3.310 \(\int x^5 (f+g x^2) \log (c (d+e x^2)^p) \, dx\)

Optimal. Leaf size=142 \[ \frac{1}{6} f x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{8} g x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac{d^2 p x^2 (4 e f-3 d g)}{24 e^3}+\frac{d^3 p (4 e f-3 d g) \log \left (d+e x^2\right )}{24 e^4}+\frac{d p x^4 (4 e f-3 d g)}{48 e^2}-\frac{p x^6 (4 e f-3 d g)}{72 e}-\frac{1}{32} g p x^8 \]

[Out]

-(d^2*(4*e*f - 3*d*g)*p*x^2)/(24*e^3) + (d*(4*e*f - 3*d*g)*p*x^4)/(48*e^2) - ((4*e*f - 3*d*g)*p*x^6)/(72*e) -
(g*p*x^8)/32 + (d^3*(4*e*f - 3*d*g)*p*Log[d + e*x^2])/(24*e^4) + (f*x^6*Log[c*(d + e*x^2)^p])/6 + (g*x^8*Log[c
*(d + e*x^2)^p])/8

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Rubi [A]  time = 0.22868, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2475, 43, 2414, 12, 77} \[ \frac{1}{6} f x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{8} g x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac{d^2 p x^2 (4 e f-3 d g)}{24 e^3}+\frac{d^3 p (4 e f-3 d g) \log \left (d+e x^2\right )}{24 e^4}+\frac{d p x^4 (4 e f-3 d g)}{48 e^2}-\frac{p x^6 (4 e f-3 d g)}{72 e}-\frac{1}{32} g p x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d^2*(4*e*f - 3*d*g)*p*x^2)/(24*e^3) + (d*(4*e*f - 3*d*g)*p*x^4)/(48*e^2) - ((4*e*f - 3*d*g)*p*x^6)/(72*e) -
(g*p*x^8)/32 + (d^3*(4*e*f - 3*d*g)*p*Log[d + e*x^2])/(24*e^4) + (f*x^6*Log[c*(d + e*x^2)^p])/6 + (g*x^8*Log[c
*(d + e*x^2)^p])/8

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 2414

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*(x_)^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = IntHide[x^m*(f + g*x^r)^q, x]}, Dist[a + b*Log[c*(d + e*x)^n], u, x] - Dist[b*e*n, Int[SimplifyI
ntegrand[u/(d + e*x), x], x], x] /; InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q, r}, x]
&& IntegerQ[m] && IntegerQ[q] && IntegerQ[r]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int x^5 \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 x^2 (f+g x) \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )\\ &=\frac{1}{6} f x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{8} g x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac{1}{2} (e p) \operatorname{Subst}\left (\int \frac{x^3 (4 f+3 g x)}{12 (d+e x)} \, dx,x,x^2\right )\\ &=\frac{1}{6} f x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{8} g x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac{1}{24} (e p) \operatorname{Subst}\left (\int \frac{x^3 (4 f+3 g x)}{d+e x} \, dx,x,x^2\right )\\ &=\frac{1}{6} f x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{8} g x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac{1}{24} (e p) \operatorname{Subst}\left (\int \left (-\frac{d^2 (-4 e f+3 d g)}{e^4}+\frac{d (-4 e f+3 d g) x}{e^3}+\frac{(4 e f-3 d g) x^2}{e^2}+\frac{3 g x^3}{e}+\frac{d^3 (-4 e f+3 d g)}{e^4 (d+e x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{d^2 (4 e f-3 d g) p x^2}{24 e^3}+\frac{d (4 e f-3 d g) p x^4}{48 e^2}-\frac{(4 e f-3 d g) p x^6}{72 e}-\frac{1}{32} g p x^8+\frac{d^3 (4 e f-3 d g) p \log \left (d+e x^2\right )}{24 e^4}+\frac{1}{6} f x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{8} g x^8 \log \left (c \left (d+e x^2\right )^p\right )\\ \end{align*}

Mathematica [A]  time = 0.0412418, size = 170, normalized size = 1.2 \[ \frac{1}{6} f x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{8} g x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac{d^2 f p x^2}{6 e^2}+\frac{d^3 f p \log \left (d+e x^2\right )}{6 e^3}-\frac{d^2 g p x^4}{16 e^2}+\frac{d^3 g p x^2}{8 e^3}-\frac{d^4 g p \log \left (d+e x^2\right )}{8 e^4}+\frac{d f p x^4}{12 e}+\frac{d g p x^6}{24 e}-\frac{1}{18} f p x^6-\frac{1}{32} g p x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d^2*f*p*x^2)/(6*e^2) + (d^3*g*p*x^2)/(8*e^3) + (d*f*p*x^4)/(12*e) - (d^2*g*p*x^4)/(16*e^2) - (f*p*x^6)/18 +
(d*g*p*x^6)/(24*e) - (g*p*x^8)/32 + (d^3*f*p*Log[d + e*x^2])/(6*e^3) - (d^4*g*p*Log[d + e*x^2])/(8*e^4) + (f*x
^6*Log[c*(d + e*x^2)^p])/6 + (g*x^8*Log[c*(d + e*x^2)^p])/8

