3.2.68 \(\int x^{-1-\frac {3 n}{2}} (b x^n)^{3/2} \, dx\) [168]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, 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} \begin {gather*} b x^{-n/2} \log (x) \sqrt {b x^n} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 19, normalized size = 0.95 \begin {gather*} x^{-3 n/2} \left (b x^n\right )^{3/2} \log (x) \end {gather*}

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 = 0.04, size = 23, normalized size = 1.15

method	result	size
risch	\(b \,x^{-\frac {n}{2}} \sqrt {b \,x^{n}}\, \ln \left (x \right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (18) = 36\).
time = 2.74, size = 110, normalized size = 5.50 \begin {gather*} \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 {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{- \frac {3 n}{2} - 1} \left (b x^{n}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {{\left (b\,x^n\right )}^{3/2}}{x^{\frac {3\,n}{2}+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^(3/2)/x^((3*n)/2 + 1),x)

[Out]

int((b*x^n)^(3/2)/x^((3*n)/2 + 1), x)

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