Integrand size = 13, antiderivative size = 106 \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=\frac {35 b^2}{12 a^3 (a+b x)^{3/2}}-\frac {1}{2 a x^2 (a+b x)^{3/2}}+\frac {7 b}{4 a^2 x (a+b x)^{3/2}}+\frac {35 b^2}{4 a^4 \sqrt {a+b x}}-\frac {35 b^2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
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Time = 0.02 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {44, 53, 65, 214} \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=-\frac {35 b^2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}}+\frac {35 b^2}{4 a^4 \sqrt {a+b x}}+\frac {35 b^2}{12 a^3 (a+b x)^{3/2}}+\frac {7 b}{4 a^2 x (a+b x)^{3/2}}-\frac {1}{2 a x^2 (a+b x)^{3/2}} \]
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Rule 44
Rule 53
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 a x^2 (a+b x)^{3/2}}-\frac {(7 b) \int \frac {1}{x^2 (a+b x)^{5/2}} \, dx}{4 a} \\ & = -\frac {1}{2 a x^2 (a+b x)^{3/2}}+\frac {7 b}{4 a^2 x (a+b x)^{3/2}}+\frac {\left (35 b^2\right ) \int \frac {1}{x (a+b x)^{5/2}} \, dx}{8 a^2} \\ & = \frac {35 b^2}{12 a^3 (a+b x)^{3/2}}-\frac {1}{2 a x^2 (a+b x)^{3/2}}+\frac {7 b}{4 a^2 x (a+b x)^{3/2}}+\frac {\left (35 b^2\right ) \int \frac {1}{x (a+b x)^{3/2}} \, dx}{8 a^3} \\ & = \frac {35 b^2}{12 a^3 (a+b x)^{3/2}}-\frac {1}{2 a x^2 (a+b x)^{3/2}}+\frac {7 b}{4 a^2 x (a+b x)^{3/2}}+\frac {35 b^2}{4 a^4 \sqrt {a+b x}}+\frac {\left (35 b^2\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{8 a^4} \\ & = \frac {35 b^2}{12 a^3 (a+b x)^{3/2}}-\frac {1}{2 a x^2 (a+b x)^{3/2}}+\frac {7 b}{4 a^2 x (a+b x)^{3/2}}+\frac {35 b^2}{4 a^4 \sqrt {a+b x}}+\frac {(35 b) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{4 a^4} \\ & = \frac {35 b^2}{12 a^3 (a+b x)^{3/2}}-\frac {1}{2 a x^2 (a+b x)^{3/2}}+\frac {7 b}{4 a^2 x (a+b x)^{3/2}}+\frac {35 b^2}{4 a^4 \sqrt {a+b x}}-\frac {35 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=\frac {-6 a^3+21 a^2 b x+140 a b^2 x^2+105 b^3 x^3}{12 a^4 x^2 (a+b x)^{3/2}}-\frac {35 b^2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
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Time = 0.14 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.66
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-11 b x +2 a \right )}{4 a^{4} x^{2}}+\frac {b^{2} \left (-\frac {70 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {48}{\sqrt {b x +a}}+\frac {16 a}{3 \left (b x +a \right )^{\frac {3}{2}}}\right )}{8 a^{4}}\) | \(70\) |
pseudoelliptic | \(-\frac {35 \left (\left (b x +a \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b^{2} x^{2}-\sqrt {a}\, b^{3} x^{3}-\frac {4 a^{\frac {3}{2}} b^{2} x^{2}}{3}-\frac {a^{\frac {5}{2}} b x}{5}+\frac {2 a^{\frac {7}{2}}}{35}\right )}{4 \left (b x +a \right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{2}}\) | \(77\) |
derivativedivides | \(2 b^{2} \left (\frac {3}{a^{4} \sqrt {b x +a}}+\frac {1}{3 a^{3} \left (b x +a \right )^{\frac {3}{2}}}-\frac {\frac {-\frac {11 \left (b x +a \right )^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {b x +a}}{8}}{b^{2} x^{2}}+\frac {35 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{4}}\right )\) | \(81\) |
default | \(2 b^{2} \left (\frac {3}{a^{4} \sqrt {b x +a}}+\frac {1}{3 a^{3} \left (b x +a \right )^{\frac {3}{2}}}-\frac {\frac {-\frac {11 \left (b x +a \right )^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {b x +a}}{8}}{b^{2} x^{2}}+\frac {35 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{4}}\right )\) | \(81\) |
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Time = 0.24 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.41 \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=\left [\frac {105 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (105 \, a b^{3} x^{3} + 140 \, a^{2} b^{2} x^{2} + 21 \, a^{3} b x - 6 \, a^{4}\right )} \sqrt {b x + a}}{24 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}}, \frac {105 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (105 \, a b^{3} x^{3} + 140 \, a^{2} b^{2} x^{2} + 21 \, a^{3} b x - 6 \, a^{4}\right )} \sqrt {b x + a}}{12 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (99) = 198\).
Time = 14.86 (sec) , antiderivative size = 464, normalized size of antiderivative = 4.38 \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=- \frac {6 a^{\frac {89}{2}} b^{75} x^{75}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {155}{2}} \sqrt {\frac {a}{b x} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {157}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {21 a^{\frac {87}{2}} b^{76} x^{76}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {155}{2}} \sqrt {\frac {a}{b x} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {157}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {140 a^{\frac {85}{2}} b^{77} x^{77}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {155}{2}} \sqrt {\frac {a}{b x} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {157}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {105 a^{\frac {83}{2}} b^{78} x^{78}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {155}{2}} \sqrt {\frac {a}{b x} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {157}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {105 a^{42} b^{\frac {155}{2}} x^{\frac {155}{2}} \sqrt {\frac {a}{b x} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {155}{2}} \sqrt {\frac {a}{b x} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {157}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {105 a^{41} b^{\frac {157}{2}} x^{\frac {157}{2}} \sqrt {\frac {a}{b x} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {155}{2}} \sqrt {\frac {a}{b x} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {157}{2}} \sqrt {\frac {a}{b x} + 1}} \]
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Time = 0.29 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=\frac {105 \, {\left (b x + a\right )}^{3} b^{2} - 175 \, {\left (b x + a\right )}^{2} a b^{2} + 56 \, {\left (b x + a\right )} a^{2} b^{2} + 8 \, a^{3} b^{2}}{12 \, {\left ({\left (b x + a\right )}^{\frac {7}{2}} a^{4} - 2 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} + {\left (b x + a\right )}^{\frac {3}{2}} a^{6}\right )}} + \frac {35 \, b^{2} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{8 \, a^{\frac {9}{2}}} \]
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Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=\frac {35 \, b^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{4}} + \frac {2 \, {\left (9 \, {\left (b x + a\right )} b^{2} + a b^{2}\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4}} + \frac {11 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2} - 13 \, \sqrt {b x + a} a b^{2}}{4 \, a^{4} b^{2} x^{2}} \]
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Time = 0.15 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=\frac {\frac {2\,b^2}{3\,a}-\frac {175\,b^2\,{\left (a+b\,x\right )}^2}{12\,a^3}+\frac {35\,b^2\,{\left (a+b\,x\right )}^3}{4\,a^4}+\frac {14\,b^2\,\left (a+b\,x\right )}{3\,a^2}}{{\left (a+b\,x\right )}^{7/2}-2\,a\,{\left (a+b\,x\right )}^{5/2}+a^2\,{\left (a+b\,x\right )}^{3/2}}-\frac {35\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{4\,a^{9/2}} \]
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