3.3.88 \(\int (f+g x^3)^3 \log (c (d+e x^2)^p) \, dx\) [288]

Optimal. Leaf size=366 \[ -2 f^3 p x+\frac {6 d^3 f g^2 p x}{7 e^3}+\frac {3 d f^2 g p x^2}{4 e}-\frac {d^4 g^3 p x^2}{10 e^4}-\frac {2 d^2 f g^2 p x^3}{7 e^2}-\frac {3}{8} f^2 g p x^4+\frac {d^3 g^3 p x^4}{20 e^3}+\frac {6 d f g^2 p x^5}{35 e}-\frac {d^2 g^3 p x^6}{30 e^2}-\frac {6}{49} f g^2 p x^7+\frac {d g^3 p x^8}{40 e}-\frac {1}{50} g^3 p x^{10}+\frac {2 \sqrt {d} f^3 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {6 d^{7/2} f g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}-\frac {3 d^2 f^2 g p \log \left (d+e x^2\right )}{4 e^2}+\frac {d^5 g^3 p \log \left (d+e x^2\right )}{10 e^5}+f^3 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {3}{4} f^2 g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log \left (c \left (d+e x^2\right )^p\right ) \]

[Out]

-2*f^3*p*x+6/7*d^3*f*g^2*p*x/e^3+3/4*d*f^2*g*p*x^2/e-1/10*d^4*g^3*p*x^2/e^4-2/7*d^2*f*g^2*p*x^3/e^2-3/8*f^2*g*
p*x^4+1/20*d^3*g^3*p*x^4/e^3+6/35*d*f*g^2*p*x^5/e-1/30*d^2*g^3*p*x^6/e^2-6/49*f*g^2*p*x^7+1/40*d*g^3*p*x^8/e-1
/50*g^3*p*x^10-6/7*d^(7/2)*f*g^2*p*arctan(x*e^(1/2)/d^(1/2))/e^(7/2)-3/4*d^2*f^2*g*p*ln(e*x^2+d)/e^2+1/10*d^5*
g^3*p*ln(e*x^2+d)/e^5+f^3*x*ln(c*(e*x^2+d)^p)+3/4*f^2*g*x^4*ln(c*(e*x^2+d)^p)+3/7*f*g^2*x^7*ln(c*(e*x^2+d)^p)+
1/10*g^3*x^10*ln(c*(e*x^2+d)^p)+2*f^3*p*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)/e^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.21, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {2521, 2498, 327, 211, 2504, 2442, 45, 2505, 308} \begin {gather*} -\frac {6 d^{7/2} f g^2 p \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}+\frac {2 \sqrt {d} f^3 p \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+f^3 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {3}{4} f^2 g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log \left (c \left (d+e x^2\right )^p\right )+\frac {d^5 g^3 p \log \left (d+e x^2\right )}{10 e^5}-\frac {d^4 g^3 p x^2}{10 e^4}+\frac {6 d^3 f g^2 p x}{7 e^3}+\frac {d^3 g^3 p x^4}{20 e^3}-\frac {3 d^2 f^2 g p \log \left (d+e x^2\right )}{4 e^2}-\frac {2 d^2 f g^2 p x^3}{7 e^2}-\frac {d^2 g^3 p x^6}{30 e^2}+\frac {3 d f^2 g p x^2}{4 e}+\frac {6 d f g^2 p x^5}{35 e}+\frac {d g^3 p x^8}{40 e}-2 f^3 p x-\frac {3}{8} f^2 g p x^4-\frac {6}{49} f g^2 p x^7-\frac {1}{50} g^3 p x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*f^3*p*x + (6*d^3*f*g^2*p*x)/(7*e^3) + (3*d*f^2*g*p*x^2)/(4*e) - (d^4*g^3*p*x^2)/(10*e^4) - (2*d^2*f*g^2*p*x
^3)/(7*e^2) - (3*f^2*g*p*x^4)/8 + (d^3*g^3*p*x^4)/(20*e^3) + (6*d*f*g^2*p*x^5)/(35*e) - (d^2*g^3*p*x^6)/(30*e^
2) - (6*f*g^2*p*x^7)/49 + (d*g^3*p*x^8)/(40*e) - (g^3*p*x^10)/50 + (2*Sqrt[d]*f^3*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]
])/Sqrt[e] - (6*d^(7/2)*f*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(7*e^(7/2)) - (3*d^2*f^2*g*p*Log[d + e*x^2])/(4*e
^2) + (d^5*g^3*p*Log[d + e*x^2])/(10*e^5) + f^3*x*Log[c*(d + e*x^2)^p] + (3*f^2*g*x^4*Log[c*(d + e*x^2)^p])/4
+ (3*f*g^2*x^7*Log[c*(d + e*x^2)^p])/7 + (g^3*x^10*Log[c*(d + e*x^2)^p])/10

