Integrand size = 20, antiderivative size = 192 \[ \int x^{5/2} \sqrt {a+b x} (A+B x) \, dx=\frac {a^3 (10 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{128 b^4}-\frac {a^2 (10 A b-7 a B) x^{3/2} \sqrt {a+b x}}{192 b^3}+\frac {a (10 A b-7 a B) x^{5/2} \sqrt {a+b x}}{240 b^2}+\frac {(10 A b-7 a B) x^{7/2} \sqrt {a+b x}}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac {a^4 (10 A b-7 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{9/2}} \]
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Time = 0.06 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223, 212} \[ \int x^{5/2} \sqrt {a+b x} (A+B x) \, dx=-\frac {a^4 (10 A b-7 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{9/2}}+\frac {a^3 \sqrt {x} \sqrt {a+b x} (10 A b-7 a B)}{128 b^4}-\frac {a^2 x^{3/2} \sqrt {a+b x} (10 A b-7 a B)}{192 b^3}+\frac {a x^{5/2} \sqrt {a+b x} (10 A b-7 a B)}{240 b^2}+\frac {x^{7/2} \sqrt {a+b x} (10 A b-7 a B)}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b} \]
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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)^{3/2}}{5 b}+\frac {\left (5 A b-\frac {7 a B}{2}\right ) \int x^{5/2} \sqrt {a+b x} \, dx}{5 b} \\ & = \frac {(10 A b-7 a B) x^{7/2} \sqrt {a+b x}}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}+\frac {(a (10 A b-7 a B)) \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx}{80 b} \\ & = \frac {a (10 A b-7 a B) x^{5/2} \sqrt {a+b x}}{240 b^2}+\frac {(10 A b-7 a B) x^{7/2} \sqrt {a+b x}}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac {\left (a^2 (10 A b-7 a B)\right ) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{96 b^2} \\ & = -\frac {a^2 (10 A b-7 a B) x^{3/2} \sqrt {a+b x}}{192 b^3}+\frac {a (10 A b-7 a B) x^{5/2} \sqrt {a+b x}}{240 b^2}+\frac {(10 A b-7 a B) x^{7/2} \sqrt {a+b x}}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}+\frac {\left (a^3 (10 A b-7 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{128 b^3} \\ & = \frac {a^3 (10 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{128 b^4}-\frac {a^2 (10 A b-7 a B) x^{3/2} \sqrt {a+b x}}{192 b^3}+\frac {a (10 A b-7 a B) x^{5/2} \sqrt {a+b x}}{240 b^2}+\frac {(10 A b-7 a B) x^{7/2} \sqrt {a+b x}}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac {\left (a^4 (10 A b-7 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{256 b^4} \\ & = \frac {a^3 (10 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{128 b^4}-\frac {a^2 (10 A b-7 a B) x^{3/2} \sqrt {a+b x}}{192 b^3}+\frac {a (10 A b-7 a B) x^{5/2} \sqrt {a+b x}}{240 b^2}+\frac {(10 A b-7 a B) x^{7/2} \sqrt {a+b x}}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac {\left (a^4 (10 A b-7 a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{128 b^4} \\ & = \frac {a^3 (10 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{128 b^4}-\frac {a^2 (10 A b-7 a B) x^{3/2} \sqrt {a+b x}}{192 b^3}+\frac {a (10 A b-7 a B) x^{5/2} \sqrt {a+b x}}{240 b^2}+\frac {(10 A b-7 a B) x^{7/2} \sqrt {a+b x}}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac {\left (a^4 (10 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 )}{128 b^4} \\ & = \frac {a^3 (10 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{128 b^4}-\frac {a^2 (10 A b-7 a B) x^{3/2} \sqrt {a+b x}}{192 b^3}+\frac {a (10 A b-7 a B) x^{5/2} \sqrt {a+b x}}{240 b^2}+\frac {(10 A b-7 a B) x^{7/2} \sqrt {a+b x}}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac {a^4 (10 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{9/2}} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.92 \[ \int x^{5/2} \sqrt {a+b x} (A+B x) \, dx=\frac {\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (-105 a^4 B+16 a b^3 x^2 (5 A+3 B x)+96 b^4 x^3 (5 A+4 B x)+10 a^3 b (15 A+7 B x)-4 a^2 b^2 x (25 A+14 B x)\right )+300 a^4 A b \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )+210 a^5 B \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{1920 b^{9/2}} \]
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Time = 0.50 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {\left (384 B \,x^{4} b^{4}+480 A \,x^{3} b^{4}+48 B \,x^{3} a \,b^{3}+80 A \,x^{2} a \,b^{3}-56 B \,x^{2} a^{2} b^{2}-100 A x \,a^{2} b^{2}+70 B x \,a^{3} b +150 A \,a^{3} b -105 B \,a^{4}\right ) \sqrt {x}\, \sqrt {b x +a}}{1920 b^{4}}-\frac {a^{4} \left (10 A b -7 B a \right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{256 b^{\frac {9}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(159\) |
