3.191 \(\int (d+e x)^3 \log (c (a+b x^3)^p) \, dx\)

Optimal. Leaf size=320 \[ -\frac{\sqrt [3]{a} p \left (-6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3+4 b d^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 b^{4/3}}+\frac{\sqrt [3]{a} p \left (-6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3+4 b d^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 b^{4/3}}-\frac{\sqrt{3} \sqrt [3]{a} p \left (6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3+4 b d^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{4 b^{4/3}}+\frac{(d+e x)^4 \log \left (c \left (a+b x^3\right )^p\right )}{4 e}-\frac{d p \left (b d^3-4 a e^3\right ) \log \left (a+b x^3\right )}{4 b e}-\frac{3 p x \left (4 b d^3-a e^3\right )}{4 b}-\frac{9}{4} d^2 e p x^2-d e^2 p x^3-\frac{3}{16} e^3 p x^4 \]

[Out]

(-3*(4*b*d^3 - a*e^3)*p*x)/(4*b) - (9*d^2*e*p*x^2)/4 - d*e^2*p*x^3 - (3*e^3*p*x^4)/16 - (Sqrt[3]*a^(1/3)*(4*b*
d^3 + 6*a^(1/3)*b^(2/3)*d^2*e - a*e^3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(4*b^(4/3)) + (a^(
1/3)*(4*b*d^3 - 6*a^(1/3)*b^(2/3)*d^2*e - a*e^3)*p*Log[a^(1/3) + b^(1/3)*x])/(4*b^(4/3)) - (a^(1/3)*(4*b*d^3 -
6*a^(1/3)*b^(2/3)*d^2*e - a*e^3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(8*b^(4/3)) - (d*(b*d^3 -
4*a*e^3)*p*Log[a + b*x^3])/(4*b*e) + ((d + e*x)^4*Log[c*(a + b*x^3)^p])/(4*e)

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Rubi [A]  time = 0.741717, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {2463, 1836, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{\sqrt [3]{a} p \left (-6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3+4 b d^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 b^{4/3}}+\frac{\sqrt [3]{a} p \left (-6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3+4 b d^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 b^{4/3}}-\frac{\sqrt{3} \sqrt [3]{a} p \left (6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3+4 b d^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{4 b^{4/3}}+\frac{(d+e x)^4 \log \left (c \left (a+b x^3\right )^p\right )}{4 e}-\frac{d p \left (b d^3-4 a e^3\right ) \log \left (a+b x^3\right )}{4 b e}-\frac{3 p x \left (4 b d^3-a e^3\right )}{4 b}-\frac{9}{4} d^2 e p x^2-d e^2 p x^3-\frac{3}{16} e^3 p x^4 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*Log[c*(a + b*x^3)^p],x]

[Out]

(-3*(4*b*d^3 - a*e^3)*p*x)/(4*b) - (9*d^2*e*p*x^2)/4 - d*e^2*p*x^3 - (3*e^3*p*x^4)/16 - (Sqrt[3]*a^(1/3)*(4*b*
d^3 + 6*a^(1/3)*b^(2/3)*d^2*e - a*e^3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(4*b^(4/3)) + (a^(
1/3)*(4*b*d^3 - 6*a^(1/3)*b^(2/3)*d^2*e - a*e^3)*p*Log[a^(1/3) + b^(1/3)*x])/(4*b^(4/3)) - (a^(1/3)*(4*b*d^3 -
6*a^(1/3)*b^(2/3)*d^2*e - a*e^3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(8*b^(4/3)) - (d*(b*d^3 -
4*a*e^3)*p*Log[a + b*x^3])/(4*b*e) + ((d + e*x)^4*Log[c*(a + b*x^3)^p])/(4*e)

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> Simp[((
f + g*x)^(r + 1)*(a + b*Log[c*(d + e*x^n)^p]))/(g*(r + 1)), x] - Dist[(b*e*n*p)/(g*(r + 1)), Int[(x^(n - 1)*(f
+ g*x)^(r + 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x] && (IGtQ[r, 0] || RationalQ[n
]) && NeQ[r, -1]

