Integrand size = 24, antiderivative size = 182 \[ \int \frac {x^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {a^3 x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^2 x^2 (a+b x)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a x^3 (a+b x)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^4 (a+b x)}{4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^4 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {660, 45} \[ \int \frac {x^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {x^4 (a+b x)}{4 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a x^3 (a+b x)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^2 x^2 (a+b x)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^4 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^3 x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x\right ) \int \frac {x^4}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {\left (a b+b^2 x\right ) \int \left (-\frac {a^3}{b^5}+\frac {a^2 x}{b^4}-\frac {a x^2}{b^3}+\frac {x^3}{b^2}+\frac {a^4}{b^5 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {a^3 x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^2 x^2 (a+b x)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a x^3 (a+b x)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^4 (a+b x)}{4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^4 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.75 \[ \int \frac {x^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {\frac {b x \left (12 a^3-6 a^2 b x+4 a b^2 x^2-3 b^3 x^3\right ) \left (\sqrt {a^2} b x+a \left (\sqrt {a^2}-\sqrt {(a+b x)^2}\right )\right )}{a^2+a b x-\sqrt {a^2} \sqrt {(a+b x)^2}}-24 a^4 \text {arctanh}\left (\frac {b x}{\sqrt {a^2}-\sqrt {(a+b x)^2}}\right )}{12 b^5} \]
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Time = 2.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.37
method | result | size |
default | \(\frac {\left (b x +a \right ) \left (3 b^{4} x^{4}-4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+12 a^{4} \ln \left (b x +a \right )-12 a^{3} b x \right )}{12 \sqrt {\left (b x +a \right )^{2}}\, b^{5}}\) | \(67\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {1}{4} b^{3} x^{4}-\frac {1}{3} a \,b^{2} x^{3}+\frac {1}{2} a^{2} b \,x^{2}-a^{3} x \right )}{\left (b x +a \right ) b^{4}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{4} \ln \left (b x +a \right )}{\left (b x +a \right ) b^{5}}\) | \(84\) |
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Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.29 \[ \int \frac {x^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {3 \, b^{4} x^{4} - 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a^{3} b x + 12 \, a^{4} \log \left (b x + a\right )}{12 \, b^{5}} \]
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Time = 1.13 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.15 \[ \int \frac {x^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\begin {cases} \frac {a^{4} \left (\frac {a}{b} + x\right ) \log {\left (\frac {a}{b} + x \right )}}{b^{4} \sqrt {b^{2} \left (\frac {a}{b} + x\right )^{2}}} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (- \frac {25 a^{3}}{12 b^{5}} + \frac {13 a^{2} x}{12 b^{4}} - \frac {7 a x^{2}}{12 b^{3}} + \frac {x^{3}}{4 b^{2}}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {a^{8} \sqrt {a^{2} + 2 a b x} - \frac {4 a^{6} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {6 a^{4} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} - \frac {4 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {9}{2}}}{9}}{16 a^{5} b^{5}} & \text {for}\: a b \neq 0 \\\frac {x^{5}}{5 \sqrt {a^{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.63 \[ \int \frac {x^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} x^{3}}{4 \, b^{2}} + \frac {13 \, a^{2} x^{2}}{12 \, b^{3}} - \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a x^{2}}{12 \, b^{3}} - \frac {13 \, a^{3} x}{6 \, b^{4}} + \frac {a^{4} \log \left (x + \frac {a}{b}\right )}{b^{5}} + \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3}}{6 \, b^{5}} \]
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Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.46 \[ \int \frac {x^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {a^{4} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (b x + a\right )}{b^{5}} + \frac {3 \, b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b x^{2} \mathrm {sgn}\left (b x + a\right ) - 12 \, a^{3} x \mathrm {sgn}\left (b x + a\right )}{12 \, b^{4}} \]
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Timed out. \[ \int \frac {x^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {x^4}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]
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