Integrand size = 29, antiderivative size = 107 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=-\frac {2 a^4 A}{5 x^{5/2}}-\frac {2 a^3 (4 A b+a B)}{3 x^{3/2}}-\frac {4 a^2 b (3 A b+2 a B)}{\sqrt {x}}+4 a b^2 (2 A b+3 a B) \sqrt {x}+\frac {2}{3} b^3 (A b+4 a B) x^{3/2}+\frac {2}{5} b^4 B x^{5/2} \]
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Time = 0.04 (sec) , antiderivative size = 107, 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} \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=-\frac {2 a^4 A}{5 x^{5/2}}-\frac {2 a^3 (a B+4 A b)}{3 x^{3/2}}-\frac {4 a^2 b (2 a B+3 A b)}{\sqrt {x}}+\frac {2}{3} b^3 x^{3/2} (4 a B+A b)+4 a b^2 \sqrt {x} (3 a B+2 A b)+\frac {2}{5} b^4 B x^{5/2} \]
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Rule 27
Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^4 (A+B x)}{x^{7/2}} \, dx \\ & = \int \left (\frac {a^4 A}{x^{7/2}}+\frac {a^3 (4 A b+a B)}{x^{5/2}}+\frac {2 a^2 b (3 A b+2 a B)}{x^{3/2}}+\frac {2 a b^2 (2 A b+3 a B)}{\sqrt {x}}+b^3 (A b+4 a B) \sqrt {x}+b^4 B x^{3/2}\right ) \, dx \\ & = -\frac {2 a^4 A}{5 x^{5/2}}-\frac {2 a^3 (4 A b+a B)}{3 x^{3/2}}-\frac {4 a^2 b (3 A b+2 a B)}{\sqrt {x}}+4 a b^2 (2 A b+3 a B) \sqrt {x}+\frac {2}{3} b^3 (A b+4 a B) x^{3/2}+\frac {2}{5} b^4 B x^{5/2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=\frac {2 \left (90 a^2 b^2 x^2 (-A+B x)+20 a b^3 x^3 (3 A+B x)-20 a^3 b x (A+3 B x)+b^4 x^4 (5 A+3 B x)-a^4 (3 A+5 B x)\right )}{15 x^{5/2}} \]
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Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {2 b^{4} B \,x^{\frac {5}{2}}}{5}+\frac {2 A \,b^{4} x^{\frac {3}{2}}}{3}+\frac {8 B a \,b^{3} x^{\frac {3}{2}}}{3}+8 A a \,b^{3} \sqrt {x}+12 B \,a^{2} b^{2} \sqrt {x}-\frac {2 a^{4} A}{5 x^{\frac {5}{2}}}-\frac {2 a^{3} \left (4 A b +B a \right )}{3 x^{\frac {3}{2}}}-\frac {4 a^{2} b \left (3 A b +2 B a \right )}{\sqrt {x}}\) | \(95\) |
default | \(\frac {2 b^{4} B \,x^{\frac {5}{2}}}{5}+\frac {2 A \,b^{4} x^{\frac {3}{2}}}{3}+\frac {8 B a \,b^{3} x^{\frac {3}{2}}}{3}+8 A a \,b^{3} \sqrt {x}+12 B \,a^{2} b^{2} \sqrt {x}-\frac {2 a^{4} A}{5 x^{\frac {5}{2}}}-\frac {2 a^{3} \left (4 A b +B a \right )}{3 x^{\frac {3}{2}}}-\frac {4 a^{2} b \left (3 A b +2 B a \right )}{\sqrt {x}}\) | \(95\) |
gosper | \(-\frac {2 \left (-3 b^{4} B \,x^{5}-5 A \,b^{4} x^{4}-20 x^{4} B \,b^{3} a -60 a A \,b^{3} x^{3}-90 x^{3} B \,a^{2} b^{2}+90 a^{2} A \,b^{2} x^{2}+60 x^{2} B \,a^{3} b +20 a^{3} A b x +5 a^{4} B x +3 A \,a^{4}\right )}{15 x^{\frac {5}{2}}}\) | \(100\) |
trager | \(-\frac {2 \left (-3 b^{4} B \,x^{5}-5 A \,b^{4} x^{4}-20 x^{4} B \,b^{3} a -60 a A \,b^{3} x^{3}-90 x^{3} B \,a^{2} b^{2}+90 a^{2} A \,b^{2} x^{2}+60 x^{2} B \,a^{3} b +20 a^{3} A b x +5 a^{4} B x +3 A \,a^{4}\right )}{15 x^{\frac {5}{2}}}\) | \(100\) |
risch | \(-\frac {2 \left (-3 b^{4} B \,x^{5}-5 A \,b^{4} x^{4}-20 x^{4} B \,b^{3} a -60 a A \,b^{3} x^{3}-90 x^{3} B \,a^{2} b^{2}+90 a^{2} A \,b^{2} x^{2}+60 x^{2} B \,a^{3} b +20 a^{3} A b x +5 a^{4} B x +3 A \,a^{4}\right )}{15 x^{\frac {5}{2}}}\) | \(100\) |
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Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=\frac {2 \, {\left (3 \, B b^{4} x^{5} - 3 \, A a^{4} + 5 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 30 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} - 30 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} - 5 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \]
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Time = 0.38 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.32 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=- \frac {2 A a^{4}}{5 x^{\frac {5}{2}}} - \frac {8 A a^{3} b}{3 x^{\frac {3}{2}}} - \frac {12 A a^{2} b^{2}}{\sqrt {x}} + 8 A a b^{3} \sqrt {x} + \frac {2 A b^{4} x^{\frac {3}{2}}}{3} - \frac {2 B a^{4}}{3 x^{\frac {3}{2}}} - \frac {8 B a^{3} b}{\sqrt {x}} + 12 B a^{2} b^{2} \sqrt {x} + \frac {8 B a b^{3} x^{\frac {3}{2}}}{3} + \frac {2 B b^{4} x^{\frac {5}{2}}}{5} \]
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Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=\frac {2}{5} \, B b^{4} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac {3}{2}} + 4 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \sqrt {x} - \frac {2 \, {\left (3 \, A a^{4} + 30 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 5 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \]
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Time = 0.32 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=\frac {2}{5} \, B b^{4} x^{\frac {5}{2}} + \frac {8}{3} \, B a b^{3} x^{\frac {3}{2}} + \frac {2}{3} \, A b^{4} x^{\frac {3}{2}} + 12 \, B a^{2} b^{2} \sqrt {x} + 8 \, A a b^{3} \sqrt {x} - \frac {2 \, {\left (60 \, B a^{3} b x^{2} + 90 \, A a^{2} b^{2} x^{2} + 5 \, B a^{4} x + 20 \, A a^{3} b x + 3 \, A a^{4}\right )}}{15 \, x^{\frac {5}{2}}} \]
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Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.89 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=x^{3/2}\,\left (\frac {2\,A\,b^4}{3}+\frac {8\,B\,a\,b^3}{3}\right )-\frac {x\,\left (\frac {2\,B\,a^4}{3}+\frac {8\,A\,b\,a^3}{3}\right )+\frac {2\,A\,a^4}{5}+x^2\,\left (8\,B\,a^3\,b+12\,A\,a^2\,b^2\right )}{x^{5/2}}+\frac {2\,B\,b^4\,x^{5/2}}{5}+4\,a\,b^2\,\sqrt {x}\,\left (2\,A\,b+3\,B\,a\right ) \]
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