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

________________________________________________________________________________________

Rubi [A]  time = 0.0016462, antiderivative size = 18, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0018206, 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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 14, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( x \right ) }{{x}^{5}} \left ( b{x}^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

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)

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Fricas [A]  time = 1.20085, size = 35, normalized size = 1.94 \begin{align*} \frac{\sqrt{b x^{2}} b^{2} \log \left (x\right )}{x} \end{align*}

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b x^{2}\right )^{\frac{5}{2}}}{x^{6}}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.73555, size = 12, normalized size = 0.67 \begin{align*} b^{\frac{5}{2}} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (x\right ) \end{align*}

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)