Integrand size = 19, antiderivative size = 117 \[ \int \frac {1}{x^{7/2} \sqrt {b x+c x^2}} \, dx=-\frac {\sqrt {b x+c x^2}}{3 b x^{7/2}}+\frac {5 c \sqrt {b x+c x^2}}{12 b^2 x^{5/2}}-\frac {5 c^2 \sqrt {b x+c x^2}}{8 b^3 x^{3/2}}+\frac {5 c^3 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{7/2}} \]
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Time = 0.03 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {686, 674, 213} \[ \int \frac {1}{x^{7/2} \sqrt {b x+c x^2}} \, dx=\frac {5 c^3 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{7/2}}-\frac {5 c^2 \sqrt {b x+c x^2}}{8 b^3 x^{3/2}}+\frac {5 c \sqrt {b x+c x^2}}{12 b^2 x^{5/2}}-\frac {\sqrt {b x+c x^2}}{3 b x^{7/2}} \]
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Rule 213
Rule 674
Rule 686
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {b x+c x^2}}{3 b x^{7/2}}-\frac {(5 c) \int \frac {1}{x^{5/2} \sqrt {b x+c x^2}} \, dx}{6 b} \\ & = -\frac {\sqrt {b x+c x^2}}{3 b x^{7/2}}+\frac {5 c \sqrt {b x+c x^2}}{12 b^2 x^{5/2}}+\frac {\left (5 c^2\right ) \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx}{8 b^2} \\ & = -\frac {\sqrt {b x+c x^2}}{3 b x^{7/2}}+\frac {5 c \sqrt {b x+c x^2}}{12 b^2 x^{5/2}}-\frac {5 c^2 \sqrt {b x+c x^2}}{8 b^3 x^{3/2}}-\frac {\left (5 c^3\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{16 b^3} \\ & = -\frac {\sqrt {b x+c x^2}}{3 b x^{7/2}}+\frac {5 c \sqrt {b x+c x^2}}{12 b^2 x^{5/2}}-\frac {5 c^2 \sqrt {b x+c x^2}}{8 b^3 x^{3/2}}-\frac {\left (5 c^3\right ) \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{8 b^3} \\ & = -\frac {\sqrt {b x+c x^2}}{3 b x^{7/2}}+\frac {5 c \sqrt {b x+c x^2}}{12 b^2 x^{5/2}}-\frac {5 c^2 \sqrt {b x+c x^2}}{8 b^3 x^{3/2}}+\frac {5 c^3 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{7/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^{7/2} \sqrt {b x+c x^2}} \, dx=\frac {-\sqrt {b} \left (8 b^3-2 b^2 c x+5 b c^2 x^2+15 c^3 x^3\right )+15 c^3 x^3 \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )}{24 b^{7/2} x^{5/2} \sqrt {x (b+c x)}} \]
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Time = 2.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.70
method | result | size |
risch | \(-\frac {\left (c x +b \right ) \left (15 c^{2} x^{2}-10 b c x +8 b^{2}\right )}{24 b^{3} x^{\frac {5}{2}} \sqrt {x \left (c x +b \right )}}+\frac {5 c^{3} \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) \sqrt {c x +b}\, \sqrt {x}}{8 b^{\frac {7}{2}} \sqrt {x \left (c x +b \right )}}\) | \(82\) |
default | \(\frac {\sqrt {x \left (c x +b \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) c^{3} x^{3}-15 c^{2} x^{2} \sqrt {b}\, \sqrt {c x +b}+10 b^{\frac {3}{2}} c x \sqrt {c x +b}-8 b^{\frac {5}{2}} \sqrt {c x +b}\right )}{24 b^{\frac {7}{2}} x^{\frac {7}{2}} \sqrt {c x +b}}\) | \(90\) |
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Time = 0.25 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.49 \[ \int \frac {1}{x^{7/2} \sqrt {b x+c x^2}} \, dx=\left [\frac {15 \, \sqrt {b} c^{3} x^{4} \log \left (-\frac {c x^{2} + 2 \, b x + 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) - 2 \, {\left (15 \, b c^{2} x^{2} - 10 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{48 \, b^{4} x^{4}}, -\frac {15 \, \sqrt {-b} c^{3} x^{4} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (15 \, b c^{2} x^{2} - 10 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{24 \, b^{4} x^{4}}\right ] \]
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\[ \int \frac {1}{x^{7/2} \sqrt {b x+c x^2}} \, dx=\int \frac {1}{x^{\frac {7}{2}} \sqrt {x \left (b + c x\right )}}\, dx \]
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\[ \int \frac {1}{x^{7/2} \sqrt {b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x} x^{\frac {7}{2}}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^{7/2} \sqrt {b x+c x^2}} \, dx=-\frac {\frac {15 \, c^{4} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{3}} + \frac {15 \, {\left (c x + b\right )}^{\frac {5}{2}} c^{4} - 40 \, {\left (c x + b\right )}^{\frac {3}{2}} b c^{4} + 33 \, \sqrt {c x + b} b^{2} c^{4}}{b^{3} c^{3} x^{3}}}{24 \, c} \]
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Timed out. \[ \int \frac {1}{x^{7/2} \sqrt {b x+c x^2}} \, dx=\int \frac {1}{x^{7/2}\,\sqrt {c\,x^2+b\,x}} \,d x \]
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