Integrand size = 18, antiderivative size = 153 \[ \int x^3 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=\frac {a^7 p \sqrt {x}}{4 b^7}-\frac {a^6 p x}{8 b^6}+\frac {a^5 p x^{3/2}}{12 b^5}-\frac {a^4 p x^2}{16 b^4}+\frac {a^3 p x^{5/2}}{20 b^3}-\frac {a^2 p x^3}{24 b^2}+\frac {a p x^{7/2}}{28 b}-\frac {p x^4}{32}-\frac {a^8 p \log \left (a+b \sqrt {x}\right )}{4 b^8}+\frac {1}{4} x^4 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \]
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Time = 0.08 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2504, 2442, 45} \[ \int x^3 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=-\frac {a^8 p \log \left (a+b \sqrt {x}\right )}{4 b^8}+\frac {a^7 p \sqrt {x}}{4 b^7}-\frac {a^6 p x}{8 b^6}+\frac {a^5 p x^{3/2}}{12 b^5}-\frac {a^4 p x^2}{16 b^4}+\frac {a^3 p x^{5/2}}{20 b^3}-\frac {a^2 p x^3}{24 b^2}+\frac {1}{4} x^4 \log \left (c \left (a+b \sqrt {x}\right )^p\right )+\frac {a p x^{7/2}}{28 b}-\frac {p x^4}{32} \]
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Rule 45
Rule 2442
Rule 2504
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^7 \log \left (c (a+b x)^p\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {1}{4} x^4 \log \left (c \left (a+b \sqrt {x}\right )^p\right )-\frac {1}{4} (b p) \text {Subst}\left (\int \frac {x^8}{a+b x} \, dx,x,\sqrt {x}\right ) \\ & = \frac {1}{4} x^4 \log \left (c \left (a+b \sqrt {x}\right )^p\right )-\frac {1}{4} (b p) \text {Subst}\left (\int \left (-\frac {a^7}{b^8}+\frac {a^6 x}{b^7}-\frac {a^5 x^2}{b^6}+\frac {a^4 x^3}{b^5}-\frac {a^3 x^4}{b^4}+\frac {a^2 x^5}{b^3}-\frac {a x^6}{b^2}+\frac {x^7}{b}+\frac {a^8}{b^8 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^7 p \sqrt {x}}{4 b^7}-\frac {a^6 p x}{8 b^6}+\frac {a^5 p x^{3/2}}{12 b^5}-\frac {a^4 p x^2}{16 b^4}+\frac {a^3 p x^{5/2}}{20 b^3}-\frac {a^2 p x^3}{24 b^2}+\frac {a p x^{7/2}}{28 b}-\frac {p x^4}{32}-\frac {a^8 p \log \left (a+b \sqrt {x}\right )}{4 b^8}+\frac {1}{4} x^4 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.87 \[ \int x^3 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=-\frac {p \left (-840 a^7 b \sqrt {x}+420 a^6 b^2 x-280 a^5 b^3 x^{3/2}+210 a^4 b^4 x^2-168 a^3 b^5 x^{5/2}+140 a^2 b^6 x^3-120 a b^7 x^{7/2}+105 b^8 x^4+840 a^8 \log \left (a+b \sqrt {x}\right )\right )}{3360 b^8}+\frac {1}{4} x^4 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \]
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Time = 0.52 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.79
method | result | size |
parts | \(\frac {x^{4} \ln \left (c \left (a +b \sqrt {x}\right )^{p}\right )}{4}-\frac {p b \left (-\frac {2 \left (-\frac {x^{4} b^{7}}{8}+\frac {a \,x^{\frac {7}{2}} b^{6}}{7}-\frac {a^{2} x^{3} b^{5}}{6}+\frac {a^{3} x^{\frac {5}{2}} b^{4}}{5}-\frac {a^{4} x^{2} b^{3}}{4}+\frac {a^{5} x^{\frac {3}{2}} b^{2}}{3}-\frac {b \,a^{6} x}{2}+a^{7} \sqrt {x}\right )}{b^{8}}+\frac {2 a^{8} \ln \left (a +b \sqrt {x}\right )}{b^{9}}\right )}{8}\) | \(121\) |
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Time = 0.32 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.84 \[ \int x^3 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=-\frac {105 \, b^{8} p x^{4} - 840 \, b^{8} x^{4} \log \left (c\right ) + 140 \, a^{2} b^{6} p x^{3} + 210 \, a^{4} b^{4} p x^{2} + 420 \, a^{6} b^{2} p x - 840 \, {\left (b^{8} p x^{4} - a^{8} p\right )} \log \left (b \sqrt {x} + a\right ) - 8 \, {\left (15 \, a b^{7} p x^{3} + 21 \, a^{3} b^{5} p x^{2} + 35 \, a^{5} b^{3} p x + 105 \, a^{7} b p\right )} \sqrt {x}}{3360 \, b^{8}} \]
