Integrand size = 20, antiderivative size = 144 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{5/2}} \, dx=-\frac {2 A}{5 a x^{5/2} (a+b x)^{3/2}}-\frac {2 (8 A b-5 a B)}{15 a^2 x^{3/2} (a+b x)^{3/2}}-\frac {4 (8 A b-5 a B)}{5 a^3 x^{3/2} \sqrt {a+b x}}+\frac {16 (8 A b-5 a B) \sqrt {a+b x}}{15 a^4 x^{3/2}}-\frac {32 b (8 A b-5 a B) \sqrt {a+b x}}{15 a^5 \sqrt {x}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{5/2}} \, dx=-\frac {32 b \sqrt {a+b x} (8 A b-5 a B)}{15 a^5 \sqrt {x}}+\frac {16 \sqrt {a+b x} (8 A b-5 a B)}{15 a^4 x^{3/2}}-\frac {4 (8 A b-5 a B)}{5 a^3 x^{3/2} \sqrt {a+b x}}-\frac {2 (8 A b-5 a B)}{15 a^2 x^{3/2} (a+b x)^{3/2}}-\frac {2 A}{5 a x^{5/2} (a+b x)^{3/2}} \]
[In]
[Out]
Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A}{5 a x^{5/2} (a+b x)^{3/2}}+\frac {\left (2 \left (-4 A b+\frac {5 a B}{2}\right )\right ) \int \frac {1}{x^{5/2} (a+b x)^{5/2}} \, dx}{5 a} \\ & = -\frac {2 A}{5 a x^{5/2} (a+b x)^{3/2}}-\frac {2 (8 A b-5 a B)}{15 a^2 x^{3/2} (a+b x)^{3/2}}-\frac {(2 (8 A b-5 a B)) \int \frac {1}{x^{5/2} (a+b x)^{3/2}} \, dx}{5 a^2} \\ & = -\frac {2 A}{5 a x^{5/2} (a+b x)^{3/2}}-\frac {2 (8 A b-5 a B)}{15 a^2 x^{3/2} (a+b x)^{3/2}}-\frac {4 (8 A b-5 a B)}{5 a^3 x^{3/2} \sqrt {a+b x}}-\frac {(8 (8 A b-5 a B)) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{5 a^3} \\ & = -\frac {2 A}{5 a x^{5/2} (a+b x)^{3/2}}-\frac {2 (8 A b-5 a B)}{15 a^2 x^{3/2} (a+b x)^{3/2}}-\frac {4 (8 A b-5 a B)}{5 a^3 x^{3/2} \sqrt {a+b x}}+\frac {16 (8 A b-5 a B) \sqrt {a+b x}}{15 a^4 x^{3/2}}+\frac {(16 b (8 A b-5 a B)) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{15 a^4} \\ & = -\frac {2 A}{5 a x^{5/2} (a+b x)^{3/2}}-\frac {2 (8 A b-5 a B)}{15 a^2 x^{3/2} (a+b x)^{3/2}}-\frac {4 (8 A b-5 a B)}{5 a^3 x^{3/2} \sqrt {a+b x}}+\frac {16 (8 A b-5 a B) \sqrt {a+b x}}{15 a^4 x^{3/2}}-\frac {32 b (8 A b-5 a B) \sqrt {a+b x}}{15 a^5 \sqrt {x}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.65 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{5/2}} \, dx=-\frac {2 \left (128 A b^4 x^4+24 a^2 b^2 x^2 (2 A-5 B x)+16 a b^3 x^3 (12 A-5 B x)+a^4 (3 A+5 B x)-2 a^3 b x (4 A+15 B x)\right )}{15 a^5 x^{5/2} (a+b x)^{3/2}} \]
[In]
[Out]
Time = 0.53 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.67
method | result | size |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (73 A \,b^{2} x^{2}-40 B a b \,x^{2}-14 a A b x +5 a^{2} B x +3 a^{2} A \right )}{15 a^{5} x^{\frac {5}{2}}}-\frac {2 b^{2} \left (11 A \,b^{2} x -8 B a b x +12 a b A -9 a^{2} B \right ) \sqrt {x}}{3 \left (b x +a \right )^{\frac {3}{2}} a^{5}}\) | \(97\) |
gosper | \(-\frac {2 \left (128 A \,b^{4} x^{4}-80 B a \,b^{3} x^{4}+192 A a \,b^{3} x^{3}-120 B \,a^{2} b^{2} x^{3}+48 A \,a^{2} b^{2} x^{2}-30 B \,a^{3} b \,x^{2}-8 A \,a^{3} b x +5 B \,a^{4} x +3 A \,a^{4}\right )}{15 x^{\frac {5}{2}} \left (b x +a \right )^{\frac {3}{2}} a^{5}}\) | \(101\) |
