Integrand size = 19, antiderivative size = 75 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^7} \, dx=-\frac {c \sqrt {b x^2+c x^4}}{x^2}-\frac {\left (b x^2+c x^4\right )^{3/2}}{3 x^6}+c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2043, 676, 634, 212} \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^7} \, dx=c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )-\frac {c \sqrt {b x^2+c x^4}}{x^2}-\frac {\left (b x^2+c x^4\right )^{3/2}}{3 x^6} \]
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Rule 212
Rule 634
Rule 676
Rule 2043
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x^4} \, dx,x,x^2\right ) \\ & = -\frac {\left (b x^2+c x^4\right )^{3/2}}{3 x^6}+\frac {1}{2} c \text {Subst}\left (\int \frac {\sqrt {b x+c x^2}}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {c \sqrt {b x^2+c x^4}}{x^2}-\frac {\left (b x^2+c x^4\right )^{3/2}}{3 x^6}+\frac {1}{2} c^2 \text {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right ) \\ & = -\frac {c \sqrt {b x^2+c x^4}}{x^2}-\frac {\left (b x^2+c x^4\right )^{3/2}}{3 x^6}+c^2 \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right ) \\ & = -\frac {c \sqrt {b x^2+c x^4}}{x^2}-\frac {\left (b x^2+c x^4\right )^{3/2}}{3 x^6}+c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.13 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^7} \, dx=-\frac {\sqrt {x^2 \left (b+c x^2\right )} \left (\sqrt {b+c x^2} \left (b+4 c x^2\right )+3 c^{3/2} x^3 \log \left (-\sqrt {c} x+\sqrt {b+c x^2}\right )\right )}{3 x^4 \sqrt {b+c x^2}} \]
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Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97
method | result | size |
risch | \(-\frac {\left (4 c \,x^{2}+b \right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{3 x^{4}}+\frac {c^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+b}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{x \sqrt {c \,x^{2}+b}}\) | \(73\) |
pseudoelliptic | \(\frac {3 x^{4} \left (-\ln \left (2\right )+\ln \left (\frac {2 c \,x^{2}+2 \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {c}+b}{\sqrt {c}}\right )\right ) c^{\frac {3}{2}}-2 \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (4 c \,x^{2}+b \right )}{6 x^{4}}\) | \(74\) |
default | \(\frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (2 c^{\frac {5}{2}} \left (c \,x^{2}+b \right )^{\frac {3}{2}} x^{4}+3 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b}\, b \,x^{4}-2 c^{\frac {3}{2}} \left (c \,x^{2}+b \right )^{\frac {5}{2}} x^{2}+3 \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+b}\right ) b^{2} c^{2} x^{3}-\left (c \,x^{2}+b \right )^{\frac {5}{2}} b \sqrt {c}\right )}{3 x^{6} \left (c \,x^{2}+b \right )^{\frac {3}{2}} b^{2} \sqrt {c}}\) | \(129\) |
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Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.80 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^7} \, dx=\left [\frac {3 \, c^{\frac {3}{2}} x^{4} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, \sqrt {c x^{4} + b x^{2}} {\left (4 \, c x^{2} + b\right )}}{6 \, x^{4}}, -\frac {3 \, \sqrt {-c} c x^{4} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + \sqrt {c x^{4} + b x^{2}} {\left (4 \, c x^{2} + b\right )}}{3 \, x^{4}}\right ] \]
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\[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^7} \, dx=\int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{7}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.19 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^7} \, dx=\frac {1}{2} \, c^{\frac {3}{2}} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - \frac {7 \, \sqrt {c x^{4} + b x^{2}} c}{6 \, x^{2}} - \frac {\sqrt {c x^{4} + b x^{2}} b}{6 \, x^{4}} - \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{6 \, x^{6}} \]
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Time = 0.42 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.63 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^7} \, dx=-\frac {1}{2} \, c^{\frac {3}{2}} \log \left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2}\right ) \mathrm {sgn}\left (x\right ) + \frac {4 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} b c^{\frac {3}{2}} \mathrm {sgn}\left (x\right ) - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} b^{2} c^{\frac {3}{2}} \mathrm {sgn}\left (x\right ) + 2 \, b^{3} c^{\frac {3}{2}} \mathrm {sgn}\left (x\right )\right )}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{3}} \]
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Timed out. \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^7} \, dx=\int \frac {{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^7} \,d x \]
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