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

Optimal. Leaf size=231 \[ f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{d^2 f g p \log \left (d+e x^2\right )}{2 e^2}-\frac{2 d^2 g^2 p x^3}{21 e^2}+\frac{2 d^3 g^2 p x}{7 e^3}-\frac{2 d^{7/2} g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}+\frac{2 \sqrt{d} f^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{d f g p x^2}{2 e}+\frac{2 d g^2 p x^5}{35 e}-2 f^2 p x-\frac{1}{4} f g p x^4-\frac{2}{49} g^2 p x^7 \]

[Out]

-2*f^2*p*x + (2*d^3*g^2*p*x)/(7*e^3) + (d*f*g*p*x^2)/(2*e) - (2*d^2*g^2*p*x^3)/(21*e^2) - (f*g*p*x^4)/4 + (2*d
*g^2*p*x^5)/(35*e) - (2*g^2*p*x^7)/49 + (2*Sqrt[d]*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*d^(7/2)*g^2
*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(7*e^(7/2)) - (d^2*f*g*p*Log[d + e*x^2])/(2*e^2) + f^2*x*Log[c*(d + e*x^2)^p]
+ (f*g*x^4*Log[c*(d + e*x^2)^p])/2 + (g^2*x^7*Log[c*(d + e*x^2)^p])/7

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Rubi [A]  time = 0.176745, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {2471, 2448, 321, 205, 2454, 2395, 43, 2455, 302} \[ f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{d^2 f g p \log \left (d+e x^2\right )}{2 e^2}-\frac{2 d^2 g^2 p x^3}{21 e^2}+\frac{2 d^3 g^2 p x}{7 e^3}-\frac{2 d^{7/2} g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}+\frac{2 \sqrt{d} f^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{d f g p x^2}{2 e}+\frac{2 d g^2 p x^5}{35 e}-2 f^2 p x-\frac{1}{4} f g p x^4-\frac{2}{49} g^2 p x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*f^2*p*x + (2*d^3*g^2*p*x)/(7*e^3) + (d*f*g*p*x^2)/(2*e) - (2*d^2*g^2*p*x^3)/(21*e^2) - (f*g*p*x^4)/4 + (2*d
*g^2*p*x^5)/(35*e) - (2*g^2*p*x^7)/49 + (2*Sqrt[d]*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*d^(7/2)*g^2
*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(7*e^(7/2)) - (d^2*f*g*p*Log[d + e*x^2])/(2*e^2) + f^2*x*Log[c*(d + e*x^2)^p]
+ (f*g*x^4*Log[c*(d + e*x^2)^p])/2 + (g^2*x^7*Log[c*(d + e*x^2)^p])/7

Rule 2471

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol]
:> With[{t = ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; Free
Q[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[
q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))

Rule 2448

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 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])

Rule 2455

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rubi steps

\begin{align*} \int \left (f+g x^3\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f^2 \log \left (c \left (d+e x^2\right )^p\right )+2 f g x^3 \log \left (c \left (d+e x^2\right )^p\right )+g^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f^2 \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx+(2 f g) \int x^3 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+g^2 \int x^6 \log \left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+(f g) \operatorname{Subst}\left (\int x \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )-\left (2 e f^2 p\right ) \int \frac{x^2}{d+e x^2} \, dx-\frac{1}{7} \left (2 e g^2 p\right ) \int \frac{x^8}{d+e x^2} \, dx\\ &=-2 f^2 p x+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+\left (2 d f^2 p\right ) \int \frac{1}{d+e x^2} \, dx-\frac{1}{2} (e f g p) \operatorname{Subst}\left (\int \frac{x^2}{d+e x} \, dx,x,x^2\right )-\frac{1}{7} \left (2 e g^2 p\right ) \int \left (-\frac{d^3}{e^4}+\frac{d^2 x^2}{e^3}-\frac{d x^4}{e^2}+\frac{x^6}{e}+\frac{d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx\\ &=-2 f^2 p x+\frac{2 d^3 g^2 p x}{7 e^3}-\frac{2 d^2 g^2 p x^3}{21 e^2}+\frac{2 d g^2 p x^5}{35 e}-\frac{2}{49} g^2 p x^7+\frac{2 \sqrt{d} f^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{1}{2} (e f g 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 )-\frac{\left (2 d^4 g^2 p\right ) \int \frac{1}{d+e x^2} \, dx}{7 e^3}\\ &=-2 f^2 p x+\frac{2 d^3 g^2 p x}{7 e^3}+\frac{d f g p x^2}{2 e}-\frac{2 d^2 g^2 p x^3}{21 e^2}-\frac{1}{4} f g p x^4+\frac{2 d g^2 p x^5}{35 e}-\frac{2}{49} g^2 p x^7+\frac{2 \sqrt{d} f^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{2 d^{7/2} g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}-\frac{d^2 f g p \log \left (d+e x^2\right )}{2 e^2}+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )\\ \end{align*}