default | \(-\frac {\sqrt {x}\, \sqrt {b x +a}\, \left (-768 B \,b^{\frac {9}{2}} x^{4} \sqrt {x \left (b x +a \right )}-960 A \,b^{\frac {9}{2}} x^{3} \sqrt {x \left (b x +a \right )}-96 B a \,b^{\frac {7}{2}} x^{3} \sqrt {x \left (b x +a \right )}-160 A a \,b^{\frac {7}{2}} x^{2} \sqrt {x \left (b x +a \right )}+112 B \,a^{2} b^{\frac {5}{2}} x^{2} \sqrt {x \left (b x +a \right )}+200 A \,b^{\frac {5}{2}} \sqrt {x \left (b x +a \right )}\, a^{2} x -140 B \,b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a^{3} x +150 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{4} b -300 A \,b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a^{3}-105 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{5}+210 B \sqrt {b}\, \sqrt {x \left (b x +a \right )}\, a^{4}\right )}{3840 b^{\frac {9}{2}} \sqrt {x \left (b x +a \right )}}\) | \(260\) |
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Time = 0.23 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.54 \[ \int x^{5/2} \sqrt {a+b x} (A+B x) \, dx=\left [-\frac {15 \, {\left (7 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (384 \, B b^{5} x^{4} - 105 \, B a^{4} b + 150 \, A a^{3} b^{2} + 48 \, {\left (B a b^{4} + 10 \, A b^{5}\right )} x^{3} - 8 \, {\left (7 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3840 \, b^{5}}, -\frac {15 \, {\left (7 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (384 \, B b^{5} x^{4} - 105 \, B a^{4} b + 150 \, A a^{3} b^{2} + 48 \, {\left (B a b^{4} + 10 \, A b^{5}\right )} x^{3} - 8 \, {\left (7 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{1920 \, b^{5}}\right ] \]
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Time = 127.16 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.86 \[ \int x^{5/2} \sqrt {a+b x} (A+B x) \, dx=\frac {5 A a^{\frac {7}{2}} \sqrt {x}}{64 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {5 A a^{\frac {5}{2}} x^{\frac {3}{2}}}{192 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {A a^{\frac {3}{2}} x^{\frac {5}{2}}}{96 b \sqrt {1 + \frac {b x}{a}}} + \frac {7 A \sqrt {a} x^{\frac {7}{2}}}{24 \sqrt {1 + \frac {b x}{a}}} - \frac {5 A a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {7}{2}}} + \frac {A b x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} - \frac {7 B a^{\frac {9}{2}} \sqrt {x}}{128 b^{4} \sqrt {1 + \frac {b x}{a}}} - \frac {7 B a^{\frac {7}{2}} x^{\frac {3}{2}}}{384 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {7 B a^{\frac {5}{2}} x^{\frac {5}{2}}}{960 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {B a^{\frac {3}{2}} x^{\frac {7}{2}}}{240 b \sqrt {1 + \frac {b x}{a}}} + \frac {9 B \sqrt {a} x^{\frac {9}{2}}}{40 \sqrt {1 + \frac {b x}{a}}} + \frac {7 B a^{5} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{128 b^{\frac {9}{2}}} + \frac {B b x^{\frac {11}{2}}}{5 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \]
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Time = 0.20 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.26 \[ \int x^{5/2} \sqrt {a+b x} (A+B x) \, dx=\frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B x^{2}}{5 \, b} - \frac {7 \, \sqrt {b x^{2} + a x} B a^{3} x}{64 \, b^{3}} - \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a x}{40 \, b^{2}} + \frac {5 \, \sqrt {b x^{2} + a x} A a^{2} x}{32 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A x}{4 \, b} + \frac {7 \, B a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {9}{2}}} - \frac {5 \, A a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {7}{2}}} - \frac {7 \, \sqrt {b x^{2} + a x} B a^{4}}{128 \, b^{4}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{2}}{48 \, b^{3}} + \frac {5 \, \sqrt {b x^{2} + a x} A a^{3}}{64 \, b^{3}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{24 \, b^{2}} \]
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Timed out. \[ \int x^{5/2} \sqrt {a+b x} (A+B x) \, dx=\text {Timed out} \]
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Timed out. \[ \int x^{5/2} \sqrt {a+b x} (A+B x) \, dx=\int x^{5/2}\,\left (A+B\,x\right )\,\sqrt {a+b\,x} \,d x \]
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