Integrand size = 20, antiderivative size = 167 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2 (A b-a B) x^{7/2}}{a b \sqrt {a+b x}}-\frac {5 a (6 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{8 b^4}+\frac {5 (6 A b-7 a B) x^{3/2} \sqrt {a+b x}}{12 b^3}-\frac {(6 A b-7 a B) x^{5/2} \sqrt {a+b x}}{3 a b^2}+\frac {5 a^2 (6 A b-7 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{9/2}} \]
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Time = 0.05 (sec) , antiderivative size = 167, 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 = {79, 52, 65, 223, 212} \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {5 a^2 (6 A b-7 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{9/2}}-\frac {5 a \sqrt {x} \sqrt {a+b x} (6 A b-7 a B)}{8 b^4}+\frac {5 x^{3/2} \sqrt {a+b x} (6 A b-7 a B)}{12 b^3}-\frac {x^{5/2} \sqrt {a+b x} (6 A b-7 a B)}{3 a b^2}+\frac {2 x^{7/2} (A b-a B)}{a b \sqrt {a+b x}} \]
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Rule 52
Rule 65
Rule 79
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {2 (A b-a B) x^{7/2}}{a b \sqrt {a+b x}}-\frac {\left (2 \left (3 A b-\frac {7 a B}{2}\right )\right ) \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx}{a b} \\ & = \frac {2 (A b-a B) x^{7/2}}{a b \sqrt {a+b x}}-\frac {(6 A b-7 a B) x^{5/2} \sqrt {a+b x}}{3 a b^2}+\frac {(5 (6 A b-7 a B)) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{6 b^2} \\ & = \frac {2 (A b-a B) x^{7/2}}{a b \sqrt {a+b x}}+\frac {5 (6 A b-7 a B) x^{3/2} \sqrt {a+b x}}{12 b^3}-\frac {(6 A b-7 a B) x^{5/2} \sqrt {a+b x}}{3 a b^2}-\frac {(5 a (6 A b-7 a B)) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{8 b^3} \\ & = \frac {2 (A b-a B) x^{7/2}}{a b \sqrt {a+b x}}-\frac {5 a (6 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{8 b^4}+\frac {5 (6 A b-7 a B) x^{3/2} \sqrt {a+b x}}{12 b^3}-\frac {(6 A b-7 a B) x^{5/2} \sqrt {a+b x}}{3 a b^2}+\frac {\left (5 a^2 (6 A b-7 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{16 b^4} \\ & = \frac {2 (A b-a B) x^{7/2}}{a b \sqrt {a+b x}}-\frac {5 a (6 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{8 b^4}+\frac {5 (6 A b-7 a B) x^{3/2} \sqrt {a+b x}}{12 b^3}-\frac {(6 A b-7 a B) x^{5/2} \sqrt {a+b x}}{3 a b^2}+\frac {\left (5 a^2 (6 A b-7 a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{8 b^4} \\ & = \frac {2 (A b-a B) x^{7/2}}{a b \sqrt {a+b x}}-\frac {5 a (6 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{8 b^4}+\frac {5 (6 A b-7 a B) x^{3/2} \sqrt {a+b x}}{12 b^3}-\frac {(6 A b-7 a B) x^{5/2} \sqrt {a+b x}}{3 a b^2}+\frac {\left (5 a^2 (6 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 )}{8 b^4} \\ & = \frac {2 (A b-a B) x^{7/2}}{a b \sqrt {a+b x}}-\frac {5 a (6 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{8 b^4}+\frac {5 (6 A b-7 a B) x^{3/2} \sqrt {a+b x}}{12 b^3}-\frac {(6 A b-7 a B) x^{5/2} \sqrt {a+b x}}{3 a b^2}+\frac {5 a^2 (6 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{9/2}} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.75 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {\sqrt {x} \left (105 a^3 B+4 b^3 x^2 (3 A+2 B x)-2 a b^2 x (15 A+7 B x)+a^2 (-90 A b+35 b B x)\right )}{24 b^4 \sqrt {a+b x}}+\frac {5 a^2 (-6 A b+7 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )}{4 b^{9/2}} \]
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Time = 0.52 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.11
method | result | size |
risch | \(-\frac {\left (-8 b^{2} B \,x^{2}-12 A \,b^{2} x +22 B a b x +42 a b A -57 a^{2} B \right ) \sqrt {x}\, \sqrt {b x +a}}{24 b^{4}}+\frac {a^{2} \left (30 A \sqrt {b}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )-\frac {35 B a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{\sqrt {b}}-\frac {32 \left (A b -B a \right ) \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{b \left (x +\frac {a}{b}\right )}\right ) \sqrt {x \left (b x +a \right )}}{16 b^{4} \sqrt {x}\, \sqrt {b x +a}}\) | \(186\) |
