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

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

[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

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Rubi [A]  time = 0.307897, antiderivative size = 366, normalized size of antiderivative = 1., 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 = {2471, 2448, 321, 205, 2454, 2395, 43, 2455, 302} \[ \frac{3}{4} f^2 g x^4 \log \left (c \left (d+e x^2\right )^p\right )+f^3 x \log \left (c \left (d+e x^2\right )^p\right )+\frac{3}{7} f g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{10} g^3 x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac{3 d^2 f^2 g p \log \left (d+e x^2\right )}{4 e^2}-\frac{2 d^2 f g^2 p x^3}{7 e^2}+\frac{6 d^3 f g^2 p x}{7 e^3}-\frac{6 d^{7/2} f g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}-\frac{d^2 g^3 p x^6}{30 e^2}+\frac{d^3 g^3 p x^4}{20 e^3}-\frac{d^4 g^3 p x^2}{10 e^4}+\frac{d^5 g^3 p \log \left (d+e x^2\right )}{10 e^5}+\frac{3 d f^2 g p x^2}{4 e}+\frac{2 \sqrt{d} f^3 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{6 d f g^2 p x^5}{35 e}+\frac{d g^3 p x^8}{40 e}-\frac{3}{8} f^2 g p x^4-2 f^3 p x-\frac{6}{49} f g^2 p x^7-\frac{1}{50} g^3 p x^{10} \]

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 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 )^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 ) \operatorname{Subst}\left (\int x \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )+\frac{1}{2} g^3 \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \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 (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 ) \operatorname{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 = 0.2487, size = 258, normalized size = 0.7 \[ \frac{210 e^5 x \left (105 f^2 g x^3+140 f^3+60 f g^2 x^6+14 g^3 x^9\right ) \log \left (c \left (d+e x^2\right )^p\right )-e p x \left (140 d^2 e^2 g^2 x^2 \left (60 f+7 g x^3\right )-210 d^3 e g^2 \left (120 f+7 g x^3\right )+2940 d^4 g^3 x-105 d e^3 g x \left (210 f^2+48 f g x^3+7 g^2 x^6\right )+3 e^4 \left (3675 f^2 g x^3+19600 f^3+1200 f g^2 x^6+196 g^3 x^9\right )\right )+1470 d^2 g p \left (2 d^3 g^2-15 e^3 f^2\right ) \log \left (d+e x^2\right )-8400 \sqrt{d} e^{3/2} f p \left (3 d^3 g^2-7 e^3 f^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{29400 e^5} \]

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)

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Maple [C]  time = 0.874, size = 1311, normalized size = 3.6 \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)^3*ln(c*(e*x^2+d)^p),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.16653, size = 1611, normalized size = 4.4 \begin{align*} \left [-\frac{588 \, e^{5} g^{3} p x^{10} - 735 \, d e^{4} g^{3} p x^{8} + 3600 \, e^{5} f g^{2} p x^{7} + 980 \, d^{2} e^{3} g^{3} p x^{6} - 5040 \, d e^{4} f g^{2} p x^{5} + 8400 \, d^{2} e^{3} f g^{2} p x^{3} + 735 \,{\left (15 \, e^{5} f^{2} g - 2 \, d^{3} e^{2} g^{3}\right )} p x^{4} - 1470 \,{\left (15 \, d e^{4} f^{2} g - 2 \, d^{4} e g^{3}\right )} p x^{2} + 4200 \,{\left (7 \, e^{5} f^{3} - 3 \, d^{3} e^{2} f 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 ) + 8400 \,{\left (7 \, e^{5} f^{3} - 3 \, d^{3} e^{2} f g^{2}\right )} p x - 210 \,{\left (14 \, e^{5} g^{3} p x^{10} + 60 \, e^{5} f g^{2} p x^{7} + 105 \, e^{5} f^{2} g p x^{4} + 140 \, e^{5} f^{3} p x - 7 \,{\left (15 \, d^{2} e^{3} f^{2} g - 2 \, d^{5} g^{3}\right )} p\right )} \log \left (e x^{2} + d\right ) - 210 \,{\left (14 \, e^{5} g^{3} x^{10} + 60 \, e^{5} f g^{2} x^{7} + 105 \, e^{5} f^{2} g x^{4} + 140 \, e^{5} f^{3} x\right )} \log \left (c\right )}{29400 \, e^{5}}, -\frac{588 \, e^{5} g^{3} p x^{10} - 735 \, d e^{4} g^{3} p x^{8} + 3600 \, e^{5} f g^{2} p x^{7} + 980 \, d^{2} e^{3} g^{3} p x^{6} - 5040 \, d e^{4} f g^{2} p x^{5} + 8400 \, d^{2} e^{3} f g^{2} p x^{3} + 735 \,{\left (15 \, e^{5} f^{2} g - 2 \, d^{3} e^{2} g^{3}\right )} p x^{4} - 1470 \,{\left (15 \, d e^{4} f^{2} g - 2 \, d^{4} e g^{3}\right )} p x^{2} - 8400 \,{\left (7 \, e^{5} f^{3} - 3 \, d^{3} e^{2} f g^{2}\right )} p \sqrt{\frac{d}{e}} \arctan \left (\frac{e x \sqrt{\frac{d}{e}}}{d}\right ) + 8400 \,{\left (7 \, e^{5} f^{3} - 3 \, d^{3} e^{2} f g^{2}\right )} p x - 210 \,{\left (14 \, e^{5} g^{3} p x^{10} + 60 \, e^{5} f g^{2} p x^{7} + 105 \, e^{5} f^{2} g p x^{4} + 140 \, e^{5} f^{3} p x - 7 \,{\left (15 \, d^{2} e^{3} f^{2} g - 2 \, d^{5} g^{3}\right )} p\right )} \log \left (e x^{2} + d\right ) - 210 \,{\left (14 \, e^{5} g^{3} x^{10} + 60 \, e^{5} f g^{2} x^{7} + 105 \, e^{5} f^{2} g x^{4} + 140 \, e^{5} f^{3} x\right )} \log \left (c\right )}{29400 \, e^{5}}\right ] \end{align*}

