3.1.20 \(\int \frac {\log (c (a+b x^3)^p)}{x^2} \, dx\) [20]

Optimal. Leaf size=133 \[ -\frac {\sqrt {3} \sqrt [3]{b} p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}-\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac {\sqrt [3]{b} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{x} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2505, 298, 31, 648, 631, 210, 642} \begin {gather*} \frac {\sqrt [3]{b} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [3]{b} p \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{x}-\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x^3)^p]/x^2,x]

[Out]

-((Sqrt[3]*b^(1/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(1/3)) - (b^(1/3)*p*Log[a^(1/3) + b^
(1/3)*x])/a^(1/3) + (b^(1/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*a^(1/3)) - Log[c*(a + b*x^3)
^p]/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 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

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 2505

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]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.00, size = 47, normalized size = 0.35 \begin {gather*} \frac {3 b p x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {b x^3}{a}\right )}{2 a}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x^3)^p]/x^2,x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.25, size = 184, normalized size = 1.38

method	result	size
risch	\(-\frac {\ln \left (\left (x^{3} b +a \right )^{p}\right )}{x}+\frac {-i \pi \,\mathrm {csgn}\left (i \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )+i \pi \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{3}-i \pi \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3} a +b \,p^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a -3 b \,p^{3}\right ) x +a p \,\textit {\_R}^{2}\right )\right ) x -2 \ln \left (c \right )}{2 x}\)	\(184\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 119, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, b p {\left (\frac {2 \, \sqrt {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 )}{b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )} - \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.38, size = 126, normalized size = 0.95 \begin {gather*} \frac {2 \, \sqrt {3} p x \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - p x \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) + 2 \, p x \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) - 2 \, p \log \left (b x^{3} + a\right ) - 2 \, \log \left (c\right )}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*sqrt(3)*p*x*(-b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(-b/a)^(1/3) + 1/3*sqrt(3)) - p*x*(-b/a)^(1/3)*log(b*x^2
- a*x*(-b/a)^(2/3) - a*(-b/a)^(1/3)) + 2*p*x*(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/3)) - 2*p*log(b*x^3 + a) - 2*l
og(c))/x

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Sympy [A]
time = 117.59, size = 165, normalized size = 1.24 \begin {gather*} \begin {cases} - \frac {\log {\left (0^{p} c \right )}}{x} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3 p}{x} - \frac {\log {\left (c \left (b x^{3}\right )^{p} \right )}}{x} & \text {for}\: a = 0 \\- \frac {\log {\left (a^{p} c \right )}}{x} & \text {for}\: b = 0 \\- \frac {\log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{x} + \frac {3 b p \left (- \frac {a}{b}\right )^{\frac {2}{3}} \log {\left (4 x^{2} + 4 x \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{2 a} - \frac {\sqrt {3} b p \left (- \frac {a}{b}\right )^{\frac {2}{3}} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{a} - \frac {b \left (- \frac {a}{b}\right )^{\frac {2}{3}} \log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{a} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-log(0**p*c)/x, Eq(a, 0) & Eq(b, 0)), (-3*p/x - log(c*(b*x**3)**p)/x, Eq(a, 0)), (-log(a**p*c)/x, E
q(b, 0)), (-log(c*(a + b*x**3)**p)/x + 3*b*p*(-a/b)**(2/3)*log(4*x**2 + 4*x*(-a/b)**(1/3) + 4*(-a/b)**(2/3))/(
2*a) - sqrt(3)*b*p*(-a/b)**(2/3)*atan(2*sqrt(3)*x/(3*(-a/b)**(1/3)) + sqrt(3)/3)/a - b*(-a/b)**(2/3)*log(c*(a
+ b*x**3)**p)/a, True))

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Giac [A]
time = 5.59, size = 137, normalized size = 1.03 \begin {gather*} -\frac {b p \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} 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 )}{a b} - \frac {p \log \left (b x^{3} + a\right )}{x} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} p \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, a b} - \frac {\log \left (c\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.81, size = 149, normalized size = 1.12 \begin {gather*} \frac {{\left (-b\right )}^{1/3}\,p\,\ln \left (a^{1/3}\,{\left (-b\right )}^{8/3}+b^3\,x\right )}{a^{1/3}}-\frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{x}+\frac {{\left (-b\right )}^{1/3}\,p\,\ln \left (9\,b^3\,p^2\,x+9\,a^{1/3}\,{\left (-b\right )}^{8/3}\,p^2\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{1/3}}-\frac {{\left (-b\right )}^{1/3}\,p\,\ln \left (9\,b^3\,p^2\,x+9\,a^{1/3}\,{\left (-b\right )}^{8/3}\,p^2\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b*x^3)^p)/x^2,x)

[Out]

((-b)^(1/3)*p*log(a^(1/3)*(-b)^(8/3) + b^3*x))/a^(1/3) - log(c*(a + b*x^3)^p)/x + ((-b)^(1/3)*p*log(9*b^3*p^2*
x + 9*a^(1/3)*(-b)^(8/3)*p^2*((3^(1/2)*1i)/2 - 1/2)^2)*((3^(1/2)*1i)/2 - 1/2))/a^(1/3) - ((-b)^(1/3)*p*log(9*b
^3*p^2*x + 9*a^(1/3)*(-b)^(8/3)*p^2*((3^(1/2)*1i)/2 + 1/2)^2)*((3^(1/2)*1i)/2 + 1/2))/a^(1/3)

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