Integrand size = 20, antiderivative size = 192 \[ \int x^{3/2} (a+b x)^{3/2} (A+B x) \, dx=-\frac {3 a^3 (2 A b-a B) \sqrt {x} \sqrt {a+b x}}{128 b^3}+\frac {a^2 (2 A b-a B) x^{3/2} \sqrt {a+b x}}{64 b^2}+\frac {a (2 A b-a B) x^{5/2} \sqrt {a+b x}}{16 b}+\frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {3 a^4 (2 A b-a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{7/2}} \]
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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} \[ \int x^{3/2} (a+b x)^{3/2} (A+B x) \, dx=\frac {3 a^4 (2 A b-a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{7/2}}-\frac {3 a^3 \sqrt {x} \sqrt {a+b x} (2 A b-a B)}{128 b^3}+\frac {a^2 x^{3/2} \sqrt {a+b x} (2 A b-a B)}{64 b^2}+\frac {a x^{5/2} \sqrt {a+b x} (2 A b-a B)}{16 b}+\frac {x^{5/2} (a+b x)^{3/2} (2 A b-a B)}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b} \]
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Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {\left (5 A b-\frac {5 a B}{2}\right ) \int x^{3/2} (a+b x)^{3/2} \, dx}{5 b} \\ & = \frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {(3 a (2 A b-a B)) \int x^{3/2} \sqrt {a+b x} \, dx}{16 b} \\ & = \frac {a (2 A b-a B) x^{5/2} \sqrt {a+b x}}{16 b}+\frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {\left (a^2 (2 A b-a B)\right ) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{32 b} \\ & = \frac {a^2 (2 A b-a B) x^{3/2} \sqrt {a+b x}}{64 b^2}+\frac {a (2 A b-a B) x^{5/2} \sqrt {a+b x}}{16 b}+\frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}-\frac {\left (3 a^3 (2 A b-a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{128 b^2} \\ & = -\frac {3 a^3 (2 A b-a B) \sqrt {x} \sqrt {a+b x}}{128 b^3}+\frac {a^2 (2 A b-a B) x^{3/2} \sqrt {a+b x}}{64 b^2}+\frac {a (2 A b-a B) x^{5/2} \sqrt {a+b x}}{16 b}+\frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {\left (3 a^4 (2 A b-a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{256 b^3} \\ & = -\frac {3 a^3 (2 A b-a B) \sqrt {x} \sqrt {a+b x}}{128 b^3}+\frac {a^2 (2 A b-a B) x^{3/2} \sqrt {a+b x}}{64 b^2}+\frac {a (2 A b-a B) x^{5/2} \sqrt {a+b x}}{16 b}+\frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {\left (3 a^4 (2 A b-a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{128 b^3} \\ & = -\frac {3 a^3 (2 A b-a B) \sqrt {x} \sqrt {a+b x}}{128 b^3}+\frac {a^2 (2 A b-a B) x^{3/2} \sqrt {a+b x}}{64 b^2}+\frac {a (2 A b-a B) x^{5/2} \sqrt {a+b x}}{16 b}+\frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {\left (3 a^4 (2 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 )}{128 b^3} \\ & = -\frac {3 a^3 (2 A b-a B) \sqrt {x} \sqrt {a+b x}}{128 b^3}+\frac {a^2 (2 A b-a B) x^{3/2} \sqrt {a+b x}}{64 b^2}+\frac {a (2 A b-a B) x^{5/2} \sqrt {a+b x}}{16 b}+\frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {3 a^4 (2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{7/2}} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.74 \[ \int x^{3/2} (a+b x)^{3/2} (A+B x) \, dx=\frac {\sqrt {x} \sqrt {a+b x} \left (15 a^4 B-10 a^3 b (3 A+B x)+4 a^2 b^2 x (5 A+2 B x)+32 b^4 x^3 (5 A+4 B x)+16 a b^3 x^2 (15 A+11 B x)\right )}{640 b^3}+\frac {3 a^4 (-2 A b+a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )}{64 b^{7/2}} \]
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Time = 0.50 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.83
method | result | size |
risch | \(-\frac {\left (-128 B \,x^{4} b^{4}-160 A \,x^{3} b^{4}-176 B \,x^{3} a \,b^{3}-240 A \,x^{2} a \,b^{3}-8 B \,x^{2} a^{2} b^{2}-20 A x \,a^{2} b^{2}+10 B x \,a^{3} b +30 A \,a^{3} b -15 B \,a^{4}\right ) \sqrt {x}\, \sqrt {b x +a}}{640 b^{3}}+\frac {3 a^{4} \left (2 A b -B a \right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{256 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(159\) |