Mathematica [A]  time = 0.207592, size = 178, normalized size = 0.77 \[ \frac{1}{14} x \left (14 f^2+7 f g x^3+2 g^2 x^6\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac{p x \left (-280 d^2 e g^2 x^2+840 d^3 g^2+42 d e^2 g x \left (35 f+4 g x^3\right )-15 e^3 \left (392 f^2+49 f g x^3+8 g^2 x^6\right )\right )}{2940 e^3}-\frac{2 \sqrt{d} p \left (d^3 g^2-7 e^3 f^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}-\frac{d^2 f g p \log \left (d+e x^2\right )}{2 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(p*x*(840*d^3*g^2 - 280*d^2*e*g^2*x^2 + 42*d*e^2*g*x*(35*f + 4*g*x^3) - 15*e^3*(392*f^2 + 49*f*g*x^3 + 8*g^2*x
^6)))/(2940*e^3) - (2*Sqrt[d]*(-7*e^3*f^2 + d^3*g^2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(7*e^(7/2)) - (d^2*f*g*p*L
og[d + e*x^2])/(2*e^2) + (x*(14*f^2 + 7*f*g*x^3 + 2*g^2*x^6)*Log[c*(d + e*x^2)^p])/14

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Maple [C]  time = 0.723, size = 869, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*ln(c)*g^2*x^7+2/7*d^3*g^2*p*x/e^3-2/21*d^2*g^2*p*x^3/e^2+2/35*d*g^2*p*x^5/e+ln(c)*f^2*x-1/14*I*Pi*g^2*x^7*
csgn(I*c*(e*x^2+d)^p)^3+(1/7*g^2*x^7+1/2*f*g*x^4+f^2*x)*ln((e*x^2+d)^p)-2/49*g^2*p*x^7-1/4*f*g*p*x^4-2*f^2*p*x
+1/2*ln(c)*f*g*x^4+1/7/e^4*p*ln(-d^4*g^2+7*d*e^3*f^2-(-d^7*e*g^4+14*d^4*e^4*f^2*g^2-49*d*e^7*f^4)^(1/2)*x)*(-d
^7*e*g^4+14*d^4*e^4*f^2*g^2-49*d*e^7*f^4)^(1/2)-1/7/e^4*p*ln(-d^4*g^2+7*d*e^3*f^2+(-d^7*e*g^4+14*d^4*e^4*f^2*g
^2-49*d*e^7*f^4)^(1/2)*x)*(-d^7*e*g^4+14*d^4*e^4*f^2*g^2-49*d*e^7*f^4)^(1/2)-1/14*I*Pi*g^2*x^7*csgn(I*(e*x^2+d
)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+1/4*I*Pi*f*g*x^4*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+1/4*I*Pi*f*g*x^4*csgn(
I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-1/4*I*Pi*f*g*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-1/
2*I*Pi*f^2*csgn(I*c*(e*x^2+d)^p)^3*x+1/2*d*f*g*p*x^2/e+1/2*I*Pi*f^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^
2*x+1/2*I*Pi*f^2*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)*x-1/2/e^2*p*ln(-d^4*g^2+7*d*e^3*f^2-(-d^7*e*g^4+14*d^4*e^4*
f^2*g^2-49*d*e^7*f^4)^(1/2)*x)*d^2*f*g-1/4*I*Pi*f*g*x^4*csgn(I*c*(e*x^2+d)^p)^3+1/14*I*Pi*g^2*x^7*csgn(I*(e*x^
2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+1/14*I*Pi*g^2*x^7*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-1/2/e^2*p*ln(-d^4*g^2+7*d*
e^3*f^2+(-d^7*e*g^4+14*d^4*e^4*f^2*g^2-49*d*e^7*f^4)^(1/2)*x)*d^2*f*g-1/2*I*Pi*f^2*csgn(I*(e*x^2+d)^p)*csgn(I*
c*(e*x^2+d)^p)*csgn(I*c)*x

