Integrand size = 20, antiderivative size = 150 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx=-\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac {2 (8 A b-15 a B) (a+b x)^{7/2}}{195 a^2 x^{13/2}}-\frac {4 b (8 A b-15 a B) (a+b x)^{7/2}}{715 a^3 x^{11/2}}+\frac {16 b^2 (8 A b-15 a B) (a+b x)^{7/2}}{6435 a^4 x^{9/2}}-\frac {32 b^3 (8 A b-15 a B) (a+b x)^{7/2}}{45045 a^5 x^{7/2}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 150, 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} \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx=-\frac {32 b^3 (a+b x)^{7/2} (8 A b-15 a B)}{45045 a^5 x^{7/2}}+\frac {16 b^2 (a+b x)^{7/2} (8 A b-15 a B)}{6435 a^4 x^{9/2}}-\frac {4 b (a+b x)^{7/2} (8 A b-15 a B)}{715 a^3 x^{11/2}}+\frac {2 (a+b x)^{7/2} (8 A b-15 a B)}{195 a^2 x^{13/2}}-\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}} \]
[In]
[Out]
Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac {\left (2 \left (-4 A b+\frac {15 a B}{2}\right )\right ) \int \frac {(a+b x)^{5/2}}{x^{15/2}} \, dx}{15 a} \\ & = -\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac {2 (8 A b-15 a B) (a+b x)^{7/2}}{195 a^2 x^{13/2}}+\frac {(2 b (8 A b-15 a B)) \int \frac {(a+b x)^{5/2}}{x^{13/2}} \, dx}{65 a^2} \\ & = -\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac {2 (8 A b-15 a B) (a+b x)^{7/2}}{195 a^2 x^{13/2}}-\frac {4 b (8 A b-15 a B) (a+b x)^{7/2}}{715 a^3 x^{11/2}}-\frac {\left (8 b^2 (8 A b-15 a B)\right ) \int \frac {(a+b x)^{5/2}}{x^{11/2}} \, dx}{715 a^3} \\ & = -\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac {2 (8 A b-15 a B) (a+b x)^{7/2}}{195 a^2 x^{13/2}}-\frac {4 b (8 A b-15 a B) (a+b x)^{7/2}}{715 a^3 x^{11/2}}+\frac {16 b^2 (8 A b-15 a B) (a+b x)^{7/2}}{6435 a^4 x^{9/2}}+\frac {\left (16 b^3 (8 A b-15 a B)\right ) \int \frac {(a+b x)^{5/2}}{x^{9/2}} \, dx}{6435 a^4} \\ & = -\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac {2 (8 A b-15 a B) (a+b x)^{7/2}}{195 a^2 x^{13/2}}-\frac {4 b (8 A b-15 a B) (a+b x)^{7/2}}{715 a^3 x^{11/2}}+\frac {16 b^2 (8 A b-15 a B) (a+b x)^{7/2}}{6435 a^4 x^{9/2}}-\frac {32 b^3 (8 A b-15 a B) (a+b x)^{7/2}}{45045 a^5 x^{7/2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.63 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx=-\frac {2 (a+b x)^{7/2} \left (128 A b^4 x^4+168 a^2 b^2 x^2 (6 A+5 B x)+231 a^4 (13 A+15 B x)-16 a b^3 x^3 (28 A+15 B x)-42 a^3 b x (44 A+45 B x)\right )}{45045 a^5 x^{15/2}} \]
[In]
[Out]
Time = 1.46 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.67
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (128 A \,b^{4} x^{4}-240 B a \,b^{3} x^{4}-448 A a \,b^{3} x^{3}+840 B \,a^{2} b^{2} x^{3}+1008 A \,a^{2} b^{2} x^{2}-1890 B \,a^{3} b \,x^{2}-1848 A \,a^{3} b x +3465 B \,a^{4} x +3003 A \,a^{4}\right )}{45045 x^{\frac {15}{2}} a^{5}}\) | \(101\) |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (128 A \,b^{6} x^{6}-240 B a \,b^{5} x^{6}-192 A a \,b^{5} x^{5}+360 B \,a^{2} b^{4} x^{5}+240 A \,a^{2} b^{4} x^{4}-450 B \,a^{3} b^{3} x^{4}-280 A \,a^{3} b^{3} x^{3}+525 B \,a^{4} b^{2} x^{3}+315 A \,a^{4} b^{2} x^{2}+5040 B \,a^{5} b \,x^{2}+4158 A \,a^{5} b x +3465 B \,a^{6} x +3003 A \,a^{6}\right )}{45045 x^{\frac {15}{2}} a^{5}}\) | \(149\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (128 A \,b^{7} x^{7}-240 B a \,b^{6} x^{7}-64 A a \,b^{6} x^{6}+120 B \,a^{2} b^{5} x^{6}+48 A \,a^{2} b^{5} x^{5}-90 B \,a^{3} b^{4} x^{5}-40 A \,a^{3} b^{4} x^{4}+75 B \,a^{4} b^{3} x^{4}+35 A \,a^{4} b^{3} x^{3}+5565 B \,a^{5} b^{2} x^{3}+4473 A \,a^{5} b^{2} x^{2}+8505 B \,a^{6} b \,x^{2}+7161 A \,a^{6} b x +3465 B \,a^{7} x +3003 A \,a^{7}\right )}{45045 x^{\frac {15}{2}} a^{5}}\) | \(173\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx=-\frac {2 \, {\left (3003 \, A a^{7} - 16 \, {\left (15 \, B a b^{6} - 8 \, A b^{7}\right )} x^{7} + 8 \, {\left (15 \, B a^{2} b^{5} - 8 \, A a b^{6}\right )} x^{6} - 6 \, {\left (15 \, B a^{3} b^{4} - 8 \, A a^{2} b^{5}\right )} x^{5} + 5 \, {\left (15 \, B a^{4} b^{3} - 8 \, A a^{3} b^{4}\right )} x^{4} + 35 \, {\left (159 \, B a^{5} b^{2} + A a^{4} b^{3}\right )} x^{3} + 63 \, {\left (135 \, B a^{6} b + 71 \, A a^{5} b^{2}\right )} x^{2} + 231 \, {\left (15 \, B a^{7} + 31 \, A a^{6} b\right )} x\right )} \sqrt {b x + a}}{45045 \, a^{5} x^{\frac {15}{2}}} \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (120) = 240\).
