Integrand size = 20, antiderivative size = 159 \[ \int x^{3/2} \sqrt {a+b x} (A+B x) \, dx=-\frac {a^2 (8 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{64 b^3}+\frac {a (8 A b-5 a B) x^{3/2} \sqrt {a+b x}}{96 b^2}+\frac {(8 A b-5 a B) x^{5/2} \sqrt {a+b x}}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b}+\frac {a^3 (8 A b-5 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{7/2}} \]
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Time = 0.05 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223, 212} \[ \int x^{3/2} \sqrt {a+b x} (A+B x) \, dx=\frac {a^3 (8 A b-5 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{7/2}}-\frac {a^2 \sqrt {x} \sqrt {a+b x} (8 A b-5 a B)}{64 b^3}+\frac {a x^{3/2} \sqrt {a+b x} (8 A b-5 a B)}{96 b^2}+\frac {x^{5/2} \sqrt {a+b x} (8 A b-5 a B)}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b} \]
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Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {B x^{5/2} (a+b x)^{3/2}}{4 b}+\frac {\left (4 A b-\frac {5 a B}{2}\right ) \int x^{3/2} \sqrt {a+b x} \, dx}{4 b} \\ & = \frac {(8 A b-5 a B) x^{5/2} \sqrt {a+b x}}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b}+\frac {(a (8 A b-5 a B)) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{48 b} \\ & = \frac {a (8 A b-5 a B) x^{3/2} \sqrt {a+b x}}{96 b^2}+\frac {(8 A b-5 a B) x^{5/2} \sqrt {a+b x}}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b}-\frac {\left (a^2 (8 A b-5 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{64 b^2} \\ & = -\frac {a^2 (8 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{64 b^3}+\frac {a (8 A b-5 a B) x^{3/2} \sqrt {a+b x}}{96 b^2}+\frac {(8 A b-5 a B) x^{5/2} \sqrt {a+b x}}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b}+\frac {\left (a^3 (8 A b-5 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{128 b^3} \\ & = -\frac {a^2 (8 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{64 b^3}+\frac {a (8 A b-5 a B) x^{3/2} \sqrt {a+b x}}{96 b^2}+\frac {(8 A b-5 a B) x^{5/2} \sqrt {a+b x}}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b}+\frac {\left (a^3 (8 A b-5 a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{64 b^3} \\ & = -\frac {a^2 (8 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{64 b^3}+\frac {a (8 A b-5 a B) x^{3/2} \sqrt {a+b x}}{96 b^2}+\frac {(8 A b-5 a B) x^{5/2} \sqrt {a+b x}}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b}+\frac {\left (a^3 (8 A b-5 a B)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^3} \\ & = -\frac {a^2 (8 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{64 b^3}+\frac {a (8 A b-5 a B) x^{3/2} \sqrt {a+b x}}{96 b^2}+\frac {(8 A b-5 a B) x^{5/2} \sqrt {a+b x}}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b}+\frac {a^3 (8 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{7/2}} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.79 \[ \int x^{3/2} \sqrt {a+b x} (A+B x) \, dx=\frac {\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (15 a^3 B+8 a b^2 x (2 A+B x)+16 b^3 x^2 (4 A+3 B x)-2 a^2 b (12 A+5 B x)\right )+6 a^3 (-8 A b+5 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )}{192 b^{7/2}} \]
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Time = 1.42 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.85
method | result | size |
risch | \(-\frac {\left (-48 b^{3} B \,x^{3}-64 A \,b^{3} x^{2}-8 B a \,b^{2} x^{2}-16 a \,b^{2} A x +10 a^{2} b B x +24 a^{2} b A -15 a^{3} B \right ) \sqrt {x}\, \sqrt {b x +a}}{192 b^{3}}+\frac {a^{3} \left (8 A b -5 B a \right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{128 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(135\) |
default | \(\frac {\sqrt {x}\, \sqrt {b x +a}\, \left (96 B \,b^{\frac {7}{2}} x^{3} \sqrt {x \left (b x +a \right )}+128 A \,b^{\frac {7}{2}} x^{2} \sqrt {x \left (b x +a \right )}+16 B a \,b^{\frac {5}{2}} x^{2} \sqrt {x \left (b x +a \right )}+32 A \sqrt {x \left (b x +a \right )}\, b^{\frac {5}{2}} a x -20 B \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{2} x +24 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{3} b -48 A \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{2}-15 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{4}+30 B \sqrt {x \left (b x +a \right )}\, \sqrt {b}\, a^{3}\right )}{384 b^{\frac {7}{2}} \sqrt {x \left (b x +a \right )}}\) | \(218\) |
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Time = 0.24 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.55 \[ \int x^{3/2} \sqrt {a+b x} (A+B x) \, dx=\left [-\frac {3 \, {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (48 \, B b^{4} x^{3} + 15 \, B a^{3} b - 24 \, A a^{2} b^{2} + 8 \, {\left (B a b^{3} + 8 \, A b^{4}\right )} x^{2} - 2 \, {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{384 \, b^{4}}, \frac {3 \, {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (48 \, B b^{4} x^{3} + 15 \, B a^{3} b - 24 \, A a^{2} b^{2} + 8 \, {\left (B a b^{3} + 8 \, A b^{4}\right )} x^{2} - 2 \, {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{192 \, b^{4}}\right ] \]
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Time = 24.29 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.86 \[ \int x^{3/2} \sqrt {a+b x} (A+B x) \, dx=- \frac {A a^{\frac {5}{2}} \sqrt {x}}{8 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {A a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b \sqrt {1 + \frac {b x}{a}}} + \frac {5 A \sqrt {a} x^{\frac {5}{2}}}{12 \sqrt {1 + \frac {b x}{a}}} + \frac {A a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {5}{2}}} + \frac {A b x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} + \frac {5 B a^{\frac {7}{2}} \sqrt {x}}{64 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {5 B a^{\frac {5}{2}} x^{\frac {3}{2}}}{192 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {B a^{\frac {3}{2}} x^{\frac {5}{2}}}{96 b \sqrt {1 + \frac {b x}{a}}} + \frac {7 B \sqrt {a} x^{\frac {7}{2}}}{24 \sqrt {1 + \frac {b x}{a}}} - \frac {5 B a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {7}{2}}} + \frac {B b x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \]
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Time = 0.22 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.25 \[ \int x^{3/2} \sqrt {a+b x} (A+B x) \, dx=\frac {5 \, \sqrt {b x^{2} + a x} B a^{2} x}{32 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B x}{4 \, b} - \frac {\sqrt {b x^{2} + a x} A a x}{4 \, b} - \frac {5 \, B a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {7}{2}}} + \frac {A a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {5}{2}}} + \frac {5 \, \sqrt {b x^{2} + a x} B a^{3}}{64 \, b^{3}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{24 \, b^{2}} - \frac {\sqrt {b x^{2} + a x} A a^{2}}{8 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{3 \, b} \]
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Timed out. \[ \int x^{3/2} \sqrt {a+b x} (A+B x) \, dx=\text {Timed out} \]
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Timed out. \[ \int x^{3/2} \sqrt {a+b x} (A+B x) \, dx=\int x^{3/2}\,\left (A+B\,x\right )\,\sqrt {a+b\,x} \,d x \]
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