Optimal. Leaf size=193 \[ -\frac {b^2 p^2}{6 a^2 x^2}-\frac {b^3 p^2 \log (x)}{a^3}+\frac {b^3 p^2 \log \left (a+b x^2\right )}{6 a^3}-\frac {b p \log \left (c \left (a+b x^2\right )^p\right )}{6 a x^4}+\frac {b^2 p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3 x^2}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac {b^3 p \log \left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{3 a^3}-\frac {b^3 p^2 \text {Li}_2\left (\frac {a}{a+b x^2}\right )}{3 a^3} \]
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Rubi [A]
time = 0.24, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {2504, 2445,
2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \begin {gather*} -\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^3 p^2 \log \left (a+b x^2\right )}{6 a^3}-\frac {b^3 p^2 \log (x)}{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 {\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 46
Rule 2351
Rule 2356
Rule 2379
Rule 2389
Rule 2438
Rule 2445
Rule 2458
Rule 2504
Rubi steps
\begin {align*} \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx &=\frac {1}{2} \text {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) \text {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 \text {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 \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 0.04, size = 205, normalized size = 1.06 \begin {gather*} -\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 )}{2 a^3}-\frac {b p \log \left (c \left (a+b x^2\right )^p\right )}{6 a x^4}+\frac {b^2 p \log \left (c \left (a+b x^2\right )^p\right )}{3 a^2 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 (\frac {a+b x^2}{a}\right )}{3 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.35, size = 1289, normalized size = 6.68
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1289\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 173, normalized size = 0.90 \begin {gather*} -\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 {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 \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{x^{7}}\, 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 = -0.01 \begin {gather*} \int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2}{x^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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