3.196 \(\int \frac{\log (c (a+b x^3)^p)}{(d+e x)^2} \, dx\)

Optimal. Leaf size=292 \[ -\frac{\sqrt [3]{a} \sqrt [3]{b} p \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \left (b d^3-a e^3\right )}-\frac{\sqrt{3} \sqrt [3]{a} \sqrt [3]{b} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3} e^2+\sqrt [3]{a} \sqrt [3]{b} d e+b^{2/3} d^2}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{e (d+e x)}+\frac{b d^2 p \log \left (a+b x^3\right )}{e \left (b d^3-a e^3\right )}-\frac{3 b d^2 p \log (d+e x)}{e \left (b d^3-a e^3\right )}+\frac{\sqrt [3]{a} \sqrt [3]{b} p \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b d^3-a e^3} \]

[Out]

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

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Rubi [A]  time = 0.549462, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2463, 6725, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{\sqrt [3]{a} \sqrt [3]{b} p \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \left (b d^3-a e^3\right )}-\frac{\sqrt{3} \sqrt [3]{a} \sqrt [3]{b} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3} e^2+\sqrt [3]{a} \sqrt [3]{b} d e+b^{2/3} d^2}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{e (d+e x)}+\frac{b d^2 p \log \left (a+b x^3\right )}{e \left (b d^3-a e^3\right )}-\frac{3 b d^2 p \log (d+e x)}{e \left (b d^3-a e^3\right )}+\frac{\sqrt [3]{a} \sqrt [3]{b} p \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b d^3-a e^3} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x^3)^p]/(d + e*x)^2,x]

[Out]

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

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 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

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

Mathematica [C]  time = 0.645978, size = 202, normalized size = 0.69 \[ -\frac{\frac{b^{2/3} d p \left (\sqrt [3]{a} e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \sqrt [3]{b} d \log \left (a+b x^3\right )-2 \sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{a} e \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+6 \sqrt [3]{b} d \log (d+e x)\right )+3 b e^2 p x^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{b x^3}{a}\right )}{2 b d^3-2 a e^3}+\frac{\log \left (c \left (a+b x^3\right )^p\right )}{d+e x}}{e} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x^3)^p]/(d + e*x)^2,x]

[Out]

-(((3*b*e^2*p*x^2*Hypergeometric2F1[2/3, 1, 5/3, -((b*x^3)/a)] + b^(2/3)*d*p*(2*Sqrt[3]*a^(1/3)*e*ArcTan[(1 -
(2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*a^(1/3)*e*Log[a^(1/3) + b^(1/3)*x] + 6*b^(1/3)*d*Log[d + e*x] + a^(1/3)*e*
Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 2*b^(1/3)*d*Log[a + b*x^3]))/(2*b*d^3 - 2*a*e^3) + Log[c*(a +
b*x^3)^p]/(d + e*x))/e)

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Maple [C]  time = 0.539, size = 1068, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x^3+a)^p)/(e*x+d)^2,x)

[Out]

