30 \(\int \frac {(b x^2)^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=18 \[ \frac {b^2 \sqrt {b x^2} \log (x)}{x} \]

[Out]

b^2*ln(x)*(b*x^2)^(1/2)/x

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Rubi [A]  time = 0.00, antiderivative size = 18, 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, 29} \[ \frac {b^2 \sqrt {b x^2} \log (x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2)^(5/2)/x^6,x]

[Out]

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

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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin {align*} \int \frac {\left (b x^2\right )^{5/2}}{x^6} \, dx &=\frac {\left (b^2 \sqrt {b x^2}\right ) \int \frac {1}{x} \, dx}{x}\\ &=\frac {b^2 \sqrt {b x^2} \log (x)}{x}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.83 \[ \frac {\left (b x^2\right )^{5/2} \log (x)}{x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2)^(5/2)/x^6,x]

[Out]

((b*x^2)^(5/2)*Log[x])/x^5

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fricas [A]  time = 0.61, size = 16, normalized size = 0.89 \[ \frac {\sqrt {b x^{2}} b^{2} \log \relax (x)}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(5/2)/x^6,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 9, normalized size = 0.50 \[ b^{\frac {5}{2}} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(5/2)/x^6,x, algorithm="giac")

[Out]

b^(5/2)*log(abs(x))*sgn(x)

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maple [A]  time = 0.00, size = 14, normalized size = 0.78 \[ \frac {\left (b \,x^{2}\right )^{\frac {5}{2}} \ln \relax (x )}{x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(5/2)/x^6,x)

[Out]

(b*x^2)^(5/2)/x^5*ln(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(5/2)/x^6,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 [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {{\left (b\,x^2\right )}^{5/2}}{x^6} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(5/2)/x^6,x)

[Out]

int((b*x^2)^(5/2)/x^6, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x^{2}\right )^{\frac {5}{2}}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2)**(5/2)/x**6,x)

[Out]

Integral((b*x**2)**(5/2)/x**6, x)

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