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

Optimal. Leaf size=51 \[ 6 b p \text {Int}\left (\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2},x\right )-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x} \]

[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)

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Rubi [A]  time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx \]

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*}

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Mathematica [A]  time = 0.83, size = 505, normalized size = 9.90 \[ \frac {p^3 \left (-96 \sqrt {a} \sqrt {1-\frac {a}{a+b x^2}} \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {a}{b x^2+a}\right )-48 \sqrt {a} \sqrt {1-\frac {a}{a+b x^2}} \log \left (a+b x^2\right ) \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {a}{b x^2+a}\right )-2 \log ^2\left (a+b x^2\right ) \left (\sqrt {a} \log \left (a+b x^2\right )+6 \sqrt {a+b x^2} \sqrt {1-\frac {a}{a+b x^2}} \sin ^{-1}\left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right )\right )\right )}{2 \sqrt {a} x}+3 p^2 \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right ) \left (-\frac {\log ^2\left (a+b x^2\right )}{x}+\frac {4 \sqrt {b} \left (i \text {Li}_2\left (\frac {\sqrt {b} x+i \sqrt {a}}{\sqrt {b} x-i \sqrt {a}}\right )+\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (\log \left (a+b x^2\right )+2 \log \left (\frac {2 i}{-\frac {\sqrt {b} x}{\sqrt {a}}+i}\right )+i \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right )\right )}{\sqrt {a}}\right )-\frac {3 p \log \left (a+b x^2\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2}{x}-\frac {\left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^3}{x}+\frac {6 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2}{\sqrt {a}} \]

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])

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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{2}}, x\right ) \]

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)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{2}}\,{d x} \]

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)

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maple [A]  time = 7.52, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )^{3}}{x^{2}}\, dx \]

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)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{x^{2}}\, dx \]

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)

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