Integrand size = 13, antiderivative size = 68 \[ \int \frac {1}{x^{7/2} (a+b x)} \, dx=-\frac {2}{5 a x^{5/2}}+\frac {2 b}{3 a^2 x^{3/2}}-\frac {2 b^2}{a^3 \sqrt {x}}-\frac {2 b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \]
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Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {53, 65, 211} \[ \int \frac {1}{x^{7/2} (a+b x)} \, dx=-\frac {2 b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}}-\frac {2 b^2}{a^3 \sqrt {x}}+\frac {2 b}{3 a^2 x^{3/2}}-\frac {2}{5 a x^{5/2}} \]
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Rule 53
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{5 a x^{5/2}}-\frac {b \int \frac {1}{x^{5/2} (a+b x)} \, dx}{a} \\ & = -\frac {2}{5 a x^{5/2}}+\frac {2 b}{3 a^2 x^{3/2}}+\frac {b^2 \int \frac {1}{x^{3/2} (a+b x)} \, dx}{a^2} \\ & = -\frac {2}{5 a x^{5/2}}+\frac {2 b}{3 a^2 x^{3/2}}-\frac {2 b^2}{a^3 \sqrt {x}}-\frac {b^3 \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{a^3} \\ & = -\frac {2}{5 a x^{5/2}}+\frac {2 b}{3 a^2 x^{3/2}}-\frac {2 b^2}{a^3 \sqrt {x}}-\frac {\left (2 b^3\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a^3} \\ & = -\frac {2}{5 a x^{5/2}}+\frac {2 b}{3 a^2 x^{3/2}}-\frac {2 b^2}{a^3 \sqrt {x}}-\frac {2 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^{7/2} (a+b x)} \, dx=-\frac {2 \left (3 a^2-5 a b x+15 b^2 x^2\right )}{15 a^3 x^{5/2}}-\frac {2 b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \]
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Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78
method | result | size |
risch | \(-\frac {2 \left (15 b^{2} x^{2}-5 a b x +3 a^{2}\right )}{15 a^{3} x^{\frac {5}{2}}}-\frac {2 b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a^{3} \sqrt {a b}}\) | \(53\) |
derivativedivides | \(-\frac {2 b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a^{3} \sqrt {a b}}-\frac {2}{5 a \,x^{\frac {5}{2}}}-\frac {2 b^{2}}{a^{3} \sqrt {x}}+\frac {2 b}{3 a^{2} x^{\frac {3}{2}}}\) | \(54\) |
default | \(-\frac {2 b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a^{3} \sqrt {a b}}-\frac {2}{5 a \,x^{\frac {5}{2}}}-\frac {2 b^{2}}{a^{3} \sqrt {x}}+\frac {2 b}{3 a^{2} x^{\frac {3}{2}}}\) | \(54\) |
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Time = 0.23 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.12 \[ \int \frac {1}{x^{7/2} (a+b x)} \, dx=\left [\frac {15 \, b^{2} x^{3} \sqrt {-\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) - 2 \, {\left (15 \, b^{2} x^{2} - 5 \, a b x + 3 \, a^{2}\right )} \sqrt {x}}{15 \, a^{3} x^{3}}, \frac {2 \, {\left (15 \, b^{2} x^{3} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (15 \, b^{2} x^{2} - 5 \, a b x + 3 \, a^{2}\right )} \sqrt {x}\right )}}{15 \, a^{3} x^{3}}\right ] \]
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Time = 9.82 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.85 \[ \int \frac {1}{x^{7/2} (a+b x)} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{7 b x^{\frac {7}{2}}} & \text {for}\: a = 0 \\- \frac {2}{5 a x^{\frac {5}{2}}} & \text {for}\: b = 0 \\- \frac {2}{5 a x^{\frac {5}{2}}} + \frac {2 b}{3 a^{2} x^{\frac {3}{2}}} - \frac {b^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{a^{3} \sqrt {- \frac {a}{b}}} + \frac {b^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{a^{3} \sqrt {- \frac {a}{b}}} - \frac {2 b^{2}}{a^{3} \sqrt {x}} & \text {otherwise} \end {cases} \]
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Time = 0.33 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^{7/2} (a+b x)} \, dx=-\frac {2 \, b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {2 \, {\left (15 \, b^{2} x^{2} - 5 \, a b x + 3 \, a^{2}\right )}}{15 \, a^{3} x^{\frac {5}{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^{7/2} (a+b x)} \, dx=-\frac {2 \, b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {2 \, {\left (15 \, b^{2} x^{2} - 5 \, a b x + 3 \, a^{2}\right )}}{15 \, a^{3} x^{\frac {5}{2}}} \]
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Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^{7/2} (a+b x)} \, dx=-\frac {\frac {2}{5\,a}+\frac {2\,b^2\,x^2}{a^3}-\frac {2\,b\,x}{3\,a^2}}{x^{5/2}}-\frac {2\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \]
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