Integrand size = 13, antiderivative size = 101 \[ \int \frac {x^{5/2}}{(a+b x)^3} \, dx=\frac {2 \sqrt {x} \left (\frac {5 a x}{b^2}+\frac {x^2}{b}\right )}{(a+b x)^2}-\frac {15 a^2 \left (\sqrt {x} \left (-\frac {1}{2 b (a+b x)^2}+\frac {1}{4 a b (a+b x)}\right )+\frac {\arctan \left (\frac {b x}{a}\right )}{4 a b \sqrt {a b}}\right )}{b^2} \]
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Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.81, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {43, 52, 65, 211} \[ \int \frac {x^{5/2}}{(a+b x)^3} \, dx=-\frac {15 \sqrt {a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{7/2}}-\frac {5 x^{3/2}}{4 b^2 (a+b x)}-\frac {x^{5/2}}{2 b (a+b x)^2}+\frac {15 \sqrt {x}}{4 b^3} \]
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Rule 43
Rule 52
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{5/2}}{2 b (a+b x)^2}+\frac {5 \int \frac {x^{3/2}}{(a+b x)^2} \, dx}{4 b} \\ & = -\frac {x^{5/2}}{2 b (a+b x)^2}-\frac {5 x^{3/2}}{4 b^2 (a+b x)}+\frac {15 \int \frac {\sqrt {x}}{a+b x} \, dx}{8 b^2} \\ & = \frac {15 \sqrt {x}}{4 b^3}-\frac {x^{5/2}}{2 b (a+b x)^2}-\frac {5 x^{3/2}}{4 b^2 (a+b x)}-\frac {(15 a) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 b^3} \\ & = \frac {15 \sqrt {x}}{4 b^3}-\frac {x^{5/2}}{2 b (a+b x)^2}-\frac {5 x^{3/2}}{4 b^2 (a+b x)}-\frac {(15 a) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 b^3} \\ & = \frac {15 \sqrt {x}}{4 b^3}-\frac {x^{5/2}}{2 b (a+b x)^2}-\frac {5 x^{3/2}}{4 b^2 (a+b x)}-\frac {15 \sqrt {a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{7/2}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.69 \[ \int \frac {x^{5/2}}{(a+b x)^3} \, dx=\frac {\sqrt {x} \left (15 a^2+25 a b x+8 b^2 x^2\right )}{4 b^3 (a+b x)^2}-\frac {15 \sqrt {a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{7/2}} \]
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Time = 0.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.55
method | result | size |
derivativedivides | \(\frac {2 \sqrt {x}}{b^{3}}-\frac {2 a \left (\frac {-\frac {9 b \,x^{\frac {3}{2}}}{8}-\frac {7 a \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {15 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{3}}\) | \(56\) |
default | \(\frac {2 \sqrt {x}}{b^{3}}-\frac {2 a \left (\frac {-\frac {9 b \,x^{\frac {3}{2}}}{8}-\frac {7 a \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {15 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{3}}\) | \(56\) |
risch | \(\frac {2 \sqrt {x}}{b^{3}}-\frac {a \left (\frac {-\frac {9 b \,x^{\frac {3}{2}}}{4}-\frac {7 a \sqrt {x}}{4}}{\left (b x +a \right )^{2}}+\frac {15 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{b^{3}}\) | \(57\) |
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Time = 0.25 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.98 \[ \int \frac {x^{5/2}}{(a+b x)^3} \, dx=\left [\frac {15 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) + 2 \, {\left (8 \, b^{2} x^{2} + 25 \, a b x + 15 \, a^{2}\right )} \sqrt {x}}{8 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac {15 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - {\left (8 \, b^{2} x^{2} + 25 \, a b x + 15 \, a^{2}\right )} \sqrt {x}}{4 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (82) = 164\).
Time = 16.89 (sec) , antiderivative size = 683, normalized size of antiderivative = 6.76 \[ \int \frac {x^{5/2}}{(a+b x)^3} \, dx=\begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {7}{2}}}{7 a^{3}} & \text {for}\: b = 0 \\\frac {2 \sqrt {x}}{b^{3}} & \text {for}\: a = 0 \\- \frac {15 a^{3} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {15 a^{3} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {30 a^{2} b \sqrt {x} \sqrt {- \frac {a}{b}}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} - \frac {30 a^{2} b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {30 a^{2} b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {50 a b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} - \frac {15 a b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {15 a b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {16 b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.72 \[ \int \frac {x^{5/2}}{(a+b x)^3} \, dx=\frac {9 \, a b x^{\frac {3}{2}} + 7 \, a^{2} \sqrt {x}}{4 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} - \frac {15 \, a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{3}} + \frac {2 \, \sqrt {x}}{b^{3}} \]
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Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.58 \[ \int \frac {x^{5/2}}{(a+b x)^3} \, dx=-\frac {15 \, a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{3}} + \frac {2 \, \sqrt {x}}{b^{3}} + \frac {9 \, a b x^{\frac {3}{2}} + 7 \, a^{2} \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} b^{3}} \]
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Time = 0.14 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.68 \[ \int \frac {x^{5/2}}{(a+b x)^3} \, dx=\frac {\frac {7\,a^2\,\sqrt {x}}{4}+\frac {9\,a\,b\,x^{3/2}}{4}}{a^2\,b^3+2\,a\,b^4\,x+b^5\,x^2}+\frac {2\,\sqrt {x}}{b^3}-\frac {15\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,b^{7/2}} \]
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