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

Optimal. Leaf size=50 \[ 6 b p \text{Unintegrable}\left (\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2},x\right )-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0450119, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[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*} \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) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.781377, size = 505, normalized size = 10.1 \[ \frac{p^3 \left (-96 \sqrt{a} \sqrt{1-\frac{a}{a+b x^2}} \, _4F_3\left (\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{a}{b x^2+a}\right )-48 \sqrt{a} \sqrt{1-\frac{a}{a+b x^2}} \log \left (a+b x^2\right ) \, _3F_2\left (\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{a}{b x^2+a}\right )-2 \log ^2\left (a+b x^2\right ) \left (\sqrt{a} \log \left (a+b x^2\right )+6 \sqrt{a+b x^2} \sqrt{1-\frac{a}{a+b x^2}} \sin ^{-1}\left (\frac{\sqrt{a}}{\sqrt{a+b x^2}}\right )\right )\right )}{2 \sqrt{a} x}+3 p^2 \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right ) \left (-\frac{\log ^2\left (a+b x^2\right )}{x}+\frac{4 \sqrt{b} \left (i \text{PolyLog}\left (2,\frac{\sqrt{b} x+i \sqrt{a}}{\sqrt{b} x-i \sqrt{a}}\right )+\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (\log \left (a+b x^2\right )+2 \log \left (\frac{2 i}{-\frac{\sqrt{b} x}{\sqrt{a}}+i}\right )+i \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right )\right )}{\sqrt{a}}\right )-\frac{3 p \log \left (a+b x^2\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2}{x}-\frac{\left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^3}{x}+\frac{6 \sqrt{b} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2}{\sqrt{a}} \]

Antiderivative was successfully verified.

[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 [A]  time = 2.409, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{3}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

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^2+a)^p)^3/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)