3.320 \(\int \frac{(f+g x^2) \log (c (d+e x^2)^p)}{x^2} \, dx\)

Optimal. Leaf size=72 \[ -\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{x}+g x \log \left (c \left (d+e x^2\right )^p\right )+\frac{2 p (d g+e f) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e}}-2 g p x \]

[Out]

-2*g*p*x + (2*(e*f + d*g)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*Sqrt[e]) - (f*Log[c*(d + e*x^2)^p])/x + g*x*
Log[c*(d + e*x^2)^p]

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Rubi [A]  time = 0.0838046, antiderivative size = 93, normalized size of antiderivative = 1.29, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2476, 2448, 321, 205, 2455} \[ -\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{x}+g x \log \left (c \left (d+e x^2\right )^p\right )+\frac{2 \sqrt{e} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d}}+\frac{2 \sqrt{d} g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-2 g p x \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^2,x]

[Out]

-2*g*p*x + (2*Sqrt[e]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + (2*Sqrt[d]*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/S
qrt[e] - (f*Log[c*(d + e*x^2)^p])/x + g*x*Log[c*(d + e*x^2)^p]

Rule 2476

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.0457904, size = 62, normalized size = 0.86 \[ \left (g x-\frac{f}{x}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac{2 p (d g+e f) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e}}-2 g p x \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^2,x]

[Out]

-2*g*p*x + (2*(e*f + d*g)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*Sqrt[e]) + (-(f/x) + g*x)*Log[c*(d + e*x^2)^
p]

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Maple [C]  time = 0.586, size = 427, normalized size = 5.9 \begin{align*} -{\frac{ \left ( -g{x}^{2}+f \right ) \ln \left ( \left ( e{x}^{2}+d \right ) ^{p} \right ) }{x}}+{\frac{1}{2\,dex} \left ( i\pi \,g{x}^{2}{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}de-i\pi \,g{x}^{2}{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \right ) de-i\pi \,g{x}^{2} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}de+i\pi \,g{x}^{2} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) de-i\pi \,def{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}+i\pi \,def{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \right ) +i\pi \,def \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}-i\pi \,def \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,\ln \left ( c \right ) g{x}^{2}de+2\,\sqrt{-de}p\ln \left ( -\sqrt{-de}x+d \right ) gdx+2\,\sqrt{-de}p\ln \left ( -\sqrt{-de}x+d \right ) fex-2\,\sqrt{-de}p\ln \left ( -\sqrt{-de}x-d \right ) gdx-2\,\sqrt{-de}p\ln \left ( -\sqrt{-de}x-d \right ) fex-4\,dgp{x}^{2}e-2\,\ln \left ( c \right ) def \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x^2+f)*ln(c*(e*x^2+d)^p)/x^2,x)

[Out]

-(-g*x^2+f)/x*ln((e*x^2+d)^p)+1/2*(I*Pi*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2*d*e-I*Pi*g*x^2*csgn(
I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)*d*e-I*Pi*g*x^2*csgn(I*c*(e*x^2+d)^p)^3*d*e+I*Pi*g*x^2*csgn(I*c*
(e*x^2+d)^p)^2*csgn(I*c)*d*e-I*Pi*d*e*f*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+I*Pi*d*e*f*csgn(I*(e*x^2+d
)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+I*Pi*d*e*f*csgn(I*c*(e*x^2+d)^p)^3-I*Pi*d*e*f*csgn(I*c*(e*x^2+d)^p)^2*csg
n(I*c)+2*ln(c)*g*x^2*d*e+2*(-d*e)^(1/2)*p*ln(-(-d*e)^(1/2)*x+d)*g*d*x+2*(-d*e)^(1/2)*p*ln(-(-d*e)^(1/2)*x+d)*f
*e*x-2*(-d*e)^(1/2)*p*ln(-(-d*e)^(1/2)*x-d)*g*d*x-2*(-d*e)^(1/2)*p*ln(-(-d*e)^(1/2)*x-d)*f*e*x-4*d*g*p*x^2*e-2
*ln(c)*d*e*f)/d/e/x

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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((g*x^2+f)*log(c*(e*x^2+d)^p)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.50089, size = 436, normalized size = 6.06 \begin{align*} \left [-\frac{2 \, d e g p x^{2} + \sqrt{-d e}{\left (e f + d g\right )} p x \log \left (\frac{e x^{2} - 2 \, \sqrt{-d e} x - d}{e x^{2} + d}\right ) -{\left (d e g p x^{2} - d e f p\right )} \log \left (e x^{2} + d\right ) -{\left (d e g x^{2} - d e f\right )} \log \left (c\right )}{d e x}, -\frac{2 \, d e g p x^{2} - 2 \, \sqrt{d e}{\left (e f + d g\right )} p x \arctan \left (\frac{\sqrt{d e} x}{d}\right ) -{\left (d e g p x^{2} - d e f p\right )} \log \left (e x^{2} + d\right ) -{\left (d e g x^{2} - d e f\right )} \log \left (c\right )}{d e x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^2,x, algorithm="fricas")