Rule 45

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 211

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

Rule 308

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]

Rule 327

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 2442

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 2498

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 2504

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 2505

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 2521

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]))

Rubi steps

\begin {align*} \int \left (f+g x^3\right )^3 \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f^3 \log \left (c \left (d+e x^2\right )^p\right )+3 f^2 g x^3 \log \left (c \left (d+e x^2\right )^p\right )+3 f g^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+g^3 x^9 \log \left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f^3 \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\left (3 f^2 g\right ) \int x^3 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\left (3 f g^2\right ) \int x^6 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+g^3 \int x^9 \log \left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=f^3 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} \left (3 f^2 g\right ) \text {Subst}\left (\int x \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )+\frac {1}{2} g^3 \text {Subst}\left (\int x^4 \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )-\left (2 e f^3 p\right ) \int \frac {x^2}{d+e x^2} \, dx-\frac {1}{7} \left (6 e f g^2 p\right ) \int \frac {x^8}{d+e x^2} \, dx\\ &=-2 f^3 p x+f^3 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {3}{4} f^2 g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log \left (c \left (d+e x^2\right )^p\right )+\left (2 d f^3 p\right ) \int \frac {1}{d+e x^2} \, dx-\frac {1}{4} \left (3 e f^2 g p\right ) \text {Subst}\left (\int \frac {x^2}{d+e x} \, dx,x,x^2\right )-\frac {1}{7} \left (6 e f 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-\frac {1}{10} \left (e g^3 p\right ) \text {Subst}\left (\int \frac {x^5}{d+e x} \, dx,x,x^2\right )\\ &=-2 f^3 p x+\frac {6 d^3 f g^2 p x}{7 e^3}-\frac {2 d^2 f g^2 p x^3}{7 e^2}+\frac {6 d f g^2 p x^5}{35 e}-\frac {6}{49} f g^2 p x^7+\frac {2 \sqrt {d} f^3 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+f^3 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {3}{4} f^2 g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{4} \left (3 e f^2 g p\right ) \text {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 (6 d^4 f g^2 p\right ) \int \frac {1}{d+e x^2} \, dx}{7 e^3}-\frac {1}{10} \left (e g^3 p\right ) \text {Subst}\left (\int \left (\frac {d^4}{e^5}-\frac {d^3 x}{e^4}+\frac {d^2 x^2}{e^3}-\frac {d x^3}{e^2}+\frac {x^4}{e}-\frac {d^5}{e^5 (d+e x)}\right ) \, dx,x,x^2\right )\\ &=-2 f^3 p x+\frac {6 d^3 f g^2 p x}{7 e^3}+\frac {3 d f^2 g p x^2}{4 e}-\frac {d^4 g^3 p x^2}{10 e^4}-\frac {2 d^2 f g^2 p x^3}{7 e^2}-\frac {3}{8} f^2 g p x^4+\frac {d^3 g^3 p x^4}{20 e^3}+\frac {6 d f g^2 p x^5}{35 e}-\frac {d^2 g^3 p x^6}{30 e^2}-\frac {6}{49} f g^2 p x^7+\frac {d g^3 p x^8}{40 e}-\frac {1}{50} g^3 p x^{10}+\frac {2 \sqrt {d} f^3 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {6 d^{7/2} f g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}-\frac {3 d^2 f^2 g p \log \left (d+e x^2\right )}{4 e^2}+\frac {d^5 g^3 p \log \left (d+e x^2\right )}{10 e^5}+f^3 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {3}{4} f^2 g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log \left (c \left (d+e x^2\right )^p\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.18, size = 258, normalized size = 0.70 \begin {gather*} \frac {-e p x \left (2940 d^4 g^3 x+140 d^2 e^2 g^2 x^2 \left (60 f+7 g x^3\right )-210 d^3 e g^2 \left (120 f+7 g x^3\right )-105 d e^3 g x \left (210 f^2+48 f g x^3+7 g^2 x^6\right )+3 e^4 \left (19600 f^3+3675 f^2 g x^3+1200 f g^2 x^6+196 g^3 x^9\right )\right )-8400 \sqrt {d} e^{3/2} f \left (-7 e^3 f^2+3 d^3 g^2\right ) p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+1470 d^2 g \left (-15 e^3 f^2+2 d^3 g^2\right ) p \log \left (d+e x^2\right )+210 e^5 x \left (140 f^3+105 f^2 g x^3+60 f g^2 x^6+14 g^3 x^9\right ) \log \left (c \left (d+e x^2\right )^p\right )}{29400 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(e*p*x*(2940*d^4*g^3*x + 140*d^2*e^2*g^2*x^2*(60*f + 7*g*x^3) - 210*d^3*e*g^2*(120*f + 7*g*x^3) - 105*d*e^3*
g*x*(210*f^2 + 48*f*g*x^3 + 7*g^2*x^6) + 3*e^4*(19600*f^3 + 3675*f^2*g*x^3 + 1200*f*g^2*x^6 + 196*g^3*x^9))) -
8400*Sqrt[d]*e^(3/2)*f*(-7*e^3*f^2 + 3*d^3*g^2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + 1470*d^2*g*(-15*e^3*f^2 + 2*d
^3*g^2)*p*Log[d + e*x^2] + 210*e^5*x*(140*f^3 + 105*f^2*g*x^3 + 60*f*g^2*x^6 + 14*g^3*x^9)*Log[c*(d + e*x^2)^p
])/(29400*e^5)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.99, size = 1311, normalized size = 3.58

