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

Optimal. Leaf size=229 \[ -\frac{\sqrt [3]{a} p \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3}}+\frac{\sqrt [3]{a} p \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3}}-\frac{\sqrt{3} \sqrt [3]{a} p \left (\sqrt [3]{a} e+2 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 b^{2/3}}+\frac{(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}-\frac{d^2 p \log \left (a+b x^3\right )}{2 e}-3 d p x-\frac{3}{4} e p x^2 \]

[Out]

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

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Rubi [A]  time = 0.31725, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {2463, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{\sqrt [3]{a} p \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3}}+\frac{\sqrt [3]{a} p \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3}}-\frac{\sqrt{3} \sqrt [3]{a} p \left (\sqrt [3]{a} e+2 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 b^{2/3}}+\frac{(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}-\frac{d^2 p \log \left (a+b x^3\right )}{2 e}-3 d p x-\frac{3}{4} e p x^2 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*Log[c*(a + b*x^3)^p],x]

[Out]

-3*d*p*x - (3*e*p*x^2)/4 - (Sqrt[3]*a^(1/3)*(2*b^(1/3)*d + a^(1/3)*e)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3
]*a^(1/3))])/(2*b^(2/3)) + (a^(1/3)*(2*b^(1/3)*d - a^(1/3)*e)*p*Log[a^(1/3) + b^(1/3)*x])/(2*b^(2/3)) - (a^(1/
3)*(2*b^(1/3)*d - a^(1/3)*e)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(4*b^(2/3)) - (d^2*p*Log[a + b*
x^3])/(2*e) + ((d + e*x)^2*Log[c*(a + b*x^3)^p])/(2*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 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) \log \left (c \left (a+b x^3\right )^p\right ) \, dx &=\frac{(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}-\frac{(3 b p) \int \frac{x^2 (d+e x)^2}{a+b x^3} \, dx}{2 e}\\ &=\frac{(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}-\frac{(3 b p) \int \left (\frac{2 d e}{b}+\frac{e^2 x}{b}-\frac{2 a d e+a e^2 x-b d^2 x^2}{b \left (a+b x^3\right )}\right ) \, dx}{2 e}\\ &=-3 d p x-\frac{3}{4} e p x^2+\frac{(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}+\frac{(3 p) \int \frac{2 a d e+a e^2 x-b d^2 x^2}{a+b x^3} \, dx}{2 e}\\ &=-3 d p x-\frac{3}{4} e p x^2+\frac{(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}+\frac{(3 p) \int \frac{2 a d e+a e^2 x}{a+b x^3} \, dx}{2 e}-\frac{\left (3 b d^2 p\right ) \int \frac{x^2}{a+b x^3} \, dx}{2 e}\\ &=-3 d p x-\frac{3}{4} e p x^2-\frac{d^2 p \log \left (a+b x^3\right )}{2 e}+\frac{(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}+\frac{p \int \frac{\sqrt [3]{a} \left (4 a \sqrt [3]{b} d e+a^{4/3} e^2\right )+\sqrt [3]{b} \left (-2 a \sqrt [3]{b} d e+a^{4/3} e^2\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^{2/3} \sqrt [3]{b} e}+\frac{1}{2} \left (\sqrt [3]{a} \left (2 d-\frac{\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) p\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx\\ &=-3 d p x-\frac{3}{4} e p x^2+\frac{\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3}}-\frac{d^2 p \log \left (a+b x^3\right )}{2 e}+\frac{(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}-\frac{\left (\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\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}{4 b^{2/3}}+\frac{1}{4} \left (3 a^{2/3} \left (2 d+\frac{\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) p\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx\\ &=-3 d p x-\frac{3}{4} e p x^2+\frac{\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3}}-\frac{\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3}}-\frac{d^2 p \log \left (a+b x^3\right )}{2 e}+\frac{(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}+\frac{\left (3 \sqrt [3]{a} \left (2 \sqrt [3]{b} d+\sqrt [3]{a} e\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 )}{2 b^{2/3}}\\ &=-3 d p x-\frac{3}{4} e p x^2-\frac{\sqrt{3} \sqrt [3]{a} \left (2 \sqrt [3]{b} d+\sqrt [3]{a} e\right ) p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 b^{2/3}}+\frac{\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3}}-\frac{\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3}}-\frac{d^2 p \log \left (a+b x^3\right )}{2 e}+\frac{(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}\\ \end{align*}

Mathematica [C]  time = 0.0681739, size = 204, normalized size = 0.89 \[ -\frac{\sqrt [3]{a} d p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+\frac{\sqrt{3} \sqrt [3]{a} d p \tan ^{-1}\left (\frac{2 b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b}}{\sqrt{3} \sqrt [3]{a} \sqrt [3]{b}}\right )}{\sqrt [3]{b}}+d x \log \left (c \left (a+b x^3\right )^p\right )+\frac{1}{2} e x^2 \log \left (c \left (a+b x^3\right )^p\right )+\frac{\sqrt [3]{a} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac{3}{4} e p x^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{b x^3}{a}\right )-3 d p x-\frac{3}{4} e p x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-3*d*p*x - (3*e*p*x^2)/4 + (Sqrt[3]*a^(1/3)*d*p*ArcTan[(-(a^(1/3)*b^(1/3)) + 2*b^(2/3)*x)/(Sqrt[3]*a^(1/3)*b^(
1/3))])/b^(1/3) + (3*e*p*x^2*Hypergeometric2F1[2/3, 1, 5/3, -((b*x^3)/a)])/4 + (a^(1/3)*d*p*Log[a^(1/3) + b^(1
/3)*x])/b^(1/3) - (a^(1/3)*d*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*b^(1/3)) + d*x*Log[c*(a + b*
x^3)^p] + (e*x^2*Log[c*(a + b*x^3)^p])/2

