Integrand size = 20, antiderivative size = 70 \[ \int \frac {x (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (b B-A c) x}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 (b B-4 A c) (b+2 c x)}{3 b^3 c \sqrt {b x+c x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 70, 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 = {791, 627} \[ \int \frac {x (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 (b+2 c x) (b B-4 A c)}{3 b^3 c \sqrt {b x+c x^2}}-\frac {2 x (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
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Rule 627
Rule 791
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b B-A c) x}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {\left (2 \left (-\frac {b^2 B}{2}+2 A b c\right )\right ) \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 c} \\ & = -\frac {2 (b B-A c) x}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 (b B-4 A c) (b+2 c x)}{3 b^3 c \sqrt {b x+c x^2}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.79 \[ \int \frac {x (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {x \left (2 b B x (3 b+2 c x)-2 A \left (3 b^2+12 b c x+8 c^2 x^2\right )\right )}{3 b^3 (x (b+c x))^{3/2}} \]
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Time = 0.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (-B x +A \right ) b^{2}+4 c \left (-\frac {B x}{6}+A \right ) x b +\frac {8 A \,c^{2} x^{2}}{3}\right )}{\sqrt {x \left (c x +b \right )}\, \left (c x +b \right ) b^{3}}\) | \(53\) |
gosper | \(-\frac {2 x^{2} \left (c x +b \right ) \left (8 A \,c^{2} x^{2}-2 B b c \,x^{2}+12 A b c x -3 b^{2} B x +3 A \,b^{2}\right )}{3 b^{3} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\) | \(62\) |
trager | \(-\frac {2 \left (8 A \,c^{2} x^{2}-2 B b c \,x^{2}+12 A b c x -3 b^{2} B x +3 A \,b^{2}\right ) \sqrt {c \,x^{2}+b x}}{3 b^{3} \left (c x +b \right )^{2} x}\) | \(64\) |
risch | \(-\frac {2 A \left (c x +b \right )}{b^{3} \sqrt {x \left (c x +b \right )}}-\frac {2 \left (5 A \,c^{2} x -2 B b c x +6 A b c -3 B \,b^{2}\right ) x}{3 \sqrt {x \left (c x +b \right )}\, \left (c x +b \right ) b^{3}}\) | \(69\) |
default | \(B \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )+A \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )\) | \(168\) |
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Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.10 \[ \int \frac {x (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (3 \, A b^{2} - 2 \, {\left (B b c - 4 \, A c^{2}\right )} x^{2} - 3 \, {\left (B b^{2} - 4 \, A b c\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{3} c^{2} x^{3} + 2 \, b^{4} c x^{2} + b^{5} x\right )}} \]
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\[ \int \frac {x (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {x \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.59 \[ \int \frac {x (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {4 \, B x}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, A x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {2 \, B x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {16 \, A c x}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {8 \, A}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, B}{3 \, \sqrt {c x^{2} + b x} b c} \]
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\[ \int \frac {x (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} x}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 10.59 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.90 \[ \int \frac {x (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2\,\sqrt {c\,x^2+b\,x}\,\left (-3\,B\,b^2\,x+3\,A\,b^2-2\,B\,b\,c\,x^2+12\,A\,b\,c\,x+8\,A\,c^2\,x^2\right )}{3\,b^3\,x\,{\left (b+c\,x\right )}^2} \]
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