default | \(\frac {\left (16 B \,b^{\frac {7}{2}} x^{3} \sqrt {x \left (b x +a \right )}+24 A \,b^{\frac {7}{2}} x^{2} \sqrt {x \left (b x +a \right )}-28 B a \,b^{\frac {5}{2}} x^{2} \sqrt {x \left (b x +a \right )}+90 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{2} b^{2} x -60 A \sqrt {x \left (b x +a \right )}\, b^{\frac {5}{2}} a x -105 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{3} b x +70 B \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{2} x +90 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{3} b -180 A \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{2}-105 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{4}+210 B \sqrt {x \left (b x +a \right )}\, \sqrt {b}\, a^{3}\right ) \sqrt {x}}{48 b^{\frac {9}{2}} \sqrt {x \left (b x +a \right )}\, \sqrt {b x +a}}\) | \(288\) |
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Time = 0.24 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.83 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{3/2}} \, dx=\left [-\frac {15 \, {\left (7 \, B a^{4} - 6 \, A a^{3} b + {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (8 \, B b^{4} x^{3} + 105 \, B a^{3} b - 90 \, A a^{2} b^{2} - 2 \, {\left (7 \, B a b^{3} - 6 \, A b^{4}\right )} x^{2} + 5 \, {\left (7 \, B a^{2} b^{2} - 6 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{48 \, {\left (b^{6} x + a b^{5}\right )}}, \frac {15 \, {\left (7 \, B a^{4} - 6 \, A a^{3} b + {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (8 \, B b^{4} x^{3} + 105 \, B a^{3} b - 90 \, A a^{2} b^{2} - 2 \, {\left (7 \, B a b^{3} - 6 \, A b^{4}\right )} x^{2} + 5 \, {\left (7 \, B a^{2} b^{2} - 6 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{24 \, {\left (b^{6} x + a b^{5}\right )}}\right ] \]
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Time = 76.71 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.46 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{3/2}} \, dx=A \left (- \frac {15 a^{\frac {3}{2}} \sqrt {x}}{4 b^{3} \sqrt {1 + \frac {b x}{a}}} - \frac {5 \sqrt {a} x^{\frac {3}{2}}}{4 b^{2} \sqrt {1 + \frac {b x}{a}}} + \frac {15 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {7}{2}}} + \frac {x^{\frac {5}{2}}}{2 \sqrt {a} b \sqrt {1 + \frac {b x}{a}}}\right ) + B \left (\frac {35 a^{\frac {5}{2}} \sqrt {x}}{8 b^{4} \sqrt {1 + \frac {b x}{a}}} + \frac {35 a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b^{3} \sqrt {1 + \frac {b x}{a}}} - \frac {7 \sqrt {a} x^{\frac {5}{2}}}{12 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {35 a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {9}{2}}} + \frac {x^{\frac {7}{2}}}{3 \sqrt {a} b \sqrt {1 + \frac {b x}{a}}}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.27 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {B x^{4}}{3 \, \sqrt {b x^{2} + a x} b} - \frac {7 \, B a x^{3}}{12 \, \sqrt {b x^{2} + a x} b^{2}} + \frac {A x^{3}}{2 \, \sqrt {b x^{2} + a x} b} + \frac {35 \, B a^{2} x^{2}}{24 \, \sqrt {b x^{2} + a x} b^{3}} - \frac {5 \, A a x^{2}}{4 \, \sqrt {b x^{2} + a x} b^{2}} + \frac {35 \, B a^{3} x}{8 \, \sqrt {b x^{2} + a x} b^{4}} - \frac {15 \, A a^{2} x}{4 \, \sqrt {b x^{2} + a x} b^{3}} - \frac {35 \, B a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {9}{2}}} + \frac {15 \, A a^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, b^{\frac {7}{2}}} \]
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Time = 15.61 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.24 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {1}{24} \, \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} B {\left | b \right |}}{b^{6}} - \frac {19 \, B a b^{17} {\left | b \right |} - 6 \, A b^{18} {\left | b \right |}}{b^{23}}\right )} + \frac {3 \, {\left (29 \, B a^{2} b^{17} {\left | b \right |} - 18 \, A a b^{18} {\left | b \right |}\right )}}{b^{23}}\right )} + \frac {5 \, {\left (7 \, B a^{3} {\left | b \right |} - 6 \, A a^{2} b {\left | b \right |}\right )} \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{16 \, b^{\frac {11}{2}}} + \frac {4 \, {\left (B a^{4} {\left | b \right |} - A a^{3} b {\left | b \right |}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} b^{\frac {9}{2}}} \]
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Timed out. \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{3/2}} \, dx=\int \frac {x^{5/2}\,\left (A+B\,x\right )}{{\left (a+b\,x\right )}^{3/2}} \,d x \]
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