\(\int \frac {(f+g x^2)^2 \log (c (d+e x^2)^p)}{x} \, dx\) [326]

Optimal result

Integrand size = 25, antiderivative size = 153 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=-f g p x^2+\frac {d g^2 p x^2}{4 e}-\frac {1}{8} g^2 p x^4-\frac {d^2 g^2 p \log \left (d+e x^2\right )}{4 e^2}+\frac {1}{4} g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {f g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {1}{2} f^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f^2 p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right ) \]

[Out]

-f*g*p*x^2+1/4*d*g^2*p*x^2/e-1/8*g^2*p*x^4-1/4*d^2*g^2*p*ln(e*x^2+d)/e^2+1/4*g^2*x^4*ln(c*(e*x^2+d)^p)+f*g*(e*
x^2+d)*ln(c*(e*x^2+d)^p)/e+1/2*f^2*ln(-e*x^2/d)*ln(c*(e*x^2+d)^p)+1/2*f^2*p*polylog(2,1+e*x^2/d)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2525, 45, 2463, 2436, 2332, 2441, 2352, 2442} \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\frac {1}{2} f^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {f g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {1}{4} g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )-\frac {d^2 g^2 p \log \left (d+e x^2\right )}{4 e^2}+\frac {1}{2} f^2 p \operatorname {PolyLog}\left (2,\frac {e x^2}{d}+1\right )+\frac {d g^2 p x^2}{4 e}-f g p x^2-\frac {1}{8} g^2 p x^4 \]

[In]

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

[Out]

-(f*g*p*x^2) + (d*g^2*p*x^2)/(4*e) - (g^2*p*x^4)/8 - (d^2*g^2*p*Log[d + e*x^2])/(4*e^2) + (g^2*x^4*Log[c*(d +
e*x^2)^p])/4 + (f*g*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e + (f^2*Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p])/2 + (f^
2*p*PolyLog[2, 1 + (e*x^2)/d])/2

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2436

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

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2463

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

Rule 2525

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(f+g x)^2 \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (2 f g \log \left (c (d+e x)^p\right )+\frac {f^2 \log \left (c (d+e x)^p\right )}{x}+g^2 x \log \left (c (d+e x)^p\right )\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{2} f^2 \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right )+(f g) \text {Subst}\left (\int \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )+\frac {1}{2} g^2 \text {Subst}\left (\int x \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{4} g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {(f g) \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{e}-\frac {1}{2} \left (e f^2 p\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^2\right )-\frac {1}{4} \left (e g^2 p\right ) \text {Subst}\left (\int \frac {x^2}{d+e x} \, dx,x,x^2\right ) \\ & = -f g p x^2+\frac {1}{4} g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {f g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {1}{2} f^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f^2 p \text {Li}_2\left (1+\frac {e x^2}{d}\right )-\frac {1}{4} \left (e g^2 p\right ) \text {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx,x,x^2\right ) \\ & = -f g p x^2+\frac {d g^2 p x^2}{4 e}-\frac {1}{8} g^2 p x^4-\frac {d^2 g^2 p \log \left (d+e x^2\right )}{4 e^2}+\frac {1}{4} g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {f g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {1}{2} f^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f^2 p \text {Li}_2\left (1+\frac {e x^2}{d}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.79 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\frac {-e g p x^2 \left (8 e f-2 d g+e g x^2\right )-2 d^2 g^2 p \log \left (d+e x^2\right )+2 e \left (g \left (4 d f+4 e f x^2+e g x^4\right )+2 e f^2 \log \left (-\frac {e x^2}{d}\right )\right ) \log \left (c \left (d+e x^2\right )^p\right )+4 e^2 f^2 p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right )}{8 e^2} \]

[In]

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

[Out]

(-(e*g*p*x^2*(8*e*f - 2*d*g + e*g*x^2)) - 2*d^2*g^2*p*Log[d + e*x^2] + 2*e*(g*(4*d*f + 4*e*f*x^2 + e*g*x^4) +
2*e*f^2*Log[-((e*x^2)/d)])*Log[c*(d + e*x^2)^p] + 4*e^2*f^2*p*PolyLog[2, 1 + (e*x^2)/d])/(8*e^2)

Maple [A] (verified)

Time = 1.15 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.37

method	result	size
parts	\(\frac {g^{2} x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{4}+\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f g \,x^{2}+\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f^{2} \ln \left (x \right )-\frac {p e \left (g \left (\frac {\frac {1}{2} e g \,x^{4}-d g \,x^{2}+4 f e \,x^{2}}{2 e^{2}}+\frac {d \left (d g -4 e f \right ) \ln \left (e \,x^{2}+d \right )}{2 e^{3}}\right )+4 f^{2} \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e}\right )\right )}{2}\)	\(209\)
risch	\(\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) g^{2} x^{4}}{4}+\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f g \,x^{2}+\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f^{2} \ln \left (x \right )-\frac {g^{2} p \,x^{4}}{8}+\frac {d \,g^{2} p \,x^{2}}{4 e}-f g p \,x^{2}-\frac {d^{2} g^{2} p \ln \left (e \,x^{2}+d \right )}{4 e^{2}}+\frac {p g d \ln \left (e \,x^{2}+d \right ) f}{e}-p \,f^{2} \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p \,f^{2} \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p \,f^{2} \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p \,f^{2} \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {g^{2} x^{4}}{4}+f g \,x^{2}+f^{2} \ln \left (x \right )\right )\)	\(359\)

[In]

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

[Out]

1/4*g^2*x^4*ln(c*(e*x^2+d)^p)+ln(c*(e*x^2+d)^p)*f*g*x^2+ln(c*(e*x^2+d)^p)*f^2*ln(x)-1/2*p*e*(g*(1/2/e^2*(1/2*e
*g*x^4-d*g*x^2+4*f*e*x^2)+1/2*d*(d*g-4*e*f)/e^3*ln(e*x^2+d))+4*f^2*(1/2*ln(x)*(ln((-e*x+(-d*e)^(1/2))/(-d*e)^(
1/2))+ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2)))/e+1/2*(dilog((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+dilog((e*x+(-d*e)^(1
/2))/(-d*e)^(1/2)))/e))

Fricas [F]

\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x} \,d x } \]

[In]

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

[Out]

integral((g^2*x^4 + 2*f*g*x^2 + f^2)*log((e*x^2 + d)^p*c)/x, x)

Sympy [F]

\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\int \frac {\left (f + g x^{2}\right )^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{x}\, dx \]

[In]

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

[Out]

Integral((f + g*x**2)**2*log(c*(d + e*x**2)**p)/x, x)

Maxima [F]

\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,{\left (g\,x^2+f\right )}^2}{x} \,d x \]

[In]

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

[Out]

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