Integrand size = 20, antiderivative size = 126 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {x}} \, dx=\frac {a (6 A b-a B) \sqrt {x} \sqrt {a+b x}}{8 b}+\frac {(6 A b-a B) \sqrt {x} (a+b x)^{3/2}}{12 b}+\frac {B \sqrt {x} (a+b x)^{5/2}}{3 b}+\frac {a^2 (6 A b-a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223, 212} \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {x}} \, dx=\frac {a^2 (6 A b-a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{3/2}}+\frac {\sqrt {x} (a+b x)^{3/2} (6 A b-a B)}{12 b}+\frac {a \sqrt {x} \sqrt {a+b x} (6 A b-a B)}{8 b}+\frac {B \sqrt {x} (a+b x)^{5/2}}{3 b} \]
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Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {B \sqrt {x} (a+b x)^{5/2}}{3 b}+\frac {\left (3 A b-\frac {a B}{2}\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx}{3 b} \\ & = \frac {(6 A b-a B) \sqrt {x} (a+b x)^{3/2}}{12 b}+\frac {B \sqrt {x} (a+b x)^{5/2}}{3 b}+\frac {(a (6 A b-a B)) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx}{8 b} \\ & = \frac {a (6 A b-a B) \sqrt {x} \sqrt {a+b x}}{8 b}+\frac {(6 A b-a B) \sqrt {x} (a+b x)^{3/2}}{12 b}+\frac {B \sqrt {x} (a+b x)^{5/2}}{3 b}+\frac {\left (a^2 (6 A b-a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{16 b} \\ & = \frac {a (6 A b-a B) \sqrt {x} \sqrt {a+b x}}{8 b}+\frac {(6 A b-a B) \sqrt {x} (a+b x)^{3/2}}{12 b}+\frac {B \sqrt {x} (a+b x)^{5/2}}{3 b}+\frac {\left (a^2 (6 A b-a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{8 b} \\ & = \frac {a (6 A b-a B) \sqrt {x} \sqrt {a+b x}}{8 b}+\frac {(6 A b-a B) \sqrt {x} (a+b x)^{3/2}}{12 b}+\frac {B \sqrt {x} (a+b x)^{5/2}}{3 b}+\frac {\left (a^2 (6 A b-a B)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{8 b} \\ & = \frac {a (6 A b-a B) \sqrt {x} \sqrt {a+b x}}{8 b}+\frac {(6 A b-a B) \sqrt {x} (a+b x)^{3/2}}{12 b}+\frac {B \sqrt {x} (a+b x)^{5/2}}{3 b}+\frac {a^2 (6 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{3/2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.79 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {x}} \, dx=\frac {\sqrt {x} \sqrt {a+b x} \left (30 a A b+3 a^2 B+12 A b^2 x+14 a b B x+8 b^2 B x^2\right )}{24 b}+\frac {a^2 (-6 A b+a B) \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{8 b^{3/2}} \]
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Time = 0.50 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.88
method | result | size |
risch | \(\frac {\left (8 b^{2} B \,x^{2}+12 A \,b^{2} x +14 B a b x +30 a b A +3 a^{2} B \right ) \sqrt {x}\, \sqrt {b x +a}}{24 b}+\frac {a^{2} \left (6 A b -B a \right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{16 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(111\) |
default | \(\frac {\sqrt {b x +a}\, \sqrt {x}\, \left (16 B \,b^{\frac {5}{2}} x^{2} \sqrt {x \left (b x +a \right )}+24 A \,b^{\frac {5}{2}} \sqrt {x \left (b x +a \right )}\, x +28 B \,b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a x +18 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{2} b +60 A \,b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a -3 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{3}+6 B \sqrt {b}\, \sqrt {x \left (b x +a \right )}\, a^{2}\right )}{48 b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}}\) | \(176\) |
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Time = 0.24 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.57 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {x}} \, dx=\left [-\frac {3 \, {\left (B a^{3} - 6 \, A a^{2} b\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (8 \, B b^{3} x^{2} + 3 \, B a^{2} b + 30 \, A a b^{2} + 2 \, {\left (7 \, B a b^{2} + 6 \, A b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{48 \, b^{2}}, \frac {3 \, {\left (B a^{3} - 6 \, A a^{2} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (8 \, B b^{3} x^{2} + 3 \, B a^{2} b + 30 \, A a b^{2} + 2 \, {\left (7 \, B a b^{2} + 6 \, A b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{24 \, b^{2}}\right ] \]
