Optimal. Leaf size=54 \[ \frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}+\frac {x (b c-a d)}{a b \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 54, 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 = {385, 217, 206} \[ \frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}+\frac {x (b c-a d)}{a b \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 385
Rubi steps
\begin {align*} \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {(b c-a d) x}{a b \sqrt {a+b x^2}}+\frac {d \int \frac {1}{\sqrt {a+b x^2}} \, dx}{b}\\ &=\frac {(b c-a d) x}{a b \sqrt {a+b x^2}}+\frac {d \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{b}\\ &=\frac {(b c-a d) x}{a b \sqrt {a+b x^2}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 70, normalized size = 1.30 \[ \frac {a^{3/2} d \sqrt {\frac {b x^2}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+\sqrt {b} x (b c-a d)}{a b^{3/2} \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 167, normalized size = 3.09 \[ \left [\frac {2 \, {\left (b^{2} c - a b d\right )} \sqrt {b x^{2} + a} x + {\left (a b d x^{2} + a^{2} d\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right )}{2 \, {\left (a b^{3} x^{2} + a^{2} b^{2}\right )}}, \frac {{\left (b^{2} c - a b d\right )} \sqrt {b x^{2} + a} x - {\left (a b d x^{2} + a^{2} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right )}{a b^{3} x^{2} + a^{2} b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 50, normalized size = 0.93 \[ -\frac {d \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {3}{2}}} + \frac {{\left (b c - a d\right )} x}{\sqrt {b x^{2} + a} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 54, normalized size = 1.00 \[ \frac {c x}{\sqrt {b \,x^{2}+a}\, a}-\frac {d x}{\sqrt {b \,x^{2}+a}\, b}+\frac {d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 46, normalized size = 0.85 \[ \frac {c x}{\sqrt {b x^{2} + a} a} - \frac {d x}{\sqrt {b x^{2} + a} b} + \frac {d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.12, size = 53, normalized size = 0.98 \[ \frac {d\,\ln \left (\sqrt {b}\,x+\sqrt {b\,x^2+a}\right )}{b^{3/2}}+\frac {c\,x}{a\,\sqrt {b\,x^2+a}}-\frac {d\,x}{b\,\sqrt {b\,x^2+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.17, size = 60, normalized size = 1.11 \[ d \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {x}{\sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + \frac {c x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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