3.324 \(\int x^3 (f+g x^2)^2 \log (c (d+e x^2)^p) \, dx\)

Optimal. Leaf size=210 \[ \frac{1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac{d^2 p \left (3 d^2 g^2-8 d e f g+6 e^2 f^2\right ) \log \left (d+e x^2\right )}{24 e^4}+\frac{d p x^2 (e f-d g)^2}{2 e^3}-\frac{g p \left (d+e x^2\right )^3 (2 e f-3 d g)}{18 e^4}-\frac{p \left (d+e x^2\right )^2 (e f-3 d g) (e f-d g)}{8 e^4}-\frac{g^2 p \left (d+e x^2\right )^4}{32 e^4} \]

[Out]

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

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Rubi [A]  time = 0.360555, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2475, 43, 2414, 12, 893} \[ \frac{1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac{d^2 p \left (3 d^2 g^2-8 d e f g+6 e^2 f^2\right ) \log \left (d+e x^2\right )}{24 e^4}+\frac{d p x^2 (e f-d g)^2}{2 e^3}-\frac{g p \left (d+e x^2\right )^3 (2 e f-3 d g)}{18 e^4}-\frac{p \left (d+e x^2\right )^2 (e f-3 d g) (e f-d g)}{8 e^4}-\frac{g^2 p \left (d+e x^2\right )^4}{32 e^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(f + g*x^2)^2*Log[c*(d + e*x^2)^p],x]

[Out]

(d*(e*f - d*g)^2*p*x^2)/(2*e^3) - ((e*f - 3*d*g)*(e*f - d*g)*p*(d + e*x^2)^2)/(8*e^4) - (g*(2*e*f - 3*d*g)*p*(
d + e*x^2)^3)/(18*e^4) - (g^2*p*(d + e*x^2)^4)/(32*e^4) - (d^2*(6*e^2*f^2 - 8*d*e*f*g + 3*d^2*g^2)*p*Log[d + e
*x^2])/(24*e^4) + (f^2*x^4*Log[c*(d + e*x^2)^p])/4 + (f*g*x^6*Log[c*(d + e*x^2)^p])/3 + (g^2*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 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (f+g x)^2 \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )\\ &=\frac{1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac{1}{2} (e p) \operatorname{Subst}\left (\int \frac{x^2 \left (6 f^2+8 f g x+3 g^2 x^2\right )}{12 (d+e x)} \, dx,x,x^2\right )\\ &=\frac{1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac{1}{24} (e p) \operatorname{Subst}\left (\int \frac{x^2 \left (6 f^2+8 f g x+3 g^2 x^2\right )}{d+e x} \, dx,x,x^2\right )\\ &=\frac{1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac{1}{24} (e p) \operatorname{Subst}\left (\int \left (-\frac{12 d (-e f+d g)^2}{e^4}+\frac{d^2 \left (6 e^2 f^2-8 d e f g+3 d^2 g^2\right )}{e^4 (d+e x)}+\frac{6 (e f-3 d g) (e f-d g) (d+e x)}{e^4}+\frac{4 g (2 e f-3 d g) (d+e x)^2}{e^4}+\frac{3 g^2 (d+e x)^3}{e^4}\right ) \, dx,x,x^2\right )\\ &=\frac{d (e f-d g)^2 p x^2}{2 e^3}-\frac{(e f-3 d g) (e f-d g) p \left (d+e x^2\right )^2}{8 e^4}-\frac{g (2 e f-3 d g) p \left (d+e x^2\right )^3}{18 e^4}-\frac{g^2 p \left (d+e x^2\right )^4}{32 e^4}-\frac{d^2 \left (6 e^2 f^2-8 d e f g+3 d^2 g^2\right ) p \log \left (d+e x^2\right )}{24 e^4}+\frac{1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )\\ \end{align*}

Mathematica [A]  time = 0.133462, size = 173, normalized size = 0.82 \[ \frac{12 e^4 x^4 \left (6 f^2+8 f g x^2+3 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )+e p x^2 \left (-6 d^2 e g \left (16 f+3 g x^2\right )+36 d^3 g^2+12 d e^2 \left (6 f^2+4 f g x^2+g^2 x^4\right )-e^3 x^2 \left (36 f^2+32 f g x^2+9 g^2 x^4\right )\right )-12 d^2 p \left (3 d^2 g^2-8 d e f g+6 e^2 f^2\right ) \log \left (d+e x^2\right )}{288 e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(f + g*x^2)^2*Log[c*(d + e*x^2)^p],x]

[Out]

(e*p*x^2*(36*d^3*g^2 - 6*d^2*e*g*(16*f + 3*g*x^2) + 12*d*e^2*(6*f^2 + 4*f*g*x^2 + g^2*x^4) - e^3*x^2*(36*f^2 +
32*f*g*x^2 + 9*g^2*x^4)) - 12*d^2*(6*e^2*f^2 - 8*d*e*f*g + 3*d^2*g^2)*p*Log[d + e*x^2] + 12*e^4*x^4*(6*f^2 +
8*f*g*x^2 + 3*g^2*x^4)*Log[c*(d + e*x^2)^p])/(288*e^4)

