Integrand size = 20, antiderivative size = 169 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt {a+b x}}+\frac {5 (4 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{4 b^4}-\frac {5 (4 A b-7 a B) x^{3/2} \sqrt {a+b x}}{6 a b^3}-\frac {5 a (4 A b-7 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{9/2}} \]
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Time = 0.05 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {79, 49, 52, 65, 223, 212} \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{5/2}} \, dx=-\frac {5 a (4 A b-7 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{9/2}}+\frac {5 \sqrt {x} \sqrt {a+b x} (4 A b-7 a B)}{4 b^4}-\frac {5 x^{3/2} \sqrt {a+b x} (4 A b-7 a B)}{6 a b^3}+\frac {2 x^{5/2} (4 A b-7 a B)}{3 a b^2 \sqrt {a+b x}}+\frac {2 x^{7/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]
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Rule 49
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}}{3 a b (a+b x)^{3/2}}-\frac {\left (2 \left (2 A b-\frac {7 a B}{2}\right )\right ) \int \frac {x^{5/2}}{(a+b x)^{3/2}} \, dx}{3 a b} \\ & = \frac {2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt {a+b x}}-\frac {(5 (4 A b-7 a B)) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{3 a b^2} \\ & = \frac {2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt {a+b x}}-\frac {5 (4 A b-7 a B) x^{3/2} \sqrt {a+b x}}{6 a b^3}+\frac {(5 (4 A b-7 a B)) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{4 b^3} \\ & = \frac {2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt {a+b x}}+\frac {5 (4 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{4 b^4}-\frac {5 (4 A b-7 a B) x^{3/2} \sqrt {a+b x}}{6 a b^3}-\frac {(5 a (4 A b-7 a B)) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{8 b^4} \\ & = \frac {2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt {a+b x}}+\frac {5 (4 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{4 b^4}-\frac {5 (4 A b-7 a B) x^{3/2} \sqrt {a+b x}}{6 a b^3}-\frac {(5 a (4 A b-7 a B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{4 b^4} \\ & = \frac {2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt {a+b x}}+\frac {5 (4 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{4 b^4}-\frac {5 (4 A b-7 a B) x^{3/2} \sqrt {a+b x}}{6 a b^3}-\frac {(5 a (4 A b-7 a B)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^4} \\ & = \frac {2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt {a+b x}}+\frac {5 (4 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{4 b^4}-\frac {5 (4 A b-7 a B) x^{3/2} \sqrt {a+b x}}{6 a b^3}-\frac {5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{9/2}} \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.72 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {x} \left (-105 a^3 B+a b^2 x (80 A-21 B x)+20 a^2 b (3 A-7 B x)+6 b^3 x^2 (2 A+B x)\right )}{12 b^4 (a+b x)^{3/2}}+\frac {5 a (-4 A b+7 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{2 b^{9/2}} \]
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Time = 0.54 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.53
method | result | size |
risch | \(\frac {\left (2 b B x +4 A b -11 B a \right ) \sqrt {x}\, \sqrt {b x +a}}{4 b^{4}}-\frac {a \left (20 A \sqrt {b}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )-\frac {35 B a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{\sqrt {b}}-\frac {16 \left (3 A b -4 B a \right ) \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{b \left (x +\frac {a}{b}\right )}+\frac {8 a^{2} \left (A b -B a \right ) \left (\frac {2 \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{3 a \left (x +\frac {a}{b}\right )^{2}}+\frac {4 b \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{3 a^{2} \left (x +\frac {a}{b}\right )}\right )}{b^{2}}\right ) \sqrt {x \left (b x +a \right )}}{8 b^{4} \sqrt {x}\, \sqrt {b x +a}}\) | \(259\) |
default | \(-\frac {\left (-12 B \,b^{\frac {7}{2}} x^{3} \sqrt {x \left (b x +a \right )}+60 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a \,b^{3} x^{2}-24 A \,b^{\frac {7}{2}} x^{2} \sqrt {x \left (b x +a \right )}-105 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{2} b^{2} x^{2}+42 B a \,b^{\frac {5}{2}} x^{2} \sqrt {x \left (b x +a \right )}+120 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{2} b^{2} x -160 A \sqrt {x \left (b x +a \right )}\, b^{\frac {5}{2}} a x -210 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{3} b x +280 B \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{2} x +60 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{3} b -120 A \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{2}-105 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{4}+210 B \sqrt {x \left (b x +a \right )}\, \sqrt {b}\, a^{3}\right ) \sqrt {x}}{24 b^{\frac {9}{2}} \sqrt {x \left (b x +a \right )}\, \left (b x +a \right )^{\frac {3}{2}}}\) | \(362\) |
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Time = 0.25 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.21 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\left [-\frac {15 \, {\left (7 \, B a^{4} - 4 \, A a^{3} b + {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{2} + 2 \, {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (6 \, B b^{4} x^{3} - 105 \, B a^{3} b + 60 \, A a^{2} b^{2} - 3 \, {\left (7 \, B a b^{3} - 4 \, A b^{4}\right )} x^{2} - 20 \, {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{24 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}, -\frac {15 \, {\left (7 \, B a^{4} - 4 \, A a^{3} b + {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{2} + 2 \, {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (6 \, B b^{4} x^{3} - 105 \, B a^{3} b + 60 \, A a^{2} b^{2} - 3 \, {\left (7 \, B a b^{3} - 4 \, A b^{4}\right )} x^{2} - 20 \, {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{12 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}\right ] \]
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Timed out. \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (135) = 270\).
Time = 0.25 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.06 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{5/2}} \, dx=-\frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B a}{b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{2}}{6 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} + \frac {5 \, \sqrt {b x^{2} + a x} B a^{3}}{6 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{2 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{4 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} - \frac {5 \, \sqrt {b x^{2} + a x} A a^{2}}{6 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} - \frac {115 \, \sqrt {b x^{2} + a x} B a^{2}}{12 \, {\left (b^{5} x + a b^{4}\right )}} + \frac {35 \, \sqrt {b x^{2} + a x} A a}{6 \, {\left (b^{4} x + a b^{3}\right )}} + \frac {35 \, B a^{2} \log \left (2 \, x + \frac {a}{b} + \frac {2 \, \sqrt {b x^{2} + a x}}{\sqrt {b}}\right )}{8 \, b^{\frac {9}{2}}} - \frac {5 \, A a \log \left (2 \, x + \frac {a}{b} + \frac {2 \, \sqrt {b x^{2} + a x}}{\sqrt {b}}\right )}{2 \, b^{\frac {7}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (135) = 270\).
Time = 15.42 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.98 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {1}{4} \, \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} B {\left | b \right |}}{b^{6}} - \frac {13 \, B a b^{11} {\left | b \right |} - 4 \, A b^{12} {\left | b \right |}}{b^{17}}\right )} - \frac {5 \, {\left (7 \, B a^{2} {\left | b \right |} - 4 \, A a b {\left | b \right |}\right )} \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{8 \, b^{\frac {11}{2}}} - \frac {4 \, {\left (12 \, B a^{3} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} {\left | b \right |} + 18 \, B a^{4} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b {\left | b \right |} - 9 \, A a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b {\left | b \right |} + 10 \, B a^{5} b^{2} {\left | b \right |} - 12 \, A a^{3} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{2} {\left | b \right |} - 7 \, A a^{4} b^{3} {\left | b \right |}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{\frac {9}{2}}} \]
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Timed out. \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\int \frac {x^{5/2}\,\left (A+B\,x\right )}{{\left (a+b\,x\right )}^{5/2}} \,d x \]
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