Optimal. Leaf size=290 \[ 6 a p \text {Int}\left (\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2},x\right )+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac {24 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}+x \log ^3\left (c \left (a+b x^2\right )^p\right )-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {24 i \sqrt {a} p^3 \text {Li}_2\left (1-\frac {2 \sqrt {a}}{i \sqrt {b} x+\sqrt {a}}\right )}{\sqrt {b}}-\frac {24 i \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}+\frac {48 \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {48 \sqrt {a} p^3 \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-48 p^3 x \]
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Rubi [A] time = 0.43, 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 = {} \[ \int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx &=x \log ^3\left (c \left (a+b x^2\right )^p\right )-(6 b p) \int \frac {x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=x \log ^3\left (c \left (a+b x^2\right )^p\right )-(6 b p) \int \left (\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac {a \log ^2\left (c \left (a+b x^2\right )^p\right )}{b \left (a+b x^2\right )}\right ) \, dx\\ &=x \log ^3\left (c \left (a+b x^2\right )^p\right )-(6 p) \int \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\left (24 b p^2\right ) \int \frac {x^2 \log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\left (24 b p^2\right ) \int \left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{b}-\frac {a \log \left (c \left (a+b x^2\right )^p\right )}{b \left (a+b x^2\right )}\right ) \, dx\\ &=-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\left (24 p^2\right ) \int \log \left (c \left (a+b x^2\right )^p\right ) \, dx-\left (24 a p^2\right ) \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac {24 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx-\left (48 b p^3\right ) \int \frac {x^2}{a+b x^2} \, dx+\left (48 a b p^3\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \left (a+b x^2\right )} \, dx\\ &=-48 p^3 x+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac {24 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\left (48 a p^3\right ) \int \frac {1}{a+b x^2} \, dx+\left (48 \sqrt {a} \sqrt {b} p^3\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a+b x^2} \, dx\\ &=-48 p^3 x+\frac {48 \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {24 i \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac {24 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx-\left (48 p^3\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{i-\frac {\sqrt {b} x}{\sqrt {a}}} \, dx\\ &=-48 p^3 x+\frac {48 \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {24 i \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}-\frac {48 \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac {24 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\left (48 p^3\right ) \int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {b} x}{\sqrt {a}}}\right )}{1+\frac {b x^2}{a}} \, dx\\ &=-48 p^3 x+\frac {48 \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {24 i \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}-\frac {48 \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac {24 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx-\frac {\left (48 i \sqrt {a} p^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {b} x}{\sqrt {a}}}\right )}{\sqrt {b}}\\ &=-48 p^3 x+\frac {48 \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {24 i \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}-\frac {48 \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac {24 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac {24 i \sqrt {a} p^3 \text {Li}_2\left (1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}+(6 a 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 = 3.53, size = 789, normalized size = 2.72 \[ \frac {p^3 \left (-6 \sqrt {-a^2} \sqrt {\frac {b x^2}{a+b x^2}} \left (8 \sqrt {a} \, _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 )+\log \left (a+b x^2\right ) \left (4 \sqrt {a} \, _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 )+\sqrt {a+b x^2} \log \left (a+b x^2\right ) \sin ^{-1}\left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right )\right )\right )-48 \sqrt {-a^2} \sqrt {\frac {b x^2}{a+b x^2}} \sqrt {a+b x^2} \sin ^{-1}\left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right )+6 (-a)^{3/2} \sqrt {-\frac {b x^2}{a}} \left (-4 \text {Li}_2\left (\frac {1}{2}-\frac {1}{2} \sqrt {-\frac {b x^2}{a}}\right )+\log ^2\left (\frac {b x^2}{a}+1\right )+2 \log ^2\left (\frac {1}{2} \left (\sqrt {-\frac {b x^2}{a}}+1\right )\right )-4 \log \left (\frac {1}{2} \left (\sqrt {-\frac {b x^2}{a}}+1\right )\right ) \log \left (\frac {b x^2}{a}+1\right )\right )+\sqrt {-a} b x^2 \left (\log ^3\left (a+b x^2\right )-6 \log ^2\left (a+b x^2\right )+24 \log \left (a+b x^2\right )-48\right )+24 a \sqrt {b x^2} \left (\log \left (a+b x^2\right )-\log \left (\frac {b x^2}{a}+1\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b x^2}}{\sqrt {-a}}\right )\right )}{\sqrt {-a} b x}-\frac {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 (4 i \sqrt {a} \text {Li}_2\left (\frac {\sqrt {b} x+i \sqrt {a}}{\sqrt {b} x-i \sqrt {a}}\right )+\sqrt {b} x \left (\log ^2\left (a+b x^2\right )-4 \log \left (a+b x^2\right )+8\right )+4 \sqrt {a} \left (\log \left (a+b x^2\right )+2 \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )-2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+4 i \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2\right )}{\sqrt {b}}+3 p x \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 \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2 \left (\log \left (c \left (a+b x^2\right )^p\right )+p \left (-\log \left (a+b x^2\right )\right )-6 p\right )+\frac {6 \sqrt {a} 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 {b}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}, x\right ) \]
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 \[ \int \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.73, size = 0, normalized size = 0.00 \[ \int \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ p^{3} x \log \left (b x^{2} + a\right )^{3} + \int \frac {b x^{2} \log \relax (c)^{3} + a \log \relax (c)^{3} - 3 \, {\left ({\left (2 \, p^{3} - p^{2} \log \relax (c)\right )} b x^{2} - a p^{2} \log \relax (c)\right )} \log \left (b x^{2} + a\right )^{2} + 3 \, {\left (b p x^{2} \log \relax (c)^{2} + a p \log \relax (c)^{2}\right )} \log \left (b x^{2} + a\right )}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3 \,d x \]
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 \[ \int \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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