Rule 1836

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq =
Coeff[Pq, x, q]}, Dist[1/(b*(m + q + n*p + 1)), Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a
*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + Simp[(Pqq*(c*x)^(m + q - n + 1)*(a + b*x^n)^(p + 1)
)/(b*c^(q - n + 1)*(m + q + n*p + 1)), x]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || Integ
erQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]

Rule 1887

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rule 1871

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int (d+e x)^3 \log \left (c \left (a+b x^3\right )^p\right ) \, dx &=\frac{(d+e x)^4 \log \left (c \left (a+b x^3\right )^p\right )}{4 e}-\frac{(3 b p) \int \frac{x^2 (d+e x)^4}{a+b x^3} \, dx}{4 e}\\ &=-\frac{3}{16} e^3 p x^4+\frac{(d+e x)^4 \log \left (c \left (a+b x^3\right )^p\right )}{4 e}-\frac{(3 p) \int \frac{x^2 \left (4 b d^4+4 e \left (4 b d^3-a e^3\right ) x+24 b d^2 e^2 x^2+16 b d e^3 x^3\right )}{a+b x^3} \, dx}{16 e}\\ &=-d e^2 p x^3-\frac{3}{16} e^3 p x^4+\frac{(d+e x)^4 \log \left (c \left (a+b x^3\right )^p\right )}{4 e}-\frac{p \int \frac{x^2 \left (12 b d \left (b d^3-4 a e^3\right )+12 b e \left (4 b d^3-a e^3\right ) x+72 b^2 d^2 e^2 x^2\right )}{a+b x^3} \, dx}{16 b e}\\ &=-d e^2 p x^3-\frac{3}{16} e^3 p x^4+\frac{(d+e x)^4 \log \left (c \left (a+b x^3\right )^p\right )}{4 e}-\frac{p \int \left (12 e \left (4 b d^3-a e^3\right )+72 b d^2 e^2 x-\frac{12 \left (a e \left (4 b d^3-a e^3\right )+6 a b d^2 e^2 x-b d \left (b d^3-4 a e^3\right ) x^2\right )}{a+b x^3}\right ) \, dx}{16 b e}\\ &=-\frac{3 \left (4 b d^3-a e^3\right ) p x}{4 b}-\frac{9}{4} d^2 e p x^2-d e^2 p x^3-\frac{3}{16} e^3 p x^4+\frac{(d+e x)^4 \log \left (c \left (a+b x^3\right )^p\right )}{4 e}+\frac{(3 p) \int \frac{a e \left (4 b d^3-a e^3\right )+6 a b d^2 e^2 x-b d \left (b d^3-4 a e^3\right ) x^2}{a+b x^3} \, dx}{4 b e}\\ &=-\frac{3 \left (4 b d^3-a e^3\right ) p x}{4 b}-\frac{9}{4} d^2 e p x^2-d e^2 p x^3-\frac{3}{16} e^3 p x^4+\frac{(d+e x)^4 \log \left (c \left (a+b x^3\right )^p\right )}{4 e}+\frac{(3 p) \int \frac{a e \left (4 b d^3-a e^3\right )+6 a b d^2 e^2 x}{a+b x^3} \, dx}{4 b e}-\frac{\left (3 d \left (b d^3-4 a e^3\right ) p\right ) \int \frac{x^2}{a+b x^3} \, dx}{4 e}\\ &=-\frac{3 \left (4 b d^3-a e^3\right ) p x}{4 b}-\frac{9}{4} d^2 e p x^2-d e^2 p x^3-\frac{3}{16} e^3 p x^4-\frac{d \left (b d^3-4 a e^3\right ) p \log \left (a+b x^3\right )}{4 b e}+\frac{(d+e x)^4 \log \left (c \left (a+b x^3\right )^p\right )}{4 e}+\frac{p \int \frac{\sqrt [3]{a} \left (6 a^{4/3} b d^2 e^2+2 a \sqrt [3]{b} e \left (4 b d^3-a e^3\right )\right )+\sqrt [3]{b} \left (6 a^{4/3} b d^2 e^2-a \sqrt [3]{b} e \left (4 b d^3-a e^3\right )\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{4 a^{2/3} b^{4/3} e}+\frac{\left (\sqrt [3]{a} \left (4 b d^3-6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3\right ) p\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{4 b}\\ &=-\frac{3 \left (4 b d^3-a e^3\right ) p x}{4 b}-\frac{9}{4} d^2 e p x^2-d e^2 p x^3-\frac{3}{16} e^3 p x^4+\frac{\sqrt [3]{a} \left (4 b d^3-6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 b^{4/3}}-\frac{d \left (b d^3-4 a e^3\right ) p \log \left (a+b x^3\right )}{4 b e}+\frac{(d+e x)^4 \log \left (c \left (a+b x^3\right )^p\right )}{4 e}-\frac{\left (\sqrt [3]{a} \left (4 b d^3-6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3\right ) p\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{8 b^{4/3}}+\frac{\left (3 a^{2/3} \left (4 b d^3+6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3\right ) p\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{8 b}\\ &=-\frac{3 \left (4 b d^3-a e^3\right ) p x}{4 b}-\frac{9}{4} d^2 e p x^2-d e^2 p x^3-\frac{3}{16} e^3 p x^4+\frac{\sqrt [3]{a} \left (4 b d^3-6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 b^{4/3}}-\frac{\sqrt [3]{a} \left (4 b d^3-6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 b^{4/3}}-\frac{d \left (b d^3-4 a e^3\right ) p \log \left (a+b x^3\right )}{4 b e}+\frac{(d+e x)^4 \log \left (c \left (a+b x^3\right )^p\right )}{4 e}+\frac{\left (3 \sqrt [3]{a} \left (4 b d^3+6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3\right ) p\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{4 b^{4/3}}\\ &=-\frac{3 \left (4 b d^3-a e^3\right ) p x}{4 b}-\frac{9}{4} d^2 e p x^2-d e^2 p x^3-\frac{3}{16} e^3 p x^4-\frac{\sqrt{3} \sqrt [3]{a} \left (4 b d^3+6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3\right ) p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{4 b^{4/3}}+\frac{\sqrt [3]{a} \left (4 b d^3-6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 b^{4/3}}-\frac{\sqrt [3]{a} \left (4 b d^3-6 \sqrt [3]{a} b^{2/3} d^2 e-a e^3\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 b^{4/3}}-\frac{d \left (b d^3-4 a e^3\right ) p \log \left (a+b x^3\right )}{4 b e}+\frac{(d+e x)^4 \log \left (c \left (a+b x^3\right )^p\right )}{4 e}\\ \end{align*}