-1/e/(e*x+d)*ln((b*x^3+a)^p)+1/2*(I*Pi*a*e^3*csgn(I*c*(b*x^3+a)^p)^3-I*Pi*b*d^3*csgn(I*c*(b*x^3+a)^p)^3+I*Pi*a
*e^3*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)*csgn(I*c)-I*Pi*a*e^3*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)^
2+I*Pi*b*d^3*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)^2-I*Pi*b*d^3*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)*
csgn(I*c)+I*Pi*b*d^3*csgn(I*c*(b*x^3+a)^p)^2*csgn(I*c)-I*Pi*a*e^3*csgn(I*c*(b*x^3+a)^p)^2*csgn(I*c)+2*sum(_R*l
n(((-4*a*e^7-2*b*d^3*e^4)*_R^3-3*b*d^2*e^3*p*_R^2+8*b*d*e^2*p^2*_R-3*b*e*p^3)*x+(-5*a*d*e^6-b*d^4*e^3)*_R^3+(a
*e^5*p-b*d^3*e^2*p)*_R^2+5*b*d^2*e*p^2*_R-3*b*d*p^3),_R=RootOf((a*e^6-b*d^3*e^3)*_Z^3+3*b*d^2*e^2*p*_Z^2-3*b*d
*e*p^2*_Z+b*p^3))*a*e^5*x-2*sum(_R*ln(((-4*a*e^7-2*b*d^3*e^4)*_R^3-3*b*d^2*e^3*p*_R^2+8*b*d*e^2*p^2*_R-3*b*e*p
^3)*x+(-5*a*d*e^6-b*d^4*e^3)*_R^3+(a*e^5*p-b*d^3*e^2*p)*_R^2+5*b*d^2*e*p^2*_R-3*b*d*p^3),_R=RootOf((a*e^6-b*d^
3*e^3)*_Z^3+3*b*d^2*e^2*p*_Z^2-3*b*d*e*p^2*_Z+b*p^3))*b*d^3*e^2*x+6*ln(-e*x-d)*b*d^2*e*p*x+2*sum(_R*ln(((-4*a*
e^7-2*b*d^3*e^4)*_R^3-3*b*d^2*e^3*p*_R^2+8*b*d*e^2*p^2*_R-3*b*e*p^3)*x+(-5*a*d*e^6-b*d^4*e^3)*_R^3+(a*e^5*p-b*
d^3*e^2*p)*_R^2+5*b*d^2*e*p^2*_R-3*b*d*p^3),_R=RootOf((a*e^6-b*d^3*e^3)*_Z^3+3*b*d^2*e^2*p*_Z^2-3*b*d*e*p^2*_Z
+b*p^3))*a*d*e^4-2*sum(_R*ln(((-4*a*e^7-2*b*d^3*e^4)*_R^3-3*b*d^2*e^3*p*_R^2+8*b*d*e^2*p^2*_R-3*b*e*p^3)*x+(-5
*a*d*e^6-b*d^4*e^3)*_R^3+(a*e^5*p-b*d^3*e^2*p)*_R^2+5*b*d^2*e*p^2*_R-3*b*d*p^3),_R=RootOf((a*e^6-b*d^3*e^3)*_Z
^3+3*b*d^2*e^2*p*_Z^2-3*b*d*e*p^2*_Z+b*p^3))*b*d^4*e+6*ln(-e*x-d)*b*d^3*p-2*ln(c)*a*e^3+2*ln(c)*b*d^3)/(e*x+d)
/e/(a*e^3-b*d^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(log(c*(b*x^3+a)^p)/(e*x+d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 14.1578, size = 13520, normalized size = 46.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^3+a)^p)/(e*x+d)^2,x, algorithm="fricas")

[Out]

