Optimal. Leaf size=334 \[ \frac {3 a^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^3}+\frac {a^2 \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {3 a^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {3 a^2 p^3 x^2}{b^2}+\frac {p^2 \left (a+b x^2\right )^3 \log \left (c \left (a+b x^2\right )^p\right )}{9 b^3}-\frac {3 a p^2 \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^3}+\frac {\left (a+b x^2\right )^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 b^3}-\frac {a \left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {p \left (a+b x^2\right )^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 b^3}+\frac {3 a p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^3}-\frac {p^3 \left (a+b x^2\right )^3}{27 b^3}+\frac {3 a p^3 \left (a+b x^2\right )^2}{8 b^3} \]
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Rubi [A] time = 0.36, antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 15, 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} \[ \frac {3 a^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^3}-\frac {3 a^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {a^2 \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {3 a^2 p^3 x^2}{b^2}+\frac {p^2 \left (a+b x^2\right )^3 \log \left (c \left (a+b x^2\right )^p\right )}{9 b^3}-\frac {3 a p^2 \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^3}-\frac {p \left (a+b x^2\right )^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 b^3}+\frac {3 a p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^3}+\frac {\left (a+b x^2\right )^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 b^3}-\frac {a \left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {p^3 \left (a+b x^2\right )^3}{27 b^3}+\frac {3 a p^3 \left (a+b x^2\right )^2}{8 b^3} \]
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^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 \log ^3\left (c (a+b x)^p\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2 \log ^3\left (c (a+b x)^p\right )}{b^2}-\frac {2 a (a+b x) \log ^3\left (c (a+b x)^p\right )}{b^2}+\frac {(a+b x)^2 \log ^3\left (c (a+b x)^p\right )}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int (a+b x)^2 \log ^3\left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{2 b^2}-\frac {a \operatorname {Subst}\left (\int (a+b x) \log ^3\left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{b^2}+\frac {a^2 \operatorname {Subst}\left (\int \log ^3\left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{2 b^2}\\ &=\frac {\operatorname {Subst}\left (\int x^2 \log ^3\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^3}-\frac {a \operatorname {Subst}\left (\int x \log ^3\left (c x^p\right ) \, dx,x,a+b x^2\right )}{b^3}+\frac {a^2 \operatorname {Subst}\left (\int \log ^3\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^3}\\ &=\frac {a^2 \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {a \left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {\left (a+b x^2\right )^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 b^3}-\frac {p \operatorname {Subst}\left (\int x^2 \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^3}+\frac {(3 a p) \operatorname {Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^3}-\frac {\left (3 a^2 p\right ) \operatorname {Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^3}\\ &=-\frac {3 a^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {3 a p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^3}-\frac {p \left (a+b x^2\right )^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 b^3}+\frac {a^2 \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {a \left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {\left (a+b x^2\right )^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 b^3}+\frac {p^2 \operatorname {Subst}\left (\int x^2 \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{3 b^3}-\frac {\left (3 a p^2\right ) \operatorname {Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^3}+\frac {\left (3 a^2 p^2\right ) \operatorname {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{b^3}\\ &=-\frac {3 a^2 p^3 x^2}{b^2}+\frac {3 a p^3 \left (a+b x^2\right )^2}{8 b^3}-\frac {p^3 \left (a+b x^2\right )^3}{27 b^3}+\frac {3 a^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^3}-\frac {3 a p^2 \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^3}+\frac {p^2 \left (a+b x^2\right )^3 \log \left (c \left (a+b x^2\right )^p\right )}{9 b^3}-\frac {3 a^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {3 a p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^3}-\frac {p \left (a+b x^2\right )^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 b^3}+\frac {a^2 \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {a \left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {\left (a+b x^2\right )^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 b^3}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 309, normalized size = 0.93 \[ \frac {11 a^3 p^2 \log \left (c \left (a+b x^2\right )^p\right )}{6 b^3}+\frac {a^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 b^3}-\frac {11 a^3 p \log ^2\left (c \left (a+b x^2\right )^p\right )}{12 b^3}+\frac {19 a^3 p^3 \log \left (a+b x^2\right )}{36 b^3}+\frac {11 a^2 p^2 x^2 \log \left (c \left (a+b x^2\right )^p\right )}{6 b^2}-\frac {a^2 p x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}-\frac {85 a^2 p^3 x^2}{36 b^2}+\frac {1}{9} p^2 x^6 \log \left (c \left (a+b x^2\right )^p\right )-\frac {5 a p^2 x^4 \log \left (c \left (a+b x^2\right )^p\right )}{12 b}+\frac {1}{6} x^6 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac {1}{6} p x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {a p x^4 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b}+\frac {19 a p^3 x^4}{72 b}-\frac {1}{27} p^3 x^6 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 359, normalized size = 1.07 \[ -\frac {8 \, b^{3} p^{3} x^{6} - 36 \, b^{3} x^{6} \log \relax (c)^{3} - 57 \, a b^{2} p^{3} x^{4} + 510 \, a^{2} b p^{3} x^{2} - 36 \, {\left (b^{3} p^{3} x^{6} + a^{3} p^{3}\right )} \log \left (b x^{2} + a\right )^{3} + 18 \, {\left (2 \, b^{3} p^{3} x^{6} - 3 \, a b^{2} p^{3} x^{4} + 6 \, a^{2} b p^{3} x^{2} + 11 \, a^{3} p^{3} - 6 \, {\left (b^{3} p^{2} x^{6} + a^{3} p^{2}\right )} \log \relax (c)\right )} \log \left (b x^{2} + a\right )^{2} + 18 \, {\left (2 \, b^{3} p x^{6} - 3 \, a b^{2} p x^{4} + 6 \, a^{2} b p x^{2}\right )} \log \relax (c)^{2} - 6 \, {\left (4 \, b^{3} p^{3} x^{6} - 15 \, a b^{2} p^{3} x^{4} + 66 \, a^{2} b p^{3} x^{2} + 85 \, a^{3} p^{3} + 