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

Optimal. Leaf size=178 \[ -\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x}+2 f g x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac{2 d^{3/2} g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2}}+\frac{2 \sqrt{e} f^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d}}+\frac{4 \sqrt{d} f g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{2 d g^2 p x}{3 e}-4 f g p x-\frac{2}{9} g^2 p x^3 \]

[Out]

-4*f*g*p*x + (2*d*g^2*p*x)/(3*e) - (2*g^2*p*x^3)/9 + (2*Sqrt[e]*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + (
4*Sqrt[d]*f*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*d^(3/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2
)) - (f^2*Log[c*(d + e*x^2)^p])/x + 2*f*g*x*Log[c*(d + e*x^2)^p] + (g^2*x^3*Log[c*(d + e*x^2)^p])/3

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Rubi [A]  time = 0.155254, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2476, 2448, 321, 205, 2455, 302} \[ -\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x}+2 f g x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac{2 d^{3/2} g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2}}+\frac{2 \sqrt{e} f^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d}}+\frac{4 \sqrt{d} f g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{2 d g^2 p x}{3 e}-4 f g p x-\frac{2}{9} g^2 p x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-4*f*g*p*x + (2*d*g^2*p*x)/(3*e) - (2*g^2*p*x^3)/9 + (2*Sqrt[e]*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + (
4*Sqrt[d]*f*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*d^(3/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2
)) - (f^2*Log[c*(d + e*x^2)^p])/x + 2*f*g*x*Log[c*(d + e*x^2)^p] + (g^2*x^3*Log[c*(d + e*x^2)^p])/3

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]

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

Mathematica [A]  time = 0.13557, size = 112, normalized size = 0.63 \[ \frac{1}{9} \left (\left (-\frac{9 f^2}{x}+18 f g x+3 g^2 x^3\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac{6 p \left (-d^2 g^2+6 d e f g+3 e^2 f^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{3/2}}-\frac{2 g p x \left (-3 d g+18 e f+e g x^2\right )}{e}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*g*p*x*(18*e*f - 3*d*g + e*g*x^2))/e + (6*(3*e^2*f^2 + 6*d*e*f*g - d^2*g^2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])
/(Sqrt[d]*e^(3/2)) + ((-9*f^2)/x + 18*f*g*x + 3*g^2*x^3)*Log[c*(d + e*x^2)^p])/9

