\(\int \frac {\log (c (a+b x^3)^p)}{(d+e x)^2} \, dx\) [196]

Optimal result

Integrand size = 20, antiderivative size = 292 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=-\frac {\sqrt {3} \sqrt [3]{a} \sqrt [3]{b} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{2/3} d^2+\sqrt [3]{a} \sqrt [3]{b} d e+a^{2/3} e^2}+\frac {\sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} d+\sqrt [3]{a} e\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b d^3-a e^3}-\frac {3 b d^2 p \log (d+e x)}{e \left (b d^3-a e^3\right )}-\frac {\sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} d+\sqrt [3]{a} e\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \left (b d^3-a e^3\right )}+\frac {b d^2 p \log \left (a+b x^3\right )}{e \left (b d^3-a e^3\right )}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{e (d+e x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2513, 6857, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=-\frac {\sqrt {3} \sqrt [3]{a} \sqrt [3]{b} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{2/3} e^2+\sqrt [3]{a} \sqrt [3]{b} d e+b^{2/3} d^2}-\frac {\sqrt [3]{a} \sqrt [3]{b} p \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \left (b d^3-a e^3\right )}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{e (d+e x)}+\frac {\sqrt [3]{a} \sqrt [3]{b} p \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b d^3-a e^3}+\frac {b d^2 p \log \left (a+b x^3\right )}{e \left (b d^3-a e^3\right )}-\frac {3 b d^2 p \log (d+e x)}{e \left (b d^3-a e^3\right )} \]

[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 31

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

Rule 210

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

Rule 266

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]

Rule 631

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 642

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 648

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 1874

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 1885

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 2513

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 6857

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]

Rubi steps \begin{align*} \text {integral}& = -\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 ) \text {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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.21 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.87 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=-\frac {2 \sqrt {3} \sqrt [3]{a} b^{2/3} d e p (d+e x) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+3 b e^2 p x^2 (d+e x) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b x^3}{a}\right )-2 \sqrt [3]{a} b^{2/3} d e p (d+e x) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+6 b d^2 p (d+e x) \log (d+e x)+\sqrt [3]{a} b^{2/3} d e p (d+e x) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 b d^2 p (d+e x) \log \left (a+b x^3\right )+2 \left (b d^3-a e^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{2 e \left (b d^3-a e^3\right ) (d+e x)} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.73 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.95

method	result	size
parts	\(-\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{e \left (e x +d \right )}+\frac {3 p b \left (\frac {d^{2} \ln \left (e x +d \right )}{a \,e^{3}-b \,d^{3}}+\frac {-a d e \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+a \,e^{2} \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-\frac {d^{2} \ln \left (b \,x^{3}+a \right )}{3}}{a \,e^{3}-b \,d^{3}}\right )}{e}\)	\(277\)
risch	\(\text {Expression too large to display}\)	\(1068\)

[In]

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

[Out]

