Optimal. Leaf size=61 \[ \frac {(b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{2 d^{3/2}}-\frac {b \sqrt {c+\frac {d}{x^2}}}{2 d x} \]
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Rubi [A] time = 0.04, antiderivative size = 61, 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 = {459, 335, 217, 206} \begin {gather*} \frac {(b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{2 d^{3/2}}-\frac {b \sqrt {c+\frac {d}{x^2}}}{2 d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 335
Rule 459
Rubi steps
\begin {align*} \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^2} \, dx &=-\frac {b \sqrt {c+\frac {d}{x^2}}}{2 d x}+\frac {(-b c+2 a d) \int \frac {1}{\sqrt {c+\frac {d}{x^2}} x^2} \, dx}{2 d}\\ &=-\frac {b \sqrt {c+\frac {d}{x^2}}}{2 d x}-\frac {(-b c+2 a d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )}{2 d}\\ &=-\frac {b \sqrt {c+\frac {d}{x^2}}}{2 d x}-\frac {(-b c+2 a d) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {1}{\sqrt {c+\frac {d}{x^2}} x}\right )}{2 d}\\ &=-\frac {b \sqrt {c+\frac {d}{x^2}}}{2 d x}+\frac {(b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )}{2 d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 80, normalized size = 1.31 \begin {gather*} \frac {x^2 \sqrt {c x^2+d} (b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {c x^2+d}}{\sqrt {d}}\right )-b \sqrt {d} \left (c x^2+d\right )}{2 d^{3/2} x^3 \sqrt {c+\frac {d}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 82, normalized size = 1.34 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (\frac {(b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {c x^2+d}}{\sqrt {d}}\right )}{2 d^{3/2}}-\frac {b \sqrt {c x^2+d}}{2 d x^2}\right )}{\sqrt {c x^2+d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 144, normalized size = 2.36 \begin {gather*} \left [-\frac {{\left (b c - 2 \, a d\right )} \sqrt {d} x \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \, b d \sqrt {\frac {c x^{2} + d}{x^{2}}}}{4 \, d^{2} x}, -\frac {{\left (b c - 2 \, a d\right )} \sqrt {-d} x \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + b d \sqrt {\frac {c x^{2} + d}{x^{2}}}}{2 \, d^{2} x}\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 [B] time = 0.06, size = 105, normalized size = 1.72 \begin {gather*} -\frac {\sqrt {c \,x^{2}+d}\, \left (2 a \,d^{2} x^{2} \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )-b c d \,x^{2} \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )+\sqrt {c \,x^{2}+d}\, b \,d^{\frac {3}{2}}\right )}{2 \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, d^{\frac {5}{2}} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.18, size = 121, normalized size = 1.98 \begin {gather*} -\frac {1}{4} \, {\left (\frac {2 \, \sqrt {c + \frac {d}{x^{2}}} c x}{{\left (c + \frac {d}{x^{2}}\right )} d x^{2} - d^{2}} + \frac {c \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{d^{\frac {3}{2}}}\right )} b + \frac {a \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{2 \, \sqrt {d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.53, size = 94, normalized size = 1.54 \begin {gather*} \left \{\begin {array}{cl} -\frac {3\,a\,x^2+b}{3\,\sqrt {c}\,x^3} & \text {\ if\ \ }d=0\\ \frac {b\,c\,\ln \left (2\,\sqrt {c+\frac {d}{x^2}}+\frac {2\,\sqrt {d}}{x}\right )}{2\,d^{3/2}}-\frac {b\,\sqrt {c+\frac {d}{x^2}}}{2\,d\,x}-\frac {a\,\ln \left (\sqrt {c+\frac {d}{x^2}}+\frac {\sqrt {d}}{x}\right )}{\sqrt {d}} & \text {\ if\ \ }d\neq 0 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.92, size = 66, normalized size = 1.08 \begin {gather*} - \frac {a \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{\sqrt {d}} - \frac {b \sqrt {c} \sqrt {1 + \frac {d}{c x^{2}}}}{2 d x} + \frac {b c \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{2 d^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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