Optimal. Leaf size=60 \[ -\frac{2 b^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac{2 b p}{3 a x} \]
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Rubi [A] time = 0.0295252, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2455, 325, 205} \[ -\frac{2 b^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac{2 b p}{3 a x} \]
Antiderivative was successfully verified.
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Rule 2455
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx &=-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac{1}{3} (2 b p) \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx\\ &=-\frac{2 b p}{3 a x}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac{\left (2 b^2 p\right ) \int \frac{1}{a+b x^2} \, dx}{3 a}\\ &=-\frac{2 b p}{3 a x}-\frac{2 b^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{3 x^3}\\ \end{align*}
Mathematica [C] time = 0.0030129, size = 49, normalized size = 0.82 \[ -\frac{\log \left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac{2 b p \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{b x^2}{a}\right )}{3 a x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.34, size = 211, normalized size = 3.5 \begin{align*} -{\frac{\ln \left ( \left ( b{x}^{2}+a \right ) ^{p} \right ) }{3\,{x}^{3}}}-{\frac{i\pi \,a{\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{\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 \left ({\it csgn} \left ( ic \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{3}+i\pi \,a \left ({\it csgn} \left ( ic \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -2\,\sum _{{\it \_R}={\it RootOf} \left ({a}^{3}{{\it \_Z}}^{2}+{b}^{3}{p}^{2} \right ) }{\it \_R}\,\ln \left ( \left ( 3\,{a}^{3}{{\it \_R}}^{2}+2\,{b}^{3}{p}^{2} \right ) x+{a}^{2}bp{\it \_R} \right ) a{x}^{3}+4\,{x}^{2}pb+2\,\ln \left ( c \right ) a}{6\,a{x}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85948, size = 311, normalized size = 5.18 \begin{align*} \left [\frac{b p x^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 2 \, b p x^{2} - a p \log \left (b x^{2} + a\right ) - a \log \left (c\right )}{3 \, a x^{3}}, -\frac{2 \, b p x^{3} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 2 \, b p x^{2} + a p \log \left (b x^{2} + a\right ) + a \log \left (c\right )}{3 \, a x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 178.799, size = 774, normalized size = 12.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.228, size = 78, normalized size = 1.3 \begin{align*} -\frac{2 \, b^{2} p \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{3 \, \sqrt{a b} a} - \frac{p \log \left (b x^{2} + a\right )}{3 \, x^{3}} - \frac{2 \, b p x^{2} + a \log \left (c\right )}{3 \, a x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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