Optimal. Leaf size=59 \[ \frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2 c^{3/2}}+\frac {a x^2 \sqrt {c+\frac {d}{x^2}}}{2 c} \]
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Rubi [A] time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {446, 78, 63, 208} \begin {gather*} \frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2 c^{3/2}}+\frac {a x^2 \sqrt {c+\frac {d}{x^2}}}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\sqrt {c+\frac {d}{x^2}}} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {a+b x}{x^2 \sqrt {c+d x}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {a \sqrt {c+\frac {d}{x^2}} x^2}{2 c}-\frac {\left (b c-\frac {a d}{2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,\frac {1}{x^2}\right )}{2 c}\\ &=\frac {a \sqrt {c+\frac {d}{x^2}} x^2}{2 c}-\frac {\left (b c-\frac {a d}{2}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+\frac {d}{x^2}}\right )}{c d}\\ &=\frac {a \sqrt {c+\frac {d}{x^2}} x^2}{2 c}+\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 79, normalized size = 1.34 \begin {gather*} \frac {\sqrt {c x^2+d} (2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {c x^2+d}}\right )+a \sqrt {c} x \left (c x^2+d\right )}{2 c^{3/2} x \sqrt {c+\frac {d}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 67, normalized size = 1.14 \begin {gather*} \frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {\frac {c x^2+d}{x^2}}}{\sqrt {c}}\right )}{2 c^{3/2}}+\frac {a x^2 \sqrt {\frac {c x^2+d}{x^2}}}{2 c} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 146, normalized size = 2.47 \begin {gather*} \left [\frac {2 \, a c x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - {\left (2 \, b c - a d\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right )}{4 \, c^{2}}, \frac {a c x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - {\left (2 \, b c - a d\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right )}{2 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 88, normalized size = 1.49 \begin {gather*} \frac {\sqrt {c x^{4} + d x^{2}} a}{2 \, c} - \frac {{\left (2 \, b c - a d\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )} \sqrt {c} - d \right |}\right )}{4 \, c^{\frac {3}{2}}} + \frac {2 \, b c \log \left ({\left | d \right |}\right ) - a d \log \left ({\left | d \right |}\right )}{4 \, c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 90, normalized size = 1.53 \begin {gather*} \frac {\sqrt {c \,x^{2}+d}\, \left (-a c d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )+2 b \,c^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )+\sqrt {c \,x^{2}+d}\, a \,c^{\frac {3}{2}} x \right )}{2 \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, c^{\frac {5}{2}} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.20, size = 109, normalized size = 1.85 \begin {gather*} \frac {1}{4} \, a {\left (\frac {2 \, \sqrt {c + \frac {d}{x^{2}}} d}{{\left (c + \frac {d}{x^{2}}\right )} c - c^{2}} + \frac {d \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} - \frac {b \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{2 \, \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.08, size = 59, normalized size = 1.00 \begin {gather*} \frac {b\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {a\,x^2\,\sqrt {c+\frac {d}{x^2}}}{2\,c}-\frac {a\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2\,c^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 84.41, size = 66, normalized size = 1.12 \begin {gather*} \frac {a \sqrt {d} x \sqrt {\frac {c x^{2}}{d} + 1}}{2 c} - \frac {a d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{2 c^{\frac {3}{2}}} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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