Optimal. Leaf size=145 \[ \frac {\left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac {a \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}-\frac {p \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^2}+\frac {a p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^2}+\frac {p^2 \left (a+b x^2\right )^2}{8 b^2}-\frac {a p^2 x^2}{b} \]
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Rubi [A] time = 0.15, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \frac {\left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac {a \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}-\frac {p \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^2}+\frac {a p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^2}+\frac {p^2 \left (a+b x^2\right )^2}{8 b^2}-\frac {a p^2 x^2}{b} \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2296
Rule 2304
Rule 2305
Rule 2389
Rule 2390
Rule 2401
Rule 2454
Rubi steps
\begin {align*} \int x^3 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x \log ^2\left (c (a+b x)^p\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a \log ^2\left (c (a+b x)^p\right )}{b}+\frac {(a+b x) \log ^2\left (c (a+b x)^p\right )}{b}\right ) \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int (a+b x) \log ^2\left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{2 b}-\frac {a \operatorname {Subst}\left (\int \log ^2\left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{2 b}\\ &=\frac {\operatorname {Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^2}-\frac {a \operatorname {Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^2}\\ &=-\frac {a \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}+\frac {\left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac {p \operatorname {Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^2}+\frac {(a p) \operatorname {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{b^2}\\ &=-\frac {a p^2 x^2}{b}+\frac {p^2 \left (a+b x^2\right )^2}{8 b^2}+\frac {a p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^2}-\frac {p \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac {a \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}+\frac {\left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 105, normalized size = 0.72 \[ \frac {-2 \left (a^2-b^2 x^4\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+2 p \left (2 a^2+2 a b x^2-b^2 x^4\right ) \log \left (c \left (a+b x^2\right )^p\right )+2 a^2 p^2 \log \left (a+b x^2\right )+b p^2 x^2 \left (b x^2-6 a\right )}{8 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 148, normalized size = 1.02 \[ \frac {b^{2} p^{2} x^{4} + 2 \, b^{2} x^{4} \log \relax (c)^{2} - 6 \, a b p^{2} x^{2} + 2 \, {\left (b^{2} p^{2} x^{4} - a^{2} p^{2}\right )} \log \left (b x^{2} + a\right )^{2} - 2 \, {\left (b^{2} p^{2} x^{4} - 2 \, a b p^{2} x^{2} - 3 \, a^{2} p^{2} - 2 \, {\left (b^{2} p x^{4} - a^{2} p\right )} \log \relax (c)\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (b^{2} p x^{4} - 2 \, a b p x^{2}\right )} \log \relax (c)}{8 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 207, normalized size = 1.43 \[ \frac {\frac {{\left (2 \, {\left (b x^{2} + a\right )}^{2} \log \left (b x^{2} + a\right )^{2} - 4 \, {\left (b x^{2} + a\right )} a \log \left (b x^{2} + a\right )^{2} - 2 \, {\left (b x^{2} + a\right )}^{2} \log \left (b x^{2} + a\right ) + 8 \, {\left (b x^{2} + a\right )} a \log \left (b x^{2} + a\right ) + {\left (b x^{2} + a\right )}^{2} - 8 \, {\left (b x^{2} + a\right )} a\right )} p^{2}}{b} + \frac {2 \, {\left (2 \, {\left (b x^{2} + a\right )}^{2} \log \left (b x^{2} + a\right ) - 4 \, {\left (b x^{2} + a\right )} a \log \left (b x^{2} + a\right ) - {\left (b x^{2} + a\right )}^{2} + 4 \, {\left (b x^{2} + a\right )} a\right )} p \log \relax (c)}{b} + \frac {2 \, {\left ({\left (b x^{2} + a\right )}^{2} - 2 \, {\left (b x^{2} + a\right )} a\right )} \log \relax (c)^{2}}{b}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.47, size = 1242, normalized size = 8.57 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 120, normalized size = 0.83 \[ \frac {1}{4} \, x^{4} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2} - \frac {1}{4} \, b p {\left (\frac {2 \, a^{2} \log \left (b x^{2} + a\right )}{b^{3}} + \frac {b x^{4} - 2 \, a x^{2}}{b^{2}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) + \frac {{\left (b^{2} x^{4} - 6 \, a b x^{2} + 2 \, a^{2} \log \left (b x^{2} + a\right )^{2} + 6 \, a^{2} \log \left (b x^{2} + a\right )\right )} p^{2}}{8 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 100, normalized size = 0.69 \[ \frac {p^2\,x^4}{8}-\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )\,\left (\frac {p\,x^4}{4}-\frac {a\,p\,x^2}{2\,b}\right )+{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2\,\left (\frac {x^4}{4}-\frac {a^2}{4\,b^2}\right )-\frac {3\,a\,p^2\,x^2}{4\,b}+\frac {3\,a^2\,p^2\,\ln \left (b\,x^2+a\right )}{4\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.64, size = 209, normalized size = 1.44 \[ \begin {cases} - \frac {a^{2} p^{2} \log {\left (a + b x^{2} \right )}^{2}}{4 b^{2}} + \frac {3 a^{2} p^{2} \log {\left (a + b x^{2} \right )}}{4 b^{2}} - \frac {a^{2} p \log {\relax (c )} \log {\left (a + b x^{2} \right )}}{2 b^{2}} + \frac {a p^{2} x^{2} \log {\left (a + b x^{2} \right )}}{2 b} - \frac {3 a p^{2} x^{2}}{4 b} + \frac {a p x^{2} \log {\relax (c )}}{2 b} + \frac {p^{2} x^{4} \log {\left (a + b x^{2} \right )}^{2}}{4} - \frac {p^{2} x^{4} \log {\left (a + b x^{2} \right )}}{4} + \frac {p^{2} x^{4}}{8} + \frac {p x^{4} \log {\relax (c )} \log {\left (a + b x^{2} \right )}}{2} - \frac {p x^{4} \log {\relax (c )}}{4} + \frac {x^{4} \log {\relax (c )}^{2}}{4} & \text {for}\: b \neq 0 \\\frac {x^{4} \log {\left (a^{p} c \right )}^{2}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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