1/4*(2*(b*d^4*e - a*d*e^4 + (b*d^3*e^2 - a*e^5)*x)*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*d^4*p^2/(b*d^3*e - a*e^
4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*sqrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^
3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3) - (b^
3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3
- a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3)*(I*sqrt(3) + 1))*log(3/2*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*
d^4*p^2/(b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*sqrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 -
3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3
- a*e^3)^2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4))
+ 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3)*(I*sqrt(3) + 1))*b*d^2*e*p + b*e*p^2*x
- 2*b*d*p^2 - 1/4*(b*d^3*e^2 - a*e^5)*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*d^4*p^2/(b*d^3*e - a*e^4)^2 - b*d*p
^2/(b*d^3*e^2 - a*e^5))*(-I*sqrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^
5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3) - (b^3*d^6*p^3/(b
*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) +
1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3)*(I*sqrt(3) + 1))^2) - 4*(b*d^3 - a*e^3)*p*log(b*x^3 + a) + (6*b*d^2*e*p*x
+ 6*b*d^3*p - (b*d^4*e - a*d*e^4 + (b*d^3*e^2 - a*e^5)*x)*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*d^4*p^2/(b*d^3*
e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*sqrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^
3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/
3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b
*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3)*(I*sqrt(3) + 1)) + sqrt(3)*(b*d^4*e - a*d*e^4 + (b*d^
3*e^2 - a*e^5)*x)*sqrt(-((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*d^4*p^2/(
b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*sqrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*
d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^
2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*
p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3)*(I*sqrt(3) + 1))^2 - 4*(b^2*d^5*e - a*b*d^2*e^4
)*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*d^4*p^2/(b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*sqrt(3) +
1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*
d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/(
(b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3)*
(I*sqrt(3) + 1))*p + 4*(b^2*d^4 - 4*a*b*d*e^3)*p^2)/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)))*log(-3/2*(2*b*d^
2*p/(b*d^3*e - a*e^4) - (b^2*d^4*p^2/(b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*sqrt(3) + 1)/(b^3*
d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 -
a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^
2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3)*(I*sqrt(3
) + 1))*b*d^2*e*p + 2*b*e*p^2*x + 2*b*d*p^2 + 1/4*(b*d^3*e^2 - a*e^5)*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*d^4*
p^2/(b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*sqrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2
*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*
e^3)^2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1
/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3)*(I*sqrt(3) + 1))^2 + 1/4*sqrt(3)*(b*d^3*e^
2 - a*e^5)*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*d^4*p^2/(b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*
sqrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*
b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*
d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^
2)^(1/3)*(I*sqrt(3) + 1))*sqrt(-((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*d
^4*p^2/(b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*sqrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 -
3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 -
a*e^3)^2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4))
+ 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3)*(I*sqrt(3) + 1))^2 - 4*(b^2*d^5*e - a*b
*d^2*e^4)*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*d^4*p^2/(b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*s
qrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b
*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d
^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2
)^(1/3)*(I*sqrt(3) + 1))*p + 4*(b^2*d^4 - 4*a*b*d*e^3)*p^2)/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8))) + (6*b*d
^2*e*p*x + 6*b*d^3*p - (b*d^4*e - a*d*e^4 + (b*d^3*e^2 - a*e^5)*x)*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*d^4*p^2
/(b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*sqrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^
2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3
)^2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*
b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3)*(I*sqrt(3) + 1)) - sqrt(3)*(b*d^4*e - a*d*e^4
+ (b*d^3*e^2 - a*e^5)*x)*sqrt(-((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*d
^4*p^2/(b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*sqrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 -
3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 -
a*e^3)^2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4))
+ 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3)*(I*sqrt(3) + 1))^2 - 4*(b^2*d^5*e - a*b
*d^2*e^4)*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*d^4*p^2/(b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*s
qrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b
*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d
^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2
)^(1/3)*(I*sqrt(3) + 1))*p + 4*(b^2*d^4 - 4*a*b*d*e^3)*p^2)/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)))*log(-3/2
*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*d^4*p^2/(b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*sqrt(3) +
1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d
^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((
b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3)*(
I*sqrt(3) + 1))*b*d^2*e*p + 2*b*e*p^2*x + 2*b*d*p^2 + 1/4*(b*d^3*e^2 - a*e^5)*(2*b*d^2*p/(b*d^3*e - a*e^4) - (
b^2*d^4*p^2/(b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*sqrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)
^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*
d^3 - a*e^3)^2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e
^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3)*(I*sqrt(3) + 1))^2 - 1/4*sqrt(3)*(
b*d^3*e^2 - a*e^5)*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*d^4*p^2/(b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^
5))*(-I*sqrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)
) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 -
3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 -
a*e^3)^2)^(1/3)*(I*sqrt(3) + 1))*sqrt(-((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(2*b*d^2*p/(b*d^3*e - a*e^4)
- (b^2*d^4*p^2/(b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5))*(-I*sqrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e
^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/
(b*d^3 - a*e^3)^2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e -
a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3)*(I*sqrt(3) + 1))^2 - 4*(b^2*d^5
*e - a*b*d^2*e^4)*(2*b*d^2*p/(b*d^3*e - a*e^4) - (b^2*d^4*p^2/(b*d^3*e - a*e^4)^2 - b*d*p^2/(b*d^3*e^2 - a*e^5
))*(-I*sqrt(3) + 1)/(b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4))
+ 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 - a*e^3)^2)^(1/3) - (b^3*d^6*p^3/(b*d^3*e - a*e^4)^3 - 3
/2*b^2*d^3*p^3/((b*d^3*e^2 - a*e^5)*(b*d^3*e - a*e^4)) + 1/2*b*p^3/(b*d^3*e^3 - a*e^6) + 1/2*a*b*p^3/(b*d^3 -
a*e^3)^2)^(1/3)*(I*sqrt(3) + 1))*p + 4*(b^2*d^4 - 4*a*b*d*e^3)*p^2)/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)))
- 12*(b*d^2*e*p*x + b*d^3*p)*log(e*x + d) - 4*(b*d^3 - a*e^3)*log(c))/(b*d^4*e - a*d*e^4 + (b*d^3*e^2 - a*e^5)
*x)