Mathematica [C]  time = 0.485492, size = 264, normalized size = 0.82 \[ \frac{\frac{\sqrt [3]{a} e p \left (a e^3-4 b d^3\right ) \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{2 b^{4/3}}+\frac{\sqrt [3]{a} e p \left (4 b d^3-a e^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{4/3}}+(d+e x)^4 \log \left (c \left (a+b x^3\right )^p\right )+9 d^2 e^2 p x^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{b x^3}{a}\right )-\frac{d p \left (b d^3-4 a e^3\right ) \log \left (a+b x^3\right )}{b}+\frac{3 e p x \left (a e^3-4 b d^3\right )}{b}-9 d^2 e^2 p x^2-4 d e^3 p x^3-\frac{3}{4} e^4 p x^4}{4 e} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*Log[c*(a + b*x^3)^p],x]

[Out]

((3*e*(-4*b*d^3 + a*e^3)*p*x)/b - 9*d^2*e^2*p*x^2 - 4*d*e^3*p*x^3 - (3*e^4*p*x^4)/4 + 9*d^2*e^2*p*x^2*Hypergeo
metric2F1[2/3, 1, 5/3, -((b*x^3)/a)] + (a^(1/3)*e*(4*b*d^3 - a*e^3)*p*Log[a^(1/3) + b^(1/3)*x])/b^(4/3) + (a^(
1/3)*e*(-4*b*d^3 + a*e^3)*p*(2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + Log[a^(2/3) - a^(1/3)*b^(
1/3)*x + b^(2/3)*x^2]))/(2*b^(4/3)) - (d*(b*d^3 - 4*a*e^3)*p*Log[a + b*x^3])/b + (d + e*x)^4*Log[c*(a + b*x^3)
^p])/(4*e)

