Integrand size = 29, antiderivative size = 151 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{9/2}} \, dx=-\frac {2 a^6 A}{7 x^{7/2}}-\frac {2 a^5 (6 A b+a B)}{5 x^{5/2}}-\frac {2 a^4 b (5 A b+2 a B)}{x^{3/2}}-\frac {10 a^3 b^2 (4 A b+3 a B)}{\sqrt {x}}+10 a^2 b^3 (3 A b+4 a B) \sqrt {x}+2 a b^4 (2 A b+5 a B) x^{3/2}+\frac {2}{5} b^5 (A b+6 a B) x^{5/2}+\frac {2}{7} b^6 B x^{7/2} \]
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Time = 0.06 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 77} \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{9/2}} \, dx=-\frac {2 a^6 A}{7 x^{7/2}}-\frac {2 a^5 (a B+6 A b)}{5 x^{5/2}}-\frac {2 a^4 b (2 a B+5 A b)}{x^{3/2}}-\frac {10 a^3 b^2 (3 a B+4 A b)}{\sqrt {x}}+10 a^2 b^3 \sqrt {x} (4 a B+3 A b)+\frac {2}{5} b^5 x^{5/2} (6 a B+A b)+2 a b^4 x^{3/2} (5 a B+2 A b)+\frac {2}{7} b^6 B x^{7/2} \]
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Rule 27
Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^6 (A+B x)}{x^{9/2}} \, dx \\ & = \int \left (\frac {a^6 A}{x^{9/2}}+\frac {a^5 (6 A b+a B)}{x^{7/2}}+\frac {3 a^4 b (5 A b+2 a B)}{x^{5/2}}+\frac {5 a^3 b^2 (4 A b+3 a B)}{x^{3/2}}+\frac {5 a^2 b^3 (3 A b+4 a B)}{\sqrt {x}}+3 a b^4 (2 A b+5 a B) \sqrt {x}+b^5 (A b+6 a B) x^{3/2}+b^6 B x^{5/2}\right ) \, dx \\ & = -\frac {2 a^6 A}{7 x^{7/2}}-\frac {2 a^5 (6 A b+a B)}{5 x^{5/2}}-\frac {2 a^4 b (5 A b+2 a B)}{x^{3/2}}-\frac {10 a^3 b^2 (4 A b+3 a B)}{\sqrt {x}}+10 a^2 b^3 (3 A b+4 a B) \sqrt {x}+2 a b^4 (2 A b+5 a B) x^{3/2}+\frac {2}{5} b^5 (A b+6 a B) x^{5/2}+\frac {2}{7} b^6 B x^{7/2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.81 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{9/2}} \, dx=\frac {2 \left (700 a^3 b^3 x^3 (-A+B x)+175 a^2 b^4 x^4 (3 A+B x)-175 a^4 b^2 x^2 (A+3 B x)+14 a b^5 x^5 (5 A+3 B x)-14 a^5 b x (3 A+5 B x)+b^6 x^6 (7 A+5 B x)-a^6 (5 A+7 B x)\right )}{35 x^{7/2}} \]
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Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {2 b^{6} B \,x^{\frac {7}{2}}}{7}+\frac {2 A \,b^{6} x^{\frac {5}{2}}}{5}+\frac {12 B a \,b^{5} x^{\frac {5}{2}}}{5}+4 A a \,b^{5} x^{\frac {3}{2}}+10 B \,a^{2} b^{4} x^{\frac {3}{2}}+30 A \,a^{2} b^{4} \sqrt {x}+40 B \,a^{3} b^{3} \sqrt {x}-\frac {2 a^{4} b \left (5 A b +2 B a \right )}{x^{\frac {3}{2}}}-\frac {2 a^{5} \left (6 A b +B a \right )}{5 x^{\frac {5}{2}}}-\frac {2 a^{6} A}{7 x^{\frac {7}{2}}}-\frac {10 a^{3} b^{2} \left (4 A b +3 B a \right )}{\sqrt {x}}\) | \(139\) |
default | \(\frac {2 b^{6} B \,x^{\frac {7}{2}}}{7}+\frac {2 A \,b^{6} x^{\frac {5}{2}}}{5}+\frac {12 B a \,b^{5} x^{\frac {5}{2}}}{5}+4 A a \,b^{5} x^{\frac {3}{2}}+10 B \,a^{2} b^{4} x^{\frac {3}{2}}+30 A \,a^{2} b^{4} \sqrt {x}+40 B \,a^{3} b^{3} \sqrt {x}-\frac {2 a^{4} b \left (5 A b +2 B a \right )}{x^{\frac {3}{2}}}-\frac {2 a^{5} \left (6 A b +B a \right )}{5 x^{\frac {5}{2}}}-\frac {2 a^{6} A}{7 x^{\frac {7}{2}}}-\frac {10 a^{3} b^{2} \left (4 A b +3 B a \right )}{\sqrt {x}}\) | \(139\) |
gosper | \(-\frac {2 \left (-5 b^{6} B \,x^{7}-7 A \,b^{6} x^{6}-42 x^{6} B a \,b^{5}-70 a A \,b^{5} x^{5}-175 x^{5} B \,b^{4} a^{2}-525 a^{2} A \,b^{4} x^{4}-700 x^{4} B \,a^{3} b^{3}+700 a^{3} A \,b^{3} x^{3}+525 x^{3} B \,a^{4} b^{2}+175 a^{4} A \,b^{2} x^{2}+70 x^{2} B \,a^{5} b +42 a^{5} A b x +7 x B \,a^{6}+5 A \,a^{6}\right )}{35 x^{\frac {7}{2}}}\) | \(148\) |
trager | \(-\frac {2 \left (-5 b^{6} B \,x^{7}-7 A \,b^{6} x^{6}-42 x^{6} B a \,b^{5}-70 a A \,b^{5} x^{5}-175 x^{5} B \,b^{4} a^{2}-525 a^{2} A \,b^{4} x^{4}-700 x^{4} B \,a^{3} b^{3}+700 a^{3} A \,b^{3} x^{3}+525 x^{3} B \,a^{4} b^{2}+175 a^{4} A \,b^{2} x^{2}+70 x^{2} B \,a^{5} b +42 a^{5} A b x +7 x B \,a^{6}+5 A \,a^{6}\right )}{35 x^{\frac {7}{2}}}\) | \(148\) |