method	result	size
risch	\(\text {Expression too large to display}\)	\(1311\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7/e^5*p*ln(-3*d^4*e*f*g^2+7*d*e^4*f^3+(-9*d^7*e^3*f^2*g^4+42*d^4*e^6*f^4*g^2-49*d*e^9*f^6)^(1/2)*x)*(-9*d^7
*e^3*f^2*g^4+42*d^4*e^6*f^4*g^2-49*d*e^9*f^6)^(1/2)+1/7/e^5*p*ln(-3*d^4*e*f*g^2+7*d*e^4*f^3-(-9*d^7*e^3*f^2*g^
4+42*d^4*e^6*f^4*g^2-49*d*e^9*f^6)^(1/2)*x)*(-9*d^7*e^3*f^2*g^4+42*d^4*e^6*f^4*g^2-49*d*e^9*f^6)^(1/2)-1/10*d^
4*g^3*p*x^2/e^4+1/20*d^3*g^3*p*x^4/e^3-1/30*d^2*g^3*p*x^6/e^2+1/40*d*g^3*p*x^8/e-3/8*I*Pi*f^2*g*x^4*csgn(I*(e*
x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-3/14*I*Pi*f*g^2*x^7*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I
*c)+3/4*ln(c)*f^2*g*x^4+3/7*ln(c)*f*g^2*x^7-2*f^3*p*x-3/4/e^2*p*ln(-3*d^4*e*f*g^2+7*d*e^4*f^3-(-9*d^7*e^3*f^2*
g^4+42*d^4*e^6*f^4*g^2-49*d*e^9*f^6)^(1/2)*x)*d^2*f^2*g-3/4/e^2*p*ln(-3*d^4*e*f*g^2+7*d*e^4*f^3+(-9*d^7*e^3*f^
2*g^4+42*d^4*e^6*f^4*g^2-49*d*e^9*f^6)^(1/2)*x)*d^2*f^2*g+1/10*ln(c)*g^3*x^10+ln(c)*f^3*x+6/35*d*f*g^2*p*x^5/e
+1/10/e^5*p*ln(-3*d^4*e*f*g^2+7*d*e^4*f^3-(-9*d^7*e^3*f^2*g^4+42*d^4*e^6*f^4*g^2-49*d*e^9*f^6)^(1/2)*x)*d^5*g^
3+1/10/e^5*p*ln(-3*d^4*e*f*g^2+7*d*e^4*f^3+(-9*d^7*e^3*f^2*g^4+42*d^4*e^6*f^4*g^2-49*d*e^9*f^6)^(1/2)*x)*d^5*g
^3-1/2*I*Pi*f^3*csgn(I*c*(e*x^2+d)^p)^3*x-1/20*I*Pi*g^3*x^10*csgn(I*c*(e*x^2+d)^p)^3+6/7*d^3*f*g^2*p*x/e^3+3/4
*d*f^2*g*p*x^2/e-2/7*d^2*f*g^2*p*x^3/e^2+3/8*I*Pi*f^2*g*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+3/14*I
*Pi*f*g^2*x^7*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+3/14*I*Pi*f*g^2*x^7*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c
)-1/2*I*Pi*f^3*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)*x+(1/10*g^3*x^10+3/7*f*g^2*x^7+3/4*f^2*g*x^
4+f^3*x)*ln((e*x^2+d)^p)-1/50*g^3*p*x^10+3/8*I*Pi*f^2*g*x^4*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-1/20*I*Pi*g^3*x^
10*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+1/20*I*Pi*g^3*x^10*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+
d)^p)^2+1/2*I*Pi*f^3*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2*x+1/20*I*Pi*g^3*x^10*csgn(I*c*(e*x^2+d)^p)^2*
csgn(I*c)+1/2*I*Pi*f^3*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)*x-3/14*I*Pi*f*g^2*x^7*csgn(I*c*(e*x^2+d)^p)^3-3/8*I*P
i*f^2*g*x^4*csgn(I*c*(e*x^2+d)^p)^3-3/8*f^2*g*p*x^4-6/49*f*g^2*p*x^7

