Optimal. Leaf size=182 \[ \frac {\sqrt {2} \sqrt {b x^2+2} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right ),1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}}+\frac {x \sqrt {b x^2+2}}{\sqrt {d x^2+3}}-\frac {\sqrt {2} \sqrt {b x^2+2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}} \]
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Rubi [A] time = 0.08, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {422, 418, 492, 411} \[ \frac {x \sqrt {b x^2+2}}{\sqrt {d x^2+3}}+\frac {\sqrt {2} \sqrt {b x^2+2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}}-\frac {\sqrt {2} \sqrt {b x^2+2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 422
Rule 492
Rubi steps
\begin {align*} \int \frac {\sqrt {2+b x^2}}{\sqrt {3+d x^2}} \, dx &=2 \int \frac {1}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx+b \int \frac {x^2}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx\\ &=\frac {x \sqrt {2+b x^2}}{\sqrt {3+d x^2}}+\frac {\sqrt {2} \sqrt {2+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}}-3 \int \frac {\sqrt {2+b x^2}}{\left (3+d x^2\right )^{3/2}} \, dx\\ &=\frac {x \sqrt {2+b x^2}}{\sqrt {3+d x^2}}-\frac {\sqrt {2} \sqrt {2+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}}+\frac {\sqrt {2} \sqrt {2+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 0.20 \[ \frac {\sqrt {2} E\left (\sin ^{-1}\left (\frac {\sqrt {-d} x}{\sqrt {3}}\right )|\frac {3 b}{2 d}\right )}{\sqrt {-d}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{2} + 2}}{\sqrt {d x^{2} + 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b x^{2} + 2}}{\sqrt {d x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 37, normalized size = 0.20 \[ \frac {\sqrt {2}\, \EllipticE \left (\frac {\sqrt {3}\, \sqrt {-d}\, x}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right )}{\sqrt {-d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b x^{2} + 2}}{\sqrt {d x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {b\,x^2+2}}{\sqrt {d\,x^2+3}} \,d x \]
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 \[ \int \frac {\sqrt {b x^{2} + 2}}{\sqrt {d x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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