Optimal. Leaf size=75 \[ c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )-\frac {c \sqrt {b x^2+c x^4}}{x^2}-\frac {\left (b x^2+c x^4\right )^{3/2}}{3 x^6} \]
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Rubi [A] time = 0.10, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2018, 662, 620, 206} \[ c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )-\frac {c \sqrt {b x^2+c x^4}}{x^2}-\frac {\left (b x^2+c x^4\right )^{3/2}}{3 x^6} \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 662
Rule 2018
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^7} \, dx &=\frac {1}{2} \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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*}
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Mathematica [C] time = 0.02, size = 56, normalized size = 0.75 \[ -\frac {b \sqrt {x^2 \left (b+c x^2\right )} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {c x^2}{b}\right )}{3 x^4 \sqrt {\frac {c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 135, normalized size = 1.80 \[ \left [\frac {3 \, c^{\frac {3}{2}} x^{4} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, \sqrt {c x^{4} + b x^{2}} {\left (4 \, c x^{2} + b\right )}}{6 \, x^{4}}, -\frac {3 \, \sqrt {-c} c x^{4} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + \sqrt {c x^{4} + b x^{2}} {\left (4 \, c x^{2} + b\right )}}{3 \, x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 122, normalized size = 1.63 \[ -\frac {1}{2} \, c^{\frac {3}{2}} \log \left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2}\right ) \mathrm {sgn}\relax (x) + \frac {4 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} b c^{\frac {3}{2}} \mathrm {sgn}\relax (x) - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} b^{2} c^{\frac {3}{2}} \mathrm {sgn}\relax (x) + 2 \, b^{3} c^{\frac {3}{2}} \mathrm {sgn}\relax (x)\right )}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 129, normalized size = 1.72 \[ \frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (3 b^{2} c^{2} x^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )+3 \sqrt {c \,x^{2}+b}\, b \,c^{\frac {5}{2}} x^{4}+2 \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {5}{2}} x^{4}-2 \left (c \,x^{2}+b \right )^{\frac {5}{2}} c^{\frac {3}{2}} x^{2}-\left (c \,x^{2}+b \right )^{\frac {5}{2}} b \sqrt {c}\right )}{3 \left (c \,x^{2}+b \right )^{\frac {3}{2}} b^{2} \sqrt {c}\, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.51, size = 89, normalized size = 1.19 \[ \frac {1}{2} \, c^{\frac {3}{2}} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - \frac {7 \, \sqrt {c x^{4} + b x^{2}} c}{6 \, x^{2}} - \frac {\sqrt {c x^{4} + b x^{2}} b}{6 \, x^{4}} - \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^7} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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