Optimal. Leaf size=72 \[ \frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{2 a d}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )}{2 a^{3/2} d} \]
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Rubi [A] time = 0.05, antiderivative size = 72, 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 = {1591, 242, 51, 63, 208} \[ \frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{2 a d}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )}{2 a^{3/2} d} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 242
Rule 1591
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b}{x}}} \, dx,x,c+d x^2\right )}{2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\frac {1}{c+d x^2}\right )}{2 d}\\ &=\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{2 a d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{c+d x^2}\right )}{4 a d}\\ &=\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{2 a d}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{c+d x^2}}\right )}{2 a d}\\ &=\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{2 a d}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )}{2 a^{3/2} d}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 70, normalized size = 0.97 \[ \frac {\sqrt {a} \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}-b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )}{2 a^{3/2} d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 267, normalized size = 3.71 \[ \left [\frac {\sqrt {a} b \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} - 4 \, {\left (2 \, a d^{2} x^{4} + {\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt {a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \, {\left (a d x^{2} + a c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, a^{2} d}, \frac {\sqrt {-a} b \arctan \left (\frac {{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt {-a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) + 2 \, {\left (a d x^{2} + a c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{4 \, a^{2} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.51, size = 138, normalized size = 1.92 \[ \frac {b \log \left ({\left | -2 \, a c d - 2 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} \sqrt {a} {\left | d \right |} - b d \right |}\right )}{4 \, a^{\frac {3}{2}} {\left | d \right |} \mathrm {sgn}\left (d x^{2} + c\right )} + \frac {\sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}}{2 \, a d \mathrm {sgn}\left (d x^{2} + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 184, normalized size = 2.56 \[ \frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-b d \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )+2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\right )}{4 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {a \,d^{2}}\, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.81, size = 129, normalized size = 1.79 \[ -\frac {b \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} d - \frac {{\left (a d x^{2} + a c + b\right )} a d}{d x^{2} + c}\right )}} + \frac {b \log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right )}{4 \, a^{\frac {3}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.34, size = 111, normalized size = 1.54 \[ \frac {\sqrt {\frac {a\,\left (d\,x^2+c\right )}{b}+1}\,\left (d\,x^2+c\right )\,\left (\frac {3\,\sqrt {b}\,\sqrt {b+a\,\left (d\,x^2+c\right )}}{2\,a\,\left (d\,x^2+c\right )}+\frac {b^{3/2}\,\mathrm {asin}\left (\frac {\sqrt {a}\,\sqrt {d\,x^2+c}\,1{}\mathrm {i}}{\sqrt {b}}\right )\,3{}\mathrm {i}}{2\,a^{3/2}\,{\left (d\,x^2+c\right )}^{3/2}}\right )}{3\,d\,\sqrt {a+\frac {b}{d\,x^2+c}}} \]
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 \[ \int \frac {x}{\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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