3.338 \(\int \frac{x^5 \log (c (d+e x^2)^p)}{f+g x^2} \, dx\)

Optimal. Leaf size=188 \[ \frac{f^2 p \text{PolyLog}\left (2,-\frac{g \left (d+e x^2\right )}{e f-d g}\right )}{2 g^3}+\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^3}-\frac{f \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}+\frac{x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac{d^2 p \log \left (d+e x^2\right )}{4 e^2 g}+\frac{d p x^2}{4 e g}+\frac{f p x^2}{2 g^2}-\frac{p x^4}{8 g} \]

[Out]

(f*p*x^2)/(2*g^2) + (d*p*x^2)/(4*e*g) - (p*x^4)/(8*g) - (d^2*p*Log[d + e*x^2])/(4*e^2*g) + (x^4*Log[c*(d + e*x
^2)^p])/(4*g) - (f*(d + e*x^2)*Log[c*(d + e*x^2)^p])/(2*e*g^2) + (f^2*Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))
/(e*f - d*g)])/(2*g^3) + (f^2*p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/(2*g^3)

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Rubi [A]  time = 0.275934, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {2475, 43, 2416, 2389, 2295, 2395, 2394, 2393, 2391} \[ \frac{f^2 p \text{PolyLog}\left (2,-\frac{g \left (d+e x^2\right )}{e f-d g}\right )}{2 g^3}+\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^3}-\frac{f \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}+\frac{x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac{d^2 p \log \left (d+e x^2\right )}{4 e^2 g}+\frac{d p x^2}{4 e g}+\frac{f p x^2}{2 g^2}-\frac{p x^4}{8 g} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*Log[c*(d + e*x^2)^p])/(f + g*x^2),x]

[Out]

(f*p*x^2)/(2*g^2) + (d*p*x^2)/(4*e*g) - (p*x^4)/(8*g) - (d^2*p*Log[d + e*x^2])/(4*e^2*g) + (x^4*Log[c*(d + e*x
^2)^p])/(4*g) - (f*(d + e*x^2)*Log[c*(d + e*x^2)^p])/(2*e*g^2) + (f^2*Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))
/(e*f - d*g)])/(2*g^3) + (f^2*p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/(2*g^3)

Rule 2475

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])

Rule 43

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 2416

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 2389

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 2295

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

Rule 2395

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 2394

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 2393

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

Rule 2391

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

Rubi steps

\begin{align*} \int \frac{x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2 \log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{f \log \left (c (d+e x)^p\right )}{g^2}+\frac{x \log \left (c (d+e x)^p\right )}{g}+\frac{f^2 \log \left (c (d+e x)^p\right )}{g^2 (f+g x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{f \operatorname{Subst}\left (\int \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 g^2}+\frac{f^2 \operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^2\right )}{2 g^2}+\frac{\operatorname{Subst}\left (\int x \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 g}\\ &=\frac{x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}+\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^3}-\frac{f \operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e g^2}-\frac{\left (e f^2 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx,x,x^2\right )}{2 g^3}-\frac{(e p) \operatorname{Subst}\left (\int \frac{x^2}{d+e x} \, dx,x,x^2\right )}{4 g}\\ &=\frac{f p x^2}{2 g^2}+\frac{x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac{f \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}+\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^3}-\frac{\left (f^2 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x^2\right )}{2 g^3}-\frac{(e p) \operatorname{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 )}{4 g}\\ &=\frac{f p x^2}{2 g^2}+\frac{d p x^2}{4 e g}-\frac{p x^4}{8 g}-\frac{d^2 p \log \left (d+e x^2\right )}{4 e^2 g}+\frac{x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac{f \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}+\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^3}+\frac{f^2 p \text{Li}_2\left (-\frac{g \left (d+e x^2\right )}{e f-d g}\right )}{2 g^3}\\ \end{align*}