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Maple [C]  time = 0.585, size = 643, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(g*x^2+f)^2*ln(c*(e*x^2+d)^p),x)

[Out]

-1/6*I*Pi*f*g*x^6*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+1/24/e*d*g^2*p*x^6-1/16/e^2*d^2*g^2*p*x^
4+1/8/e^3*d^3*g^2*p*x^2+1/4/e*d*f^2*p*x^2-1/8/e^4*ln(e*x^2+d)*d^4*g^2*p-1/4/e^2*ln(e*x^2+d)*d^2*f^2*p-1/16*I*P
i*g^2*x^8*csgn(I*c*(e*x^2+d)^p)^3-1/8*I*Pi*f^2*x^4*csgn(I*c*(e*x^2+d)^p)^3-1/9*f*g*p*x^6+1/3*ln(c)*f*g*x^6-1/3
2*g^2*p*x^8-1/8*f^2*p*x^4+1/8*ln(c)*g^2*x^8+1/4*ln(c)*f^2*x^4+(1/8*g^2*x^8+1/3*f*g*x^6+1/4*f^2*x^4)*ln((e*x^2+
d)^p)+1/6/e*d*f*g*p*x^4-1/3/e^2*d^2*f*g*p*x^2+1/3/e^3*ln(e*x^2+d)*d^3*f*g*p+1/8*I*Pi*f^2*x^4*csgn(I*(e*x^2+d)^
p)*csgn(I*c*(e*x^2+d)^p)^2+1/8*I*Pi*f^2*x^4*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+1/16*I*Pi*g^2*x^8*csgn(I*(e*x^2+
d)^p)*csgn(I*c*(e*x^2+d)^p)^2+1/16*I*Pi*g^2*x^8*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-1/6*I*Pi*f*g*x^6*csgn(I*c*(e
*x^2+d)^p)^3+1/6*I*Pi*f*g*x^6*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-1/8*I*Pi*f^2*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*
(e*x^2+d)^p)*csgn(I*c)-1/16*I*Pi*g^2*x^8*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+1/6*I*Pi*f*g*x^6*
csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2

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Maxima [A]  time = 1.02316, size = 250, normalized size = 1.19 \begin{align*} -\frac{1}{288} \, e p{\left (\frac{9 \, e^{3} g^{2} x^{8} + 4 \,{\left (8 \, e^{3} f g - 3 \, d e^{2} g^{2}\right )} x^{6} + 6 \,{\left (6 \, e^{3} f^{2} - 8 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} x^{4} - 12 \,{\left (6 \, d e^{2} f^{2} - 8 \, d^{2} e f g + 3 \, d^{3} g^{2}\right )} x^{2}}{e^{4}} + \frac{12 \,{\left (6 \, d^{2} e^{2} f^{2} - 8 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} \log \left (e x^{2} + d\right )}{e^{5}}\right )} + \frac{1}{24} \,{\left (3 \, g^{2} x^{8} + 8 \, f g x^{6} + 6 \, f^{2} x^{4}\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^3*(g*x^2+f)^2*log(c*(e*x^2+d)^p),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.7163, size = 477, normalized size = 2.27 \begin{align*} -\frac{9 \, e^{4} g^{2} p x^{8} + 4 \,{\left (8 \, e^{4} f g - 3 \, d e^{3} g^{2}\right )} p x^{6} + 6 \,{\left (6 \, e^{4} f^{2} - 8 \, d e^{3} f g + 3 \, d^{2} e^{2} g^{2}\right )} p x^{4} - 12 \,{\left (6 \, d e^{3} f^{2} - 8 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} p x^{2} - 12 \,{\left (3 \, e^{4} g^{2} p x^{8} + 8 \, e^{4} f g p x^{6} + 6 \, e^{4} f^{2} p x^{4} -{\left (6 \, d^{2} e^{2} f^{2} - 8 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} p\right )} \log \left (e x^{2} + d\right ) - 12 \,{\left (3 \, e^{4} g^{2} x^{8} + 8 \, e^{4} f g x^{6} + 6 \, e^{4} f^{2} x^{4}\right )} \log \left (c\right )}{288 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(g*x^2+f)^2*log(c*(e*x^2+d)^p),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [B]  time = 1.31719, size = 564, normalized size = 2.69 \begin{align*} \frac{1}{288} \,{\left (36 \, g^{2} x^{8} e \log \left (c\right ) + 96 \, f g x^{6} e \log \left (c\right ) + 36 \,{\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 )} f^{2} p e^{\left (-1\right )} + 72 \,{\left ({\left (x^{2} e + d\right )}^{2} - 2 \,{\left (x^{2} e + d\right )} d\right )} f^{2} e^{\left (-1\right )} \log \left (c\right ) + 16 \,{\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 g 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^{2} p\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(g*x^2+f)^2*log(c*(e*x^2+d)^p),x, algorithm="giac")

[Out]

1/288*(36*g^2*x^8*e*log(c) + 96*f*g*x^6*e*log(c) + 36*(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)*f^2*p*e^(-1) + 72*((x^2*e + d)^2 - 2*(x^2*e + d)*d)*f^2*e^(-1)*log
(c) + 16*(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*g*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^2*p)*e^(-1)