Integrand size = 20, antiderivative size = 225 \[ \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx=\frac {a^4 (12 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{512 b^4}-\frac {a^3 (12 A b-7 a B) x^{3/2} \sqrt {a+b x}}{768 b^3}+\frac {a^2 (12 A b-7 a B) x^{5/2} \sqrt {a+b x}}{960 b^2}+\frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}-\frac {a^5 (12 A b-7 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{9/2}} \]
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Time = 0.07 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223, 212} \[ \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx=-\frac {a^5 (12 A b-7 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{9/2}}+\frac {a^4 \sqrt {x} \sqrt {a+b x} (12 A b-7 a B)}{512 b^4}-\frac {a^3 x^{3/2} \sqrt {a+b x} (12 A b-7 a B)}{768 b^3}+\frac {a^2 x^{5/2} \sqrt {a+b x} (12 A b-7 a B)}{960 b^2}+\frac {a x^{7/2} \sqrt {a+b x} (12 A b-7 a B)}{160 b}+\frac {x^{7/2} (a+b x)^{3/2} (12 A b-7 a B)}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b} \]
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Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {B x^{7/2} (a+b x)^{5/2}}{6 b}+\frac {\left (6 A b-\frac {7 a B}{2}\right ) \int x^{5/2} (a+b x)^{3/2} \, dx}{6 b} \\ & = \frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}+\frac {(a (12 A b-7 a B)) \int x^{5/2} \sqrt {a+b x} \, dx}{40 b} \\ & = \frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}+\frac {\left (a^2 (12 A b-7 a B)\right ) \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx}{320 b} \\ & = \frac {a^2 (12 A b-7 a B) x^{5/2} \sqrt {a+b x}}{960 b^2}+\frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}-\frac {\left (a^3 (12 A b-7 a B)\right ) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{384 b^2} \\ & = -\frac {a^3 (12 A b-7 a B) x^{3/2} \sqrt {a+b x}}{768 b^3}+\frac {a^2 (12 A b-7 a B) x^{5/2} \sqrt {a+b x}}{960 b^2}+\frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}+\frac {\left (a^4 (12 A b-7 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{512 b^3} \\ & = \frac {a^4 (12 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{512 b^4}-\frac {a^3 (12 A b-7 a B) x^{3/2} \sqrt {a+b x}}{768 b^3}+\frac {a^2 (12 A b-7 a B) x^{5/2} \sqrt {a+b x}}{960 b^2}+\frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}-\frac {\left (a^5 (12 A b-7 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{1024 b^4} \\ & = \frac {a^4 (12 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{512 b^4}-\frac {a^3 (12 A b-7 a B) x^{3/2} \sqrt {a+b x}}{768 b^3}+\frac {a^2 (12 A b-7 a B) x^{5/2} \sqrt {a+b x}}{960 b^2}+\frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}-\frac {\left (a^5 (12 A b-7 a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{512 b^4} \\ & = \frac {a^4 (12 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{512 b^4}-\frac {a^3 (12 A b-7 a B) x^{3/2} \sqrt {a+b x}}{768 b^3}+\frac {a^2 (12 A b-7 a B) x^{5/2} \sqrt {a+b x}}{960 b^2}+\frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}-\frac {\left (a^5 (12 A b-7 a B)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^4} \\ & = \frac {a^4 (12 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{512 b^4}-\frac {a^3 (12 A b-7 a B) x^{3/2} \sqrt {a+b x}}{768 b^3}+\frac {a^2 (12 A b-7 a B) x^{5/2} \sqrt {a+b x}}{960 b^2}+\frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}-\frac {a^5 (12 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{9/2}} \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.86 \[ \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx=\frac {\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (-105 a^5 B+48 a^2 b^3 x^2 (2 A+B x)+256 b^5 x^4 (6 A+5 B x)-8 a^3 b^2 x (15 A+7 B x)+10 a^4 b (18 A+7 B x)+64 a b^4 x^3 (33 A+26 B x)\right )+360 a^5 A b \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )+210 a^6 B \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{7680 b^{9/2}} \]
