Optimal. Leaf size=45 \[ x \log \left (c \left (a+b x^2\right )^p\right )+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-2 p x \]
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Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2448, 321, 205} \[ x \log \left (c \left (a+b x^2\right )^p\right )+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-2 p x \]
Antiderivative was successfully verified.
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Rule 205
Rule 321
Rule 2448
Rubi steps
\begin {align*} \int \log \left (c \left (a+b x^2\right )^p\right ) \, dx &=x \log \left (c \left (a+b x^2\right )^p\right )-(2 b p) \int \frac {x^2}{a+b x^2} \, dx\\ &=-2 p x+x \log \left (c \left (a+b x^2\right )^p\right )+(2 a p) \int \frac {1}{a+b x^2} \, dx\\ &=-2 p x+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+x \log \left (c \left (a+b x^2\right )^p\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 45, normalized size = 1.00 \[ x \log \left (c \left (a+b x^2\right )^p\right )+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-2 p x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 107, normalized size = 2.38 \[ \left [p x \log \left (b x^{2} + a\right ) + p \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 2 \, p x + x \log \relax (c), p x \log \left (b x^{2} + a\right ) + 2 \, p \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 2 \, p x + x \log \relax (c)\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 41, normalized size = 0.91 \[ p x \log \left (b x^{2} + a\right ) + \frac {2 \, a p \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} - {\left (2 \, p - \log \relax (c)\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 38, normalized size = 0.84 \[ \frac {2 a p \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}-2 p x +x \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 45, normalized size = 1.00 \[ 2 \, b p {\left (\frac {a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} - \frac {x}{b}\right )} + x \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 37, normalized size = 0.82 \[ x\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )-2\,p\,x+\frac {2\,\sqrt {a}\,p\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.84, size = 90, normalized size = 2.00 \[ \begin {cases} \frac {i \sqrt {a} p \log {\left (a + b x^{2} \right )}}{b \sqrt {\frac {1}{b}}} - \frac {2 i \sqrt {a} p \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + x \right )}}{b \sqrt {\frac {1}{b}}} + p x \log {\left (a + b x^{2} \right )} - 2 p x + x \log {\relax (c )} & \text {for}\: b \neq 0 \\x \log {\left (a^{p} c \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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