18 \, {\left (b^{3} p x^{6} + a^{3} p\right )} \log \relax (c)^{2} - 6 \, {\left (2 \, b^{3} p^{2} x^{6} - 3 \, a b^{2} p^{2} x^{4} + 6 \, a^{2} b p^{2} x^{2} + 11 \, a^{3} p^{2}\right )} \log \relax (c)\right )} \log \left (b x^{2} + a\right ) - 6 \, {\left (4 \, b^{3} p^{2} x^{6} - 15 \, a b^{2} p^{2} x^{4} + 66 \, a^{2} b p^{2} x^{2}\right )} \log \relax (c)}{216 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 595, normalized size = 1.78 \[ \frac {36 \, b x^{6} \log \relax (c)^{3} + {\left (\frac {36 \, {\left (b x^{2} + a\right )}^{3} \log \left (b x^{2} + a\right )^{3}}{b^{2}} - \frac {108 \, {\left (b x^{2} + a\right )}^{2} a \log \left (b x^{2} + a\right )^{3}}{b^{2}} + \frac {108 \, {\left (b x^{2} + a\right )} a^{2} \log \left (b x^{2} + a\right )^{3}}{b^{2}} - \frac {36 \, {\left (b x^{2} + a\right )}^{3} \log \left (b x^{2} + a\right )^{2}}{b^{2}} + \frac {162 \, {\left (b x^{2} + a\right )}^{2} a \log \left (b x^{2} + a\right )^{2}}{b^{2}} - \frac {324 \, {\left (b x^{2} + a\right )} a^{2} \log \left (b x^{2} + a\right )^{2}}{b^{2}} + \frac {24 \, {\left (b x^{2} + a\right )}^{3} \log \left (b x^{2} + a\right )}{b^{2}} - \frac {162 \, {\left (b x^{2} + a\right )}^{2} a \log \left (b x^{2} + a\right )}{b^{2}} + \frac {648 \, {\left (b x^{2} + a\right )} a^{2} \log \left (b x^{2} + a\right )}{b^{2}} - \frac {8 \, {\left (b x^{2} + a\right )}^{3}}{b^{2}} + \frac {81 \, {\left (b x^{2} + a\right )}^{2} a}{b^{2}} - \frac {648 \, {\left (b x^{2} + a\right )} a^{2}}{b^{2}}\right )} p^{3} + 6 \, {\left (\frac {18 \, {\left (b x^{2} + a\right )}^{3} \log \left (b x^{2} + a\right )^{2}}{b^{2}} - \frac {54 \, {\left (b x^{2} + a\right )}^{2} a \log \left (b x^{2} + a\right )^{2}}{b^{2}} + \frac {54 \, {\left (b x^{2} + a\right )} a^{2} \log \left (b x^{2} + a\right )^{2}}{b^{2}} - \frac {12 \, {\left (b x^{2} + a\right )}^{3} \log \left (b x^{2} + a\right )}{b^{2}} + \frac {54 \, {\left (b x^{2} + a\right )}^{2} a \log \left (b x^{2} + a\right )}{b^{2}} - \frac {108 \, {\left (b x^{2} + a\right )} a^{2} \log \left (b x^{2} + a\right )}{b^{2}} + \frac {4 \, {\left (b x^{2} + a\right )}^{3}}{b^{2}} - \frac {27 \, {\left (b x^{2} + a\right )}^{2} a}{b^{2}} + \frac {108 \, {\left (b x^{2} + a\right )} a^{2}}{b^{2}}\right )} p^{2} \log \relax (c) + 18 \, {\left (\frac {6 \, {\left (b x^{2} + a\right )}^{3} \log \left (b x^{2} + a\right )}{b^{2}} - \frac {18 \, {\left (b x^{2} + a\right )}^{2} a \log \left (b x^{2} + a\right )}{b^{2}} + \frac {18 \, {\left (b x^{2} + a\right )} a^{2} \log \left (b x^{2} + a\right )}{b^{2}} - \frac {2 \, {\left (b x^{2} + a\right )}^{3}}{b^{2}} + \frac {9 \, {\left (b x^{2} + a\right )}^{2} a}{b^{2}} - \frac {18 \, {\left (b x^{2} + a\right )} a^{2}}{b^{2}}\right )} p \log \relax (c)^{2}}{216 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.13, size = 5905, normalized size = 17.68 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 239, normalized size = 0.72 \[ \frac {1}{6} \, x^{6} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3} + \frac {1}{12} \, b p {\left (\frac {6 \, a^{3} \log \left (b x^{2} + a\right )}{b^{4}} - \frac {2 \, b^{2} x^{6} - 3 \, a b x^{4} + 6 \, a^{2} x^{2}}{b^{3}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2} - \frac {1}{216} \, b