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.02251, size = 1027, normalized size = 4.45 \begin{align*} \left [-\frac{120 \, e^{3} g^{2} p x^{7} - 168 \, d e^{2} g^{2} p x^{5} + 735 \, e^{3} f g p x^{4} + 280 \, d^{2} e g^{2} p x^{3} - 1470 \, d e^{2} f g p x^{2} + 420 \,{\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p \sqrt{-\frac{d}{e}} \log \left (\frac{e x^{2} - 2 \, e x \sqrt{-\frac{d}{e}} - d}{e x^{2} + d}\right ) + 840 \,{\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p x - 210 \,{\left (2 \, e^{3} g^{2} p x^{7} + 7 \, e^{3} f g p x^{4} + 14 \, e^{3} f^{2} p x - 7 \, d^{2} e f g p\right )} \log \left (e x^{2} + d\right ) - 210 \,{\left (2 \, e^{3} g^{2} x^{7} + 7 \, e^{3} f g x^{4} + 14 \, e^{3} f^{2} x\right )} \log \left (c\right )}{2940 \, e^{3}}, -\frac{120 \, e^{3} g^{2} p x^{7} - 168 \, d e^{2} g^{2} p x^{5} + 735 \, e^{3} f g p x^{4} + 280 \, d^{2} e g^{2} p x^{3} - 1470 \, d e^{2} f g p x^{2} - 840 \,{\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p \sqrt{\frac{d}{e}} \arctan \left (\frac{e x \sqrt{\frac{d}{e}}}{d}\right ) + 840 \,{\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p x - 210 \,{\left (2 \, e^{3} g^{2} p x^{7} + 7 \, e^{3} f g p x^{4} + 14 \, e^{3} f^{2} p x - 7 \, d^{2} e f g p\right )} \log \left (e x^{2} + d\right ) - 210 \,{\left (2 \, e^{3} g^{2} x^{7} + 7 \, e^{3} f g x^{4} + 14 \, e^{3} f^{2} x\right )} \log \left (c\right )}{2940 \, e^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2940*(120*e^3*g^2*p*x^7 - 168*d*e^2*g^2*p*x^5 + 735*e^3*f*g*p*x^4 + 280*d^2*e*g^2*p*x^3 - 1470*d*e^2*f*g*p
*x^2 + 420*(7*e^3*f^2 - d^3*g^2)*p*sqrt(-d/e)*log((e*x^2 - 2*e*x*sqrt(-d/e) - d)/(e*x^2 + d)) + 840*(7*e^3*f^2
- d^3*g^2)*p*x - 210*(2*e^3*g^2*p*x^7 + 7*e^3*f*g*p*x^4 + 14*e^3*f^2*p*x - 7*d^2*e*f*g*p)*log(e*x^2 + d) - 21
0*(2*e^3*g^2*x^7 + 7*e^3*f*g*x^4 + 14*e^3*f^2*x)*log(c))/e^3, -1/2940*(120*e^3*g^2*p*x^7 - 168*d*e^2*g^2*p*x^5
+ 735*e^3*f*g*p*x^4 + 280*d^2*e*g^2*p*x^3 - 1470*d*e^2*f*g*p*x^2 - 840*(7*e^3*f^2 - d^3*g^2)*p*sqrt(d/e)*arct
an(e*x*sqrt(d/e)/d) + 840*(7*e^3*f^2 - d^3*g^2)*p*x - 210*(2*e^3*g^2*p*x^7 + 7*e^3*f*g*p*x^4 + 14*e^3*f^2*p*x
- 7*d^2*e*f*g*p)*log(e*x^2 + d) - 210*(2*e^3*g^2*x^7 + 7*e^3*f*g*x^4 + 14*e^3*f^2*x)*log(c))/e^3]

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

[Out]

Timed out

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Giac [A]  time = 1.22128, size = 304, normalized size = 1.32 \begin{align*} -\frac{1}{2} \, d^{2} f g p e^{\left (-2\right )} \log \left (x^{2} e + d\right ) - \frac{2 \,{\left (d^{4} g^{2} p - 7 \, d f^{2} p e^{3}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{7}{2}\right )}}{7 \, \sqrt{d}} + \frac{1}{2940} \,{\left (420 \, g^{2} p x^{7} e^{3} \log \left (x^{2} e + d\right ) - 120 \, g^{2} p x^{7} e^{3} + 420 \, g^{2} x^{7} e^{3} \log \left (c\right ) + 168 \, d g^{2} p x^{5} e^{2} - 280 \, d^{2} g^{2} p x^{3} e + 1470 \, f g p x^{4} e^{3} \log \left (x^{2} e + d\right ) - 735 \, f g p x^{4} e^{3} + 1470 \, f g x^{4} e^{3} \log \left (c\right ) + 840 \, d^{3} g^{2} p x + 1470 \, d f g p x^{2} e^{2} + 2940 \, f^{2} p x e^{3} \log \left (x^{2} e + d\right ) - 5880 \, f^{2} p x e^{3} + 2940 \, f^{2} x e^{3} \log \left (c\right )\right )} e^{\left (-3\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*d^2*f*g*p*e^(-2)*log(x^2*e + d) - 2/7*(d^4*g^2*p - 7*d*f^2*p*e^3)*arctan(x*e^(1/2)/sqrt(d))*e^(-7/2)/sqrt
(d) + 1/2940*(420*g^2*p*x^7*e^3*log(x^2*e + d) - 120*g^2*p*x^7*e^3 + 420*g^2*x^7*e^3*log(c) + 168*d*g^2*p*x^5*
e^2 - 280*d^2*g^2*p*x^3*e + 1470*f*g*p*x^4*e^3*log(x^2*e + d) - 735*f*g*p*x^4*e^3 + 1470*f*g*x^4*e^3*log(c) +
840*d^3*g^2*p*x + 1470*d*f*g*p*x^2*e^2 + 2940*f^2*p*x*e^3*log(x^2*e + d) - 5880*f^2*p*x*e^3 + 2940*f^2*x*e^3*l
og(c))*e^(-3)