Integrand size = 13, antiderivative size = 95 \[ \int \frac {x^{7/2}}{(a+b x)^3} \, dx=-\frac {35 a \sqrt {x}}{4 b^4}+\frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b (a+b x)^2}-\frac {7 x^{5/2}}{4 b^2 (a+b x)}+\frac {35 a^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}} \]
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Time = 0.02 (sec) , antiderivative size = 95, 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 = {43, 52, 65, 211} \[ \int \frac {x^{7/2}}{(a+b x)^3} \, dx=\frac {35 a^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}}-\frac {35 a \sqrt {x}}{4 b^4}-\frac {7 x^{5/2}}{4 b^2 (a+b x)}-\frac {x^{7/2}}{2 b (a+b x)^2}+\frac {35 x^{3/2}}{12 b^3} \]
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Rule 43
Rule 52
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{7/2}}{2 b (a+b x)^2}+\frac {7 \int \frac {x^{5/2}}{(a+b x)^2} \, dx}{4 b} \\ & = -\frac {x^{7/2}}{2 b (a+b x)^2}-\frac {7 x^{5/2}}{4 b^2 (a+b x)}+\frac {35 \int \frac {x^{3/2}}{a+b x} \, dx}{8 b^2} \\ & = \frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b (a+b x)^2}-\frac {7 x^{5/2}}{4 b^2 (a+b x)}-\frac {(35 a) \int \frac {\sqrt {x}}{a+b x} \, dx}{8 b^3} \\ & = -\frac {35 a \sqrt {x}}{4 b^4}+\frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b (a+b x)^2}-\frac {7 x^{5/2}}{4 b^2 (a+b x)}+\frac {\left (35 a^2\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 b^4} \\ & = -\frac {35 a \sqrt {x}}{4 b^4}+\frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b (a+b x)^2}-\frac {7 x^{5/2}}{4 b^2 (a+b x)}+\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 b^4} \\ & = -\frac {35 a \sqrt {x}}{4 b^4}+\frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b (a+b x)^2}-\frac {7 x^{5/2}}{4 b^2 (a+b x)}+\frac {35 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.85 \[ \int \frac {x^{7/2}}{(a+b x)^3} \, dx=\frac {\sqrt {x} \left (-105 a^3-175 a^2 b x-56 a b^2 x^2+8 b^3 x^3\right )}{12 b^4 (a+b x)^2}+\frac {35 a^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}} \]
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Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69
method | result | size |
risch | \(-\frac {2 \left (-b x +9 a \right ) \sqrt {x}}{3 b^{4}}+\frac {a^{2} \left (\frac {-\frac {13 b \,x^{\frac {3}{2}}}{4}-\frac {11 a \sqrt {x}}{4}}{\left (b x +a \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{b^{4}}\) | \(66\) |
derivativedivides | \(-\frac {2 \left (-\frac {b \,x^{\frac {3}{2}}}{3}+3 a \sqrt {x}\right )}{b^{4}}+\frac {2 a^{2} \left (\frac {-\frac {13 b \,x^{\frac {3}{2}}}{8}-\frac {11 a \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{4}}\) | \(68\) |
default | \(-\frac {2 \left (-\frac {b \,x^{\frac {3}{2}}}{3}+3 a \sqrt {x}\right )}{b^{4}}+\frac {2 a^{2} \left (\frac {-\frac {13 b \,x^{\frac {3}{2}}}{8}-\frac {11 a \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{4}}\) | \(68\) |
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Time = 0.24 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.39 \[ \int \frac {x^{7/2}}{(a+b x)^3} \, dx=\left [\frac {105 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) + 2 \, {\left (8 \, b^{3} x^{3} - 56 \, a b^{2} x^{2} - 175 \, a^{2} b x - 105 \, a^{3}\right )} \sqrt {x}}{24 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac {105 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (8 \, b^{3} x^{3} - 56 \, a b^{2} x^{2} - 175 \, a^{2} b x - 105 \, a^{3}\right )} \sqrt {x}}{12 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (87) = 174\).
Time = 66.07 (sec) , antiderivative size = 762, normalized size of antiderivative = 8.02 \[ \int \frac {x^{7/2}}{(a+b x)^3} \, dx=\begin {cases} \tilde {\infty } x^{\frac {3}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {9}{2}}}{9 a^{3}} & \text {for}\: b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 b^{3}} & \text {for}\: a = 0 \\\frac {105 a^{4} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} - \frac {105 a^{4} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} - \frac {210 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} + \frac {210 a^{3} b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} - \frac {210 a^{3} b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} - \frac {350 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} + \frac {105 a^{2} b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} - \frac {105 a^{2} b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} - \frac {112 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} + \frac {16 b^{4} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \frac {x^{7/2}}{(a+b x)^3} \, dx=-\frac {13 \, a^{2} b x^{\frac {3}{2}} + 11 \, a^{3} \sqrt {x}}{4 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac {35 \, a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{4}} + \frac {2 \, {\left (b x^{\frac {3}{2}} - 9 \, a \sqrt {x}\right )}}{3 \, b^{4}} \]
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Time = 0.33 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.81 \[ \int \frac {x^{7/2}}{(a+b x)^3} \, dx=\frac {35 \, a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{4}} - \frac {13 \, a^{2} b x^{\frac {3}{2}} + 11 \, a^{3} \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} b^{4}} + \frac {2 \, {\left (b^{6} x^{\frac {3}{2}} - 9 \, a b^{5} \sqrt {x}\right )}}{3 \, b^{9}} \]
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Time = 0.13 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.85 \[ \int \frac {x^{7/2}}{(a+b x)^3} \, dx=\frac {2\,x^{3/2}}{3\,b^3}-\frac {\frac {11\,a^3\,\sqrt {x}}{4}+\frac {13\,a^2\,b\,x^{3/2}}{4}}{a^2\,b^4+2\,a\,b^5\,x+b^6\,x^2}-\frac {6\,a\,\sqrt {x}}{b^4}+\frac {35\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,b^{9/2}} \]
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