3.187 \(\int \log (c (a+b x^2)^p) \, dx\)

Optimal. Leaf size=45 \[ x \log \left (c \left (a+b x^2\right )^p\right )+\frac{2 \sqrt{a} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-2 p x \]

[Out]

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

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Rubi [A]  time = 0.0190329, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2448, 321, 205} \[ x \log \left (c \left (a+b x^2\right )^p\right )+\frac{2 \sqrt{a} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-2 p x \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x^2)^p],x]

[Out]

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

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0126797, size = 45, normalized size = 1. \[ x \log \left (c \left (a+b x^2\right )^p\right )+\frac{2 \sqrt{a} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-2 p x \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x^2)^p],x]

[Out]

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

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Maple [A]  time = 0.073, size = 38, normalized size = 0.8 \begin{align*} x\ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) -2\,px+2\,{\frac{ap}{\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.3912, size = 250, normalized size = 5.56 \begin{align*} \left [p x \log \left (b x^{2} + a\right ) + p \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 2 \, p x + x \log \left (c\right ), p x \log \left (b x^{2} + a\right ) + 2 \, p \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - 2 \, p x + x \log \left (c\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[p*x*log(b*x^2 + a) + p*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 2*p*x + x*log(c), p*x*log
(b*x^2 + a) + 2*p*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - 2*p*x + x*log(c)]

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Sympy [A]  time = 10.002, size = 90, normalized size = 2. \begin{align*} \begin{cases} \frac{i \sqrt{a} p \log{\left (a + b x^{2} \right )}}{b \sqrt{\frac{1}{b}}} - \frac{2 i \sqrt{a} p \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{b \sqrt{\frac{1}{b}}} + p x \log{\left (a + b x^{2} \right )} - 2 p x + x \log{\left (c \right )} & \text{for}\: b \neq 0 \\x \log{\left (a^{p} c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((I*sqrt(a)*p*log(a + b*x**2)/(b*sqrt(1/b)) - 2*I*sqrt(a)*p*log(-I*sqrt(a)*sqrt(1/b) + x)/(b*sqrt(1/b
)) + p*x*log(a + b*x**2) - 2*p*x + x*log(c), Ne(b, 0)), (x*log(a**p*c), True))

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Giac [A]  time = 1.27252, size = 55, normalized size = 1.22 \begin{align*} p x \log \left (b x^{2} + a\right ) + \frac{2 \, a p \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b}} -{\left (2 \, p - \log \left (c\right )\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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