3.39 \(\int x \log (c (a+\frac{b}{x^2})^p) \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{2} x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+\frac{b p \log \left (a x^2+b\right )}{2 a} \]

[Out]

(x^2*Log[c*(a + b/x^2)^p])/2 + (b*p*Log[b + a*x^2])/(2*a)

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Rubi [A]  time = 0.0144121, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2455, 263, 260} \[ \frac{1}{2} x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+\frac{b p \log \left (a x^2+b\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[x*Log[c*(a + b/x^2)^p],x]

[Out]

(x^2*Log[c*(a + b/x^2)^p])/2 + (b*p*Log[b + a*x^2])/(2*a)

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \, dx &=\frac{1}{2} x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+(b p) \int \frac{1}{\left (a+\frac{b}{x^2}\right ) x} \, dx\\ &=\frac{1}{2} x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+(b p) \int \frac{x}{b+a x^2} \, dx\\ &=\frac{1}{2} x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+\frac{b p \log \left (b+a x^2\right )}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.002538, size = 45, normalized size = 1.22 \[ \frac{1}{2} x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+\frac{b p \log \left (a+\frac{b}{x^2}\right )}{2 a}+\frac{b p \log (x)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Log[c*(a + b/x^2)^p],x]

[Out]

(b*p*Log[a + b/x^2])/(2*a) + (x^2*Log[c*(a + b/x^2)^p])/2 + (b*p*Log[x])/a

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Maple [F]  time = 0.245, size = 0, normalized size = 0. \begin{align*} \int x\ln \left ( c \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*ln(c*(a+b/x^2)^p),x)

[Out]

int(x*ln(c*(a+b/x^2)^p),x)

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Maxima [A]  time = 1.18933, size = 45, normalized size = 1.22 \begin{align*} \frac{1}{2} \, x^{2} \log \left ({\left (a + \frac{b}{x^{2}}\right )}^{p} c\right ) + \frac{b p \log \left (a x^{2} + b\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(c*(a+b/x^2)^p),x, algorithm="maxima")

[Out]

1/2*x^2*log((a + b/x^2)^p*c) + 1/2*b*p*log(a*x^2 + b)/a

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Fricas [A]  time = 2.20729, size = 100, normalized size = 2.7 \begin{align*} \frac{a p x^{2} \log \left (\frac{a x^{2} + b}{x^{2}}\right ) + a x^{2} \log \left (c\right ) + b p \log \left (a x^{2} + b\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(c*(a+b/x^2)^p),x, algorithm="fricas")

[Out]

1/2*(a*p*x^2*log((a*x^2 + b)/x^2) + a*x^2*log(c) + b*p*log(a*x^2 + b))/a

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Sympy [A]  time = 11.8473, size = 71, normalized size = 1.92 \begin{align*} \begin{cases} \frac{p x^{2} \log{\left (a + \frac{b}{x^{2}} \right )}}{2} + \frac{x^{2} \log{\left (c \right )}}{2} + \frac{b p \log{\left (a x^{2} + b \right )}}{2 a} & \text{for}\: a \neq 0 \\\frac{p x^{2} \log{\left (b \right )}}{2} - p x^{2} \log{\left (x \right )} + \frac{p x^{2}}{2} + \frac{x^{2} \log{\left (c \right )}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*ln(c*(a+b/x**2)**p),x)

[Out]

Piecewise((p*x**2*log(a + b/x**2)/2 + x**2*log(c)/2 + b*p*log(a*x**2 + b)/(2*a), Ne(a, 0)), (p*x**2*log(b)/2 -
p*x**2*log(x) + p*x**2/2 + x**2*log(c)/2, True))

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Giac [A]  time = 1.16985, size = 63, normalized size = 1.7 \begin{align*} \frac{1}{2} \, p x^{2} \log \left (a x^{2} + b\right ) - \frac{1}{2} \, p x^{2} \log \left (x^{2}\right ) + \frac{1}{2} \, x^{2} \log \left (c\right ) + \frac{b p \log \left (a x^{2} + b\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(c*(a+b/x^2)^p),x, algorithm="giac")

[Out]

1/2*p*x^2*log(a*x^2 + b) - 1/2*p*x^2*log(x^2) + 1/2*x^2*log(c) + 1/2*b*p*log(a*x^2 + b)/a