3.319 \(\int (f+g x^2) \log (c (d+e x^2)^p) \, dx\)

Optimal. Leaf size=117 \[ f x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac{2 d^{3/2} g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2}}+\frac{2 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{2 d g p x}{3 e}-2 f p x-\frac{2}{9} g p x^3 \]

[Out]

-2*f*p*x + (2*d*g*p*x)/(3*e) - (2*g*p*x^3)/9 + (2*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*d^(3/2
)*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)) + f*x*Log[c*(d + e*x^2)^p] + (g*x^3*Log[c*(d + e*x^2)^p])/3

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Rubi [A]  time = 0.0856192, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2471, 2448, 321, 205, 2455, 302} \[ f x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac{2 d^{3/2} g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2}}+\frac{2 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{2 d g p x}{3 e}-2 f p x-\frac{2}{9} g p x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*f*p*x + (2*d*g*p*x)/(3*e) - (2*g*p*x^3)/9 + (2*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*d^(3/2
)*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)) + f*x*Log[c*(d + e*x^2)^p] + (g*x^3*Log[c*(d + e*x^2)^p])/3

Rule 2471

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol]
:> With[{t = ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; Free
Q[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[
q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))

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]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rubi steps

\begin{align*} \int \left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f \log \left (c \left (d+e x^2\right )^p\right )+g x^2 \log \left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx+g \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=f x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log \left (c \left (d+e x^2\right )^p\right )-(2 e f p) \int \frac{x^2}{d+e x^2} \, dx-\frac{1}{3} (2 e g p) \int \frac{x^4}{d+e x^2} \, dx\\ &=-2 f p x+f x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log \left (c \left (d+e x^2\right )^p\right )+(2 d f p) \int \frac{1}{d+e x^2} \, dx-\frac{1}{3} (2 e g p) \int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=-2 f p x+\frac{2 d g p x}{3 e}-\frac{2}{9} g p x^3+\frac{2 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+f x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac{\left (2 d^2 g p\right ) \int \frac{1}{d+e x^2} \, dx}{3 e}\\ &=-2 f p x+\frac{2 d g p x}{3 e}-\frac{2}{9} g p x^3+\frac{2 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{2 d^{3/2} g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2}}+f x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log \left (c \left (d+e x^2\right )^p\right )\\ \end{align*}

Mathematica [A]  time = 0.035529, size = 117, normalized size = 1. \[ f x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac{2 d^{3/2} g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2}}+\frac{2 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{2 d g p x}{3 e}-2 f p x-\frac{2}{9} g p x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*f*p*x + (2*d*g*p*x)/(3*e) - (2*g*p*x^3)/9 + (2*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*d^(3/2
)*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)) + f*x*Log[c*(d + e*x^2)^p] + (g*x^3*Log[c*(d + e*x^2)^p])/3

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Maple [C]  time = 0.102, size = 416, normalized size = 3.6 \begin{align*} \left ({\frac{g{x}^{3}}{3}}+fx \right ) \ln \left ( \left ( e{x}^{2}+d \right ) ^{p} \right ) +{\frac{i}{6}}\pi \,g{x}^{3} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{6}}\pi \,g{x}^{3}{\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 ) +{\frac{i}{2}}\pi \,f{\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}x-{\frac{i}{6}}\pi \,g{x}^{3} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}-{\frac{i}{2}}\pi \,f{\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 ) x+{\frac{i}{2}}\pi \,f \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) x+{\frac{i}{6}}\pi \,g{x}^{3}{\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}-{\frac{i}{2}}\pi \,f \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}x+{\frac{\ln \left ( c \right ) g{x}^{3}}{3}}-{\frac{2\,gp{x}^{3}}{9}}-{\frac{pdg}{3\,{e}^{2}}\sqrt{-de}\ln \left ( \sqrt{-de}x-d \right ) }+{\frac{pf}{e}\sqrt{-de}\ln \left ( \sqrt{-de}x-d \right ) }+{\frac{pdg}{3\,{e}^{2}}\sqrt{-de}\ln \left ( -\sqrt{-de}x-d \right ) }-{\frac{pf}{e}\sqrt{-de}\ln \left ( -\sqrt{-de}x-d \right ) }+\ln \left ( c \right ) fx+{\frac{2\,dgpx}{3\,e}}-2\,fpx \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)

[Out]

