Optimal. Leaf size=43 \[ \frac {e \sqrt {a+c x^2}}{c}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c}} \]
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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {655, 223, 212}
\begin {gather*} \frac {d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c}}+\frac {e \sqrt {a+c x^2}}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 655
Rubi steps
\begin {align*} \int \frac {d+e x}{\sqrt {a+c x^2}} \, dx &=\frac {e \sqrt {a+c x^2}}{c}+d \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=\frac {e \sqrt {a+c x^2}}{c}+d \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )\\ &=\frac {e \sqrt {a+c x^2}}{c}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 46, normalized size = 1.07 \begin {gather*} \frac {e \sqrt {a+c x^2}}{c}-\frac {d \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{\sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 37, normalized size = 0.86
method | result | size |
default | \(\frac {e \sqrt {c \,x^{2}+a}}{c}+\frac {d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}}\) | \(37\) |
risch | \(\frac {e \sqrt {c \,x^{2}+a}}{c}+\frac {d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 30, normalized size = 0.70 \begin {gather*} \frac {d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {c}} + \frac {\sqrt {c x^{2} + a} e}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.13, size = 94, normalized size = 2.19 \begin {gather*} \left [\frac {\sqrt {c} d \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, \sqrt {c x^{2} + a} e}{2 \, c}, -\frac {\sqrt {-c} d \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - \sqrt {c x^{2} + a} e}{c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 78 vs.
\(2 (37) = 74\).
time = 0.56, size = 102, normalized size = 2.37 \begin {gather*} d \left (\begin {cases} \frac {\sqrt {- \frac {a}{c}} \operatorname {asin}{\left (x \sqrt {- \frac {c}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge c < 0 \\\frac {\sqrt {\frac {a}{c}} \operatorname {asinh}{\left (x \sqrt {\frac {c}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge c > 0 \\\frac {\sqrt {- \frac {a}{c}} \operatorname {acosh}{\left (x \sqrt {- \frac {c}{a}} \right )}}{\sqrt {- a}} & \text {for}\: c > 0 \wedge a < 0 \end {cases}\right ) + e \left (\begin {cases} \frac {x^{2}}{2 \sqrt {a}} & \text {for}\: c = 0 \\\frac {\sqrt {a + c x^{2}}}{c} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.95, size = 40, normalized size = 0.93 \begin {gather*} -\frac {d \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{\sqrt {c}} + \frac {\sqrt {c x^{2} + a} e}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.62, size = 36, normalized size = 0.84 \begin {gather*} \frac {e\,\sqrt {c\,x^2+a}}{c}+\frac {d\,\ln \left (\sqrt {c}\,x+\sqrt {c\,x^2+a}\right )}{\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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