Optimal. Leaf size=76 \[ -\frac {\left (b x^2+c x^4\right )^{3/2}}{x^4}+\frac {3}{2} c \sqrt {b x^2+c x^4}+\frac {3}{2} b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2018, 662, 664, 620, 206} \[ -\frac {\left (b x^2+c x^4\right )^{3/2}}{x^4}+\frac {3}{2} c \sqrt {b x^2+c x^4}+\frac {3}{2} b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 662
Rule 664
Rule 2018
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{x^4}+\frac {1}{2} (3 c) \operatorname {Subst}\left (\int \frac {\sqrt {b x+c x^2}}{x} \, dx,x,x^2\right )\\ &=\frac {3}{2} c \sqrt {b x^2+c x^4}-\frac {\left (b x^2+c x^4\right )^{3/2}}{x^4}+\frac {1}{4} (3 b c) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac {3}{2} c \sqrt {b x^2+c x^4}-\frac {\left (b x^2+c x^4\right )^{3/2}}{x^4}+\frac {1}{2} (3 b c) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )\\ &=\frac {3}{2} c \sqrt {b x^2+c x^4}-\frac {\left (b x^2+c x^4\right )^{3/2}}{x^4}+\frac {3}{2} b \sqrt {c} \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 = 0.01, size = 54, normalized size = 0.71 \[ -\frac {b \sqrt {x^2 \left (b+c x^2\right )} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {c x^2}{b}\right )}{x^2 \sqrt {\frac {c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 139, normalized size = 1.83 \[ \left [\frac {3 \, b \sqrt {c} x^{2} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) + 2 \, \sqrt {c x^{4} + b x^{2}} {\left (c x^{2} - 2 \, b\right )}}{4 \, x^{2}}, -\frac {3 \, b \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) - \sqrt {c x^{4} + b x^{2}} {\left (c x^{2} - 2 \, b\right )}}{2 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 79, normalized size = 1.04 \[ \frac {1}{2} \, \sqrt {c x^{2} + b} c x \mathrm {sgn}\relax (x) - \frac {3}{4} \, b \sqrt {c} \log \left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2}\right ) \mathrm {sgn}\relax (x) + \frac {2 \, b^{2} \sqrt {c} \mathrm {sgn}\relax (x)}{{\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 107, normalized size = 1.41 \[ \frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (3 b^{2} c x \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )+3 \sqrt {c \,x^{2}+b}\, b \,c^{\frac {3}{2}} x^{2}+2 \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {3}{2}} x^{2}-2 \left (c \,x^{2}+b \right )^{\frac {5}{2}} \sqrt {c}\right )}{2 \left (c \,x^{2}+b \right )^{\frac {3}{2}} b \sqrt {c}\, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 71, normalized size = 0.93 \[ \frac {3}{4} \, b \sqrt {c} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - \frac {3 \, \sqrt {c x^{4} + b x^{2}} b}{2 \, x^{2}} + \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{2 \, x^{4}} \]
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 \[ \int \frac {{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^5} \,d x \]
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 \[ \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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