risch | \(-\frac {2 \left (-5 b^{6} B \,x^{7}-7 A \,b^{6} x^{6}-42 x^{6} B a \,b^{5}-70 a A \,b^{5} x^{5}-175 x^{5} B \,b^{4} a^{2}-525 a^{2} A \,b^{4} x^{4}-700 x^{4} B \,a^{3} b^{3}+700 a^{3} A \,b^{3} x^{3}+525 x^{3} B \,a^{4} b^{2}+175 a^{4} A \,b^{2} x^{2}+70 x^{2} B \,a^{5} b +42 a^{5} A b x +7 x B \,a^{6}+5 A \,a^{6}\right )}{35 x^{\frac {7}{2}}}\) | \(148\) |
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Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.97 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{9/2}} \, dx=\frac {2 \, {\left (5 \, B b^{6} x^{7} - 5 \, A a^{6} + 7 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 35 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 175 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 175 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 35 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 7 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{35 \, x^{\frac {7}{2}}} \]
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Time = 0.69 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.34 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{9/2}} \, dx=- \frac {2 A a^{6}}{7 x^{\frac {7}{2}}} - \frac {12 A a^{5} b}{5 x^{\frac {5}{2}}} - \frac {10 A a^{4} b^{2}}{x^{\frac {3}{2}}} - \frac {40 A a^{3} b^{3}}{\sqrt {x}} + 30 A a^{2} b^{4} \sqrt {x} + 4 A a b^{5} x^{\frac {3}{2}} + \frac {2 A b^{6} x^{\frac {5}{2}}}{5} - \frac {2 B a^{6}}{5 x^{\frac {5}{2}}} - \frac {4 B a^{5} b}{x^{\frac {3}{2}}} - \frac {30 B a^{4} b^{2}}{\sqrt {x}} + 40 B a^{3} b^{3} \sqrt {x} + 10 B a^{2} b^{4} x^{\frac {3}{2}} + \frac {12 B a b^{5} x^{\frac {5}{2}}}{5} + \frac {2 B b^{6} x^{\frac {7}{2}}}{7} \]
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Time = 0.20 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.98 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{9/2}} \, dx=\frac {2}{7} \, B b^{6} x^{\frac {7}{2}} + \frac {2}{5} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac {5}{2}} + 2 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac {3}{2}} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} \sqrt {x} - \frac {2 \, {\left (5 \, A a^{6} + 175 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 35 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 7 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{35 \, x^{\frac {7}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.98 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{9/2}} \, dx=\frac {2}{7} \, B b^{6} x^{\frac {7}{2}} + \frac {12}{5} \, B a b^{5} x^{\frac {5}{2}} + \frac {2}{5} \, A b^{6} x^{\frac {5}{2}} + 10 \, B a^{2} b^{4} x^{\frac {3}{2}} + 4 \, A a b^{5} x^{\frac {3}{2}} + 40 \, B a^{3} b^{3} \sqrt {x} + 30 \, A a^{2} b^{4} \sqrt {x} - \frac {2 \, {\left (525 \, B a^{4} b^{2} x^{3} + 700 \, A a^{3} b^{3} x^{3} + 70 \, B a^{5} b x^{2} + 175 \, A a^{4} b^{2} x^{2} + 7 \, B a^{6} x + 42 \, A a^{5} b x + 5 \, A a^{6}\right )}}{35 \, x^{\frac {7}{2}}} \]
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Time = 9.99 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{9/2}} \, dx=x^{5/2}\,\left (\frac {2\,A\,b^6}{5}+\frac {12\,B\,a\,b^5}{5}\right )-\frac {x\,\left (\frac {2\,B\,a^6}{5}+\frac {12\,A\,b\,a^5}{5}\right )+\frac {2\,A\,a^6}{7}+x^2\,\left (4\,B\,a^5\,b+10\,A\,a^4\,b^2\right )+x^3\,\left (30\,B\,a^4\,b^2+40\,A\,a^3\,b^3\right )}{x^{7/2}}+\frac {2\,B\,b^6\,x^{7/2}}{7}+10\,a^2\,b^3\,\sqrt {x}\,\left (3\,A\,b+4\,B\,a\right )+2\,a\,b^4\,x^{3/2}\,\left (2\,A\,b+5\,B\,a\right ) \]
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