Integrand size = 20, antiderivative size = 114 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{3/2}} \, dx=-\frac {2 A}{5 a x^{5/2} \sqrt {a+b x}}-\frac {2 (6 A b-5 a B)}{5 a^2 x^{3/2} \sqrt {a+b x}}+\frac {8 (6 A b-5 a B) \sqrt {a+b x}}{15 a^3 x^{3/2}}-\frac {16 b (6 A b-5 a B) \sqrt {a+b x}}{15 a^4 \sqrt {x}} \]
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Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{3/2}} \, dx=-\frac {16 b \sqrt {a+b x} (6 A b-5 a B)}{15 a^4 \sqrt {x}}+\frac {8 \sqrt {a+b x} (6 A b-5 a B)}{15 a^3 x^{3/2}}-\frac {2 (6 A b-5 a B)}{5 a^2 x^{3/2} \sqrt {a+b x}}-\frac {2 A}{5 a x^{5/2} \sqrt {a+b x}} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A}{5 a x^{5/2} \sqrt {a+b x}}+\frac {\left (2 \left (-3 A b+\frac {5 a B}{2}\right )\right ) \int \frac {1}{x^{5/2} (a+b x)^{3/2}} \, dx}{5 a} \\ & = -\frac {2 A}{5 a x^{5/2} \sqrt {a+b x}}-\frac {2 (6 A b-5 a B)}{5 a^2 x^{3/2} \sqrt {a+b x}}-\frac {(4 (6 A b-5 a B)) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{5 a^2} \\ & = -\frac {2 A}{5 a x^{5/2} \sqrt {a+b x}}-\frac {2 (6 A b-5 a B)}{5 a^2 x^{3/2} \sqrt {a+b x}}+\frac {8 (6 A b-5 a B) \sqrt {a+b x}}{15 a^3 x^{3/2}}+\frac {(8 b (6 A b-5 a B)) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{15 a^3} \\ & = -\frac {2 A}{5 a x^{5/2} \sqrt {a+b x}}-\frac {2 (6 A b-5 a B)}{5 a^2 x^{3/2} \sqrt {a+b x}}+\frac {8 (6 A b-5 a B) \sqrt {a+b x}}{15 a^3 x^{3/2}}-\frac {16 b (6 A b-5 a B) \sqrt {a+b x}}{15 a^4 \sqrt {x}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.66 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{3/2}} \, dx=-\frac {2 \left (48 A b^3 x^3+8 a b^2 x^2 (3 A-5 B x)+a^3 (3 A+5 B x)-2 a^2 b x (3 A+10 B x)\right )}{15 a^4 x^{5/2} \sqrt {a+b x}} \]
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Time = 0.52 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.68
method | result | size |
gosper | \(-\frac {2 \left (48 A \,b^{3} x^{3}-40 B a \,b^{2} x^{3}+24 a A \,b^{2} x^{2}-20 B \,a^{2} b \,x^{2}-6 a^{2} A b x +5 a^{3} B x +3 a^{3} A \right )}{15 x^{\frac {5}{2}} \sqrt {b x +a}\, a^{4}}\) | \(77\) |
default | \(-\frac {2 \left (48 A \,b^{3} x^{3}-40 B a \,b^{2} x^{3}+24 a A \,b^{2} x^{2}-20 B \,a^{2} b \,x^{2}-6 a^{2} A b x +5 a^{3} B x +3 a^{3} A \right )}{15 x^{\frac {5}{2}} \sqrt {b x +a}\, a^{4}}\) | \(77\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (33 A \,b^{2} x^{2}-25 B a b \,x^{2}-9 a A b x +5 a^{2} B x +3 a^{2} A \right )}{15 a^{4} x^{\frac {5}{2}}}-\frac {2 b^{2} \left (A b -B a \right ) \sqrt {x}}{a^{4} \sqrt {b x +a}}\) | \(80\) |
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Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{3/2}} \, dx=-\frac {2 \, {\left (3 \, A a^{3} - 8 \, {\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} x^{3} - 4 \, {\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{2} + {\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{15 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (110) = 220\).
Time = 30.11 (sec) , antiderivative size = 573, normalized size of antiderivative = 5.03 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{3/2}} \, dx=A \left (- \frac {2 a^{5} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {10 a^{3} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {60 a^{2} b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {80 a b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {32 b^{\frac {29}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}}\right ) + B \left (- \frac {2 a^{3} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac {6 a^{2} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac {24 a b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac {16 b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{3/2}} \, dx=\frac {16 \, B b^{2} x}{3 \, \sqrt {b x^{2} + a x} a^{3}} - \frac {32 \, A b^{3} x}{5 \, \sqrt {b x^{2} + a x} a^{4}} + \frac {8 \, B b}{3 \, \sqrt {b x^{2} + a x} a^{2}} - \frac {16 \, A b^{2}}{5 \, \sqrt {b x^{2} + a x} a^{3}} - \frac {2 \, B}{3 \, \sqrt {b x^{2} + a x} a x} + \frac {4 \, A b}{5 \, \sqrt {b x^{2} + a x} a^{2} x} - \frac {2 \, A}{5 \, \sqrt {b x^{2} + a x} a x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (90) = 180\).
Time = 0.34 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.23 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{3/2}} \, dx=\frac {2 \, \sqrt {b x + a} {\left ({\left (b x + a\right )} {\left (\frac {{\left (25 \, B a^{6} b^{7} - 33 \, A a^{5} b^{8}\right )} {\left (b x + a\right )}}{a^{9} b^{2} {\left | b \right |}} - \frac {5 \, {\left (11 \, B a^{7} b^{7} - 15 \, A a^{6} b^{8}\right )}}{a^{9} b^{2} {\left | b \right |}}\right )} + \frac {15 \, {\left (2 \, B a^{8} b^{7} - 3 \, A a^{7} b^{8}\right )}}{a^{9} b^{2} {\left | b \right |}}\right )}}{15 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {5}{2}}} + \frac {4 \, {\left (B^{2} a^{2} b^{7} - 2 \, A B a b^{8} + A^{2} b^{9}\right )}}{{\left (B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {7}{2}} + B a^{2} b^{\frac {9}{2}} - A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {9}{2}} - A a b^{\frac {11}{2}}\right )} a^{3} {\left | b \right |}} \]
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Time = 0.97 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{3/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{5\,a\,b}+\frac {8\,x^2\,\left (6\,A\,b-5\,B\,a\right )}{15\,a^3}+\frac {x^3\,\left (96\,A\,b^3-80\,B\,a\,b^2\right )}{15\,a^4\,b}+\frac {x\,\left (10\,B\,a^3-12\,A\,a^2\,b\right )}{15\,a^4\,b}\right )}{x^{7/2}+\frac {a\,x^{5/2}}{b}} \]
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