\(\int \frac {\log ^3(c (a+b x^2)^p)}{x^2} \, dx\) [100]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x}+6 b p \text {Int}\left (\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2},x\right ) \]

[Out]

-ln(c*(b*x^2+a)^p)^3/x+6*b*p*Unintegrable(ln(c*(b*x^2+a)^p)^2/(b*x^2+a),x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx \]

[In]

Int[Log[c*(a + b*x^2)^p]^3/x^2,x]

[Out]

-(Log[c*(a + b*x^2)^p]^3/x) + 6*b*p*Defer[Int][Log[c*(a + b*x^2)^p]^2/(a + b*x^2), x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x}+(6 b p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.00 (sec) , antiderivative size = 505, normalized size of antiderivative = 28.06 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\frac {p^3 \left (-96 \sqrt {a} \sqrt {1-\frac {a}{a+b x^2}} \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right )-48 \sqrt {a} \sqrt {1-\frac {a}{a+b x^2}} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right ) \log \left (a+b x^2\right )-2 \log ^2\left (a+b x^2\right ) \left (6 \sqrt {a+b x^2} \sqrt {1-\frac {a}{a+b x^2}} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right )+\sqrt {a} \log \left (a+b x^2\right )\right )\right )}{2 \sqrt {a} x}+\frac {6 \sqrt {b} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{\sqrt {a}}-\frac {3 p \log \left (a+b x^2\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{x}-\frac {\left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^3}{x}+3 p^2 \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right ) \left (-\frac {\log ^2\left (a+b x^2\right )}{x}+\frac {4 \sqrt {b} \left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+2 \log \left (\frac {2 i}{i-\frac {\sqrt {b} x}{\sqrt {a}}}\right )+\log \left (a+b x^2\right )\right )+i \operatorname {PolyLog}\left (2,\frac {i \sqrt {a}+\sqrt {b} x}{-i \sqrt {a}+\sqrt {b} x}\right )\right )}{\sqrt {a}}\right ) \]

[In]

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

[Out]

(p^3*(-96*Sqrt[a]*Sqrt[1 - a/(a + b*x^2)]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, a/(a + b*x^
2)] - 48*Sqrt[a]*Sqrt[1 - a/(a + b*x^2)]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, a/(a + b*x^2)]*Log[a +
b*x^2] - 2*Log[a + b*x^2]^2*(6*Sqrt[a + b*x^2]*Sqrt[1 - a/(a + b*x^2)]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]] + Sqrt
[a]*Log[a + b*x^2])))/(2*Sqrt[a]*x) + (6*Sqrt[b]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-(p*Log[a + b*x^2]) + Log[c*(a
+ b*x^2)^p])^2)/Sqrt[a] - (3*p*Log[a + b*x^2]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/x - (-(p*Log[a
+ b*x^2]) + Log[c*(a + b*x^2)^p])^3/x + 3*p^2*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])*(-(Log[a + b*x^2]^2
/x) + (4*Sqrt[b]*(ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(I*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + 2*Log[(2*I)/(I - (Sqrt[b]*x)/Sq
rt[a])] + Log[a + b*x^2]) + I*PolyLog[2, (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b]*x)]))/Sqrt[a])

Maple [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

\[\int \frac {{\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{x^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

integral(log((b*x^2 + a)^p*c)^3/x^2, x)

Sympy [N/A]

Not integrable

Time = 2.95 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{x^{2}}\, dx \]

[In]

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

[Out]

Integral(log(c*(a + b*x**2)**p)**3/x**2, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.36 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3}{x^2} \,d x \]

[In]

int(log(c*(a + b*x^2)^p)^3/x^2,x)

[Out]

int(log(c*(a + b*x^2)^p)^3/x^2, x)