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

[Out]

Timed out

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Giac [A]  time = 1.33915, size = 537, normalized size = 1.84 \begin{align*} \frac{b d^{2} p \log \left ({\left | b x^{3} + a \right |}\right )}{b d^{3} e - a e^{4}} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} b p \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 )}{b^{2} d^{2} - \left (-a b^{2}\right )^{\frac{1}{3}} b d e + \left (-a b^{2}\right )^{\frac{2}{3}} e^{2}} - \frac{{\left (a b^{3} d^{4} p e^{2} - a b^{3} d^{3} p \left (-\frac{a}{b}\right )^{\frac{1}{3}} e^{3} - a^{2} b^{2} d p e^{5} + a^{2} b^{2} p \left (-\frac{a}{b}\right )^{\frac{1}{3}} e^{6}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a b^{3} d^{6} e^{2} - 2 \, a^{2} b^{2} d^{3} e^{5} + a^{3} b e^{8}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b d p - \left (-a b^{2}\right )^{\frac{2}{3}} p e\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{2 \,{\left (b^{2} d^{3} - a b e^{3}\right )}} - \frac{3 \, b d^{2} p x e \log \left (x e + d\right ) + b d^{3} p \log \left (b x^{3} + a\right ) + 3 \, b d^{3} p \log \left (x e + d\right ) + b d^{3} \log \left (c\right ) - a p e^{3} \log \left (b x^{3} + a\right ) - a e^{3} \log \left (c\right )}{b d^{3} x e^{2} + b d^{4} e - a x e^{5} - a d e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^3+a)^p)/(e*x+d)^2,x, algorithm="giac")

[Out]

b*d^2*p*log(abs(b*x^3 + a))/(b*d^3*e - a*e^4) + sqrt(3)*(-a*b^2)^(1/3)*b*p*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1
/3))/(-a/b)^(1/3))/(b^2*d^2 - (-a*b^2)^(1/3)*b*d*e + (-a*b^2)^(2/3)*e^2) - (a*b^3*d^4*p*e^2 - a*b^3*d^3*p*(-a/
b)^(1/3)*e^3 - a^2*b^2*d*p*e^5 + a^2*b^2*p*(-a/b)^(1/3)*e^6)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^3*d^
6*e^2 - 2*a^2*b^2*d^3*e^5 + a^3*b*e^8) + 1/2*((-a*b^2)^(1/3)*b*d*p - (-a*b^2)^(2/3)*p*e)*log(x^2 + x*(-a/b)^(1
/3) + (-a/b)^(2/3))/(b^2*d^3 - a*b*e^3) - (3*b*d^2*p*x*e*log(x*e + d) + b*d^3*p*log(b*x^3 + a) + 3*b*d^3*p*log
(x*e + d) + b*d^3*log(c) - a*p*e^3*log(b*x^3 + a) - a*e^3*log(c))/(b*d^3*x*e^2 + b*d^4*e - a*x*e^5 - a*d*e^4)