Optimal. Leaf size=43 \[ \frac {a \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {b \sqrt {c+\frac {d}{x^2}}}{d} \]
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Rubi [A] time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {446, 80, 63, 208} \begin {gather*} \frac {a \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {b \sqrt {c+\frac {d}{x^2}}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {a+b x}{x \sqrt {c+d x}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {b \sqrt {c+\frac {d}{x^2}}}{d}-\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {b \sqrt {c+\frac {d}{x^2}}}{d}-\frac {a \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+\frac {d}{x^2}}\right )}{d}\\ &=-\frac {b \sqrt {c+\frac {d}{x^2}}}{d}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 73, normalized size = 1.70 \begin {gather*} \frac {a d x \sqrt {c x^2+d} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {c x^2+d}}\right )-b \sqrt {c} \left (c x^2+d\right )}{\sqrt {c} d x^2 \sqrt {c+\frac {d}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 51, normalized size = 1.19 \begin {gather*} \frac {a \tanh ^{-1}\left (\frac {\sqrt {\frac {c x^2+d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {b \sqrt {\frac {c x^2+d}{x^2}}}{d} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 130, normalized size = 3.02 \begin {gather*} \left [\frac {a \sqrt {c} d \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) - 2 \, b c \sqrt {\frac {c x^{2} + d}{x^{2}}}}{2 \, c d}, -\frac {a \sqrt {-c} d \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + b c \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 70, normalized size = 1.63 \begin {gather*} -\frac {\sqrt {c \,x^{2}+d}\, \left (-a d x \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )+\sqrt {c \,x^{2}+d}\, b \sqrt {c}\right )}{\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \sqrt {c}\, d \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 54, normalized size = 1.26 \begin {gather*} -\frac {a \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{2 \, \sqrt {c}} - \frac {b \sqrt {c + \frac {d}{x^{2}}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.85, size = 35, normalized size = 0.81 \begin {gather*} \frac {a\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {b\,\sqrt {c+\frac {d}{x^2}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 22.34, size = 63, normalized size = 1.47 \begin {gather*} - \frac {a \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{c}} \sqrt {c + \frac {d}{x^{2}}}} \right )}}{c \sqrt {- \frac {1}{c}}} + \frac {b \left (\begin {cases} - \frac {1}{\sqrt {c} x^{2}} & \text {for}\: d = 0 \\- \frac {2 \sqrt {c + \frac {d}{x^{2}}}}{d} & \text {otherwise} \end {cases}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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