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Maple [C]  time = 0.567, size = 413, normalized size = 2.9 \begin{align*} \left ({\frac{g{x}^{8}}{8}}+{\frac{f{x}^{6}}{6}} \right ) \ln \left ( \left ( e{x}^{2}+d \right ) ^{p} \right ) -{\frac{i}{12}}\pi \,f{x}^{6}{\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}{12}}\pi \,f{x}^{6} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}+{\frac{i}{12}}\pi \,f{x}^{6}{\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}{16}}\pi \,g{x}^{8}{\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}{16}}\pi \,g{x}^{8} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}+{\frac{i}{12}}\pi \,f{x}^{6} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{16}}\pi \,g{x}^{8} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{16}}\pi \,g{x}^{8}{\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{\ln \left ( c \right ) g{x}^{8}}{8}}-{\frac{gp{x}^{8}}{32}}+{\frac{\ln \left ( c \right ) f{x}^{6}}{6}}+{\frac{dgp{x}^{6}}{24\,e}}-{\frac{fp{x}^{6}}{18}}-{\frac{{d}^{2}gp{x}^{4}}{16\,{e}^{2}}}+{\frac{dfp{x}^{4}}{12\,e}}+{\frac{{d}^{3}gp{x}^{2}}{8\,{e}^{3}}}-{\frac{p{d}^{2}f{x}^{2}}{6\,{e}^{2}}}-{\frac{\ln \left ( e{x}^{2}+d \right ){d}^{4}gp}{8\,{e}^{4}}}+{\frac{\ln \left ( e{x}^{2}+d \right ){d}^{3}fp}{6\,{e}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/8*g*x^8+1/6*f*x^6)*ln((e*x^2+d)^p)-1/12*I*Pi*f*x^6*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-1/12
*I*Pi*f*x^6*csgn(I*c*(e*x^2+d)^p)^3+1/12*I*Pi*f*x^6*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-1/16*I*Pi*g*x^
8*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-1/16*I*Pi*g*x^8*csgn(I*c*(e*x^2+d)^p)^3+1/12*I*Pi*f*x^6*
csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+1/16*I*Pi*g*x^8*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+1/16*I*Pi*g*x^8*csgn(I*(e*
x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+1/8*ln(c)*g*x^8-1/32*g*p*x^8+1/6*ln(c)*f*x^6+1/24/e*d*g*p*x^6-1/18*f*p*x^6-1
/16/e^2*d^2*g*p*x^4+1/12/e*d*f*p*x^4+1/8/e^3*d^3*g*p*x^2-1/6/e^2*d^2*f*p*x^2-1/8/e^4*ln(e*x^2+d)*d^4*g*p+1/6/e
^3*ln(e*x^2+d)*d^3*f*p

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Maxima [A]  time = 1.00222, size = 178, normalized size = 1.25 \begin{align*} -\frac{1}{288} \, e p{\left (\frac{9 \, e^{3} g x^{8} + 4 \,{\left (4 \, e^{3} f - 3 \, d e^{2} g\right )} x^{6} - 6 \,{\left (4 \, d e^{2} f - 3 \, d^{2} e g\right )} x^{4} + 12 \,{\left (4 \, d^{2} e f - 3 \, d^{3} g\right )} x^{2}}{e^{4}} - \frac{12 \,{\left (4 \, d^{3} e f - 3 \, d^{4} g\right )} \log \left (e x^{2} + d\right )}{e^{5}}\right )} + \frac{1}{24} \,{\left (3 \, g x^{8} + 4 \, f x^{6}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/288*e*p*((9*e^3*g*x^8 + 4*(4*e^3*f - 3*d*e^2*g)*x^6 - 6*(4*d*e^2*f - 3*d^2*e*g)*x^4 + 12*(4*d^2*e*f - 3*d^3
*g)*x^2)/e^4 - 12*(4*d^3*e*f - 3*d^4*g)*log(e*x^2 + d)/e^5) + 1/24*(3*g*x^8 + 4*f*x^6)*log((e*x^2 + d)^p*c)

