Integrand size = 24, antiderivative size = 100 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx=-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{63 a^2 b (a+b x)^6}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{315 a^3 b (a+b x)^5} \]
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Time = 0.02 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665} \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx=-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{63 a^2 b (a+b x)^6}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{315 a^3 b (a+b x)^5} \]
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Rule 665
Rule 673
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}+\frac {2 \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6} \, dx}{9 a} \\ & = -\frac {\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{63 a^2 b (a+b x)^6}+\frac {2 \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^5} \, dx}{63 a^2} \\ & = -\frac {\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{63 a^2 b (a+b x)^6}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{315 a^3 b (a+b x)^5} \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.60 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx=-\frac {(a-b x)^2 \sqrt {a^2-b^2 x^2} \left (47 a^2+14 a b x+2 b^2 x^2\right )}{315 a^3 b (a+b x)^5} \]
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Time = 2.78 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(-\frac {\left (-b x +a \right ) \left (2 b^{2} x^{2}+14 a b x +47 a^{2}\right ) \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{315 \left (b x +a \right )^{6} a^{3} b}\) | \(55\) |
trager | \(-\frac {\left (2 b^{4} x^{4}+10 a \,b^{3} x^{3}+21 a^{2} b^{2} x^{2}-80 a^{3} b x +47 a^{4}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{315 a^{3} \left (b x +a \right )^{5} b}\) | \(71\) |
default | \(\frac {-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{9 a b \left (x +\frac {a}{b}\right )^{7}}+\frac {2 b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{7 a b \left (x +\frac {a}{b}\right )^{6}}-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{35 a^{2} \left (x +\frac {a}{b}\right )^{5}}\right )}{9 a}}{b^{7}}\) | \(145\) |
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none
Time = 0.29 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.70 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx=-\frac {47 \, b^{5} x^{5} + 235 \, a b^{4} x^{4} + 470 \, a^{2} b^{3} x^{3} + 470 \, a^{3} b^{2} x^{2} + 235 \, a^{4} b x + 47 \, a^{5} + {\left (2 \, b^{4} x^{4} + 10 \, a b^{3} x^{3} + 21 \, a^{2} b^{2} x^{2} - 80 \, a^{3} b x + 47 \, a^{4}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{315 \, {\left (a^{3} b^{6} x^{5} + 5 \, a^{4} b^{5} x^{4} + 10 \, a^{5} b^{4} x^{3} + 10 \, a^{6} b^{3} x^{2} + 5 \, a^{7} b^{2} x + a^{8} b\right )}} \]
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\[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx=\int \frac {\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{\frac {3}{2}}}{\left (a + b x\right )^{7}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (88) = 176\).
Time = 0.20 (sec) , antiderivative size = 342, normalized size of antiderivative = 3.42 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx=-\frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}}}{3 \, {\left (b^{7} x^{6} + 6 \, a b^{6} x^{5} + 15 \, a^{2} b^{5} x^{4} + 20 \, a^{3} b^{4} x^{3} + 15 \, a^{4} b^{3} x^{2} + 6 \, a^{5} b^{2} x + a^{6} b\right )}} + \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}} a}{9 \, {\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{63 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{105 \, {\left (a b^{4} x^{3} + 3 \, a^{2} b^{3} x^{2} + 3 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{315 \, {\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{315 \, {\left (a^{3} b^{2} x + a^{4} b\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (88) = 176\).
Time = 0.29 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.89 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx=\frac {2 \, {\left (\frac {108 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}}{b^{2} x} + \frac {1062 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac {1638 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac {3402 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac {2520 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac {2310 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac {630 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac {315 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + 47\right )}}{315 \, a^{3} {\left (\frac {a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}}{b^{2} x} + 1\right )}^{9} {\left | b \right |}} \]
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Time = 10.44 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx=\frac {20\,\sqrt {a^2-b^2\,x^2}}{63\,b\,{\left (a+b\,x\right )}^4}-\frac {4\,a\,\sqrt {a^2-b^2\,x^2}}{9\,b\,{\left (a+b\,x\right )}^5}-\frac {\sqrt {a^2-b^2\,x^2}}{105\,a\,b\,{\left (a+b\,x\right )}^3}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{315\,a^2\,b\,{\left (a+b\,x\right )}^2}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{315\,a^3\,b\,\left (a+b\,x\right )} \]
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