Mathematica [A]  time = 0.123798, size = 143, normalized size = 0.76 \[ \frac{4 e^2 f^2 p \text{PolyLog}\left (2,\frac{g \left (d+e x^2\right )}{d g-e f}\right )+e \log \left (c \left (d+e x^2\right )^p\right ) \left (4 e f^2 \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )+2 g \left (-2 d f-2 e f x^2+e g x^4\right )\right )-2 d^2 g^2 p \log \left (d+e x^2\right )+e g p x^2 \left (2 d g+4 e f-e g x^2\right )}{8 e^2 g^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*Log[c*(d + e*x^2)^p])/(f + g*x^2),x]

[Out]

(e*g*p*x^2*(4*e*f + 2*d*g - e*g*x^2) - 2*d^2*g^2*p*Log[d + e*x^2] + e*Log[c*(d + e*x^2)^p]*(2*g*(-2*d*f - 2*e*
f*x^2 + e*g*x^4) + 4*e*f^2*Log[(e*(f + g*x^2))/(e*f - d*g)]) + 4*e^2*f^2*p*PolyLog[2, (g*(d + e*x^2))/(-(e*f)
+ d*g)])/(8*e^2*g^3)

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Maple [C]  time = 0.849, size = 902, normalized size = 4.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*ln(c*(e*x^2+d)^p)/(g*x^2+f),x)

[Out]

1/4*ln((e*x^2+d)^p)/g*x^4-1/2*ln((e*x^2+d)^p)/g^2*f*x^2+1/2*ln((e*x^2+d)^p)*f^2/g^3*ln(g*x^2+f)-1/2*p*f^2/g^3*
sum(ln(x-_alpha)*ln(g*x^2+f)-ln(x-_alpha)*(ln((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=1)-x+_alpha)/Root
Of(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=1))+ln((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=2)-x+_alpha)/R
ootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=2)))-dilog((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=1)-x+_a
lpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=1))-dilog((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=2
)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=2)),_alpha=RootOf(_Z^2*e+d))-1/8*p*x^4/g+1/4*d*p*x^2
/e/g+1/2*f*p*x^2/g^2-1/4*d^2*p*ln(e*x^2+d)/e^2/g-1/2*p/e/g^2*d*ln(e*x^2+d)*f+1/4*I*Pi*csgn(I*(e*x^2+d)^p)*csgn
(I*c*(e*x^2+d)^p)^2*f^2/g^3*ln(g*x^2+f)+1/8*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2/g*x^4-1/4*I*Pi*cs
gn(I*c*(e*x^2+d)^p)^3*f^2/g^3*ln(g*x^2+f)+1/4*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)/g^2*f*x
^2+1/8*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)/g*x^4-1/4*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2/g^2*f
*x^2+1/4*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)*f^2/g^3*ln(g*x^2+f)-1/8*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x
^2+d)^p)*csgn(I*c)/g*x^4-1/4*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)*f^2/g^3*ln(g*x^2+f)+1/4*
I*Pi*csgn(I*c*(e*x^2+d)^p)^3/g^2*f*x^2-1/4*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)/g^2*f*x^2-1/8*I*Pi*csgn(I*c*
(e*x^2+d)^p)^3/g*x^4+1/4*ln(c)/g*x^4-1/2*ln(c)/g^2*f*x^2+1/2*ln(c)*f^2/g^3*ln(g*x^2+f)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*log(c*(e*x^2+d)^p)/(g*x^2+f),x, algorithm="maxima")

[Out]

integrate(x^5*log((e*x^2 + d)^p*c)/(g*x^2 + f), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*log(c*(e*x^2+d)^p)/(g*x^2+f),x, algorithm="fricas")

[Out]

integral(x^5*log((e*x^2 + d)^p*c)/(g*x^2 + f), 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(x**5*ln(c*(e*x**2+d)**p)/(g*x**2+f),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*log(c*(e*x^2+d)^p)/(g*x^2+f),x, algorithm="giac")

[Out]

integrate(x^5*log((e*x^2 + d)^p*c)/(g*x^2 + f), x)