________________________________________________________________________________________

Maxima [A]
time = 0.55, size = 266, normalized size = 0.73 \begin {gather*} \frac {1}{29400} \, {\left (1470 \, {\left (2 \, d^{5} g^{3} - 15 \, d^{2} f^{2} g e^{3}\right )} e^{\left (-6\right )} \log \left (x^{2} e + d\right ) - \frac {8400 \, {\left (3 \, d^{4} f g^{2} - 7 \, d f^{3} e^{3}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {9}{2}\right )}}{\sqrt {d}} - {\left (588 \, g^{3} x^{10} e^{4} - 735 \, d g^{3} x^{8} e^{3} + 980 \, d^{2} g^{3} x^{6} e^{2} + 3600 \, f g^{2} x^{7} e^{4} - 5040 \, d f g^{2} x^{5} e^{3} + 8400 \, d^{2} f g^{2} x^{3} e^{2} - 735 \, {\left (2 \, d^{3} g^{3} e - 15 \, f^{2} g e^{4}\right )} x^{4} + 1470 \, {\left (2 \, d^{4} g^{3} - 15 \, d f^{2} g e^{3}\right )} x^{2} - 8400 \, {\left (3 \, d^{3} f g^{2} e - 7 \, f^{3} e^{4}\right )} x\right )} e^{\left (-5\right )}\right )} p e + \frac {1}{140} \, {\left (14 \, g^{3} x^{10} + 60 \, f g^{2} x^{7} + 105 \, f^{2} g x^{4} + 140 \, f^{3} x\right )} \log \left ({\left (x^{2} e + d\right )}^{p} c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/29400*(1470*(2*d^5*g^3 - 15*d^2*f^2*g*e^3)*e^(-6)*log(x^2*e + d) - 8400*(3*d^4*f*g^2 - 7*d*f^3*e^3)*arctan(x
*e^(1/2)/sqrt(d))*e^(-9/2)/sqrt(d) - (588*g^3*x^10*e^4 - 735*d*g^3*x^8*e^3 + 980*d^2*g^3*x^6*e^2 + 3600*f*g^2*
x^7*e^4 - 5040*d*f*g^2*x^5*e^3 + 8400*d^2*f*g^2*x^3*e^2 - 735*(2*d^3*g^3*e - 15*f^2*g*e^4)*x^4 + 1470*(2*d^4*g
^3 - 15*d*f^2*g*e^3)*x^2 - 8400*(3*d^3*f*g^2*e - 7*f^3*e^4)*x)*e^(-5))*p*e + 1/140*(14*g^3*x^10 + 60*f*g^2*x^7
+ 105*f^2*g*x^4 + 140*f^3*x)*log((x^2*e + d)^p*c)

