Integrand size = 15, antiderivative size = 164 \[ \int x^{5/2} (a+b x)^{5/2} \, dx=\frac {5 a^5 \sqrt {x} \sqrt {a+b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}-\frac {5 a^6 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{7/2}} \]
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Time = 0.04 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {52, 65, 223, 212} \[ \int x^{5/2} (a+b x)^{5/2} \, dx=-\frac {5 a^6 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{7/2}}+\frac {5 a^5 \sqrt {x} \sqrt {a+b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2} \]
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Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^{7/2} (a+b x)^{5/2}+\frac {1}{12} (5 a) \int x^{5/2} (a+b x)^{3/2} \, dx \\ & = \frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}+\frac {1}{8} a^2 \int x^{5/2} \sqrt {a+b x} \, dx \\ & = \frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}+\frac {1}{64} a^3 \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx \\ & = \frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}-\frac {\left (5 a^4\right ) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{384 b} \\ & = -\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}+\frac {\left (5 a^5\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{512 b^2} \\ & = \frac {5 a^5 \sqrt {x} \sqrt {a+b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}-\frac {\left (5 a^6\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{1024 b^3} \\ & = \frac {5 a^5 \sqrt {x} \sqrt {a+b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}-\frac {\left (5 a^6\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{512 b^3} \\ & = \frac {5 a^5 \sqrt {x} \sqrt {a+b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}-\frac {\left (5 a^6\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^3} \\ & = \frac {5 a^5 \sqrt {x} \sqrt {a+b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}-\frac {5 a^6 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{7/2}} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.72 \[ \int x^{5/2} (a+b x)^{5/2} \, dx=\frac {\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (15 a^5-10 a^4 b x+8 a^3 b^2 x^2+432 a^2 b^3 x^3+640 a b^4 x^4+256 b^5 x^5\right )+30 a^6 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )}{1536 b^{7/2}} \]
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Time = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.73
method | result | size |
risch | \(\frac {\left (256 b^{5} x^{5}+640 a \,b^{4} x^{4}+432 a^{2} b^{3} x^{3}+8 a^{3} b^{2} x^{2}-10 a^{4} b x +15 a^{5}\right ) \sqrt {x}\, \sqrt {b x +a}}{1536 b^{3}}-\frac {5 a^{6} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{1024 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(120\) |
default | \(\frac {x^{\frac {5}{2}} \left (b x +a \right )^{\frac {7}{2}}}{6 b}-\frac {5 a \left (\frac {x^{\frac {3}{2}} \left (b x +a \right )^{\frac {7}{2}}}{5 b}-\frac {3 a \left (\frac {\sqrt {x}\, \left (b x +a \right )^{\frac {7}{2}}}{4 b}-\frac {a \left (\frac {\left (b x +a \right )^{\frac {5}{2}} \sqrt {x}}{3}+\frac {5 a \left (\frac {\left (b x +a \right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 a \left (\sqrt {x}\, \sqrt {b x +a}+\frac {a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )}{12 b}\) | \(160\) |
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Time = 0.24 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.26 \[ \int x^{5/2} (a+b x)^{5/2} \, dx=\left [\frac {15 \, a^{6} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (256 \, b^{6} x^{5} + 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} + 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x + 15 \, a^{5} b\right )} \sqrt {b x + a} \sqrt {x}}{3072 \, b^{4}}, \frac {15 \, a^{6} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (256 \, b^{6} x^{5} + 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} + 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x + 15 \, a^{5} b\right )} \sqrt {b x + a} \sqrt {x}}{1536 \, b^{4}}\right ] \]
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Timed out. \[ \int x^{5/2} (a+b x)^{5/2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (118) = 236\).
Time = 0.30 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.49 \[ \int x^{5/2} (a+b x)^{5/2} \, dx=\frac {5 \, a^{6} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{1024 \, b^{\frac {7}{2}}} + \frac {\frac {15 \, \sqrt {b x + a} a^{6} b^{5}}{\sqrt {x}} - \frac {85 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6} b^{4}}{x^{\frac {3}{2}}} + \frac {198 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{6} b^{3}}{x^{\frac {5}{2}}} + \frac {198 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{6} b^{2}}{x^{\frac {7}{2}}} - \frac {85 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{6} b}{x^{\frac {9}{2}}} + \frac {15 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{6}}{x^{\frac {11}{2}}}}{1536 \, {\left (b^{9} - \frac {6 \, {\left (b x + a\right )} b^{8}}{x} + \frac {15 \, {\left (b x + a\right )}^{2} b^{7}}{x^{2}} - \frac {20 \, {\left (b x + a\right )}^{3} b^{6}}{x^{3}} + \frac {15 \, {\left (b x + a\right )}^{4} b^{5}}{x^{4}} - \frac {6 \, {\left (b x + a\right )}^{5} b^{4}}{x^{5}} + \frac {{\left (b x + a\right )}^{6} b^{3}}{x^{6}}\right )}} \]
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Timed out. \[ \int x^{5/2} (a+b x)^{5/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int x^{5/2} (a+b x)^{5/2} \, dx=\int x^{5/2}\,{\left (a+b\,x\right )}^{5/2} \,d x \]
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