Time = 0.20 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.64 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx=\frac {32 \, \sqrt {b x^{2} + a x} B b^{6}}{3003 \, a^{4} x} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{7}}{45045 \, a^{5} x} - \frac {16 \, \sqrt {b x^{2} + a x} B b^{5}}{3003 \, a^{3} x^{2}} + \frac {128 \, \sqrt {b x^{2} + a x} A b^{6}}{45045 \, a^{4} x^{2}} + \frac {4 \, \sqrt {b x^{2} + a x} B b^{4}}{1001 \, a^{2} x^{3}} - \frac {32 \, \sqrt {b x^{2} + a x} A b^{5}}{15015 \, a^{3} x^{3}} - \frac {10 \, \sqrt {b x^{2} + a x} B b^{3}}{3003 \, a x^{4}} + \frac {16 \, \sqrt {b x^{2} + a x} A b^{4}}{9009 \, a^{2} x^{4}} + \frac {5 \, \sqrt {b x^{2} + a x} B b^{2}}{1716 \, x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{3}}{1287 \, a x^{5}} - \frac {3 \, \sqrt {b x^{2} + a x} B a b}{1144 \, x^{6}} + \frac {\sqrt {b x^{2} + a x} A b^{2}}{715 \, x^{6}} - \frac {3 \, \sqrt {b x^{2} + a x} B a^{2}}{104 \, x^{7}} - \frac {\sqrt {b x^{2} + a x} A a b}{780 \, x^{7}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{8 \, x^{8}} - \frac {\sqrt {b x^{2} + a x} A a^{2}}{60 \, x^{8}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{4 \, x^{9}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{12 \, x^{9}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{5 \, x^{10}} \]
[In]
[Out]
none
Time = 0.37 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx=\frac {2 \, {\left ({\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (15 \, B a^{3} b^{14} - 8 \, A a^{2} b^{15}\right )} {\left (b x + a\right )}}{a^{7}} - \frac {15 \, {\left (15 \, B a^{4} b^{14} - 8 \, A a^{3} b^{15}\right )}}{a^{7}}\right )} + \frac {195 \, {\left (15 \, B a^{5} b^{14} - 8 \, A a^{4} b^{15}\right )}}{a^{7}}\right )} - \frac {715 \, {\left (15 \, B a^{6} b^{14} - 8 \, A a^{5} b^{15}\right )}}{a^{7}}\right )} {\left (b x + a\right )} + \frac {6435 \, {\left (B a^{7} b^{14} - A a^{6} b^{15}\right )}}{a^{7}}\right )} {\left (b x + a\right )}^{\frac {7}{2}} b}{45045 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {15}{2}} {\left | b \right |}} \]
[In]
[Out]
Time = 0.91 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a^2}{15}+\frac {x^7\,\left (256\,A\,b^7-480\,B\,a\,b^6\right )}{45045\,a^5}+\frac {2\,a\,x\,\left (31\,A\,b+15\,B\,a\right )}{195}+\frac {2\,b\,x^2\,\left (71\,A\,b+135\,B\,a\right )}{715}-\frac {2\,b^3\,x^4\,\left (8\,A\,b-15\,B\,a\right )}{9009\,a^2}+\frac {4\,b^4\,x^5\,\left (8\,A\,b-15\,B\,a\right )}{15015\,a^3}-\frac {16\,b^5\,x^6\,\left (8\,A\,b-15\,B\,a\right )}{45045\,a^4}+\frac {2\,b^2\,x^3\,\left (A\,b+159\,B\,a\right )}{1287\,a}\right )}{x^{15/2}} \]
[In]
[Out]