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 635, normalized size = 1.73 \begin {gather*} \left [-\frac {1}{29400} \, {\left (2940 \, d^{4} g^{3} p x^{2} e - 210 \, {\left (14 \, g^{3} x^{10} + 60 \, f g^{2} x^{7} + 105 \, f^{2} g x^{4} + 140 \, f^{3} x\right )} e^{5} \log \left (c\right ) - 4200 \, {\left (3 \, d^{3} f g^{2} p e^{2} - 7 \, f^{3} p e^{5}\right )} \sqrt {-d e^{\left (-1\right )}} \log \left (\frac {x^{2} e - 2 \, \sqrt {-d e^{\left (-1\right )}} x e - d}{x^{2} e + d}\right ) + 3 \, {\left (196 \, g^{3} p x^{10} + 1200 \, f g^{2} p x^{7} + 3675 \, f^{2} g p x^{4} + 19600 \, f^{3} p x\right )} e^{5} - 105 \, {\left (7 \, d g^{3} p x^{8} + 48 \, d f g^{2} p x^{5} + 210 \, d f^{2} g p x^{2}\right )} e^{4} + 140 \, {\left (7 \, d^{2} g^{3} p x^{6} + 60 \, d^{2} f g^{2} p x^{3}\right )} e^{3} - 210 \, {\left (7 \, d^{3} g^{3} p x^{4} + 120 \, d^{3} f g^{2} p x\right )} e^{2} - 210 \, {\left (14 \, d^{5} g^{3} p - 105 \, d^{2} f^{2} g p e^{3} + {\left (14 \, g^{3} p x^{10} + 60 \, f g^{2} p x^{7} + 105 \, f^{2} g p x^{4} + 140 \, f^{3} p x\right )} e^{5}\right )} \log \left (x^{2} e + d\right )\right )} e^{\left (-5\right )}, -\frac {1}{29400} \, {\left (2940 \, d^{4} g^{3} p x^{2} e + 8400 \, {\left (3 \, d^{3} f g^{2} p e^{2} - 7 \, f^{3} p e^{5}\right )} \sqrt {d} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )} - 210 \, {\left (14 \, g^{3} x^{10} + 60 \, f g^{2} x^{7} + 105 \, f^{2} g x^{4} + 140 \, f^{3} x\right )} e^{5} \log \left (c\right ) + 3 \, {\left (196 \, g^{3} p x^{10} + 1200 \, f g^{2} p x^{7} + 3675 \, f^{2} g p x^{4} + 19600 \, f^{3} p x\right )} e^{5} - 105 \, {\left (7 \, d g^{3} p x^{8} + 48 \, d f g^{2} p x^{5} + 210 \, d f^{2} g p x^{2}\right )} e^{4} + 140 \, {\left (7 \, d^{2} g^{3} p x^{6} + 60 \, d^{2} f g^{2} p x^{3}\right )} e^{3} - 210 \, {\left (7 \, d^{3} g^{3} p x^{4} + 120 \, d^{3} f g^{2} p x\right )} e^{2} - 210 \, {\left (14 \, d^{5} g^{3} p - 105 \, d^{2} f^{2} g p e^{3} + {\left (14 \, g^{3} p x^{10} + 60 \, f g^{2} p x^{7} + 105 \, f^{2} g p x^{4} + 140 \, f^{3} p x\right )} e^{5}\right )} \log \left (x^{2} e + d\right )\right )} e^{\left (-5\right )}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/29400*(2940*d^4*g^3*p*x^2*e - 210*(14*g^3*x^10 + 60*f*g^2*x^7 + 105*f^2*g*x^4 + 140*f^3*x)*e^5*log(c) - 42
00*(3*d^3*f*g^2*p*e^2 - 7*f^3*p*e^5)*sqrt(-d*e^(-1))*log((x^2*e - 2*sqrt(-d*e^(-1))*x*e - d)/(x^2*e + d)) + 3*
(196*g^3*p*x^10 + 1200*f*g^2*p*x^7 + 3675*f^2*g*p*x^4 + 19600*f^3*p*x)*e^5 - 105*(7*d*g^3*p*x^8 + 48*d*f*g^2*p
*x^5 + 210*d*f^2*g*p*x^2)*e^4 + 140*(7*d^2*g^3*p*x^6 + 60*d^2*f*g^2*p*x^3)*e^3 - 210*(7*d^3*g^3*p*x^4 + 120*d^
3*f*g^2*p*x)*e^2 - 210*(14*d^5*g^3*p - 105*d^2*f^2*g*p*e^3 + (14*g^3*p*x^10 + 60*f*g^2*p*x^7 + 105*f^2*g*p*x^4
+ 140*f^3*p*x)*e^5)*log(x^2*e + d))*e^(-5), -1/29400*(2940*d^4*g^3*p*x^2*e + 8400*(3*d^3*f*g^2*p*e^2 - 7*f^3*
p*e^5)*sqrt(d)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2) - 210*(14*g^3*x^10 + 60*f*g^2*x^7 + 105*f^2*g*x^4 + 140*f^3*
x)*e^5*log(c) + 3*(196*g^3*p*x^10 + 1200*f*g^2*p*x^7 + 3675*f^2*g*p*x^4 + 19600*f^3*p*x)*e^5 - 105*(7*d*g^3*p*
x^8 + 48*d*f*g^2*p*x^5 + 210*d*f^2*g*p*x^2)*e^4 + 140*(7*d^2*g^3*p*x^6 + 60*d^2*f*g^2*p*x^3)*e^3 - 210*(7*d^3*
g^3*p*x^4 + 120*d^3*f*g^2*p*x)*e^2 - 210*(14*d^5*g^3*p - 105*d^2*f^2*g*p*e^3 + (14*g^3*p*x^10 + 60*f*g^2*p*x^7
+ 105*f^2*g*p*x^4 + 140*f^3*p*x)*e^5)*log(x^2*e + d))*e^(-5)]

