Optimal. Leaf size=80 \[ -\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {b p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}+\frac {b p^2 \text {Li}_2\left (\frac {b x^2}{a}+1\right )}{a} \]
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Rubi [A] time = 0.08, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2454, 2397, 2394, 2315} \[ \frac {b p^2 \text {PolyLog}\left (2,\frac {b x^2}{a}+1\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {b p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2394
Rule 2397
Rule 2454
Rubi steps
\begin {align*} \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log ^2\left (c (a+b x)^p\right )}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {(b p) \operatorname {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,x^2\right )}{a}\\ &=\frac {b p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}-\frac {\left (b^2 p^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,x^2\right )}{a}\\ &=\frac {b p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {b p^2 \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 93, normalized size = 1.16 \[ -\frac {b \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 x^2}+\frac {b p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}+\frac {b p^2 \text {Li}_2\left (\frac {b x^2+a}{a}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{3}}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.38, size = 841, normalized size = 10.51 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 118, normalized size = 1.48 \[ \frac {1}{2} \, b^{2} p^{2} {\left (\frac {\log \left (b x^{2} + a\right )^{2}}{a b} - \frac {2 \, {\left (2 \, \log \left (\frac {b x^{2}}{a} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {b x^{2}}{a}\right )\right )}}{a b}\right )} - b p {\left (\frac {\log \left (b x^{2} + a\right )}{a} - \frac {\log \left (x^{2}\right )}{a}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2}{x^3} \,d x \]
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 \[ \int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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