(1/3*g*x^3+f*x)*ln((e*x^2+d)^p)+1/6*I*Pi*g*x^3*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-1/6*I*Pi*g*x^3*csgn(I*(e*x^2+
d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+1/2*I*Pi*f*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2*x-1/6*I*Pi*g*x^3*
csgn(I*c*(e*x^2+d)^p)^3-1/2*I*Pi*f*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)*x+1/2*I*Pi*f*csgn(I*c*(
e*x^2+d)^p)^2*csgn(I*c)*x+1/6*I*Pi*g*x^3*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-1/2*I*Pi*f*csgn(I*c*(e*x^
2+d)^p)^3*x+1/3*ln(c)*g*x^3-2/9*g*p*x^3-1/3/e^2*(-d*e)^(1/2)*p*ln((-d*e)^(1/2)*x-d)*d*g+1/e*(-d*e)^(1/2)*p*ln(
(-d*e)^(1/2)*x-d)*f+1/3/e^2*(-d*e)^(1/2)*p*ln(-(-d*e)^(1/2)*x-d)*d*g-1/e*(-d*e)^(1/2)*p*ln(-(-d*e)^(1/2)*x-d)*
f+ln(c)*f*x+2/3*d*g*p*x/e-2*f*p*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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.26087, size = 504, normalized size = 4.31 \begin{align*} \left [-\frac{2 \, e g p x^{3} + 3 \,{\left (3 \, e f - d g\right )} p \sqrt{-\frac{d}{e}} \log \left (\frac{e x^{2} - 2 \, e x \sqrt{-\frac{d}{e}} - d}{e x^{2} + d}\right ) + 6 \,{\left (3 \, e f - d g\right )} p x - 3 \,{\left (e g p x^{3} + 3 \, e f p x\right )} \log \left (e x^{2} + d\right ) - 3 \,{\left (e g x^{3} + 3 \, e f x\right )} \log \left (c\right )}{9 \, e}, -\frac{2 \, e g p x^{3} - 6 \,{\left (3 \, e f - d g\right )} p \sqrt{\frac{d}{e}} \arctan \left (\frac{e x \sqrt{\frac{d}{e}}}{d}\right ) + 6 \,{\left (3 \, e f - d g\right )} p x - 3 \,{\left (e g p x^{3} + 3 \, e f p x\right )} \log \left (e x^{2} + d\right ) - 3 \,{\left (e g x^{3} + 3 \, e f x\right )} \log \left (c\right )}{9 \, e}\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, algorithm="fricas")

[Out]

[-1/9*(2*e*g*p*x^3 + 3*(3*e*f - d*g)*p*sqrt(-d/e)*log((e*x^2 - 2*e*x*sqrt(-d/e) - d)/(e*x^2 + d)) + 6*(3*e*f -
d*g)*p*x - 3*(e*g*p*x^3 + 3*e*f*p*x)*log(e*x^2 + d) - 3*(e*g*x^3 + 3*e*f*x)*log(c))/e, -1/9*(2*e*g*p*x^3 - 6*
(3*e*f - d*g)*p*sqrt(d/e)*arctan(e*x*sqrt(d/e)/d) + 6*(3*e*f - d*g)*p*x - 3*(e*g*p*x^3 + 3*e*f*p*x)*log(e*x^2
+ d) - 3*(e*g*x^3 + 3*e*f*x)*log(c))/e]

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Sympy [A]  time = 35.0358, size = 228, normalized size = 1.95 \begin{align*} \begin{cases} - \frac{i d^{\frac{3}{2}} g p \log{\left (d + e x^{2} \right )}}{3 e^{2} \sqrt{\frac{1}{e}}} + \frac{2 i d^{\frac{3}{2}} g p \log{\left (- i \sqrt{d} \sqrt{\frac{1}{e}} + x \right )}}{3 e^{2} \sqrt{\frac{1}{e}}} + \frac{i \sqrt{d} f p \log{\left (d + e x^{2} \right )}}{e \sqrt{\frac{1}{e}}} - \frac{2 i \sqrt{d} f p \log{\left (- i \sqrt{d} \sqrt{\frac{1}{e}} + x \right )}}{e \sqrt{\frac{1}{e}}} + \frac{2 d g p x}{3 e} + f p x \log{\left (d + e x^{2} \right )} - 2 f p x + f x \log{\left (c \right )} + \frac{g p x^{3} \log{\left (d + e x^{2} \right )}}{3} - \frac{2 g p x^{3}}{9} + \frac{g x^{3} \log{\left (c \right )}}{3} & \text{for}\: e \neq 0 \\\left (f x + \frac{g x^{3}}{3}\right ) \log{\left (c d^{p} \right )} & \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)

[Out]

Piecewise((-I*d**(3/2)*g*p*log(d + e*x**2)/(3*e**2*sqrt(1/e)) + 2*I*d**(3/2)*g*p*log(-I*sqrt(d)*sqrt(1/e) + x)
/(3*e**2*sqrt(1/e)) + I*sqrt(d)*f*p*log(d + e*x**2)/(e*sqrt(1/e)) - 2*I*sqrt(d)*f*p*log(-I*sqrt(d)*sqrt(1/e) +
x)/(e*sqrt(1/e)) + 2*d*g*p*x/(3*e) + f*p*x*log(d + e*x**2) - 2*f*p*x + f*x*log(c) + g*p*x**3*log(d + e*x**2)/
3 - 2*g*p*x**3/9 + g*x**3*log(c)/3, Ne(e, 0)), ((f*x + g*x**3/3)*log(c*d**p), True))

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Giac [A]  time = 1.16656, size = 147, normalized size = 1.26 \begin{align*} -\frac{2 \,{\left (d^{2} g p - 3 \, d f p e\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{3}{2}\right )}}{3 \, \sqrt{d}} + \frac{1}{9} \,{\left (3 \, g p x^{3} e \log \left (x^{2} e + d\right ) - 2 \, g p x^{3} e + 3 \, g x^{3} e \log \left (c\right ) + 9 \, f p x e \log \left (x^{2} e + d\right ) + 6 \, d g p x - 18 \, f p x e + 9 \, f x e \log \left (c\right )\right )} e^{\left (-1\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, algorithm="giac")

[Out]

-2/3*(d^2*g*p - 3*d*f*p*e)*arctan(x*e^(1/2)/sqrt(d))*e^(-3/2)/sqrt(d) + 1/9*(3*g*p*x^3*e*log(x^2*e + d) - 2*g*
p*x^3*e + 3*g*x^3*e*log(c) + 9*f*p*x*e*log(x^2*e + d) + 6*d*g*p*x - 18*f*p*x*e + 9*f*x*e*log(c))*e^(-1)