Integrand size = 20, antiderivative size = 53 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{9/2}} \, dx=-\frac {2 A (a+b x)^{5/2}}{7 a x^{7/2}}+\frac {2 (2 A b-7 a B) (a+b x)^{5/2}}{35 a^2 x^{5/2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {79, 37} \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{9/2}} \, dx=\frac {2 (a+b x)^{5/2} (2 A b-7 a B)}{35 a^2 x^{5/2}}-\frac {2 A (a+b x)^{5/2}}{7 a x^{7/2}} \]
[In]
[Out]
Rule 37
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A (a+b x)^{5/2}}{7 a x^{7/2}}+\frac {\left (2 \left (-A b+\frac {7 a B}{2}\right )\right ) \int \frac {(a+b x)^{3/2}}{x^{7/2}} \, dx}{7 a} \\ & = -\frac {2 A (a+b x)^{5/2}}{7 a x^{7/2}}+\frac {2 (2 A b-7 a B) (a+b x)^{5/2}}{35 a^2 x^{5/2}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.68 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{9/2}} \, dx=-\frac {2 (a+b x)^{5/2} (5 a A-2 A b x+7 a B x)}{35 a^2 x^{7/2}} \]
[In]
[Out]
Time = 0.50 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.58
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-2 A b x +7 B a x +5 A a \right )}{35 x^{\frac {7}{2}} a^{2}}\) | \(31\) |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-2 A \,b^{2} x^{2}+7 B a b \,x^{2}+3 a A b x +7 a^{2} B x +5 a^{2} A \right )}{35 x^{\frac {7}{2}} a^{2}}\) | \(53\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-2 A \,b^{3} x^{3}+7 B a \,b^{2} x^{3}+a A \,b^{2} x^{2}+14 B \,a^{2} b \,x^{2}+8 a^{2} A b x +7 a^{3} B x +5 a^{3} A \right )}{35 x^{\frac {7}{2}} a^{2}}\) | \(76\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{9/2}} \, dx=-\frac {2 \, {\left (5 \, A a^{3} + {\left (7 \, B a b^{2} - 2 \, A b^{3}\right )} x^{3} + {\left (14 \, B a^{2} b + A a b^{2}\right )} x^{2} + {\left (7 \, B a^{3} + 8 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{35 \, a^{2} x^{\frac {7}{2}}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (49) = 98\).
Time = 12.82 (sec) , antiderivative size = 500, normalized size of antiderivative = 9.43 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{9/2}} \, dx=- \frac {30 A a^{6} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {66 A a^{5} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {34 A a^{4} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {6 A a^{3} b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {24 A a^{2} b^{\frac {17}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {16 A a b^{\frac {19}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {2 A b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{5 x^{2}} - \frac {2 A b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{15 a x} + \frac {4 A b^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}}{15 a^{2}} - \frac {2 B a \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{5 x^{2}} - \frac {4 B b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{5 x} - \frac {2 B b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{5 a} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (41) = 82\).
Time = 0.22 (sec) , antiderivative size = 176, normalized size of antiderivative = 3.32 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{9/2}} \, dx=-\frac {2 \, \sqrt {b x^{2} + a x} B b^{2}}{5 \, a x} + \frac {4 \, \sqrt {b x^{2} + a x} A b^{3}}{35 \, a^{2} x} + \frac {\sqrt {b x^{2} + a x} B b}{5 \, x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{2}}{35 \, a x^{2}} + \frac {3 \, \sqrt {b x^{2} + a x} B a}{5 \, x^{3}} + \frac {3 \, \sqrt {b x^{2} + a x} A b}{70 \, x^{3}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{x^{4}} + \frac {3 \, \sqrt {b x^{2} + a x} A a}{14 \, x^{4}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{2 \, x^{5}} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{9/2}} \, dx=-\frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}} b {\left (\frac {{\left (7 \, B a^{2} b^{6} - 2 \, A a b^{7}\right )} {\left (b x + a\right )}}{a^{3}} - \frac {7 \, {\left (B a^{3} b^{6} - A a^{2} b^{7}\right )}}{a^{3}}\right )}}{35 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {7}{2}} {\left | b \right |}} \]
[In]
[Out]
Time = 0.77 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{9/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a}{7}+\frac {x\,\left (14\,B\,a^3+16\,A\,b\,a^2\right )}{35\,a^2}-\frac {x^3\,\left (4\,A\,b^3-14\,B\,a\,b^2\right )}{35\,a^2}+\frac {2\,b\,x^2\,\left (A\,b+14\,B\,a\right )}{35\,a}\right )}{x^{7/2}} \]
[In]
[Out]