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Maple [C]  time = 0.606, size = 742, normalized size = 4.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-g^2*x^4-6*f*g*x^2+3*f^2)/x*ln((e*x^2+d)^p)-1/18*(-18*I*Pi*f*g*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)*x^2*e^2
*d-3*I*Pi*g^2*x^4*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)*e^2*d-3*I*Pi*g^2*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d
)^p)^2*e^2*d+9*I*Pi*d*e^2*f^2*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+18*I*Pi*f*g*csgn(I*c*(e*x^2+d)^p)^3*x^2*e^2*d+
3*I*Pi*g^2*x^4*csgn(I*c*(e*x^2+d)^p)^3*e^2*d-18*I*Pi*f*g*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2*x^2*e^2*d
-9*I*Pi*d*e^2*f^2*csgn(I*c*(e*x^2+d)^p)^3-6*ln(c)*g^2*x^4*e^2*d+18*I*Pi*f*g*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^
2+d)^p)*csgn(I*c)*x^2*e^2*d+3*I*Pi*g^2*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)*e^2*d+9*I*Pi*d*
e^2*f^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-9*I*Pi*d*e^2*f^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)
*csgn(I*c)+4*d*e^2*g^2*p*x^4-36*ln(c)*f*g*x^2*e^2*d-6*(-d*e)^(1/2)*d^2*p*ln(-(-d*e)^(1/2)*x-d)*g^2*x+36*(-d*e)
^(1/2)*p*ln(-(-d*e)^(1/2)*x-d)*f*g*e*d*x+18*(-d*e)^(1/2)*p*ln(-(-d*e)^(1/2)*x-d)*f^2*e^2*x+6*(-d*e)^(1/2)*d^2*
p*ln(-(-d*e)^(1/2)*x+d)*g^2*x-36*(-d*e)^(1/2)*p*ln(-(-d*e)^(1/2)*x+d)*f*g*e*d*x-18*(-d*e)^(1/2)*p*ln(-(-d*e)^(
1/2)*x+d)*f^2*e^2*x-12*d^2*e*g^2*p*x^2+72*d*e^2*f*g*p*x^2+18*ln(c)*d*e^2*f^2)/e^2/d/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)^2*log(c*(e*x^2+d)^p)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.77989, size = 790, normalized size = 4.44 \begin{align*} \left [-\frac{2 \, d e^{2} g^{2} p x^{4} - 3 \,{\left (3 \, e^{2} f^{2} + 6 \, d e f g - d^{2} g^{2}\right )} \sqrt{-d e} p x \log \left (\frac{e x^{2} + 2 \, \sqrt{-d e} x - d}{e x^{2} + d}\right ) + 6 \,{\left (6 \, d e^{2} f g - d^{2} e g^{2}\right )} p x^{2} - 3 \,{\left (d e^{2} g^{2} p x^{4} + 6 \, d e^{2} f g p x^{2} - 3 \, d e^{2} f^{2} p\right )} \log \left (e x^{2} + d\right ) - 3 \,{\left (d e^{2} g^{2} x^{4} + 6 \, d e^{2} f g x^{2} - 3 \, d e^{2} f^{2}\right )} \log \left (c\right )}{9 \, d e^{2} x}, -\frac{2 \, d e^{2} g^{2} p x^{4} - 6 \,{\left (3 \, e^{2} f^{2} + 6 \, d e f g - d^{2} g^{2}\right )} \sqrt{d e} p x \arctan \left (\frac{\sqrt{d e} x}{d}\right ) + 6 \,{\left (6 \, d e^{2} f g - d^{2} e g^{2}\right )} p x^{2} - 3 \,{\left (d e^{2} g^{2} p x^{4} + 6 \, d e^{2} f g p x^{2} - 3 \, d e^{2} f^{2} p\right )} \log \left (e x^{2} + d\right ) - 3 \,{\left (d e^{2} g^{2} x^{4} + 6 \, d e^{2} f g x^{2} - 3 \, d e^{2} f^{2}\right )} \log \left (c\right )}{9 \, d e^{2} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/9*(2*d*e^2*g^2*p*x^4 - 3*(3*e^2*f^2 + 6*d*e*f*g - d^2*g^2)*sqrt(-d*e)*p*x*log((e*x^2 + 2*sqrt(-d*e)*x - d)
/(e*x^2 + d)) + 6*(6*d*e^2*f*g - d^2*e*g^2)*p*x^2 - 3*(d*e^2*g^2*p*x^4 + 6*d*e^2*f*g*p*x^2 - 3*d*e^2*f^2*p)*lo
g(e*x^2 + d) - 3*(d*e^2*g^2*x^4 + 6*d*e^2*f*g*x^2 - 3*d*e^2*f^2)*log(c))/(d*e^2*x), -1/9*(2*d*e^2*g^2*p*x^4 -
6*(3*e^2*f^2 + 6*d*e*f*g - d^2*g^2)*sqrt(d*e)*p*x*arctan(sqrt(d*e)*x/d) + 6*(6*d*e^2*f*g - d^2*e*g^2)*p*x^2 -
3*(d*e^2*g^2*p*x^4 + 6*d*e^2*f*g*p*x^2 - 3*d*e^2*f^2*p)*log(e*x^2 + d) - 3*(d*e^2*g^2*x^4 + 6*d*e^2*f*g*x^2 -
3*d*e^2*f^2)*log(c))/(d*e^2*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.2542, size = 227, normalized size = 1.28 \begin{align*} -\frac{2 \,{\left (d^{2} g^{2} p - 6 \, d f g p e - 3 \, f^{2} p e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{3}{2}\right )}}{3 \, \sqrt{d}} + \frac{{\left (3 \, g^{2} p x^{4} e \log \left (x^{2} e + d\right ) - 2 \, g^{2} p x^{4} e + 3 \, g^{2} x^{4} e \log \left (c\right ) + 18 \, f g p x^{2} e \log \left (x^{2} e + d\right ) + 6 \, d g^{2} p x^{2} - 36 \, f g p x^{2} e + 18 \, f g x^{2} e \log \left (c\right ) - 9 \, f^{2} p e \log \left (x^{2} e + d\right ) - 9 \, f^{2} e \log \left (c\right )\right )} e^{\left (-1\right )}}{9 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(d^2*g^2*p - 6*d*f*g*p*e - 3*f^2*p*e^2)*arctan(x*e^(1/2)/sqrt(d))*e^(-3/2)/sqrt(d) + 1/9*(3*g^2*p*x^4*e*l
og(x^2*e + d) - 2*g^2*p*x^4*e + 3*g^2*x^4*e*log(c) + 18*f*g*p*x^2*e*log(x^2*e + d) + 6*d*g^2*p*x^2 - 36*f*g*p*
x^2*e + 18*f*g*x^2*e*log(c) - 9*f^2*p*e*log(x^2*e + d) - 9*f^2*e*log(c))*e^(-1)/x