Integrand size = 27, antiderivative size = 121 \[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {a (A b-a B)}{4 b^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-2 a B}{3 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {B}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 121, 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 = {784, 78} \[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {A b-2 a B}{3 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a (A b-a B)}{4 b^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {B}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 78
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {x (A+B x)}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {a (-A b+a B)}{b^7 (a+b x)^5}+\frac {A b-2 a B}{b^7 (a+b x)^4}+\frac {B}{b^7 (a+b x)^3}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {a (A b-a B)}{4 b^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-2 a B}{3 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {B}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(319\) vs. \(2(121)=242\).
Time = 0.84 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.64 \[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {x^2 \left (-3 a^2 b^5 B x^6-3 a^3 \sqrt {a^2} b^2 x^2 \sqrt {(a+b x)^2} (A+B x)+a^6 b x (4 A+B x)+a^4 \sqrt {a^2} b x \sqrt {(a+b x)^2} (2 A+3 B x)+a^7 (6 A+4 B x)-3 a b^4 x^4 \left (-A b^2 x^2+\sqrt {a^2} A \sqrt {(a+b x)^2}+\sqrt {a^2} B x \sqrt {(a+b x)^2}\right )-a^5 \left (-A b^2 x^2+6 \sqrt {a^2} A \sqrt {(a+b x)^2}+4 \sqrt {a^2} B x \sqrt {(a+b x)^2}\right )+3 \sqrt {a^2} b^3 x^3 \sqrt {(a+b x)^2} \left (A b^2 x^2+a^2 (A+B x)\right )\right )}{12 a^7 (a+b x)^3 \left (\sqrt {a^2} b x+a \left (\sqrt {a^2}-\sqrt {(a+b x)^2}\right )\right )} \]
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Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {\left (b x +a \right ) \left (6 b^{2} B \,x^{2}+4 A \,b^{2} x +4 B a b x +A b a +B \,a^{2}\right )}{12 b^{3} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(52\) |
default | \(-\frac {\left (b x +a \right ) \left (6 b^{2} B \,x^{2}+4 A \,b^{2} x +4 B a b x +A b a +B \,a^{2}\right )}{12 b^{3} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(52\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {B \,x^{2}}{2 b}-\frac {\left (A b +B a \right ) x}{3 b^{2}}-\frac {a \left (A b +B a \right )}{12 b^{3}}\right )}{\left (b x +a \right )^{5}}\) | \(54\) |
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Time = 0.41 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.66 \[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {6 \, B b^{2} x^{2} + B a^{2} + A a b + 4 \, {\left (B a b + A b^{2}\right )} x}{12 \, {\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \]
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\[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {x \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.74 \[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {A}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {B}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {2 \, B a}{3 \, b^{6} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {B a^{2}}{4 \, b^{7} {\left (x + \frac {a}{b}\right )}^{4}} + \frac {A a}{4 \, b^{6} {\left (x + \frac {a}{b}\right )}^{4}} \]
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Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.43 \[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {6 \, B b^{2} x^{2} + 4 \, B a b x + 4 \, A b^{2} x + B a^{2} + A a b}{12 \, {\left (b x + a\right )}^{4} b^{3} \mathrm {sgn}\left (b x + a\right )} \]
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Time = 10.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.51 \[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (B\,a^2+4\,B\,a\,b\,x+A\,a\,b+6\,B\,b^2\,x^2+4\,A\,b^2\,x\right )}{12\,b^3\,{\left (a+b\,x\right )}^5} \]
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