Optimal. Leaf size=76 \[ \frac {3}{2} c \sqrt {b x^2+c x^4}-\frac {\left (b x^2+c x^4\right )^{3/2}}{x^4}+\frac {3}{2} b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right ) \]
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Rubi [A]
time = 0.06, 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 = {2043, 676, 678,
634, 212} \begin {gather*} -\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 676
Rule 678
Rule 2043
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^5} \, dx &=\frac {1}{2} \text {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) \text {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) \text {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) \text {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 [A]
time = 0.09, size = 80, normalized size = 1.05 \begin {gather*} \frac {-2 b^2-b c x^2+c^2 x^4-3 b \sqrt {c} x \sqrt {b+c x^2} \log \left (-\sqrt {c} x+\sqrt {b+c x^2}\right )}{2 \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 107, normalized size = 1.41
method | result | size |
risch | \(-\frac {\left (-c \,x^{2}+2 b \right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{2 x^{2}}+\frac {3 b \sqrt {c}\, \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+b}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{2 x \sqrt {c \,x^{2}+b}}\) | \(77\) |
default | \(\frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (2 c^{\frac {3}{2}} \left (c \,x^{2}+b \right )^{\frac {3}{2}} x^{2}+3 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b}\, b \,x^{2}-2 \left (c \,x^{2}+b \right )^{\frac {5}{2}} \sqrt {c}+3 \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+b}\right ) b^{2} c x \right )}{2 x^{4} \left (c \,x^{2}+b \right )^{\frac {3}{2}} b \sqrt {c}}\) | \(107\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 71, normalized size = 0.93 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 139, normalized size = 1.83 \begin {gather*} \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 ] \end {gather*}
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 \begin {gather*} \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.52, size = 79, normalized size = 1.04 \begin {gather*} \frac {1}{2} \, \sqrt {c x^{2} + b} c x \mathrm {sgn}\left (x\right ) - \frac {3}{4} \, b \sqrt {c} \log \left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2}\right ) \mathrm {sgn}\left (x\right ) + \frac {2 \, b^{2} \sqrt {c} \mathrm {sgn}\left (x\right )}{{\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b} \end {gather*}
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 \begin {gather*} \int \frac {{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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