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Time = 1.42 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.81
method | result | size |
risch | \(\frac {\left (1280 b^{5} B \,x^{5}+1536 A \,b^{5} x^{4}+1664 B a \,b^{4} x^{4}+2112 A a \,b^{4} x^{3}+48 B \,a^{2} b^{3} x^{3}+96 A \,a^{2} b^{3} x^{2}-56 B \,a^{3} b^{2} x^{2}-120 a^{3} b^{2} A x +70 a^{4} b B x +180 a^{4} b A -105 a^{5} B \right ) \sqrt {x}\, \sqrt {b x +a}}{7680 b^{4}}-\frac {a^{5} \left (12 A b -7 B a \right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{1024 b^{\frac {9}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(183\) |
default | \(-\frac {\sqrt {x}\, \sqrt {b x +a}\, \left (-2560 B \,b^{\frac {11}{2}} x^{5} \sqrt {x \left (b x +a \right )}-3072 A \,b^{\frac {11}{2}} x^{4} \sqrt {x \left (b x +a \right )}-3328 B a \,b^{\frac {9}{2}} x^{4} \sqrt {x \left (b x +a \right )}-4224 A a \,b^{\frac {9}{2}} x^{3} \sqrt {x \left (b x +a \right )}-96 B \,a^{2} b^{\frac {7}{2}} x^{3} \sqrt {x \left (b x +a \right )}-192 A \,a^{2} b^{\frac {7}{2}} x^{2} \sqrt {x \left (b x +a \right )}+112 B \,a^{3} b^{\frac {5}{2}} x^{2} \sqrt {x \left (b x +a \right )}+240 A \sqrt {x \left (b x +a \right )}\, b^{\frac {5}{2}} a^{3} x -140 B \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{4} x +180 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{5} b -360 A \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{4}-105 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{6}+210 B \sqrt {x \left (b x +a \right )}\, \sqrt {b}\, a^{5}\right )}{15360 b^{\frac {9}{2}} \sqrt {x \left (b x +a \right )}}\) | \(302\) |
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Time = 0.25 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.52 \[ \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx=\left [-\frac {15 \, {\left (7 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (1280 \, B b^{6} x^{5} - 105 \, B a^{5} b + 180 \, A a^{4} b^{2} + 128 \, {\left (13 \, B a b^{5} + 12 \, A b^{6}\right )} x^{4} + 48 \, {\left (B a^{2} b^{4} + 44 \, A a b^{5}\right )} x^{3} - 8 \, {\left (7 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{15360 \, b^{5}}, -\frac {15 \, {\left (7 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (1280 \, B b^{6} x^{5} - 105 \, B a^{5} b + 180 \, A a^{4} b^{2} + 128 \, {\left (13 \, B a b^{5} + 12 \, A b^{6}\right )} x^{4} + 48 \, {\left (B a^{2} b^{4} + 44 \, A a b^{5}\right )} x^{3} - 8 \, {\left (7 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{7680 \, b^{5}}\right ] \]
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Timed out. \[ \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.24 \[ \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx=-\frac {7 \, \sqrt {b x^{2} + a x} B a^{4} x}{256 \, b^{3}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{2} x}{96 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a x} A a^{3} x}{64 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B x}{6 \, b} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a x}{8 \, b} + \frac {7 \, B a^{6} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{1024 \, b^{\frac {9}{2}}} - \frac {3 \, A a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {7}{2}}} - \frac {7 \, \sqrt {b x^{2} + a x} B a^{5}}{512 \, b^{4}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{3}}{192 \, b^{3}} + \frac {3 \, \sqrt {b x^{2} + a x} A a^{4}}{128 \, b^{3}} - \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} B a}{60 \, b^{2}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a^{2}}{16 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{5 \, b} \]
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Timed out. \[ \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx=\text {Timed out} \]
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Timed out. \[ \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx=\int x^{5/2}\,\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2} \,d x \]
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