Optimal. Leaf size=193 \[ -\frac{b^3 p^2 \text{PolyLog}\left (2,\frac{a}{a+b x^2}\right )}{3 a^3}+\frac{b^3 p \log \left (1-\frac{a}{a+b x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3}+\frac{b^2 p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3 x^2}-\frac{b^2 p^2}{6 a^2 x^2}+\frac{b^3 p^2 \log \left (a+b x^2\right )}{6 a^3}-\frac{b^3 p^2 \log (x)}{a^3}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac{b p \log \left (c \left (a+b x^2\right )^p\right )}{6 a x^4} \]
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Rubi [A] time = 0.407236, antiderivative size = 211, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 12, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ \frac{b^3 p^2 \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right )}{3 a^3}-\frac{b^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 a^3}+\frac{b^3 p \log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3}+\frac{b^2 p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3 x^2}-\frac{b^2 p^2}{6 a^2 x^2}+\frac{b^3 p^2 \log \left (a+b x^2\right )}{6 a^3}-\frac{b^3 p^2 \log (x)}{a^3}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac{b p \log \left (c \left (a+b x^2\right )^p\right )}{6 a x^4} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2398
Rule 2411
Rule 2347
Rule 2344
Rule 2301
Rule 2317
Rule 2391
Rule 2314
Rule 31
Rule 2319
Rule 44
Rubi steps
\begin{align*} \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log ^2\left (c (a+b x)^p\right )}{x^4} \, dx,x,x^2\right )\\ &=-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac{1}{3} (b p) \operatorname{Subst}\left (\int \frac{\log \left (c (a+b x)^p\right )}{x^3 (a+b x)} \, dx,x,x^2\right )\\ &=-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac{1}{3} p \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right )}{x \left (-\frac{a}{b}+\frac{x}{b}\right )^3} \, dx,x,a+b x^2\right )\\ &=-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac{p \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right )}{\left (-\frac{a}{b}+\frac{x}{b}\right )^3} \, dx,x,a+b x^2\right )}{3 a}-\frac{(b p) \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right )}{x \left (-\frac{a}{b}+\frac{x}{b}\right )^2} \, dx,x,a+b x^2\right )}{3 a}\\ &=-\frac{b p \log \left (c \left (a+b x^2\right )^p\right )}{6 a x^4}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac{(b p) \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right )}{\left (-\frac{a}{b}+\frac{x}{b}\right )^2} \, dx,x,a+b x^2\right )}{3 a^2}+\frac{\left (b^2 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right )}{x \left (-\frac{a}{b}+\frac{x}{b}\right )} \, dx,x,a+b x^2\right )}{3 a^2}+\frac{\left (b p^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{a}{b}+\frac{x}{b}\right )^2} \, dx,x,a+b x^2\right )}{6 a}\\ &=-\frac{b p \log \left (c \left (a+b x^2\right )^p\right )}{6 a x^4}+\frac{b^2 p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3 x^2}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac{\left (b^2 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right )}{-\frac{a}{b}+\frac{x}{b}} \, dx,x,a+b x^2\right )}{3 a^3}-\frac{\left (b^3 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right )}{x} \, dx,x,a+b x^2\right )}{3 a^3}+\frac{\left (b p^2\right ) \operatorname{Subst}\left (\int \left (\frac{b^2}{a (a-x)^2}+\frac{b^2}{a^2 (a-x)}+\frac{b^2}{a^2 x}\right ) \, dx,x,a+b x^2\right )}{6 a}-\frac{\left (b^2 p^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x}{b}} \, dx,x,a+b x^2\right )}{3 a^3}\\ &=-\frac{b^2 p^2}{6 a^2 x^2}-\frac{b^3 p^2 \log (x)}{a^3}+\frac{b^3 p^2 \log \left (a+b x^2\right )}{6 a^3}-\frac{b p \log \left (c \left (a+b x^2\right )^p\right )}{6 a x^4}+\frac{b^2 p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3 x^2}+\frac{b^3 p \log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3}-\frac{b^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 a^3}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac{\left (b^3 p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{a}\right )}{x} \, dx,x,a+b x^2\right )}{3 a^3}\\ &=-\frac{b^2 p^2}{6 a^2 x^2}-\frac{b^3 p^2 \log (x)}{a^3}+\frac{b^3 p^2 \log \left (a+b x^2\right )}{6 a^3}-\frac{b p \log \left (c \left (a+b x^2\right )^p\right )}{6 a x^4}+\frac{b^2 p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3 x^2}+\frac{b^3 p \log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3}-\frac{b^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 a^3}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac{b^3 p^2 \text{Li}_2\left (1+\frac{b x^2}{a}\right )}{3 a^3}\\ \end{align*}
Mathematica [A] time = 0.0544469, size = 205, normalized size = 1.06 \[ \frac{b^3 p^2 \text{PolyLog}\left (2,\frac{a+b x^2}{a}\right )}{3 a^3}-\frac{b^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 a^3}+\frac{b^3 p \log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3}+\frac{b^2 p \log \left (c \left (a+b x^2\right )^p\right )}{3 a^2 x^2}-\frac{b^2 p^2}{6 a^2 x^2}+\frac{b^3 p^2 \log \left (a+b x^2\right )}{2 a^3}-\frac{b^3 p^2 \log (x)}{a^3}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac{b p \log \left (c \left (a+b x^2\right )^p\right )}{6 a x^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.516, size = 1289, normalized size = 6.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05452, size = 234, normalized size = 1.21 \begin{align*} -\frac{1}{6} \, b^{2} p^{2}{\left (\frac{2 \,{\left (2 \, \log \left (\frac{b x^{2}}{a} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{b x^{2}}{a}\right )\right )} b}{a^{3}} - \frac{3 \, b \log \left (b x^{2} + a\right )}{a^{3}} - \frac{b x^{2} \log \left (b x^{2} + a\right )^{2} - 6 \, b x^{2} \log \left (x\right ) - a}{a^{3} x^{2}}\right )} - \frac{1}{6} \, b p{\left (\frac{2 \, b^{2} \log \left (b x^{2} + a\right )}{a^{3}} - \frac{2 \, b^{2} \log \left (x^{2}\right )}{a^{3}} - \frac{2 \, b x^{2} - a}{a^{2} x^{4}}\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}}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{7}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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