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*(588*e^5*g^3*p*x^10 - 735*d*e^4*g^3*p*x^8 + 3600*e^5*f*g^2*p*x^7 + 980*d^2*e^3*g^3*p*x^6 - 5040*d*e^
4*f*g^2*p*x^5 + 8400*d^2*e^3*f*g^2*p*x^3 + 735*(15*e^5*f^2*g - 2*d^3*e^2*g^3)*p*x^4 - 1470*(15*d*e^4*f^2*g - 2
*d^4*e*g^3)*p*x^2 + 4200*(7*e^5*f^3 - 3*d^3*e^2*f*g^2)*p*sqrt(-d/e)*log((e*x^2 - 2*e*x*sqrt(-d/e) - d)/(e*x^2
+ d)) + 8400*(7*e^5*f^3 - 3*d^3*e^2*f*g^2)*p*x - 210*(14*e^5*g^3*p*x^10 + 60*e^5*f*g^2*p*x^7 + 105*e^5*f^2*g*p
*x^4 + 140*e^5*f^3*p*x - 7*(15*d^2*e^3*f^2*g - 2*d^5*g^3)*p)*log(e*x^2 + d) - 210*(14*e^5*g^3*x^10 + 60*e^5*f*
g^2*x^7 + 105*e^5*f^2*g*x^4 + 140*e^5*f^3*x)*log(c))/e^5, -1/29400*(588*e^5*g^3*p*x^10 - 735*d*e^4*g^3*p*x^8 +
3600*e^5*f*g^2*p*x^7 + 980*d^2*e^3*g^3*p*x^6 - 5040*d*e^4*f*g^2*p*x^5 + 8400*d^2*e^3*f*g^2*p*x^3 + 735*(15*e^
5*f^2*g - 2*d^3*e^2*g^3)*p*x^4 - 1470*(15*d*e^4*f^2*g - 2*d^4*e*g^3)*p*x^2 - 8400*(7*e^5*f^3 - 3*d^3*e^2*f*g^2
)*p*sqrt(d/e)*arctan(e*x*sqrt(d/e)/d) + 8400*(7*e^5*f^3 - 3*d^3*e^2*f*g^2)*p*x - 210*(14*e^5*g^3*p*x^10 + 60*e
^5*f*g^2*p*x^7 + 105*e^5*f^2*g*p*x^4 + 140*e^5*f^3*p*x - 7*(15*d^2*e^3*f^2*g - 2*d^5*g^3)*p)*log(e*x^2 + d) -
210*(14*e^5*g^3*x^10 + 60*e^5*f*g^2*x^7 + 105*e^5*f^2*g*x^4 + 140*e^5*f^3*x)*log(c))/e^5]

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

[Out]

Timed out

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Giac [A]  time = 1.30057, size = 478, normalized size = 1.31 \begin{align*} \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{align*}

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)