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Time = 8.63 (sec) , antiderivative size = 394, normalized size of antiderivative = 3.13 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {x}} \, dx=A a^{\frac {3}{2}} \sqrt {x} \sqrt {1 + \frac {b x}{a}} + \frac {A a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}} + 2 A b \left (\begin {cases} - \frac {a^{2} \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x} + 2 b \sqrt {x} \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {\sqrt {x} \log {\left (\sqrt {x} \right )}}{\sqrt {b x}} & \text {otherwise} \end {cases}\right )}{8 b} + \frac {a \sqrt {x} \sqrt {a + b x}}{8 b} + \frac {x^{\frac {3}{2}} \sqrt {a + b x}}{4} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{\frac {3}{2}}}{3} & \text {otherwise} \end {cases}\right ) - \frac {B a^{\frac {5}{2}} \sqrt {x}}{8 b \sqrt {1 + \frac {b x}{a}}} - \frac {B a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 \sqrt {1 + \frac {b x}{a}}} + \frac {5 B \sqrt {a} b x^{\frac {5}{2}}}{12 \sqrt {1 + \frac {b x}{a}}} + \frac {B a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {3}{2}}} + 2 B a \left (\begin {cases} - \frac {a^{2} \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x} + 2 b \sqrt {x} \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {\sqrt {x} \log {\left (\sqrt {x} \right )}}{\sqrt {b x}} & \text {otherwise} \end {cases}\right )}{8 b} + \frac {a \sqrt {x} \sqrt {a + b x}}{8 b} + \frac {x^{\frac {3}{2}} \sqrt {a + b x}}{4} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{\frac {3}{2}}}{3} & \text {otherwise} \end {cases}\right ) + \frac {B b^{2} x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (95) = 190\).
Time = 0.22 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.24 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {x}} \, dx=\frac {1}{3} \, \sqrt {b x^{2} + a x} B b x^{2} - \frac {5}{12} \, \sqrt {b x^{2} + a x} B a x - \frac {5 \, B a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {3}{2}}} + \frac {A a^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{\sqrt {b}} + \frac {5 \, \sqrt {b x^{2} + a x} B a^{2}}{8 \, b} + \frac {{\left (2 \, B a b + A b^{2}\right )} \sqrt {b x^{2} + a x} x}{2 \, b} + \frac {3 \, {\left (2 \, B a b + A b^{2}\right )} a^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, b^{\frac {5}{2}}} - \frac {{\left (B a^{2} + 2 \, A a b\right )} a \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{2 \, b^{\frac {3}{2}}} - \frac {3 \, {\left (2 \, B a b + A b^{2}\right )} \sqrt {b x^{2} + a x} a}{4 \, b^{2}} + \frac {{\left (B a^{2} + 2 \, A a b\right )} \sqrt {b x^{2} + a x}}{b} \]
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Time = 77.58 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {x}} \, dx=\frac {{\left (\sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} B}{b^{2}} - \frac {B a b^{2} - 6 \, A b^{3}}{b^{4}}\right )} - \frac {3 \, {\left (B a^{2} b^{2} - 6 \, A a b^{3}\right )}}{b^{4}}\right )} + \frac {3 \, {\left (B a^{3} - 6 \, A a^{2} b\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right )}{b^{\frac {3}{2}}}\right )} b}{24 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {x}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}}{\sqrt {x}} \,d x \]
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