Integrand size = 20, antiderivative size = 200 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {35 a (2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{8 b^5}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {35 a^2 (2 A b-3 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{11/2}} \]
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Time = 0.06 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {79, 49, 52, 65, 223, 212} \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {35 a^2 (2 A b-3 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{11/2}}-\frac {35 a \sqrt {x} \sqrt {a+b x} (2 A b-3 a B)}{8 b^5}+\frac {35 x^{3/2} \sqrt {a+b x} (2 A b-3 a B)}{12 b^4}-\frac {7 x^{5/2} \sqrt {a+b x} (2 A b-3 a B)}{3 a b^3}+\frac {2 x^{7/2} (2 A b-3 a B)}{a b^2 \sqrt {a+b x}}+\frac {2 x^{9/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]
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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^{9/2}}{3 a b (a+b x)^{3/2}}-\frac {\left (2 \left (3 A b-\frac {9 a B}{2}\right )\right ) \int \frac {x^{7/2}}{(a+b x)^{3/2}} \, dx}{3 a b} \\ & = \frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {(7 (2 A b-3 a B)) \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx}{a b^2} \\ & = \frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {(35 (2 A b-3 a B)) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{6 b^3} \\ & = \frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}-\frac {(35 a (2 A b-3 a B)) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{8 b^4} \\ & = \frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {35 a (2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{8 b^5}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {\left (35 a^2 (2 A b-3 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{16 b^5} \\ & = \frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {35 a (2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{8 b^5}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {\left (35 a^2 (2 A b-3 a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{8 b^5} \\ & = \frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {35 a (2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{8 b^5}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {\left (35 a^2 (2 A b-3 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^5} \\ & = \frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {35 a (2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{8 b^5}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {35 a^2 (2 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{11/2}} \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.72 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {x} \left (315 a^4 B-210 a^3 b (A-2 B x)+4 b^4 x^3 (3 A+2 B x)-6 a b^3 x^2 (7 A+3 B x)+7 a^2 b^2 x (-40 A+9 B x)\right )}{24 b^5 (a+b x)^{3/2}}+\frac {35 a^2 (-2 A b+3 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )}{4 b^{11/2}} \]
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Time = 1.51 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.40
method | result | size |
risch | \(-\frac {\left (-8 b^{2} B \,x^{2}-12 A \,b^{2} x +34 B a b x +66 a b A -123 a^{2} B \right ) \sqrt {x}\, \sqrt {b x +a}}{24 b^{5}}+\frac {a^{2} \left (70 A \sqrt {b}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )-\frac {105 B a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{\sqrt {b}}-\frac {32 \left (4 A b -5 B a \right ) \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{b \left (x +\frac {a}{b}\right )}+\frac {16 a^{2} \left (A b -B a \right ) \left (\frac {2 \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{3 a \left (x +\frac {a}{b}\right )^{2}}+\frac {4 b \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{3 a^{2} \left (x +\frac {a}{b}\right )}\right )}{b^{2}}\right ) \sqrt {x \left (b x +a \right )}}{16 b^{5} \sqrt {x}\, \sqrt {b x +a}}\) | \(281\) |
default | \(\frac {\left (16 B \,b^{\frac {9}{2}} x^{4} \sqrt {x \left (b x +a \right )}+24 A \,b^{\frac {9}{2}} x^{3} \sqrt {x \left (b x +a \right )}-36 B a \,b^{\frac {7}{2}} x^{3} \sqrt {x \left (b x +a \right )}+210 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{2} b^{3} x^{2}-84 A a \,b^{\frac {7}{2}} x^{2} \sqrt {x \left (b x +a \right )}-315 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{3} b^{2} x^{2}+126 B \,a^{2} b^{\frac {5}{2}} x^{2} \sqrt {x \left (b x +a \right )}+420 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{3} b^{2} x -560 A \,b^{\frac {5}{2}} \sqrt {x \left (b x +a \right )}\, a^{2} x -630 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{4} b x +840 B \,b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a^{3} x +210 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{4} b -420 A \,b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a^{3}-315 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{5}+630 B \sqrt {b}\, \sqrt {x \left (b x +a \right )}\, a^{4}\right ) \sqrt {x}}{48 b^{\frac {11}{2}} \sqrt {x \left (b x +a \right )}\, \left (b x +a \right )^{\frac {3}{2}}}\) | \(406\) |
