Integrand size = 27, antiderivative size = 59 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{7/2}} \, dx=-\frac {2 a^2 A}{5 x^{5/2}}-\frac {2 a (2 A b+a B)}{3 x^{3/2}}-\frac {2 b (A b+2 a B)}{\sqrt {x}}+2 b^2 B \sqrt {x} \]
[Out]
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} \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{7/2}} \, dx=-\frac {2 a^2 A}{5 x^{5/2}}-\frac {2 a (a B+2 A b)}{3 x^{3/2}}-\frac {2 b (2 a B+A b)}{\sqrt {x}}+2 b^2 B \sqrt {x} \]
[In]
[Out]
Rule 27
Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^2 (A+B x)}{x^{7/2}} \, dx \\ & = \int \left (\frac {a^2 A}{x^{7/2}}+\frac {a (2 A b+a B)}{x^{5/2}}+\frac {b (A b+2 a B)}{x^{3/2}}+\frac {b^2 B}{\sqrt {x}}\right ) \, dx \\ & = -\frac {2 a^2 A}{5 x^{5/2}}-\frac {2 a (2 A b+a B)}{3 x^{3/2}}-\frac {2 b (A b+2 a B)}{\sqrt {x}}+2 b^2 B \sqrt {x} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{7/2}} \, dx=-\frac {2 \left (15 b^2 x^2 (A-B x)+10 a b x (A+3 B x)+a^2 (3 A+5 B x)\right )}{15 x^{5/2}} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(-\frac {2 a^{2} A}{5 x^{\frac {5}{2}}}-\frac {2 a \left (2 A b +B a \right )}{3 x^{\frac {3}{2}}}-\frac {2 b \left (A b +2 B a \right )}{\sqrt {x}}+2 b^{2} B \sqrt {x}\) | \(48\) |
default | \(-\frac {2 a^{2} A}{5 x^{\frac {5}{2}}}-\frac {2 a \left (2 A b +B a \right )}{3 x^{\frac {3}{2}}}-\frac {2 b \left (A b +2 B a \right )}{\sqrt {x}}+2 b^{2} B \sqrt {x}\) | \(48\) |
gosper | \(-\frac {2 \left (-15 B \,b^{2} x^{3}+15 A \,b^{2} x^{2}+30 B a b \,x^{2}+10 a A b x +5 a^{2} B x +3 A \,a^{2}\right )}{15 x^{\frac {5}{2}}}\) | \(52\) |
trager | \(-\frac {2 \left (-15 B \,b^{2} x^{3}+15 A \,b^{2} x^{2}+30 B a b \,x^{2}+10 a A b x +5 a^{2} B x +3 A \,a^{2}\right )}{15 x^{\frac {5}{2}}}\) | \(52\) |
risch | \(-\frac {2 \left (-15 B \,b^{2} x^{3}+15 A \,b^{2} x^{2}+30 B a b \,x^{2}+10 a A b x +5 a^{2} B x +3 A \,a^{2}\right )}{15 x^{\frac {5}{2}}}\) | \(52\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{7/2}} \, dx=\frac {2 \, {\left (15 \, B b^{2} x^{3} - 3 \, A a^{2} - 15 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} - 5 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \]
[In]
[Out]
Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.27 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{7/2}} \, dx=- \frac {2 A a^{2}}{5 x^{\frac {5}{2}}} - \frac {4 A a b}{3 x^{\frac {3}{2}}} - \frac {2 A b^{2}}{\sqrt {x}} - \frac {2 B a^{2}}{3 x^{\frac {3}{2}}} - \frac {4 B a b}{\sqrt {x}} + 2 B b^{2} \sqrt {x} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.88 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{7/2}} \, dx=2 \, B b^{2} \sqrt {x} - \frac {2 \, {\left (3 \, A a^{2} + 15 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.88 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{7/2}} \, dx=2 \, B b^{2} \sqrt {x} - \frac {2 \, {\left (30 \, B a b x^{2} + 15 \, A b^{2} x^{2} + 5 \, B a^{2} x + 10 \, A a b x + 3 \, A a^{2}\right )}}{15 \, x^{\frac {5}{2}}} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.88 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{7/2}} \, dx=2\,B\,b^2\,\sqrt {x}-\frac {x^2\,\left (2\,A\,b^2+4\,B\,a\,b\right )+\frac {2\,A\,a^2}{5}+x\,\left (\frac {2\,B\,a^2}{3}+\frac {4\,A\,b\,a}{3}\right )}{x^{5/2}} \]
[In]
[Out]