Integrand size = 16, antiderivative size = 93 \[ \int x \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=\frac {a^3 p \sqrt {x}}{2 b^3}-\frac {a^2 p x}{4 b^2}+\frac {a p x^{3/2}}{6 b}-\frac {p x^2}{8}-\frac {a^4 p \log \left (a+b \sqrt {x}\right )}{2 b^4}+\frac {1}{2} x^2 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \]
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Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2504, 2442, 45} \[ \int x \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=-\frac {a^4 p \log \left (a+b \sqrt {x}\right )}{2 b^4}+\frac {a^3 p \sqrt {x}}{2 b^3}-\frac {a^2 p x}{4 b^2}+\frac {1}{2} x^2 \log \left (c \left (a+b \sqrt {x}\right )^p\right )+\frac {a p x^{3/2}}{6 b}-\frac {p x^2}{8} \]
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Rule 45
Rule 2442
Rule 2504
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^3 \log \left (c (a+b x)^p\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {1}{2} x^2 \log \left (c \left (a+b \sqrt {x}\right )^p\right )-\frac {1}{2} (b p) \text {Subst}\left (\int \frac {x^4}{a+b x} \, dx,x,\sqrt {x}\right ) \\ & = \frac {1}{2} x^2 \log \left (c \left (a+b \sqrt {x}\right )^p\right )-\frac {1}{2} (b p) \text {Subst}\left (\int \left (-\frac {a^3}{b^4}+\frac {a^2 x}{b^3}-\frac {a x^2}{b^2}+\frac {x^3}{b}+\frac {a^4}{b^4 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^3 p \sqrt {x}}{2 b^3}-\frac {a^2 p x}{4 b^2}+\frac {a p x^{3/2}}{6 b}-\frac {p x^2}{8}-\frac {a^4 p \log \left (a+b \sqrt {x}\right )}{2 b^4}+\frac {1}{2} x^2 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.95 \[ \int x \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=\frac {b p \sqrt {x} \left (12 a^3-6 a^2 b \sqrt {x}+4 a b^2 x-3 b^3 x^{3/2}\right )-12 a^4 p \log \left (a+b \sqrt {x}\right )+12 b^4 x^2 \log \left (c \left (a+b \sqrt {x}\right )^p\right )}{24 b^4} \]
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Time = 0.36 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.83
method | result | size |
parts | \(\frac {x^{2} \ln \left (c \left (a +b \sqrt {x}\right )^{p}\right )}{2}-\frac {p b \left (-\frac {2 \left (-\frac {b^{3} x^{2}}{4}+\frac {a \,x^{\frac {3}{2}} b^{2}}{3}-\frac {a^{2} b x}{2}+a^{3} \sqrt {x}\right )}{b^{4}}+\frac {2 a^{4} \ln \left (a +b \sqrt {x}\right )}{b^{5}}\right )}{4}\) | \(77\) |
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none
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.86 \[ \int x \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=-\frac {3 \, b^{4} p x^{2} - 12 \, b^{4} x^{2} \log \left (c\right ) + 6 \, a^{2} b^{2} p x - 12 \, {\left (b^{4} p x^{2} - a^{4} p\right )} \log \left (b \sqrt {x} + a\right ) - 4 \, {\left (a b^{3} p x + 3 \, a^{3} b p\right )} \sqrt {x}}{24 \, b^{4}} \]
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Time = 1.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.99 \[ \int x \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=- \frac {b p \left (\frac {2 a^{4} \left (\begin {cases} \frac {\sqrt {x}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b \sqrt {x} \right )}}{b} & \text {otherwise} \end {cases}\right )}{b^{4}} - \frac {2 a^{3} \sqrt {x}}{b^{4}} + \frac {a^{2} x}{b^{3}} - \frac {2 a x^{\frac {3}{2}}}{3 b^{2}} + \frac {x^{2}}{2 b}\right )}{4} + \frac {x^{2} \log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{2} \]
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Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.82 \[ \int x \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=-\frac {1}{24} \, b p {\left (\frac {12 \, a^{4} \log \left (b \sqrt {x} + a\right )}{b^{5}} + \frac {3 \, b^{3} x^{2} - 4 \, a b^{2} x^{\frac {3}{2}} + 6 \, a^{2} b x - 12 \, a^{3} \sqrt {x}}{b^{4}}\right )} + \frac {1}{2} \, x^{2} \log \left ({\left (b \sqrt {x} + a\right )}^{p} c\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (73) = 146\).
Time = 0.35 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.84 \[ \int x \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=\frac {12 \, b x^{2} \log \left (c\right ) + {\left (\frac {12 \, {\left (b \sqrt {x} + a\right )}^{4} \log \left (b \sqrt {x} + a\right )}{b^{3}} - \frac {48 \, {\left (b \sqrt {x} + a\right )}^{3} a \log \left (b \sqrt {x} + a\right )}{b^{3}} + \frac {72 \, {\left (b \sqrt {x} + a\right )}^{2} a^{2} \log \left (b \sqrt {x} + a\right )}{b^{3}} - \frac {48 \, {\left (b \sqrt {x} + a\right )} a^{3} \log \left (b \sqrt {x} + a\right )}{b^{3}} - \frac {3 \, {\left (b \sqrt {x} + a\right )}^{4}}{b^{3}} + \frac {16 \, {\left (b \sqrt {x} + a\right )}^{3} a}{b^{3}} - \frac {36 \, {\left (b \sqrt {x} + a\right )}^{2} a^{2}}{b^{3}} + \frac {48 \, {\left (b \sqrt {x} + a\right )} a^{3}}{b^{3}}\right )} p}{24 \, b} \]
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Time = 1.40 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.78 \[ \int x \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx=\frac {x^2\,\ln \left (c\,{\left (a+b\,\sqrt {x}\right )}^p\right )}{2}-\frac {p\,x^2}{8}-\frac {a^4\,p\,\ln \left (a+b\,\sqrt {x}\right )}{2\,b^4}+\frac {a^3\,p\,\sqrt {x}}{2\,b^3}-\frac {a^2\,p\,x}{4\,b^2}+\frac {a\,p\,x^{3/2}}{6\,b} \]
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