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Time = 0.24 (sec) , antiderivative size = 424, normalized size of antiderivative = 2.12 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\left [-\frac {105 \, {\left (3 \, B a^{5} - 2 \, A a^{4} b + {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{2} + 2 \, {\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (8 \, B b^{5} x^{4} + 315 \, B a^{4} b - 210 \, A a^{3} b^{2} - 6 \, {\left (3 \, B a b^{4} - 2 \, A b^{5}\right )} x^{3} + 21 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{2} + 140 \, {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{48 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}}, \frac {105 \, {\left (3 \, B a^{5} - 2 \, A a^{4} b + {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{2} + 2 \, {\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (8 \, B b^{5} x^{4} + 315 \, B a^{4} b - 210 \, A a^{3} b^{2} - 6 \, {\left (3 \, B a b^{4} - 2 \, A b^{5}\right )} x^{3} + 21 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{2} + 140 \, {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{24 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}}\right ] \]
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Timed out. \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (162) = 324\).
Time = 0.21 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.08 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {B x^{6}}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} - \frac {3 \, B a x^{5}}{4 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} + \frac {A x^{5}}{2 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {35 \, B a^{3} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {a x}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, x}{\sqrt {b x^{2} + a x} a b} - \frac {1}{\sqrt {b x^{2} + a x} b^{2}}\right )}}{16 \, b^{3}} - \frac {35 \, A a^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {a x}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, x}{\sqrt {b x^{2} + a x} a b} - \frac {1}{\sqrt {b x^{2} + a x} b^{2}}\right )}}{24 \, b^{2}} + \frac {21 \, B a^{2} x^{4}}{8 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{3}} - \frac {7 \, A a x^{4}}{4 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} + \frac {35 \, B a^{3} x}{4 \, \sqrt {b x^{2} + a x} b^{5}} - \frac {35 \, A a^{2} x}{6 \, \sqrt {b x^{2} + a x} b^{4}} - \frac {105 \, B a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {11}{2}}} + \frac {35 \, A a^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, b^{\frac {9}{2}}} + \frac {35 \, \sqrt {b x^{2} + a x} B a^{2}}{8 \, b^{5}} - \frac {35 \, \sqrt {b x^{2} + a x} A a}{12 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (162) = 324\).
Time = 15.52 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.85 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {1}{24} \, \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} B {\left | b \right |}}{b^{7}} - \frac {25 \, B a b^{20} {\left | b \right |} - 6 \, A b^{21} {\left | b \right |}}{b^{27}}\right )} + \frac {3 \, {\left (55 \, B a^{2} b^{20} {\left | b \right |} - 26 \, A a b^{21} {\left | b \right |}\right )}}{b^{27}}\right )} + \frac {35 \, {\left (3 \, B a^{3} {\left | b \right |} - 2 \, A a^{2} b {\left | b \right |}\right )} \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{16 \, b^{\frac {13}{2}}} + \frac {4 \, {\left (15 \, B a^{4} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} {\left | b \right |} + 24 \, B a^{5} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b {\left | b \right |} - 12 \, A a^{3} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b {\left | b \right |} + 13 \, B a^{6} b^{2} {\left | b \right |} - 18 \, A a^{4} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{2} {\left | b \right |} - 10 \, A a^{5} b^{3} {\left | b \right |}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{\frac {11}{2}}} \]
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Timed out. \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\int \frac {x^{7/2}\,\left (A+B\,x\right )}{{\left (a+b\,x\right )}^{5/2}} \,d x \]
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