default | \(-\frac {2 \left (128 A \,b^{4} x^{4}-80 B a \,b^{3} x^{4}+192 A a \,b^{3} x^{3}-120 B \,a^{2} b^{2} x^{3}+48 A \,a^{2} b^{2} x^{2}-30 B \,a^{3} b \,x^{2}-8 A \,a^{3} b x +5 B \,a^{4} x +3 A \,a^{4}\right )}{15 x^{\frac {5}{2}} \left (b x +a \right )^{\frac {3}{2}} a^{5}}\) | \(101\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{5/2}} \, dx=-\frac {2 \, {\left (3 \, A a^{4} - 16 \, {\left (5 \, B a b^{3} - 8 \, A b^{4}\right )} x^{4} - 24 \, {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{3} - 6 \, {\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{2} + {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{15 \, {\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.22 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{5/2}} \, dx=-\frac {4 \, B b x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2}} + \frac {32 \, B b^{2} x}{3 \, \sqrt {b x^{2} + a x} a^{4}} + \frac {32 \, A b^{2} x}{15 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3}} - \frac {256 \, A b^{3} x}{15 \, \sqrt {b x^{2} + a x} a^{5}} - \frac {2 \, B}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a} + \frac {16 \, B b}{3 \, \sqrt {b x^{2} + a x} a^{3}} + \frac {16 \, A b}{15 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2}} - \frac {128 \, A b^{2}}{15 \, \sqrt {b x^{2} + a x} a^{4}} - \frac {2 \, A}{5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (114) = 228\).
Time = 0.42 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.37 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{5/2}} \, dx=\frac {2 \, \sqrt {b x + a} {\left ({\left (b x + a\right )} {\left (\frac {{\left (40 \, B a^{8} b^{7} - 73 \, A a^{7} b^{8}\right )} {\left (b x + a\right )}}{a^{12} b^{2} {\left | b \right |}} - \frac {5 \, {\left (17 \, B a^{9} b^{7} - 32 \, A a^{8} b^{8}\right )}}{a^{12} b^{2} {\left | b \right |}}\right )} + \frac {45 \, {\left (B a^{10} b^{7} - 2 \, A a^{9} b^{8}\right )}}{a^{12} b^{2} {\left | b \right |}}\right )}}{15 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {5}{2}}} + \frac {4 \, {\left (6 \, B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {7}{2}} + 18 \, B a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {9}{2}} - 9 \, A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {9}{2}} + 8 \, B a^{3} b^{\frac {11}{2}} - 24 \, A a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {11}{2}} - 11 \, A a^{2} b^{\frac {13}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{4} {\left | b \right |}} \]
[In]
[Out]
Time = 1.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^{5/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{5\,a\,b^2}+\frac {16\,x^3\,\left (8\,A\,b-5\,B\,a\right )}{5\,a^4}+\frac {4\,x^2\,\left (8\,A\,b-5\,B\,a\right )}{5\,a^3\,b}+\frac {x^4\,\left (256\,A\,b^4-160\,B\,a\,b^3\right )}{15\,a^5\,b^2}+\frac {x\,\left (10\,B\,a^4-16\,A\,a^3\,b\right )}{15\,a^5\,b^2}\right )}{x^{9/2}+\frac {2\,a\,x^{7/2}}{b}+\frac {a^2\,x^{5/2}}{b^2}} \]
[In]
[Out]