Integrand size = 27, antiderivative size = 59 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{5/2}} \, dx=-\frac {2 a^2 A}{3 x^{3/2}}-\frac {2 a (2 A b+a B)}{\sqrt {x}}+2 b (A b+2 a B) \sqrt {x}+\frac {2}{3} b^2 B x^{3/2} \]
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Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {27, 77} \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{5/2}} \, dx=-\frac {2 a^2 A}{3 x^{3/2}}-\frac {2 a (a B+2 A b)}{\sqrt {x}}+2 b \sqrt {x} (2 a B+A b)+\frac {2}{3} b^2 B x^{3/2} \]
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Rule 27
Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^2 (A+B x)}{x^{5/2}} \, dx \\ & = \int \left (\frac {a^2 A}{x^{5/2}}+\frac {a (2 A b+a B)}{x^{3/2}}+\frac {b (A b+2 a B)}{\sqrt {x}}+b^2 B \sqrt {x}\right ) \, dx \\ & = -\frac {2 a^2 A}{3 x^{3/2}}-\frac {2 a (2 A b+a B)}{\sqrt {x}}+2 b (A b+2 a B) \sqrt {x}+\frac {2}{3} b^2 B x^{3/2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{5/2}} \, dx=\frac {2 \left (6 a b x (-A+B x)+b^2 x^2 (3 A+B x)-a^2 (A+3 B x)\right )}{3 x^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86
method | result | size |
gosper | \(-\frac {2 \left (-B \,b^{2} x^{3}-3 A \,b^{2} x^{2}-6 B a b \,x^{2}+6 a A b x +3 a^{2} B x +A \,a^{2}\right )}{3 x^{\frac {3}{2}}}\) | \(51\) |
derivativedivides | \(\frac {2 b^{2} B \,x^{\frac {3}{2}}}{3}+2 A \,b^{2} \sqrt {x}+4 a b B \sqrt {x}-\frac {2 a^{2} A}{3 x^{\frac {3}{2}}}-\frac {2 a \left (2 A b +B a \right )}{\sqrt {x}}\) | \(51\) |
default | \(\frac {2 b^{2} B \,x^{\frac {3}{2}}}{3}+2 A \,b^{2} \sqrt {x}+4 a b B \sqrt {x}-\frac {2 a^{2} A}{3 x^{\frac {3}{2}}}-\frac {2 a \left (2 A b +B a \right )}{\sqrt {x}}\) | \(51\) |
trager | \(-\frac {2 \left (-B \,b^{2} x^{3}-3 A \,b^{2} x^{2}-6 B a b \,x^{2}+6 a A b x +3 a^{2} B x +A \,a^{2}\right )}{3 x^{\frac {3}{2}}}\) | \(51\) |
risch | \(-\frac {2 \left (-B \,b^{2} x^{3}-3 A \,b^{2} x^{2}-6 B a b \,x^{2}+6 a A b x +3 a^{2} B x +A \,a^{2}\right )}{3 x^{\frac {3}{2}}}\) | \(51\) |
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Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.85 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{5/2}} \, dx=\frac {2 \, {\left (B b^{2} x^{3} - A a^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \]
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Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{5/2}} \, dx=- \frac {2 A a^{2}}{3 x^{\frac {3}{2}}} - \frac {4 A a b}{\sqrt {x}} + 2 A b^{2} \sqrt {x} - \frac {2 B a^{2}}{\sqrt {x}} + 4 B a b \sqrt {x} + \frac {2 B b^{2} x^{\frac {3}{2}}}{3} \]
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Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{5/2}} \, dx=\frac {2}{3} \, B b^{2} x^{\frac {3}{2}} + 2 \, {\left (2 \, B a b + A b^{2}\right )} \sqrt {x} - \frac {2 \, {\left (A a^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \]
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Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{5/2}} \, dx=\frac {2}{3} \, B b^{2} x^{\frac {3}{2}} + 4 \, B a b \sqrt {x} + 2 \, A b^{2} \sqrt {x} - \frac {2 \, {\left (3 \, B a^{2} x + 6 \, A a b x + A a^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]
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Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{5/2}} \, dx=-\frac {6\,B\,a^2\,x+2\,A\,a^2-12\,B\,a\,b\,x^2+12\,A\,a\,b\,x-2\,B\,b^2\,x^3-6\,A\,b^2\,x^2}{3\,x^{3/2}} \]
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