-ln(c*(b*x^3+a)^p)/e/(e*x+d)+3*p*b/e*(d^2/(a*e^3-b*d^3)*ln(e*x+d)+(-a*d*e*(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))
-1/6/b/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/
3)*x-1)))+a*e^2*(-1/3/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/6/b/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3*
3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))-1/3*d^2*ln(b*x^3+a))/(a*e^3-b*d^3))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.25 (sec) , antiderivative size = 7010, normalized size of antiderivative = 24.01 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.07 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=-\frac {{\left (\frac {6 \, d^{2} \log \left (e x + d\right )}{b d^{3} - a e^{3}} + \frac {2 \, \sqrt {3} {\left (a e^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a d e \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{{\left (b^{2} d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (2 \, b d^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a e^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} - a d e\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b^{2} d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, {\left (b d^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a e^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + a d e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )} b p}{2 \, e} - \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{{\left (e x + d\right )} e} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.24 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=-\frac {3 \, b d^{2} p \log \left (e x + d\right )}{b d^{3} e - a e^{4}} + \frac {b d^{2} p \log \left ({\left | b x^{3} + a \right |}\right )}{b d^{3} e - a e^{4}} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} b p \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{2} d^{2} - \left (-a b^{2}\right )^{\frac {1}{3}} b d e + \left (-a b^{2}\right )^{\frac {2}{3}} e^{2}} + \frac {{\left (a b^{3} d^{3} e^{3} p \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{2} e^{6} p \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{3} d^{4} e^{2} p + a^{2} b^{2} d e^{5} p\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a b^{3} d^{6} e^{2} - 2 \, a^{2} b^{2} d^{3} e^{5} + a^{3} b e^{8}} - \frac {p \log \left (b x^{3} + a\right )}{e^{2} x + d e} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b d p - \left (-a b^{2}\right )^{\frac {2}{3}} e p\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, {\left (b^{2} d^{3} - a b e^{3}\right )}} - \frac {\log \left (c\right )}{e^{2} x + d e} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.56 (sec) , antiderivative size = 736, normalized size of antiderivative = 2.52 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=\left (\sum _{k=1}^3\ln \left (-\frac {27\,a\,b^4\,d\,p^3+27\,a\,b^4\,e\,p^3\,x+{\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )}^3\,a\,b^4\,d^4\,e^3\,9+{\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )}^3\,a^2\,b^3\,d\,e^6\,45-{\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )}^2\,a^2\,b^3\,e^5\,p\,9+{\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )}^3\,a^2\,b^3\,e^7\,x\,36+{\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )}^2\,a\,b^4\,d^3\,e^2\,p\,9+{\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )}^3\,a\,b^4\,d^3\,e^4\,x\,18-\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )\,a\,b^4\,d^2\,e\,p^2\,45-\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )\,a\,b^4\,d\,e^2\,p^2\,x\,72+{\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )}^2\,a\,b^4\,d^2\,e^3\,p\,x\,27}{e^2}\right )\,\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )\right )-\frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{x\,e^2+d\,e}+\frac {3\,b\,d^2\,p\,\ln \left (d+e\,x\right )}{a\,e^4-b\,d^3\,e} \]

[In]

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

[Out]

symsum(log(-(27*a*b^4*d*p^3 + 27*a*b^4*e*p^3*x + 9*root(b*d^3*e^3*z^3 - a*e^6*z^3 - 3*b*d^2*e^2*p*z^2 + 3*b*d*
e*p^2*z - b*p^3, z, k)^3*a*b^4*d^4*e^3 + 45*root(b*d^3*e^3*z^3 - a*e^6*z^3 - 3*b*d^2*e^2*p*z^2 + 3*b*d*e*p^2*z
- b*p^3, z, k)^3*a^2*b^3*d*e^6 - 9*root(b*d^3*e^3*z^3 - a*e^6*z^3 - 3*b*d^2*e^2*p*z^2 + 3*b*d*e*p^2*z - b*p^3
, z, k)^2*a^2*b^3*e^5*p + 36*root(b*d^3*e^3*z^3 - a*e^6*z^3 - 3*b*d^2*e^2*p*z^2 + 3*b*d*e*p^2*z - b*p^3, z, k)
^3*a^2*b^3*e^7*x + 9*root(b*d^3*e^3*z^3 - a*e^6*z^3 - 3*b*d^2*e^2*p*z^2 + 3*b*d*e*p^2*z - b*p^3, z, k)^2*a*b^4
*d^3*e^2*p + 18*root(b*d^3*e^3*z^3 - a*e^6*z^3 - 3*b*d^2*e^2*p*z^2 + 3*b*d*e*p^2*z - b*p^3, z, k)^3*a*b^4*d^3*
e^4*x - 45*root(b*d^3*e^3*z^3 - a*e^6*z^3 - 3*b*d^2*e^2*p*z^2 + 3*b*d*e*p^2*z - b*p^3, z, k)*a*b^4*d^2*e*p^2 -
72*root(b*d^3*e^3*z^3 - a*e^6*z^3 - 3*b*d^2*e^2*p*z^2 + 3*b*d*e*p^2*z - b*p^3, z, k)*a*b^4*d*e^2*p^2*x + 27*r
oot(b*d^3*e^3*z^3 - a*e^6*z^3 - 3*b*d^2*e^2*p*z^2 + 3*b*d*e*p^2*z - b*p^3, z, k)^2*a*b^4*d^2*e^3*p*x)/e^2)*roo
t(b*d^3*e^3*z^3 - a*e^6*z^3 - 3*b*d^2*e^2*p*z^2 + 3*b*d*e*p^2*z - b*p^3, z, k), k, 1, 3) - log(c*(a + b*x^3)^p
)/(d*e + e^2*x) + (3*b*d^2*p*log(d + e*x))/(a*e^4 - b*d^3*e)