Optimal. Leaf size=336 \[ \frac {184 a^2 p^2 x}{75 b^2}-\frac {64 a p^2 x^3}{225 b}+\frac {8 p^2 x^5}{125}-\frac {184 a^{5/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{75 b^{5/2}}+\frac {4 i a^{5/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{5 b^{5/2}}+\frac {8 a^{5/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{5 b^{5/2}}-\frac {4 a^2 p x \log \left (c \left (a+b x^2\right )^p\right )}{5 b^2}+\frac {4 a p x^3 \log \left (c \left (a+b x^2\right )^p\right )}{15 b}-\frac {4}{25} p x^5 \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 a^{5/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{5 b^{5/2}}+\frac {1}{5} x^5 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {4 i a^{5/2} p^2 \text {Li}_2\left (1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{5 b^{5/2}} \]
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Rubi [A]
time = 0.26, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {2507, 2526,
2498, 327, 211, 2505, 308, 2520, 12, 5040, 4964, 2449, 2352} \begin {gather*} \frac {4 i a^{5/2} p^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{5 b^{5/2}}+\frac {4 a^{5/2} p \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{5 b^{5/2}}+\frac {4 i a^{5/2} p^2 \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{5 b^{5/2}}-\frac {184 a^{5/2} p^2 \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{75 b^{5/2}}+\frac {8 a^{5/2} p^2 \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{5 b^{5/2}}-\frac {4 a^2 p x \log \left (c \left (a+b x^2\right )^p\right )}{5 b^2}+\frac {184 a^2 p^2 x}{75 b^2}+\frac {1}{5} x^5 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {4}{25} p x^5 \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 a p x^3 \log \left (c \left (a+b x^2\right )^p\right )}{15 b}-\frac {64 a p^2 x^3}{225 b}+\frac {8 p^2 x^5}{125} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 2498
Rule 2505
Rule 2507
Rule 2520
Rule 2526
Rule 4964
Rule 5040
Rubi steps
\begin {align*} \int x^4 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac {1}{5} x^5 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {1}{5} (4 b p) \int \frac {x^6 \log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=\frac {1}{5} x^5 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {1}{5} (4 b p) \int \left (\frac {a^2 \log \left (c \left (a+b x^2\right )^p\right )}{b^3}-\frac {a x^2 \log \left (c \left (a+b x^2\right )^p\right )}{b^2}+\frac {x^4 \log \left (c \left (a+b x^2\right )^p\right )}{b}-\frac {a^3 \log \left (c \left (a+b x^2\right )^p\right )}{b^3 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {1}{5} x^5 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {1}{5} (4 p) \int x^4 \log \left (c \left (a+b x^2\right )^p\right ) \, dx-\frac {\left (4 a^2 p\right ) \int \log \left (c \left (a+b x^2\right )^p\right ) \, dx}{5 b^2}+\frac {\left (4 a^3 p\right ) \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{5 b^2}+\frac {(4 a p) \int x^2 \log \left (c \left (a+b x^2\right )^p\right ) \, dx}{5 b}\\ &=-\frac {4 a^2 p x \log \left (c \left (a+b x^2\right )^p\right )}{5 b^2}+\frac {4 a p x^3 \log \left (c \left (a+b x^2\right )^p\right )}{15 b}-\frac {4}{25} p x^5 \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 a^{5/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{5 b^{5/2}}+\frac {1}{5} x^5 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {1}{15} \left (8 a p^2\right ) \int \frac {x^4}{a+b x^2} \, dx+\frac {\left (8 a^2 p^2\right ) \int \frac {x^2}{a+b x^2} \, dx}{5 b}-\frac {\left (8 a^3 p^2\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}{5 b}+\frac {1}{25} \left (8 b p^2\right ) \int \frac {x^6}{a+b x^2} \, dx\\ &=\frac {8 a^2 p^2 x}{5 b^2}-\frac {4 a^2 p x \log \left (c \left (a+b x^2\right )^p\right )}{5 b^2}+\frac {4 a p x^3 \log \left (c \left (a+b x^2\right )^p\right )}{15 b}-\frac {4}{25} p x^5 \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 a^{5/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{5 b^{5/2}}+\frac {1}{5} x^5 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {1}{15} \left (8 a p^2\right ) \int \left (-\frac {a}{b^2}+\frac {x^2}{b}+\frac {a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx-\frac {\left (8 a^3 p^2\right ) \int \frac {1}{a+b x^2} \, dx}{5 b^2}-\frac {\left (8 a^{5/2} p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a+b x^2} \, dx}{5 b^{3/2}}+\frac {1}{25} \left (8 b p^2\right ) \int \left (\frac {a^2}{b^3}-\frac {a x^2}{b^2}+\frac {x^4}{b}-\frac {a^3}{b^3 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {184 a^2 p^2 x}{75 b^2}-\frac {64 a p^2 x^3}{225 b}+\frac {8 p^2 x^5}{125}-\frac {8 a^{5/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{5 b^{5/2}}+\frac {4 i a^{5/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{5 b^{5/2}}-\frac {4 a^2 p x \log \left (c \left (a+b x^2\right )^p\right )}{5 b^2}+\frac {4 a p x^3 \log \left (c \left (a+b x^2\right )^p\right )}{15 b}-\frac {4}{25} p x^5 \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 a^{5/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{5 b^{5/2}}+\frac {1}{5} x^5 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {\left (8 a^2 p^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{i-\frac {\sqrt {b} x}{\sqrt {a}}} \, dx}{5 b^2}-\frac {\left (8 a^3 p^2\right ) \int \frac {1}{a+b x^2} \, dx}{25 b^2}-\frac {\left (8 a^3 p^2\right ) \int \frac {1}{a+b x^2} \, dx}{15 b^2}\\ &=\frac {184 a^2 p^2 x}{75 b^2}-\frac {64 a p^2 x^3}{225 b}+\frac {8 p^2 x^5}{125}-\frac {184 a^{5/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{75 b^{5/2}}+\frac {4 i a^{5/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{5 b^{5/2}}+\frac {8 a^{5/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{5 b^{5/2}}-\frac {4 a^2 p x \log \left (c \left (a+b x^2\right )^p\right )}{5 b^2}+\frac {4 a p x^3 \log \left (c \left (a+b x^2\right )^p\right )}{15 b}-\frac {4}{25} p x^5 \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 a^{5/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{5 b^{5/2}}+\frac {1}{5} x^5 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {\left (8 a^2 p^2\right ) \int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {b} x}{\sqrt {a}}}\right )}{1+\frac {b x^2}{a}} \, dx}{5 b^2}\\ &=\frac {184 a^2 p^2 x}{75 b^2}-\frac {64 a p^2 x^3}{225 b}+\frac {8 p^2 x^5}{125}-\frac {184 a^{5/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{75 b^{5/2}}+\frac {4 i a^{5/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{5 b^{5/2}}+\frac {8 a^{5/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{5 b^{5/2}}-\frac {4 a^2 p x \log \left (c \left (a+b x^2\right )^p\right )}{5 b^2}+\frac {4 a p x^3 \log \left (c \left (a+b x^2\right )^p\right )}{15 b}-\frac {4}{25} p x^5 \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 a^{5/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{5 b^{5/2}}+\frac {1}{5} x^5 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {\left (8 i a^{5/2} p^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {b} x}{\sqrt {a}}}\right )}{5 b^{5/2}}\\ &=\frac {184 a^2 p^2 x}{75 b^2}-\frac {64 a p^2 x^3}{225 b}+\frac {8 p^2 x^5}{125}-\frac {184 a^{5/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{75 b^{5/2}}+\frac {4 i a^{5/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{5 b^{5/2}}+\frac {8 a^{5/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{5 b^{5/2}}-\frac {4 a^2 p x \log \left (c \left (a+b x^2\right )^p\right )}{5 b^2}+\frac {4 a p x^3 \log \left (c \left (a+b x^2\right )^p\right )}{15 b}-\frac {4}{25} p x^5 \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 a^{5/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{5 b^{5/2}}+\frac {1}{5} x^5 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {4 i a^{5/2} p^2 \text {Li}_2\left (1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{5 b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 248, normalized size = 0.74 \begin {gather*} \frac {900 i a^{5/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2+60 a^{5/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-46 p+30 p \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )+15 \log \left (c \left (a+b x^2\right )^p\right )\right )+\sqrt {b} x \left (8 p^2 \left (345 a^2-40 a b x^2+9 b^2 x^4\right )-60 p \left (15 a^2-5 a b x^2+3 b^2 x^4\right ) \log \left (c \left (a+b x^2\right )^p\right )+225 b^2 x^4 \log ^2\left (c \left (a+b x^2\right )^p\right )\right )+900 i a^{5/2} p^2 \text {Li}_2\left (\frac {i \sqrt {a}+\sqrt {b} x}{-i \sqrt {a}+\sqrt {b} x}\right )}{1125 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int x^{4} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{4} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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