Optimal. Leaf size=78 \[ \frac{b^2 p}{6 a^2 x^2}-\frac{b^3 p \log \left (a+b x^2\right )}{6 a^3}+\frac{b^3 p \log (x)}{3 a^3}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac{b p}{12 a x^4} \]
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Rubi [A] time = 0.0601039, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2454, 2395, 44} \[ \frac{b^2 p}{6 a^2 x^2}-\frac{b^3 p \log \left (a+b x^2\right )}{6 a^3}+\frac{b^3 p \log (x)}{3 a^3}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac{b p}{12 a x^4} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2395
Rule 44
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log \left (c (a+b x)^p\right )}{x^4} \, dx,x,x^2\right )\\ &=-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac{1}{6} (b p) \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)} \, dx,x,x^2\right )\\ &=-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac{1}{6} (b p) \operatorname{Subst}\left (\int \left (\frac{1}{a x^3}-\frac{b}{a^2 x^2}+\frac{b^2}{a^3 x}-\frac{b^3}{a^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{b p}{12 a x^4}+\frac{b^2 p}{6 a^2 x^2}+\frac{b^3 p \log (x)}{3 a^3}-\frac{b^3 p \log \left (a+b x^2\right )}{6 a^3}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{6 x^6}\\ \end{align*}
Mathematica [A] time = 0.0648393, size = 68, normalized size = 0.87 \[ -\frac{\frac{b p x^2 \left (2 b^2 x^4 \log \left (a+b x^2\right )+a \left (a-2 b x^2\right )-4 b^2 x^4 \log (x)\right )}{a^3}+2 \log \left (c \left (a+b x^2\right )^p\right )}{12 x^6} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.286, size = 206, normalized size = 2.6 \begin{align*} -{\frac{\ln \left ( \left ( b{x}^{2}+a \right ) ^{p} \right ) }{6\,{x}^{6}}}-{\frac{-4\,{b}^{3}p\ln \left ( x \right ){x}^{6}+2\,{b}^{3}p\ln \left ( b{x}^{2}+a \right ){x}^{6}+i\pi \,{a}^{3}{\it csgn} \left ( i \left ( b{x}^{2}+a \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{2}-i\pi \,{a}^{3}{\it csgn} \left ( i \left ( b{x}^{2}+a \right ) ^{p} \right ){\it csgn} \left ( ic \left ( b{x}^{2}+a \right ) ^{p} \right ){\it csgn} \left ( ic \right ) -i\pi \,{a}^{3} \left ({\it csgn} \left ( ic \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{3}+i\pi \,{a}^{3} \left ({\it csgn} \left ( ic \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -2\,a{b}^{2}p{x}^{4}+{a}^{2}bp{x}^{2}+2\,\ln \left ( c \right ){a}^{3}}{12\,{a}^{3}{x}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04477, size = 93, normalized size = 1.19 \begin{align*} -\frac{1}{12} \, 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 )} - \frac{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06221, size = 163, normalized size = 2.09 \begin{align*} \frac{4 \, b^{3} p x^{6} \log \left (x\right ) + 2 \, a b^{2} p x^{4} - a^{2} b p x^{2} - 2 \, a^{3} \log \left (c\right ) - 2 \,{\left (b^{3} p x^{6} + a^{3} p\right )} \log \left (b x^{2} + a\right )}{12 \, a^{3} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 54.4347, size = 116, normalized size = 1.49 \begin{align*} \begin{cases} - \frac{p \log{\left (a + b x^{2} \right )}}{6 x^{6}} - \frac{\log{\left (c \right )}}{6 x^{6}} - \frac{b p}{12 a x^{4}} + \frac{b^{2} p}{6 a^{2} x^{2}} + \frac{b^{3} p \log{\left (x \right )}}{3 a^{3}} - \frac{b^{3} p \log{\left (a + b x^{2} \right )}}{6 a^{3}} & \text{for}\: a \neq 0 \\- \frac{p \log{\left (b \right )}}{6 x^{6}} - \frac{p \log{\left (x \right )}}{3 x^{6}} - \frac{p}{18 x^{6}} - \frac{\log{\left (c \right )}}{6 x^{6}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17609, size = 258, normalized size = 3.31 \begin{align*} -\frac{\frac{2 \, b^{4} p \log \left (b x^{2} + a\right )}{{\left (b x^{2} + a\right )}^{3} - 3 \,{\left (b x^{2} + a\right )}^{2} a + 3 \,{\left (b x^{2} + a\right )} a^{2} - a^{3}} + \frac{2 \, b^{4} p \log \left (b x^{2} + a\right )}{a^{3}} - \frac{2 \, b^{4} p \log \left (b x^{2}\right )}{a^{3}} - \frac{2 \,{\left (b x^{2} + a\right )}^{2} b^{4} p - 5 \,{\left (b x^{2} + a\right )} a b^{4} p + 3 \, a^{2} b^{4} p - 2 \, a^{2} b^{4} \log \left (c\right )}{{\left (b x^{2} + a\right )}^{3} a^{2} - 3 \,{\left (b x^{2} + a\right )}^{2} a^{3} + 3 \,{\left (b x^{2} + a\right )} a^{4} - a^{5}}}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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