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 = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, 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 205
Rule 325
Rule 2455
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*}
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Mathematica [C] time = 0.00, 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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fricas [A] time = 0.48, size = 135, normalized size = 2.25 \[ \left [\frac {b p x^{3} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 2 \, b p x^{2} - a p \log \left (b x^{2} + a\right ) - a \log \relax (c)}{3 \, a x^{3}}, -\frac {2 \, b p x^{3} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 2 \, b p x^{2} + a p \log \left (b x^{2} + a\right ) + a \log \relax (c)}{3 \, a x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 58, normalized size = 0.97 \[ -\frac {2 \, b^{2} p \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} a} - \frac {p \log \left (b x^{2} + a\right )}{3 \, x^{3}} - \frac {2 \, b p x^{2} + a \log \relax (c)}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.35, size = 211, normalized size = 3.52 \[ -\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{3 x^{3}}-\frac {-2 a \,x^{3} \RootOf \left (a^{3} \textit {\_Z}^{2}+b^{3} p^{2}\right ) \ln \left (\RootOf \left (a^{3} \textit {\_Z}^{2}+b^{3} p^{2}\right ) a^{2} b p +\left (3 \RootOf \left (a^{3} \textit {\_Z}^{2}+b^{3} p^{2}\right )^{2} a^{3}+2 b^{3} p^{2}\right ) x \right )-i \pi a \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )+i \pi a \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}+i \pi a \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}-i \pi a \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}+4 b p \,x^{2}+2 a \ln \relax (c )}{6 a \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 49, normalized size = 0.82 \[ -\frac {2}{3} \, b p {\left (\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} + \frac {1}{a x}\right )} - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 46, normalized size = 0.77 \[ -\frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{3\,x^3}-\frac {2\,b^{3/2}\,p\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{3\,a^{3/2}}-\frac {2\,b\,p}{3\,a\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 111.55, size = 763, normalized size = 12.72 \[ \begin {cases} - \frac {\log {\left (0^{p} c \right )}}{3 x^{3}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {\log {\left (a^{p} c \right )}}{3 x^{3}} & \text {for}\: b = 0 \\- \frac {p \log {\relax (b )}}{3 x^{3}} - \frac {2 p \log {\relax (x )}}{3 x^{3}} - \frac {2 p}{9 x^{3}} - \frac {\log {\relax (c )}}{3 x^{3}} & \text {for}\: a = 0 \\- \frac {i a^{\frac {5}{2}} p \sqrt {\frac {1}{b}} \log {\left (a + b x^{2} \right )}}{3 i a^{\frac {5}{2}} x^{3} \sqrt {\frac {1}{b}} + 3 i a^{\frac {3}{2}} b x^{5} \sqrt {\frac {1}{b}}} - \frac {i a^{\frac {5}{2}} \sqrt {\frac {1}{b}} \log {\relax (c )}}{3 i a^{\frac {5}{2}} x^{3} \sqrt {\frac {1}{b}} + 3 i a^{\frac {3}{2}} b x^{5} \sqrt {\frac {1}{b}}} - \frac {i a^{\frac {3}{2}} p x^{2} \sqrt {\frac {1}{b}} \log {\left (a + b x^{2} \right )}}{\frac {3 i a^{\frac {5}{2}} x^{3} \sqrt {\frac {1}{b}}}{b} + 3 i a^{\frac {3}{2}} x^{5} \sqrt {\frac {1}{b}}} - \frac {2 i a^{\frac {3}{2}} p x^{2} \sqrt {\frac {1}{b}}}{\frac {3 i a^{\frac {5}{2}} x^{3} \sqrt {\frac {1}{b}}}{b} + 3 i a^{\frac {3}{2}} x^{5} \sqrt {\frac {1}{b}}} - \frac {i a^{\frac {3}{2}} x^{2} \sqrt {\frac {1}{b}} \log {\relax (c )}}{\frac {3 i a^{\frac {5}{2}} x^{3} \sqrt {\frac {1}{b}}}{b} + 3 i a^{\frac {3}{2}} x^{5} \sqrt {\frac {1}{b}}} - \frac {2 i \sqrt {a} b p x^{4} \sqrt {\frac {1}{b}}}{\frac {3 i a^{\frac {5}{2}} x^{3} \sqrt {\frac {1}{b}}}{b} + 3 i a^{\frac {3}{2}} x^{5} \sqrt {\frac {1}{b}}} + \frac {a p x^{3} \log {\left (a + b x^{2} \right )}}{\frac {3 i a^{\frac {5}{2}} x^{3} \sqrt {\frac {1}{b}}}{b} + 3 i a^{\frac {3}{2}} x^{5} \sqrt {\frac {1}{b}}} - \frac {2 a p x^{3} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + x \right )}}{\frac {3 i a^{\frac {5}{2}} x^{3} \sqrt {\frac {1}{b}}}{b} + 3 i a^{\frac {3}{2}} x^{5} \sqrt {\frac {1}{b}}} + \frac {a x^{3} \log {\relax (c )}}{\frac {3 i a^{\frac {5}{2}} x^{3} \sqrt {\frac {1}{b}}}{b} + 3 i a^{\frac {3}{2}} x^{5} \sqrt {\frac {1}{b}}} + \frac {b p x^{5} \log {\left (a + b x^{2} \right )}}{\frac {3 i a^{\frac {5}{2}} x^{3} \sqrt {\frac {1}{b}}}{b} + 3 i a^{\frac {3}{2}} x^{5} \sqrt {\frac {1}{b}}} - \frac {2 b p x^{5} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + x \right )}}{\frac {3 i a^{\frac {5}{2}} x^{3} \sqrt {\frac {1}{b}}}{b} + 3 i a^{\frac {3}{2}} x^{5} \sqrt {\frac {1}{b}}} + \frac {b x^{5} \log {\relax (c )}}{\frac {3 i a^{\frac {5}{2}} x^{3} \sqrt {\frac {1}{b}}}{b} + 3 i a^{\frac {3}{2}} x^{5} \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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