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 3.97, size = 354, normalized size = 0.97 \begin {gather*} \frac {1}{20} \, {\left (2 \, d^{5} g^{3} p - 15 \, d^{2} f^{2} g p e^{3}\right )} e^{\left (-5\right )} \log \left (x^{2} e + d\right ) - \frac {2 \, {\left (3 \, d^{4} f g^{2} p - 7 \, d f^{3} 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}{29400} \, {\left (2940 \, g^{3} p x^{10} e^{4} \log \left (x^{2} e + d\right ) - 588 \, g^{3} p x^{10} e^{4} + 2940 \, g^{3} x^{10} e^{4} \log \left (c\right ) + 735 \, d g^{3} p x^{8} e^{3} - 980 \, d^{2} g^{3} p x^{6} e^{2} + 12600 \, f g^{2} p x^{7} e^{4} \log \left (x^{2} e + d\right ) - 3600 \, f g^{2} p x^{7} e^{4} + 1470 \, d^{3} g^{3} p x^{4} e + 12600 \, f g^{2} x^{7} e^{4} \log \left (c\right ) + 5040 \, d f g^{2} p x^{5} e^{3} - 2940 \, d^{4} g^{3} p x^{2} - 8400 \, d^{2} f g^{2} p x^{3} e^{2} + 22050 \, f^{2} g p x^{4} e^{4} \log \left (x^{2} e + d\right ) - 11025 \, f^{2} g p x^{4} e^{4} + 25200 \, d^{3} f g^{2} p x e + 22050 \, f^{2} g x^{4} e^{4} \log \left (c\right ) + 22050 \, d f^{2} g p x^{2} e^{3} + 29400 \, f^{3} p x e^{4} \log \left (x^{2} e + d\right ) - 58800 \, f^{3} p x e^{4} + 29400 \, f^{3} x e^{4} \log \left (c\right )\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(2*d^5*g^3*p - 15*d^2*f^2*g*p*e^3)*e^(-5)*log(x^2*e + d) - 2/7*(3*d^4*f*g^2*p - 7*d*f^3*p*e^3)*arctan(x*e
^(1/2)/sqrt(d))*e^(-7/2)/sqrt(d) + 1/29400*(2940*g^3*p*x^10*e^4*log(x^2*e + d) - 588*g^3*p*x^10*e^4 + 2940*g^3
*x^10*e^4*log(c) + 735*d*g^3*p*x^8*e^3 - 980*d^2*g^3*p*x^6*e^2 + 12600*f*g^2*p*x^7*e^4*log(x^2*e + d) - 3600*f
*g^2*p*x^7*e^4 + 1470*d^3*g^3*p*x^4*e + 12600*f*g^2*x^7*e^4*log(c) + 5040*d*f*g^2*p*x^5*e^3 - 2940*d^4*g^3*p*x
^2 - 8400*d^2*f*g^2*p*x^3*e^2 + 22050*f^2*g*p*x^4*e^4*log(x^2*e + d) - 11025*f^2*g*p*x^4*e^4 + 25200*d^3*f*g^2
*p*x*e + 22050*f^2*g*x^4*e^4*log(c) + 22050*d*f^2*g*p*x^2*e^3 + 29400*f^3*p*x*e^4*log(x^2*e + d) - 58800*f^3*p
*x*e^4 + 29400*f^3*x*e^4*log(c))*e^(-4)