default | \(\frac {\sqrt {x}\, \sqrt {b x +a}\, \left (256 B \,b^{\frac {9}{2}} x^{4} \sqrt {x \left (b x +a \right )}+320 A \,b^{\frac {9}{2}} x^{3} \sqrt {x \left (b x +a \right )}+352 B a \,b^{\frac {7}{2}} x^{3} \sqrt {x \left (b x +a \right )}+480 A a \,b^{\frac {7}{2}} x^{2} \sqrt {x \left (b x +a \right )}+16 B \,a^{2} b^{\frac {5}{2}} x^{2} \sqrt {x \left (b x +a \right )}+40 A \,b^{\frac {5}{2}} \sqrt {x \left (b x +a \right )}\, a^{2} x -20 B \,b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a^{3} x +30 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{4} b -60 A \,b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a^{3}-15 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{5}+30 B \sqrt {b}\, \sqrt {x \left (b x +a \right )}\, a^{4}\right )}{1280 b^{\frac {7}{2}} \sqrt {x \left (b x +a \right )}}\) | \(260\) |
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Time = 0.23 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.51 \[ \int x^{3/2} (a+b x)^{3/2} (A+B x) \, dx=\left [-\frac {15 \, {\left (B a^{5} - 2 \, A a^{4} b\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (128 \, B b^{5} x^{4} + 15 \, B a^{4} b - 30 \, A a^{3} b^{2} + 16 \, {\left (11 \, B a b^{4} + 10 \, A b^{5}\right )} x^{3} + 8 \, {\left (B a^{2} b^{3} + 30 \, A a b^{4}\right )} x^{2} - 10 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{1280 \, b^{4}}, \frac {15 \, {\left (B a^{5} - 2 \, A a^{4} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (128 \, B b^{5} x^{4} + 15 \, B a^{4} b - 30 \, A a^{3} b^{2} + 16 \, {\left (11 \, B a b^{4} + 10 \, A b^{5}\right )} x^{3} + 8 \, {\left (B a^{2} b^{3} + 30 \, A a b^{4}\right )} x^{2} - 10 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{640 \, b^{4}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (173) = 346\).
Time = 150.37 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.85 \[ \int x^{3/2} (a+b x)^{3/2} (A+B x) \, dx=- \frac {3 A a^{\frac {7}{2}} \sqrt {x}}{64 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {A a^{\frac {5}{2}} x^{\frac {3}{2}}}{64 b \sqrt {1 + \frac {b x}{a}}} + \frac {13 A a^{\frac {3}{2}} x^{\frac {5}{2}}}{32 \sqrt {1 + \frac {b x}{a}}} + \frac {5 A \sqrt {a} b x^{\frac {7}{2}}}{8 \sqrt {1 + \frac {b x}{a}}} + \frac {3 A a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {5}{2}}} + \frac {A b^{2} x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} + \frac {3 B a^{\frac {9}{2}} \sqrt {x}}{128 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {B a^{\frac {7}{2}} x^{\frac {3}{2}}}{128 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {B a^{\frac {5}{2}} x^{\frac {5}{2}}}{320 b \sqrt {1 + \frac {b x}{a}}} + \frac {23 B a^{\frac {3}{2}} x^{\frac {7}{2}}}{80 \sqrt {1 + \frac {b x}{a}}} + \frac {19 B \sqrt {a} b x^{\frac {9}{2}}}{40 \sqrt {1 + \frac {b x}{a}}} - \frac {3 B a^{5} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{128 b^{\frac {7}{2}}} + \frac {B b^{2} x^{\frac {11}{2}}}{5 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \]
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Time = 0.21 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.23 \[ \int x^{3/2} (a+b x)^{3/2} (A+B x) \, dx=\frac {1}{4} \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} A x + \frac {3 \, \sqrt {b x^{2} + a x} B a^{3} x}{64 \, b^{2}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a x}{8 \, b} - \frac {3 \, \sqrt {b x^{2} + a x} A a^{2} x}{32 \, b} - \frac {3 \, B a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {7}{2}}} + \frac {3 \, A a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {5}{2}}} + \frac {3 \, \sqrt {b x^{2} + a x} B a^{4}}{128 \, b^{3}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{2}}{16 \, b^{2}} - \frac {3 \, \sqrt {b x^{2} + a x} A a^{3}}{64 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{5 \, b} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{8 \, b} \]
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Timed out. \[ \int x^{3/2} (a+b x)^{3/2} (A+B x) \, dx=\text {Timed out} \]
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Timed out. \[ \int x^{3/2} (a+b x)^{3/2} (A+B x) \, dx=\int x^{3/2}\,\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2} \,d x \]
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