Optimal. Leaf size=58 \[ \frac {d x \sqrt {a+b x^2}}{2 b}+\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {396, 223, 212}
\begin {gather*} \frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 396
Rubi steps
\begin {align*} \int \frac {c+d x^2}{\sqrt {a+b x^2}} \, dx &=\frac {d x \sqrt {a+b x^2}}{2 b}-\frac {(-2 b c+a d) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 b}\\ &=\frac {d x \sqrt {a+b x^2}}{2 b}-\frac {(-2 b c+a d) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 b}\\ &=\frac {d x \sqrt {a+b x^2}}{2 b}+\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 59, normalized size = 1.02 \begin {gather*} \frac {d x \sqrt {a+b x^2}}{2 b}+\frac {(-2 b c+a d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 63, normalized size = 1.09
method | result | size |
risch | \(\frac {d x \sqrt {b \,x^{2}+a}}{2 b}-\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) a d}{2 b^{\frac {3}{2}}}+\frac {c \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}\) | \(62\) |
default | \(d \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )+\frac {c \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 47, normalized size = 0.81 \begin {gather*} \frac {\sqrt {b x^{2} + a} d x}{2 \, b} + \frac {c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} - \frac {a d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.55, size = 113, normalized size = 1.95 \begin {gather*} \left [\frac {2 \, \sqrt {b x^{2} + a} b d x - {\left (2 \, b c - a d\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right )}{4 \, b^{2}}, \frac {\sqrt {b x^{2} + a} b d x - {\left (2 \, b c - a d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right )}{2 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.60, size = 126, normalized size = 2.17 \begin {gather*} \frac {\sqrt {a} d x \sqrt {1 + \frac {b x^{2}}{a}}}{2 b} - \frac {a d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {3}{2}}} + c \left (\begin {cases} \frac {\sqrt {- \frac {a}{b}} \operatorname {asin}{\left (x \sqrt {- \frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b < 0 \\\frac {\sqrt {\frac {a}{b}} \operatorname {asinh}{\left (x \sqrt {\frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b > 0 \\\frac {\sqrt {- \frac {a}{b}} \operatorname {acosh}{\left (x \sqrt {- \frac {b}{a}} \right )}}{\sqrt {- a}} & \text {for}\: b > 0 \wedge a < 0 \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.82, size = 49, normalized size = 0.84 \begin {gather*} \frac {\sqrt {b x^{2} + a} d x}{2 \, b} - \frac {{\left (2 \, b c - a d\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.51, size = 86, normalized size = 1.48 \begin {gather*} \left \{\begin {array}{cl} \frac {d\,x^3+3\,c\,x}{3\,\sqrt {a}} & \text {\ if\ \ }b=0\\ \frac {c\,\ln \left (\sqrt {b}\,x+\sqrt {b\,x^2+a}\right )}{\sqrt {b}}-\frac {a\,d\,\ln \left (2\,\sqrt {b}\,x+2\,\sqrt {b\,x^2+a}\right )}{2\,b^{3/2}}+\frac {d\,x\,\sqrt {b\,x^2+a}}{2\,b} & \text {\ if\ \ }b\neq 0 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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