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Fricas [A]  time = 1.98399, size = 339, normalized size = 2.39 \begin{align*} -\frac{9 \, e^{4} g p x^{8} + 4 \,{\left (4 \, e^{4} f - 3 \, d e^{3} g\right )} p x^{6} - 6 \,{\left (4 \, d e^{3} f - 3 \, d^{2} e^{2} g\right )} p x^{4} + 12 \,{\left (4 \, d^{2} e^{2} f - 3 \, d^{3} e g\right )} p x^{2} - 12 \,{\left (3 \, e^{4} g p x^{8} + 4 \, e^{4} f p x^{6} +{\left (4 \, d^{3} e f - 3 \, d^{4} g\right )} p\right )} \log \left (e x^{2} + d\right ) - 12 \,{\left (3 \, e^{4} g x^{8} + 4 \, e^{4} f x^{6}\right )} \log \left (c\right )}{288 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/288*(9*e^4*g*p*x^8 + 4*(4*e^4*f - 3*d*e^3*g)*p*x^6 - 6*(4*d*e^3*f - 3*d^2*e^2*g)*p*x^4 + 12*(4*d^2*e^2*f -
3*d^3*e*g)*p*x^2 - 12*(3*e^4*g*p*x^8 + 4*e^4*f*p*x^6 + (4*d^3*e*f - 3*d^4*g)*p)*log(e*x^2 + d) - 12*(3*e^4*g*x
^8 + 4*e^4*f*x^6)*log(c))/e^4

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

[Out]

Timed out

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Giac [B]  time = 1.17367, size = 416, normalized size = 2.93 \begin{align*} \frac{1}{288} \,{\left (36 \, g x^{8} e \log \left (c\right ) + 48 \, f x^{6} e \log \left (c\right ) + 8 \,{\left (6 \,{\left (x^{2} e + d\right )}^{3} e^{\left (-2\right )} \log \left (x^{2} e + d\right ) - 18 \,{\left (x^{2} e + d\right )}^{2} d e^{\left (-2\right )} \log \left (x^{2} e + d\right ) + 18 \,{\left (x^{2} e + d\right )} d^{2} e^{\left (-2\right )} \log \left (x^{2} e + d\right ) - 2 \,{\left (x^{2} e + d\right )}^{3} e^{\left (-2\right )} + 9 \,{\left (x^{2} e + d\right )}^{2} d e^{\left (-2\right )} - 18 \,{\left (x^{2} e + d\right )} d^{2} e^{\left (-2\right )}\right )} f p + 3 \,{\left (12 \,{\left (x^{2} e + d\right )}^{4} e^{\left (-3\right )} \log \left (x^{2} e + d\right ) - 48 \,{\left (x^{2} e + d\right )}^{3} d e^{\left (-3\right )} \log \left (x^{2} e + d\right ) + 72 \,{\left (x^{2} e + d\right )}^{2} d^{2} e^{\left (-3\right )} \log \left (x^{2} e + d\right ) - 48 \,{\left (x^{2} e + d\right )} d^{3} e^{\left (-3\right )} \log \left (x^{2} e + d\right ) - 3 \,{\left (x^{2} e + d\right )}^{4} e^{\left (-3\right )} + 16 \,{\left (x^{2} e + d\right )}^{3} d e^{\left (-3\right )} - 36 \,{\left (x^{2} e + d\right )}^{2} d^{2} e^{\left (-3\right )} + 48 \,{\left (x^{2} e + d\right )} d^{3} e^{\left (-3\right )}\right )} g p\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/288*(36*g*x^8*e*log(c) + 48*f*x^6*e*log(c) + 8*(6*(x^2*e + d)^3*e^(-2)*log(x^2*e + d) - 18*(x^2*e + d)^2*d*e
^(-2)*log(x^2*e + d) + 18*(x^2*e + d)*d^2*e^(-2)*log(x^2*e + d) - 2*(x^2*e + d)^3*e^(-2) + 9*(x^2*e + d)^2*d*e
^(-2) - 18*(x^2*e + d)*d^2*e^(-2))*f*p + 3*(12*(x^2*e + d)^4*e^(-3)*log(x^2*e + d) - 48*(x^2*e + d)^3*d*e^(-3)
*log(x^2*e + d) + 72*(x^2*e + d)^2*d^2*e^(-3)*log(x^2*e + d) - 48*(x^2*e + d)*d^3*e^(-3)*log(x^2*e + d) - 3*(x
^2*e + d)^4*e^(-3) + 16*(x^2*e + d)^3*d*e^(-3) - 36*(x^2*e + d)^2*d^2*e^(-3) + 48*(x^2*e + d)*d^3*e^(-3))*g*p)
*e^(-1)