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

Optimal. Leaf size=94 \[ \frac{\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac{p (e f-d g)^2 \log \left (d+e x^2\right )}{4 e^2 g}-\frac{p x^2 (e f-d g)}{4 e}-\frac{p \left (f+g x^2\right )^2}{8 g} \]

[Out]

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

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Rubi [A]  time = 0.0928483, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2475, 2395, 43} \[ \frac{\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac{p (e f-d g)^2 \log \left (d+e x^2\right )}{4 e^2 g}-\frac{p x^2 (e f-d g)}{4 e}-\frac{p \left (f+g x^2\right )^2}{8 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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])

Rubi steps

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

Mathematica [A]  time = 0.0437708, size = 98, normalized size = 1.04 \[ \frac{1}{2} f \left (\frac{\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}-p x^2\right )+\frac{1}{4} g x^4 \log \left (c \left (d+e x^2\right )^p\right )-\frac{d^2 g p \log \left (d+e x^2\right )}{4 e^2}+\frac{d g p x^2}{4 e}-\frac{1}{8} g p x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.571, size = 361, normalized size = 3.8 \begin{align*} \left ({\frac{g{x}^{4}}{4}}+{\frac{f{x}^{2}}{2}} \right ) \ln \left ( \left ( e{x}^{2}+d \right ) ^{p} \right ) +{\frac{i}{8}}\pi \,g{x}^{4}{\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}{4}}\pi \,f{x}^{2} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{8}}\pi \,g{x}^{4}{\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}{4}}\pi \,f{x}^{2} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}-{\frac{i}{8}}\pi \,g{x}^{4} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}+{\frac{i}{4}}\pi \,f{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}-{\frac{i}{4}}\pi \,f{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 ) +{\frac{i}{8}}\pi \,g{x}^{4} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) g{x}^{4}}{4}}-{\frac{gp{x}^{4}}{8}}+{\frac{\ln \left ( c \right ) f{x}^{2}}{2}}+{\frac{dgp{x}^{2}}{4\,e}}-{\frac{fp{x}^{2}}{2}}-{\frac{\ln \left ( e{x}^{2}+d \right ){d}^{2}gp}{4\,{e}^{2}}}+{\frac{\ln \left ( e{x}^{2}+d \right ) dfp}{2\,e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/4*g*x^4+1/2*f*x^2)*ln((e*x^2+d)^p)+1/8*I*Pi*g*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+1/4*I*Pi*f*x^
2*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-1/8*I*Pi*g*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-1/4*I*P
i*f*x^2*csgn(I*c*(e*x^2+d)^p)^3-1/8*I*Pi*g*x^4*csgn(I*c*(e*x^2+d)^p)^3+1/4*I*Pi*f*x^2*csgn(I*(e*x^2+d)^p)*csgn
(I*c*(e*x^2+d)^p)^2-1/4*I*Pi*f*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+1/8*I*Pi*g*x^4*csgn(I*c
*(e*x^2+d)^p)^2*csgn(I*c)+1/4*ln(c)*g*x^4-1/8*g*p*x^4+1/2*ln(c)*f*x^2+1/4*d*g*p*x^2/e-1/2*f*p*x^2-1/4*d^2*g*p*
ln(e*x^2+d)/e^2+1/2/e*ln(e*x^2+d)*d*f*p

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Maxima [A]  time = 1.03011, size = 134, normalized size = 1.43 \begin{align*} -\frac{e p{\left (\frac{e g^{2} x^{4} + 2 \,{\left (2 \, e f g - d g^{2}\right )} x^{2}}{e^{2}} + \frac{2 \,{\left (e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x^{2} + d\right )}{e^{3}}\right )}}{8 \, g} + \frac{{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{4 \, g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*e*p*((e*g^2*x^4 + 2*(2*e*f*g - d*g^2)*x^2)/e^2 + 2*(e^2*f^2 - 2*d*e*f*g + d^2*g^2)*log(e*x^2 + d)/e^3)/g
+ 1/4*(g*x^2 + f)^2*log((e*x^2 + d)^p*c)/g

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Fricas [A]  time = 1.83286, size = 216, normalized size = 2.3 \begin{align*} -\frac{e^{2} g p x^{4} + 2 \,{\left (2 \, e^{2} f - d e g\right )} p x^{2} - 2 \,{\left (e^{2} g p x^{4} + 2 \, e^{2} f p x^{2} +{\left (2 \, d e f - d^{2} g\right )} p\right )} \log \left (e x^{2} + d\right ) - 2 \,{\left (e^{2} g x^{4} + 2 \, e^{2} f x^{2}\right )} \log \left (c\right )}{8 \, e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 67.4428, size = 139, normalized size = 1.48 \begin{align*} \begin{cases} - \frac{d^{2} g p \log{\left (d + e x^{2} \right )}}{4 e^{2}} + \frac{d f p \log{\left (d + e x^{2} \right )}}{2 e} + \frac{d g p x^{2}}{4 e} + \frac{f p x^{2} \log{\left (d + e x^{2} \right )}}{2} - \frac{f p x^{2}}{2} + \frac{f x^{2} \log{\left (c \right )}}{2} + \frac{g p x^{4} \log{\left (d + e x^{2} \right )}}{4} - \frac{g p x^{4}}{8} + \frac{g x^{4} \log{\left (c \right )}}{4} & \text{for}\: e \neq 0 \\\left (\frac{f x^{2}}{2} + \frac{g x^{4}}{4}\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(x*(g*x**2+f)*ln(c*(e*x**2+d)**p),x)

[Out]

Piecewise((-d**2*g*p*log(d + e*x**2)/(4*e**2) + d*f*p*log(d + e*x**2)/(2*e) + d*g*p*x**2/(4*e) + f*p*x**2*log(
d + e*x**2)/2 - f*p*x**2/2 + f*x**2*log(c)/2 + g*p*x**4*log(d + e*x**2)/4 - g*p*x**4/8 + g*x**4*log(c)/4, Ne(e
, 0)), ((f*x**2/2 + g*x**4/4)*log(c*d**p), True))

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Giac [A]  time = 1.21401, size = 200, normalized size = 2.13 \begin{align*} \frac{1}{8} \,{\left ({\left (2 \,{\left (x^{2} e + d\right )}^{2} \log \left (x^{2} e + d\right ) - 4 \,{\left (x^{2} e + d\right )} d \log \left (x^{2} e + d\right ) -{\left (x^{2} e + d\right )}^{2} + 4 \,{\left (x^{2} e + d\right )} d\right )} g p e^{\left (-1\right )} + 2 \,{\left ({\left (x^{2} e + d\right )}^{2} - 2 \,{\left (x^{2} e + d\right )} d\right )} g e^{\left (-1\right )} \log \left (c\right ) - 4 \,{\left (x^{2} e -{\left (x^{2} e + d\right )} \log \left (x^{2} e + d\right ) + d\right )} f p + 4 \,{\left (x^{2} e + d\right )} f \log \left (c\right )\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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