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Maple [C]  time = 0.783, size = 738, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*ln(c*(b*x^3+a)^p),x)

[Out]

-3*d^3*p*x+3/4/b*e^3*a*p*x-1/2*I*Pi*d^3*x*csgn(I*c*(b*x^3+a)^p)^3-1/8*I*e^3*Pi*x^4*csgn(I*c*(b*x^3+a)^p)^3+1/4
/b^2*p/e*sum((b*d*(4*a*e^3-b*d^3)*_R^2+6*a*b*d^2*e^2*_R-a^2*e^4+4*a*b*d^3*e)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a)
)-1/8*I*e^3*Pi*x^4*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)*csgn(I*c)+1/2*I*e^2*Pi*d*x^3*csgn(I*(b*x^3+a)^p)*
csgn(I*c*(b*x^3+a)^p)^2+1/2*I*e^2*Pi*d*x^3*csgn(I*c*(b*x^3+a)^p)^2*csgn(I*c)+3/4*I*e*Pi*d^2*x^2*csgn(I*(b*x^3+
a)^p)*csgn(I*c*(b*x^3+a)^p)^2+3/4*I*e*Pi*d^2*x^2*csgn(I*c*(b*x^3+a)^p)^2*csgn(I*c)-1/2*I*Pi*d^3*x*csgn(I*(b*x^
3+a)^p)*csgn(I*c*(b*x^3+a)^p)*csgn(I*c)+ln(c)*d^3*x+1/4*e^3*ln(c)*x^4+1/4*(e*x+d)^4/e*ln((b*x^3+a)^p)+3/2*e*ln
(c)*d^2*x^2+e^2*ln(c)*d*x^3-3/16*e^3*p*x^4-d*e^2*p*x^3-9/4*d^2*e*p*x^2-1/2*I*e^2*Pi*d*x^3*csgn(I*(b*x^3+a)^p)*
csgn(I*c*(b*x^3+a)^p)*csgn(I*c)-3/4*I*e*Pi*d^2*x^2*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)*csgn(I*c)+1/8*I*e
^3*Pi*x^4*csgn(I*c*(b*x^3+a)^p)^2*csgn(I*c)-1/2*I*e^2*Pi*d*x^3*csgn(I*c*(b*x^3+a)^p)^3-3/4*I*e*Pi*d^2*x^2*csgn
(I*c*(b*x^3+a)^p)^3+1/2*I*Pi*d^3*x*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)^2+1/2*I*Pi*d^3*x*csgn(I*c*(b*x^3+
a)^p)^2*csgn(I*c)+1/8*I*e^3*Pi*x^4*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)^2

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

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Fricas [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((e*x+d)^3*log(c*(b*x^3+a)^p),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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Giac [B]  time = 1.34227, size = 753, normalized size = 2.35 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*log(c*(b*x^3+a)^p),x, algorithm="giac")

[Out]

1/8*a^2*b^3*p*(2*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^4) - 2*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)
*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^5) - (-a*b^2)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^5))
*e^3 - 3/4*a*b^2*d^2*p*(2*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^2) + 2*sqrt(3)*(-a*b^2)^(2/3)*arctan(1/
3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^4) - (-a*b^2)^(2/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))
/(a*b^4))*e - 1/2*a*b*d^3*p*(2*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b) - 2*sqrt(3)*(-a*b^2)^(1/3)*arctan
(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^2) - (-a*b^2)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/
3))/(a*b^2)) + 1/16*(4*b*p*x^4*e^3*log(b*x^3 + a) + 16*b*d*p*x^3*e^2*log(b*x^3 + a) + 24*b*d^2*p*x^2*e*log(b*x
^3 + a) - 3*b*p*x^4*e^3 - 16*b*d*p*x^3*e^2 - 36*b*d^2*p*x^2*e + 16*b*d^3*p*x*log(b*x^3 + a) + 4*b*x^4*e^3*log(
c) + 16*b*d*x^3*e^2*log(c) + 24*b*d^2*x^2*e*log(c) - 48*b*d^3*p*x + 16*b*d^3*x*log(c) + 16*a*d*p*e^2*log(b*x^3
+ a) + 12*a*p*x*e^3)/b