Integrand size = 20, antiderivative size = 150 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx=-\frac {2 A (a+b x)^{5/2}}{13 a x^{13/2}}+\frac {2 (8 A b-13 a B) (a+b x)^{5/2}}{143 a^2 x^{11/2}}-\frac {4 b (8 A b-13 a B) (a+b x)^{5/2}}{429 a^3 x^{9/2}}+\frac {16 b^2 (8 A b-13 a B) (a+b x)^{5/2}}{3003 a^4 x^{7/2}}-\frac {32 b^3 (8 A b-13 a B) (a+b x)^{5/2}}{15015 a^5 x^{5/2}} \]
[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} \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx=-\frac {32 b^3 (a+b x)^{5/2} (8 A b-13 a B)}{15015 a^5 x^{5/2}}+\frac {16 b^2 (a+b x)^{5/2} (8 A b-13 a B)}{3003 a^4 x^{7/2}}-\frac {4 b (a+b x)^{5/2} (8 A b-13 a B)}{429 a^3 x^{9/2}}+\frac {2 (a+b x)^{5/2} (8 A b-13 a B)}{143 a^2 x^{11/2}}-\frac {2 A (a+b x)^{5/2}}{13 a x^{13/2}} \]
[In]
[Out]
Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A (a+b x)^{5/2}}{13 a x^{13/2}}+\frac {\left (2 \left (-4 A b+\frac {13 a B}{2}\right )\right ) \int \frac {(a+b x)^{3/2}}{x^{13/2}} \, dx}{13 a} \\ & = -\frac {2 A (a+b x)^{5/2}}{13 a x^{13/2}}+\frac {2 (8 A b-13 a B) (a+b x)^{5/2}}{143 a^2 x^{11/2}}+\frac {(6 b (8 A b-13 a B)) \int \frac {(a+b x)^{3/2}}{x^{11/2}} \, dx}{143 a^2} \\ & = -\frac {2 A (a+b x)^{5/2}}{13 a x^{13/2}}+\frac {2 (8 A b-13 a B) (a+b x)^{5/2}}{143 a^2 x^{11/2}}-\frac {4 b (8 A b-13 a B) (a+b x)^{5/2}}{429 a^3 x^{9/2}}-\frac {\left (8 b^2 (8 A b-13 a B)\right ) \int \frac {(a+b x)^{3/2}}{x^{9/2}} \, dx}{429 a^3} \\ & = -\frac {2 A (a+b x)^{5/2}}{13 a x^{13/2}}+\frac {2 (8 A b-13 a B) (a+b x)^{5/2}}{143 a^2 x^{11/2}}-\frac {4 b (8 A b-13 a B) (a+b x)^{5/2}}{429 a^3 x^{9/2}}+\frac {16 b^2 (8 A b-13 a B) (a+b x)^{5/2}}{3003 a^4 x^{7/2}}+\frac {\left (16 b^3 (8 A b-13 a B)\right ) \int \frac {(a+b x)^{3/2}}{x^{7/2}} \, dx}{3003 a^4} \\ & = -\frac {2 A (a+b x)^{5/2}}{13 a x^{13/2}}+\frac {2 (8 A b-13 a B) (a+b x)^{5/2}}{143 a^2 x^{11/2}}-\frac {4 b (8 A b-13 a B) (a+b x)^{5/2}}{429 a^3 x^{9/2}}+\frac {16 b^2 (8 A b-13 a B) (a+b x)^{5/2}}{3003 a^4 x^{7/2}}-\frac {32 b^3 (8 A b-13 a B) (a+b x)^{5/2}}{15015 a^5 x^{5/2}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.63 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx=-\frac {2 (a+b x)^{5/2} \left (128 A b^4 x^4+105 a^4 (11 A+13 B x)-70 a^3 b x (12 A+13 B x)+40 a^2 b^2 x^2 (14 A+13 B x)-16 a b^3 x^3 (20 A+13 B x)\right )}{15015 a^5 x^{13/2}} \]
[In]
[Out]
Time = 1.44 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.67
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (128 A \,b^{4} x^{4}-208 B a \,b^{3} x^{4}-320 A a \,b^{3} x^{3}+520 B \,a^{2} b^{2} x^{3}+560 A \,a^{2} b^{2} x^{2}-910 B \,a^{3} b \,x^{2}-840 A \,a^{3} b x +1365 B \,a^{4} x +1155 A \,a^{4}\right )}{15015 x^{\frac {13}{2}} a^{5}}\) | \(101\) |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (128 A \,b^{5} x^{5}-208 B a \,b^{4} x^{5}-192 a A \,b^{4} x^{4}+312 B \,a^{2} b^{3} x^{4}+240 a^{2} A \,b^{3} x^{3}-390 B \,a^{3} b^{2} x^{3}-280 a^{3} A \,b^{2} x^{2}+455 B \,a^{4} b \,x^{2}+315 a^{4} A b x +1365 a^{5} B x +1155 a^{5} A \right )}{15015 x^{\frac {13}{2}} a^{5}}\) | \(125\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (128 A \,b^{6} x^{6}-208 B a \,b^{5} x^{6}-64 A a \,b^{5} x^{5}+104 B \,a^{2} b^{4} x^{5}+48 A \,a^{2} b^{4} x^{4}-78 B \,a^{3} b^{3} x^{4}-40 A \,a^{3} b^{3} x^{3}+65 B \,a^{4} b^{2} x^{3}+35 A \,a^{4} b^{2} x^{2}+1820 B \,a^{5} b \,x^{2}+1470 A \,a^{5} b x +1365 B \,a^{6} x +1155 A \,a^{6}\right )}{15015 x^{\frac {13}{2}} a^{5}}\) | \(149\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx=-\frac {2 \, {\left (1155 \, A a^{6} - 16 \, {\left (13 \, B a b^{5} - 8 \, A b^{6}\right )} x^{6} + 8 \, {\left (13 \, B a^{2} b^{4} - 8 \, A a b^{5}\right )} x^{5} - 6 \, {\left (13 \, B a^{3} b^{3} - 8 \, A a^{2} b^{4}\right )} x^{4} + 5 \, {\left (13 \, B a^{4} b^{2} - 8 \, A a^{3} b^{3}\right )} x^{3} + 35 \, {\left (52 \, B a^{5} b + A a^{4} b^{2}\right )} x^{2} + 105 \, {\left (13 \, B a^{6} + 14 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{15015 \, a^{5} x^{\frac {13}{2}}} \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (120) = 240\).
Time = 0.21 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.09 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx=\frac {32 \, \sqrt {b x^{2} + a x} B b^{5}}{1155 \, a^{4} x} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{6}}{15015 \, a^{5} x} - \frac {16 \, \sqrt {b x^{2} + a x} B b^{4}}{1155 \, a^{3} x^{2}} + \frac {128 \, \sqrt {b x^{2} + a x} A b^{5}}{15015 \, a^{4} x^{2}} + \frac {4 \, \sqrt {b x^{2} + a x} B b^{3}}{385 \, a^{2} x^{3}} - \frac {32 \, \sqrt {b x^{2} + a x} A b^{4}}{5005 \, a^{3} x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} B b^{2}}{231 \, a x^{4}} + \frac {16 \, \sqrt {b x^{2} + a x} A b^{3}}{3003 \, a^{2} x^{4}} + \frac {\sqrt {b x^{2} + a x} B b}{132 \, x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{2}}{429 \, a x^{5}} + \frac {3 \, \sqrt {b x^{2} + a x} B a}{44 \, x^{6}} + \frac {3 \, \sqrt {b x^{2} + a x} A b}{715 \, x^{6}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{4 \, x^{7}} + \frac {3 \, \sqrt {b x^{2} + a x} A a}{65 \, x^{7}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{5 \, x^{8}} \]
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx=\frac {2 \, {\left ({\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (13 \, B a^{2} b^{12} - 8 \, A a b^{13}\right )} {\left (b x + a\right )}}{a^{6}} - \frac {13 \, {\left (13 \, B a^{3} b^{12} - 8 \, A a^{2} b^{13}\right )}}{a^{6}}\right )} + \frac {143 \, {\left (13 \, B a^{4} b^{12} - 8 \, A a^{3} b^{13}\right )}}{a^{6}}\right )} - \frac {429 \, {\left (13 \, B a^{5} b^{12} - 8 \, A a^{4} b^{13}\right )}}{a^{6}}\right )} {\left (b x + a\right )} + \frac {3003 \, {\left (B a^{6} b^{12} - A a^{5} b^{13}\right )}}{a^{6}}\right )} {\left (b x + a\right )}^{\frac {5}{2}} b}{15015 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {13}{2}} {\left | b \right |}} \]
[In]
[Out]
Time = 0.90 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a}{13}+x\,\left (\frac {28\,A\,b}{143}+\frac {2\,B\,a}{11}\right )+\frac {x^6\,\left (256\,A\,b^6-416\,B\,a\,b^5\right )}{15015\,a^5}-\frac {2\,b^2\,x^3\,\left (8\,A\,b-13\,B\,a\right )}{3003\,a^2}+\frac {4\,b^3\,x^4\,\left (8\,A\,b-13\,B\,a\right )}{5005\,a^3}-\frac {16\,b^4\,x^5\,\left (8\,A\,b-13\,B\,a\right )}{15015\,a^4}+\frac {2\,b\,x^2\,\left (A\,b+52\,B\,a\right )}{429\,a}\right )}{x^{13/2}} \]
[In]
[Out]