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 = 0.04, 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 = {1605, 248, 44,
65, 214} \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 214
Rule 248
Rule 1605
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b}{x}}} \, dx,x,c+d x^2\right )}{2 d}\\ &=-\frac {\text {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 \text {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 {\text {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 = 0.11, size = 88, normalized size = 1.22 \begin {gather*} \frac {\left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{2 a d}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right )}{2 a^{3/2} d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(183\) vs.
\(2(60)=120\).
time = 0.24, size = 184, normalized size = 2.56
method | result | size |
derivativedivides | \(\frac {\sqrt {\frac {a \left (d \,x^{2}+c \right )+b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (2 \sqrt {\left (d \,x^{2}+c \right ) \left (a \left (d \,x^{2}+c \right )+b \right )}\, \sqrt {a}-b \ln \left (\frac {2 \sqrt {\left (d \,x^{2}+c \right ) \left (a \left (d \,x^{2}+c \right )+b \right )}\, \sqrt {a}+2 a \left (d \,x^{2}+c \right )+b}{2 \sqrt {a}}\right )\right )}{4 d \sqrt {\left (d \,x^{2}+c \right ) \left (a \left (d \,x^{2}+c \right )+b \right )}\, a^{\frac {3}{2}}}\) | \(134\) |
risch | \(\frac {a d \,x^{2}+a c +b}{2 d a \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}-\frac {\ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right ) b \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{4 a \sqrt {a \,d^{2}}\, \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(169\) |
default | \(\frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-\ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}\right )}{4 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a d \sqrt {a \,d^{2}}}\) | \(184\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs.
\(2 (60) = 120\).
time = 0.51, size = 129, normalized size = 1.79 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 267, normalized size = 3.71 \begin {gather*} \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 ] \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 {x}{\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs.
\(2 (60) = 120\).
time = 4.82, size = 129, normalized size = 1.79 \begin {gather*} \frac {\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 )}{a^{\frac {3}{2}} {\left | d \right |}} + \frac {2 \, \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}}{a d}}{4 \, \mathrm {sgn}\left (d x^{2} + c\right )} \end {gather*}
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 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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