[Out]

[-(2*d*e*g*p*x^2 + sqrt(-d*e)*(e*f + d*g)*p*x*log((e*x^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)) - (d*e*g*p*x^2 - d
*e*f*p)*log(e*x^2 + d) - (d*e*g*x^2 - d*e*f)*log(c))/(d*e*x), -(2*d*e*g*p*x^2 - 2*sqrt(d*e)*(e*f + d*g)*p*x*ar
ctan(sqrt(d*e)*x/d) - (d*e*g*p*x^2 - d*e*f*p)*log(e*x^2 + d) - (d*e*g*x^2 - d*e*f)*log(c))/(d*e*x)]

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Sympy [A]  time = 62.6268, size = 262, normalized size = 3.64 \begin{align*} \begin{cases} \left (- \frac{f}{x} + g x\right ) \log{\left (0^{p} c \right )} & \text{for}\: d = 0 \wedge e = 0 \\\left (- \frac{f}{x} + g x\right ) \log{\left (c d^{p} \right )} & \text{for}\: e = 0 \\- \frac{f p \log{\left (e \right )}}{x} - \frac{2 f p \log{\left (x \right )}}{x} - \frac{2 f p}{x} - \frac{f \log{\left (c \right )}}{x} + g p x \log{\left (e \right )} + 2 g p x \log{\left (x \right )} - 2 g p x + g x \log{\left (c \right )} & \text{for}\: d = 0 \\\frac{i \sqrt{d} g p \log{\left (d + e x^{2} \right )}}{e \sqrt{\frac{1}{e}}} - \frac{2 i \sqrt{d} g p \log{\left (- i \sqrt{d} \sqrt{\frac{1}{e}} + x \right )}}{e \sqrt{\frac{1}{e}}} - \frac{f p \log{\left (d + e x^{2} \right )}}{x} - \frac{f \log{\left (c \right )}}{x} + g p x \log{\left (d + e x^{2} \right )} - 2 g p x + g x \log{\left (c \right )} + \frac{i f p \log{\left (d + e x^{2} \right )}}{\sqrt{d} \sqrt{\frac{1}{e}}} - \frac{2 i f p \log{\left (- i \sqrt{d} \sqrt{\frac{1}{e}} + x \right )}}{\sqrt{d} \sqrt{\frac{1}{e}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x**2+f)*ln(c*(e*x**2+d)**p)/x**2,x)

[Out]

Piecewise(((-f/x + g*x)*log(0**p*c), Eq(d, 0) & Eq(e, 0)), ((-f/x + g*x)*log(c*d**p), Eq(e, 0)), (-f*p*log(e)/
x - 2*f*p*log(x)/x - 2*f*p/x - f*log(c)/x + g*p*x*log(e) + 2*g*p*x*log(x) - 2*g*p*x + g*x*log(c), Eq(d, 0)), (
I*sqrt(d)*g*p*log(d + e*x**2)/(e*sqrt(1/e)) - 2*I*sqrt(d)*g*p*log(-I*sqrt(d)*sqrt(1/e) + x)/(e*sqrt(1/e)) - f*
p*log(d + e*x**2)/x - f*log(c)/x + g*p*x*log(d + e*x**2) - 2*g*p*x + g*x*log(c) + I*f*p*log(d + e*x**2)/(sqrt(
d)*sqrt(1/e)) - 2*I*f*p*log(-I*sqrt(d)*sqrt(1/e) + x)/(sqrt(d)*sqrt(1/e)), True))

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Giac [A]  time = 1.26186, size = 105, normalized size = 1.46 \begin{align*} \frac{2 \,{\left (d g p + f p e\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{1}{2}\right )}}{\sqrt{d}} + \frac{g p x^{2} \log \left (x^{2} e + d\right ) - 2 \, g p x^{2} + g x^{2} \log \left (c\right ) - f p \log \left (x^{2} e + d\right ) - f \log \left (c\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^2,x, algorithm="giac")

[Out]

2*(d*g*p + f*p*e)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/sqrt(d) + (g*p*x^2*log(x^2*e + d) - 2*g*p*x^2 + g*x^2*log
(c) - f*p*log(x^2*e + d) - f*log(c))/x