________________________________________________________________________________________

Mupad [B]
time = 3.33, size = 316, normalized size = 0.86 \begin {gather*} \frac {g^3\,x^{10}\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{10}-2\,f^3\,p\,x-\frac {g^3\,p\,x^{10}}{50}+f^3\,x\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )+\frac {3\,f^2\,g\,x^4\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{4}+\frac {3\,f\,g^2\,x^7\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{7}-\frac {3\,f^2\,g\,p\,x^4}{8}-\frac {6\,f\,g^2\,p\,x^7}{49}+\frac {d\,g^3\,p\,x^8}{40\,e}+\frac {2\,\sqrt {d}\,f^3\,p\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {d^5\,g^3\,p\,\ln \left (e\,x^2+d\right )}{10\,e^5}-\frac {d^2\,g^3\,p\,x^6}{30\,e^2}+\frac {d^3\,g^3\,p\,x^4}{20\,e^3}-\frac {d^4\,g^3\,p\,x^2}{10\,e^4}-\frac {6\,d^{7/2}\,f\,g^2\,p\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )}{7\,e^{7/2}}-\frac {3\,d^2\,f^2\,g\,p\,\ln \left (e\,x^2+d\right )}{4\,e^2}-\frac {2\,d^2\,f\,g^2\,p\,x^3}{7\,e^2}+\frac {3\,d\,f^2\,g\,p\,x^2}{4\,e}+\frac {6\,d\,f\,g^2\,p\,x^5}{35\,e}+\frac {6\,d^3\,f\,g^2\,p\,x}{7\,e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(g^3*x^10*log(c*(d + e*x^2)^p))/10 - 2*f^3*p*x - (g^3*p*x^10)/50 + f^3*x*log(c*(d + e*x^2)^p) + (3*f^2*g*x^4*l
og(c*(d + e*x^2)^p))/4 + (3*f*g^2*x^7*log(c*(d + e*x^2)^p))/7 - (3*f^2*g*p*x^4)/8 - (6*f*g^2*p*x^7)/49 + (d*g^
3*p*x^8)/(40*e) + (2*d^(1/2)*f^3*p*atan((e^(1/2)*x)/d^(1/2)))/e^(1/2) + (d^5*g^3*p*log(d + e*x^2))/(10*e^5) -
(d^2*g^3*p*x^6)/(30*e^2) + (d^3*g^3*p*x^4)/(20*e^3) - (d^4*g^3*p*x^2)/(10*e^4) - (6*d^(7/2)*f*g^2*p*atan((e^(1
/2)*x)/d^(1/2)))/(7*e^(7/2)) - (3*d^2*f^2*g*p*log(d + e*x^2))/(4*e^2) - (2*d^2*f*g^2*p*x^3)/(7*e^2) + (3*d*f^2
*g*p*x^2)/(4*e) + (6*d*f*g^2*p*x^5)/(35*e) + (6*d^3*f*g^2*p*x)/(7*e^3)

________________________________________________________________________________________