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

Optimal. Leaf size=20 \[ b x^{-n/2} \log (x) \sqrt{b x^n} \]

[Out]

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

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Rubi [A]  time = 0.0020854, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {15, 29} \[ b x^{-n/2} \log (x) \sqrt{b x^n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - (3*n)/2)*(b*x^n)^(3/2),x]

[Out]

(b*Sqrt[b*x^n]*Log[x])/x^(n/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 29

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

Rubi steps

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

Mathematica [A]  time = 0.0043352, size = 19, normalized size = 0.95 \[ x^{-3 n/2} \log (x) \left (b x^n\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*x^n)^(3/2)*Log[x])/x^((3*n)/2)

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Maple [A]  time = 0.03, size = 23, normalized size = 1.2 \begin{align*}{b\ln \left ( x \right ) \sqrt{b \left ({x}^{{\frac{n}{2}}} \right ) ^{2}} \left ({x}^{{\frac{n}{2}}} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b x^{n}\right )^{\frac{3}{2}} x^{-\frac{3}{2} \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 37.6017, size = 300, normalized size = 15. \begin{align*} \left [\frac{1}{2} \, b^{\frac{3}{2}} \log \left (\frac{b x^{4} +{\left (x^{4} - 1\right )} \sqrt{b} x^{\frac{1}{3}} x^{-\frac{1}{2} \, n - \frac{1}{3}} \sqrt{\frac{b}{x^{\frac{2}{3}} x^{-n - \frac{2}{3}}}} + b}{x^{2}}\right ), -\sqrt{-b} b \arctan \left (\frac{{\left (x^{2} + 1\right )} \sqrt{-b} x^{\frac{1}{3}} x^{-\frac{1}{2} \, n - \frac{1}{3}} \sqrt{\frac{b}{x^{\frac{2}{3}} x^{-n - \frac{2}{3}}}}}{b x^{2} - b}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*b^(3/2)*log((b*x^4 + (x^4 - 1)*sqrt(b)*x^(1/3)*x^(-1/2*n - 1/3)*sqrt(b/(x^(2/3)*x^(-n - 2/3))) + b)/x^2),
-sqrt(-b)*b*arctan((x^2 + 1)*sqrt(-b)*x^(1/3)*x^(-1/2*n - 1/3)*sqrt(b/(x^(2/3)*x^(-n - 2/3)))/(b*x^2 - b))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.26095, size = 8, normalized size = 0.4 \begin{align*} b^{\frac{3}{2}} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^(3/2)*log(x)