Integrand size = 17, antiderivative size = 94 \[ \int \frac {x^5}{\left (a x+b x^2\right )^{5/2}} \, dx=-\frac {2 x^4}{3 b \left (a x+b x^2\right )^{3/2}}-\frac {10 x^2}{3 b^2 \sqrt {a x+b x^2}}+\frac {5 \sqrt {a x+b x^2}}{b^3}-\frac {5 a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{b^{7/2}} \]
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Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {682, 654, 634, 212} \[ \int \frac {x^5}{\left (a x+b x^2\right )^{5/2}} \, dx=-\frac {5 a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{b^{7/2}}+\frac {5 \sqrt {a x+b x^2}}{b^3}-\frac {10 x^2}{3 b^2 \sqrt {a x+b x^2}}-\frac {2 x^4}{3 b \left (a x+b x^2\right )^{3/2}} \]
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Rule 212
Rule 634
Rule 654
Rule 682
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^4}{3 b \left (a x+b x^2\right )^{3/2}}+\frac {5 \int \frac {x^3}{\left (a x+b x^2\right )^{3/2}} \, dx}{3 b} \\ & = -\frac {2 x^4}{3 b \left (a x+b x^2\right )^{3/2}}-\frac {10 x^2}{3 b^2 \sqrt {a x+b x^2}}+\frac {5 \int \frac {x}{\sqrt {a x+b x^2}} \, dx}{b^2} \\ & = -\frac {2 x^4}{3 b \left (a x+b x^2\right )^{3/2}}-\frac {10 x^2}{3 b^2 \sqrt {a x+b x^2}}+\frac {5 \sqrt {a x+b x^2}}{b^3}-\frac {(5 a) \int \frac {1}{\sqrt {a x+b x^2}} \, dx}{2 b^3} \\ & = -\frac {2 x^4}{3 b \left (a x+b x^2\right )^{3/2}}-\frac {10 x^2}{3 b^2 \sqrt {a x+b x^2}}+\frac {5 \sqrt {a x+b x^2}}{b^3}-\frac {(5 a) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a x+b x^2}}\right )}{b^3} \\ & = -\frac {2 x^4}{3 b \left (a x+b x^2\right )^{3/2}}-\frac {10 x^2}{3 b^2 \sqrt {a x+b x^2}}+\frac {5 \sqrt {a x+b x^2}}{b^3}-\frac {5 a \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{b^{7/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.02 \[ \int \frac {x^5}{\left (a x+b x^2\right )^{5/2}} \, dx=\frac {x \left (\sqrt {b} x \left (15 a^2+20 a b x+3 b^2 x^2\right )+30 a \sqrt {x} (a+b x)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )\right )}{3 b^{7/2} (x (a+b x))^{3/2}} \]
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Time = 1.91 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.87
method | result | size |
pseudoelliptic | \(-\frac {5 \left (a \sqrt {x \left (b x +a \right )}\, \left (b x +a \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )-\sqrt {b}\, a^{2} x -\frac {4 x^{2} a \,b^{\frac {3}{2}}}{3}-\frac {b^{\frac {5}{2}} x^{3}}{5}\right )}{\sqrt {x \left (b x +a \right )}\, b^{\frac {7}{2}} \left (b x +a \right )}\) | \(82\) |
risch | \(\frac {x \left (b x +a \right )}{b^{3} \sqrt {x \left (b x +a \right )}}-\frac {5 a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 b^{\frac {7}{2}}}-\frac {2 a^{2} \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{3 b^{5} \left (x +\frac {a}{b}\right )^{2}}+\frac {14 a \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{3 b^{4} \left (x +\frac {a}{b}\right )}\) | \(131\) |
default | \(\frac {x^{4}}{b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {x^{2}}{b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {a \left (-\frac {x}{2 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {1}{3 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {2 \left (2 b x +a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {16 b \left (2 b x +a \right )}{3 a^{4} \sqrt {b \,x^{2}+a x}}\right )}{2 b}\right )}{4 b}\right )}{2 b}\right )}{2 b}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a x}}-\frac {a \left (-\frac {1}{b \sqrt {b \,x^{2}+a x}}+\frac {2 b x +a}{a b \sqrt {b \,x^{2}+a x}}\right )}{2 b}+\frac {\ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\) | \(268\) |
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Time = 0.25 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.35 \[ \int \frac {x^5}{\left (a x+b x^2\right )^{5/2}} \, dx=\left [\frac {15 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt {b} \log \left (2 \, b x + a - 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (3 \, b^{3} x^{2} + 20 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {b x^{2} + a x}}{6 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac {15 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x}\right ) + {\left (3 \, b^{3} x^{2} + 20 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {b x^{2} + a x}}{3 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]
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\[ \int \frac {x^5}{\left (a x+b x^2\right )^{5/2}} \, dx=\int \frac {x^{5}}{\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (78) = 156\).
Time = 0.19 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.72 \[ \int \frac {x^5}{\left (a x+b x^2\right )^{5/2}} \, dx=\frac {5 \, a x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {a x}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, x}{\sqrt {b x^{2} + a x} a b} - \frac {1}{\sqrt {b x^{2} + a x} b^{2}}\right )}}{6 \, b} + \frac {x^{4}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {10 \, a x}{3 \, \sqrt {b x^{2} + a x} b^{3}} - \frac {5 \, a \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{2 \, b^{\frac {7}{2}}} + \frac {5 \, \sqrt {b x^{2} + a x}}{3 \, b^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.53 \[ \int \frac {x^5}{\left (a x+b x^2\right )^{5/2}} \, dx=\frac {5 \, a \log \left ({\left | -2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} - a \right |}\right )}{2 \, b^{\frac {7}{2}}} + \frac {\sqrt {b x^{2} + a x}}{b^{3}} + \frac {2 \, {\left (9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{2} b + 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{3} \sqrt {b} + 7 \, a^{4}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a\right )}^{3} b^{\frac {7}{2}}} \]
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Timed out. \[ \int \frac {x^5}{\left (a x+b x^2\right )^{5/2}} \, dx=\int \frac {x^5}{{\left (b\,x^2+a\,x\right )}^{5/2}} \,d x \]
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