Integrand size = 20, antiderivative size = 113 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^{5/2}} \, dx=-\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}}-\frac {2 (2 A b-a B)}{3 a^2 \sqrt {x} (a+b x)^{3/2}}-\frac {8 (2 A b-a B)}{3 a^3 \sqrt {x} \sqrt {a+b x}}+\frac {16 (2 A b-a B) \sqrt {a+b x}}{3 a^4 \sqrt {x}} \]
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Time = 0.03 (sec) , antiderivative size = 113, 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} \[ \int \frac {A+B x}{x^{5/2} (a+b x)^{5/2}} \, dx=\frac {16 \sqrt {a+b x} (2 A b-a B)}{3 a^4 \sqrt {x}}-\frac {8 (2 A b-a B)}{3 a^3 \sqrt {x} \sqrt {a+b x}}-\frac {2 (2 A b-a B)}{3 a^2 \sqrt {x} (a+b x)^{3/2}}-\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}}+\frac {\left (2 \left (-3 A b+\frac {3 a B}{2}\right )\right ) \int \frac {1}{x^{3/2} (a+b x)^{5/2}} \, dx}{3 a} \\ & = -\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}}-\frac {2 (2 A b-a B)}{3 a^2 \sqrt {x} (a+b x)^{3/2}}-\frac {(4 (2 A b-a B)) \int \frac {1}{x^{3/2} (a+b x)^{3/2}} \, dx}{3 a^2} \\ & = -\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}}-\frac {2 (2 A b-a B)}{3 a^2 \sqrt {x} (a+b x)^{3/2}}-\frac {8 (2 A b-a B)}{3 a^3 \sqrt {x} \sqrt {a+b x}}-\frac {(8 (2 A b-a B)) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{3 a^3} \\ & = -\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}}-\frac {2 (2 A b-a B)}{3 a^2 \sqrt {x} (a+b x)^{3/2}}-\frac {8 (2 A b-a B)}{3 a^3 \sqrt {x} \sqrt {a+b x}}+\frac {16 (2 A b-a B) \sqrt {a+b x}}{3 a^4 \sqrt {x}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.62 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^{5/2}} \, dx=-\frac {2 \left (-16 A b^3 x^3-6 a^2 b x (A-2 B x)+8 a b^2 x^2 (-3 A+B x)+a^3 (A+3 B x)\right )}{3 a^4 x^{3/2} (a+b x)^{3/2}} \]
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Time = 1.49 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.64
method | result | size |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-8 A b x +3 B a x +A a \right )}{3 a^{4} x^{\frac {3}{2}}}+\frac {2 b \left (8 A \,b^{2} x -5 B a b x +9 a b A -6 a^{2} B \right ) \sqrt {x}}{3 \left (b x +a \right )^{\frac {3}{2}} a^{4}}\) | \(72\) |
gosper | \(-\frac {2 \left (-16 A \,b^{3} x^{3}+8 B a \,b^{2} x^{3}-24 a A \,b^{2} x^{2}+12 B \,a^{2} b \,x^{2}-6 a^{2} A b x +3 a^{3} B x +a^{3} A \right )}{3 x^{\frac {3}{2}} \left (b x +a \right )^{\frac {3}{2}} a^{4}}\) | \(76\) |
default | \(-\frac {2 \left (-16 A \,b^{3} x^{3}+8 B a \,b^{2} x^{3}-24 a A \,b^{2} x^{2}+12 B \,a^{2} b \,x^{2}-6 a^{2} A b x +3 a^{3} B x +a^{3} A \right )}{3 x^{\frac {3}{2}} \left (b x +a \right )^{\frac {3}{2}} a^{4}}\) | \(76\) |
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Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^{5/2}} \, dx=-\frac {2 \, {\left (A a^{3} + 8 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} x^{3} + 12 \, {\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{2} + 3 \, {\left (B a^{3} - 2 \, A a^{2} b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (110) = 220\).
Time = 105.96 (sec) , antiderivative size = 495, normalized size of antiderivative = 4.38 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^{5/2}} \, dx=A \left (- \frac {2 a^{4} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {10 a^{3} b^{\frac {21}{2}} x \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {60 a^{2} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {80 a b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {32 b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}}\right ) + B \left (- \frac {6 a^{2} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} - \frac {24 a b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} - \frac {16 b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^{5/2}} \, dx=\frac {2 \, B x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a} - \frac {16 \, B b x}{3 \, \sqrt {b x^{2} + a x} a^{3}} - \frac {4 \, A b x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2}} + \frac {32 \, A b^{2} x}{3 \, \sqrt {b x^{2} + a x} a^{4}} - \frac {8 \, B}{3 \, \sqrt {b x^{2} + a x} a^{2}} - \frac {2 \, A}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a} + \frac {16 \, A b}{3 \, \sqrt {b x^{2} + a x} a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (86) = 172\).
Time = 0.39 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.68 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^{5/2}} \, dx=-\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (3 \, B a^{4} b^{3} {\left | b \right |} - 8 \, A a^{3} b^{4} {\left | b \right |}\right )} {\left (b x + a\right )}}{a^{7} b^{2}} - \frac {3 \, {\left (B a^{5} b^{3} {\left | b \right |} - 3 \, A a^{4} b^{4} {\left | b \right |}\right )}}{a^{7} b^{2}}\right )}}{3 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}}} - \frac {4 \, {\left (3 \, B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {5}{2}} + 12 \, B a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {7}{2}} - 6 \, A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {7}{2}} + 5 \, B a^{3} b^{\frac {9}{2}} - 18 \, A a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {9}{2}} - 8 \, A a^{2} b^{\frac {11}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{3} {\left | b \right |}} \]
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Time = 0.92 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^{5/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{3\,a\,b^2}-\frac {8\,x^2\,\left (2\,A\,b-B\,a\right )}{a^3\,b}-\frac {x^3\,\left (32\,A\,b^3-16\,B\,a\,b^2\right )}{3\,a^4\,b^2}+\frac {x\,\left (6\,B\,a^3-12\,A\,a^2\,b\right )}{3\,a^4\,b^2}\right )}{x^{7/2}+\frac {2\,a\,x^{5/2}}{b}+\frac {a^2\,x^{3/2}}{b^2}} \]
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