Optimal. Leaf size=51 \[ -\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x}+6 b p \text {Int}\left (\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2},x\right ) \]
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Rubi [A]
time = 0.03, 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 = {}
\begin {gather*} \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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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] Result contains complex when optimal does not.
time = 0.58, size = 505, normalized size = 9.90 \begin {gather*} \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}} \sin ^{-1}\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 \tan ^{-1}\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 (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (i \tan ^{-1}\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 \text {Li}_2\left (\frac {i \sqrt {a}+\sqrt {b} x}{-i \sqrt {a}+\sqrt {b} x}\right )\right )}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, 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.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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