p {\left (\frac {{\left (8 \, b^{3} x^{6} - 57 \, a b^{2} x^{4} - 36 \, a^{3} \log \left (b x^{2} + a\right )^{3} + 510 \, a^{2} b x^{2} - 198 \, a^{3} \log \left (b x^{2} + a\right )^{2} - 510 \, a^{3} \log \left (b x^{2} + a\right )\right )} p^{2}}{b^{4}} - \frac {6 \, {\left (4 \, b^{3} x^{6} - 15 \, a b^{2} x^{4} + 66 \, a^{2} b x^{2} - 18 \, a^{3} \log \left (b x^{2} + a\right )^{2} - 66 \, a^{3} \log \left (b x^{2} + a\right )\right )} p \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{b^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 187, normalized size = 0.56 \[ {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3\,\left (\frac {x^6}{6}+\frac {a^3}{6\,b^3}\right )-{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2\,\left (\frac {p\,x^6}{6}+\frac {11\,a^3\,p}{12\,b^3}+\frac {a^2\,p\,x^2}{2\,b^2}-\frac {a\,p\,x^4}{4\,b}\right )-\frac {p^3\,x^6}{27}+\frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )\,\left (\frac {b\,p^2\,x^6}{3}-\frac {5\,a\,p^2\,x^4}{4}+\frac {11\,a^2\,p^2\,x^2}{2\,b}\right )}{3\,b}+\frac {19\,a\,p^3\,x^4}{72\,b}+\frac {85\,a^3\,p^3\,\ln \left (b\,x^2+a\right )}{36\,b^3}-\frac {85\,a^2\,p^3\,x^2}{36\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 37.19, size = 561, normalized size = 1.68 \[ \begin {cases} \frac {a^{3} p^{3} \log {\left (a + b x^{2} \right )}^{3}}{6 b^{3}} - \frac {11 a^{3} p^{3} \log {\left (a + b x^{2} \right )}^{2}}{12 b^{3}} + \frac {85 a^{3} p^{3} \log {\left (a + b x^{2} \right )}}{36 b^{3}} + \frac {a^{3} p^{2} \log {\relax (c )} \log {\left (a + b x^{2} \right )}^{2}}{2 b^{3}} - \frac {11 a^{3} p^{2} \log {\relax (c )} \log {\left (a + b x^{2} \right )}}{6 b^{3}} + \frac {a^{3} p \log {\relax (c )}^{2} \log {\left (a + b x^{2} \right )}}{2 b^{3}} - \frac {a^{2} p^{3} x^{2} \log {\left (a + b x^{2} \right )}^{2}}{2 b^{2}} + \frac {11 a^{2} p^{3} x^{2} \log {\left (a + b x^{2} \right )}}{6 b^{2}} - \frac {85 a^{2} p^{3} x^{2}}{36 b^{2}} - \frac {a^{2} p^{2} x^{2} \log {\relax (c )} \log {\left (a + b x^{2} \right )}}{b^{2}} + \frac {11 a^{2} p^{2} x^{2} \log {\relax (c )}}{6 b^{2}} - \frac {a^{2} p x^{2} \log {\relax (c )}^{2}}{2 b^{2}} + \frac {a p^{3} x^{4} \log {\left (a + b x^{2} \right )}^{2}}{4 b} - \frac {5 a p^{3} x^{4} \log {\left (a + b x^{2} \right )}}{12 b} + \frac {19 a p^{3} x^{4}}{72 b} + \frac {a p^{2} x^{4} \log {\relax (c )} \log {\left (a + b x^{2} \right )}}{2 b} - \frac {5 a p^{2} x^{4} \log {\relax (c )}}{12 b} + \frac {a p x^{4} \log {\relax (c )}^{2}}{4 b} + \frac {p^{3} x^{6} \log {\left (a + b x^{2} \right )}^{3}}{6} - \frac {p^{3} x^{6} \log {\left (a + b x^{2} \right )}^{2}}{6} + \frac {p^{3} x^{6} \log {\left (a + b x^{2} \right )}}{9} - \frac {p^{3} x^{6}}{27} + \frac {p^{2} x^{6} \log {\relax (c )} \log {\left (a + b x^{2} \right )}^{2}}{2} - \frac {p^{2} x^{6} \log {\relax (c )} \log {\left (a + b x^{2} \right )}}{3} + \frac {p^{2} x^{6} \log {\relax (c )}}{9} + \frac {p x^{6} \log {\relax (c )}^{2} \log {\left (a + b x^{2} \right )}}{2} - \frac {p x^{6} \log {\relax (c )}^{2}}{6} + \frac {x^{6} \log {\relax (c )}^{3}}{6} & \text {for}\: b \neq 0 \\\frac {x^{6} \log {\left (a^{p} c \right )}^{3}}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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