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Time = 12.25 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int x^3 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=- \frac {b p \left (\frac {2 a^{8} \left (\begin {cases} \frac {\sqrt {x}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b \sqrt {x} \right )}}{b} & \text {otherwise} \end {cases}\right )}{b^{8}} - \frac {2 a^{7} \sqrt {x}}{b^{8}} + \frac {a^{6} x}{b^{7}} - \frac {2 a^{5} x^{\frac {3}{2}}}{3 b^{6}} + \frac {a^{4} x^{2}}{2 b^{5}} - \frac {2 a^{3} x^{\frac {5}{2}}}{5 b^{4}} + \frac {a^{2} x^{3}}{3 b^{3}} - \frac {2 a x^{\frac {7}{2}}}{7 b^{2}} + \frac {x^{4}}{4 b}\right )}{8} + \frac {x^{4} \log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{4} \]
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Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.78 \[ \int x^3 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=\frac {1}{4} \, x^{4} \log \left ({\left (b \sqrt {x} + a\right )}^{p} c\right ) - \frac {1}{3360} \, b p {\left (\frac {840 \, a^{8} \log \left (b \sqrt {x} + a\right )}{b^{9}} + \frac {105 \, b^{7} x^{4} - 120 \, a b^{6} x^{\frac {7}{2}} + 140 \, a^{2} b^{5} x^{3} - 168 \, a^{3} b^{4} x^{\frac {5}{2}} + 210 \, a^{4} b^{3} x^{2} - 280 \, a^{5} b^{2} x^{\frac {3}{2}} + 420 \, a^{6} b x - 840 \, a^{7} \sqrt {x}}{b^{8}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (121) = 242\).
Time = 0.30 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.22 \[ \int x^3 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=\frac {840 \, b x^{4} \log \left (c\right ) + {\left (\frac {840 \, {\left (b \sqrt {x} + a\right )}^{8} \log \left (b \sqrt {x} + a\right )}{b^{7}} - \frac {6720 \, {\left (b \sqrt {x} + a\right )}^{7} a \log \left (b \sqrt {x} + a\right )}{b^{7}} + \frac {23520 \, {\left (b \sqrt {x} + a\right )}^{6} a^{2} \log \left (b \sqrt {x} + a\right )}{b^{7}} - \frac {47040 \, {\left (b \sqrt {x} + a\right )}^{5} a^{3} \log \left (b \sqrt {x} + a\right )}{b^{7}} + \frac {58800 \, {\left (b \sqrt {x} + a\right )}^{4} a^{4} \log \left (b \sqrt {x} + a\right )}{b^{7}} - \frac {47040 \, {\left (b \sqrt {x} + a\right )}^{3} a^{5} \log \left (b \sqrt {x} + a\right )}{b^{7}} + \frac {23520 \, {\left (b \sqrt {x} + a\right )}^{2} a^{6} \log \left (b \sqrt {x} + a\right )}{b^{7}} - \frac {6720 \, {\left (b \sqrt {x} + a\right )} a^{7} \log \left (b \sqrt {x} + a\right )}{b^{7}} - \frac {105 \, {\left (b \sqrt {x} + a\right )}^{8}}{b^{7}} + \frac {960 \, {\left (b \sqrt {x} + a\right )}^{7} a}{b^{7}} - \frac {3920 \, {\left (b \sqrt {x} + a\right )}^{6} a^{2}}{b^{7}} + \frac {9408 \, {\left (b \sqrt {x} + a\right )}^{5} a^{3}}{b^{7}} - \frac {14700 \, {\left (b \sqrt {x} + a\right )}^{4} a^{4}}{b^{7}} + \frac {15680 \, {\left (b \sqrt {x} + a\right )}^{3} a^{5}}{b^{7}} - \frac {11760 \, {\left (b \sqrt {x} + a\right )}^{2} a^{6}}{b^{7}} + \frac {6720 \, {\left (b \sqrt {x} + a\right )} a^{7}}{b^{7}}\right )} p}{3360 \, b} \]
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Time = 1.38 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.79 \[ \int x^3 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=\frac {x^4\,\ln \left (c\,{\left (a+b\,\sqrt {x}\right )}^p\right )}{4}-\frac {p\,x^4}{32}-\frac {a^8\,p\,\ln \left (a+b\,\sqrt {x}\right )}{4\,b^8}-\frac {a^2\,p\,x^3}{24\,b^2}-\frac {a^4\,p\,x^2}{16\,b^4}+\frac {a^3\,p\,x^{5/2}}{20\,b^3}+\frac {a^5\,p\,x^{3/2}}{12\,b^5}+\frac {a^7\,p\,\sqrt {x}}{4\,b^7}+\frac {a\,p\,x^{7/2}}{28\,b}-\frac {a^6\,p\,x}{8\,b^6} \]
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