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Maple [C]  time = 0.717, size = 335, normalized size = 1.5 \begin{align*} \left ({\frac{e{x}^{2}}{2}}+dx \right ) \ln \left ( \left ( b{x}^{3}+a \right ) ^{p} \right ) +{\frac{i}{4}}\pi \,e{x}^{2}{\it csgn} \left ( i \left ( b{x}^{3}+a \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{2}-{\frac{i}{4}}\pi \,e{x}^{2}{\it csgn} \left ( i \left ( b{x}^{3}+a \right ) ^{p} \right ){\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{4}}\pi \,e{x}^{2} \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{3}+{\frac{i}{4}}\pi \,e{x}^{2} \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{2}}\pi \,d{\it csgn} \left ( i \left ( b{x}^{3}+a \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{2}x-{\frac{i}{2}}\pi \,d{\it csgn} \left ( i \left ( b{x}^{3}+a \right ) ^{p} \right ){\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ){\it csgn} \left ( ic \right ) x-{\frac{i}{2}}\pi \,d \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{3}x+{\frac{i}{2}}\pi \,d \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) x+{\frac{\ln \left ( c \right ) e{x}^{2}}{2}}-{\frac{3\,ep{x}^{2}}{4}}+\ln \left ( c \right ) dx-3\,dpx+{\frac{ap}{2\,b}\sum _{{\it \_R}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{\frac{ \left ( e{\it \_R}+2\,d \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 13.2546, size = 5179, normalized size = 22.62 \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)*log(c*(b*x^3+a)^p),x, algorithm="fricas")

[Out]

-3/4*e*p*x^2 - 3*d*p*x + 1/4*(4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*
d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3
)/b^2)^(1/3)*(I*sqrt(3) + 1))*log(4*a*d*e^2*p^2 + 2*(4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e
^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*
b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))*b*d^2*p + 1/4*(4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(
((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*
p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))^2*b*e + (8*b*d^3 + a*e^3)*p^2*x) - 1/8*(4*
(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3
)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1) - s
qrt(3)*sqrt(-(32*a*d*e*p^2 + (4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*
d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3
)/b^2)^(1/3)*(I*sqrt(3) + 1))^2*b)/b))*log(-2*a*d*e^2*p^2 - (4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d
^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2
+ (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))*b*d^2*p - 1/8*(4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3
) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a
*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))^2*b*e + (8*b*d^3 + a*e^3)*p^2*x +
1/8*sqrt(3)*(8*b*d^2*p - (4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*
p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^
2)^(1/3)*(I*sqrt(3) + 1))*b*e)*sqrt(-(32*a*d*e*p^2 + (4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*
e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a
*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))^2*b)/b)) - 1/8*(4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)
/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*
a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1) + sqrt(3)*sqrt(-(32*a*d*e*p^2 + (4*(1/2)^
(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) -
(1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))^2*b)/b))
*log(-2*a*d*e^2*p^2 - (4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3
- a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^
(1/3)*(I*sqrt(3) + 1))*b*d^2*p - 1/8*(4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 +
(8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2
*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))^2*b*e + (8*b*d^3 + a*e^3)*p^2*x - 1/8*sqrt(3)*(8*b*d^2*p - (4*(1/2)^(2/3
)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/
2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))*b*e)*sqrt(-(
32*a*d*e*p^2 + (4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*
e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(
I*sqrt(3) + 1))^2*b)/b)) + 1/2*(e*p*x^2 + 2*d*p*x)*log(b*x^3 + a) + 1/2*(e*x^2 + 2*d*x)*log(c)

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

[Out]

Timed out

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Giac [A]  time = 1.2946, size = 406, normalized size = 1.77 \begin{align*} -\frac{1}{4} \, a b^{2} p{\left (\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a b^{2}} + \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{a b^{4}} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{a b^{4}}\right )} e - \frac{1}{2} \, a b d p{\left (\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a b} - \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{a b^{2}} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{a b^{2}}\right )} + \frac{1}{2} \, p x^{2} e \log \left (b x^{3} + a\right ) - \frac{3}{4} \, p x^{2} e + d p x \log \left (b x^{3} + a\right ) + \frac{1}{2} \, x^{2} e \log \left (c\right ) - 3 \, d p x + d x \log \left (c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a*b^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*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))
+ 1/2*p*x^2*e*log(b*x^3 + a) - 3/4*p*x^2*e + d*p*x*log(b*x^3 + a) + 1/2*x^2*e*log(c) - 3*d*p*x + d*x*log(c)