Optimal. Leaf size=191 \[ \frac {a \sqrt {d x^2+2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{\sqrt {2} \sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}+\frac {b x \sqrt {d x^2+2}}{d \sqrt {f x^2+3}}-\frac {\sqrt {2} b \sqrt {d x^2+2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{d \sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}} \]
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Rubi [A] time = 0.10, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {531, 418, 492, 411} \[ \frac {a \sqrt {d x^2+2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{\sqrt {2} \sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}+\frac {b x \sqrt {d x^2+2}}{d \sqrt {f x^2+3}}-\frac {\sqrt {2} b \sqrt {d x^2+2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{d \sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 492
Rule 531
Rubi steps
\begin {align*} \int \frac {a+b x^2}{\sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx &=a \int \frac {1}{\sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx+b \int \frac {x^2}{\sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx\\ &=\frac {b x \sqrt {2+d x^2}}{d \sqrt {3+f x^2}}+\frac {a \sqrt {2+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{\sqrt {2} \sqrt {f} \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}}-\frac {(3 b) \int \frac {\sqrt {2+d x^2}}{\left (3+f x^2\right )^{3/2}} \, dx}{d}\\ &=\frac {b x \sqrt {2+d x^2}}{d \sqrt {3+f x^2}}-\frac {\sqrt {2} b \sqrt {2+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{d \sqrt {f} \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}}+\frac {a \sqrt {2+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{\sqrt {2} \sqrt {f} \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 81, normalized size = 0.42 \[ -\frac {i \left ((a f-3 b) F\left (i \sinh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )+3 b E\left (i \sinh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )\right )}{\sqrt {3} \sqrt {d} f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )} \sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}}{d f x^{4} + {\left (3 \, d + 2 \, f\right )} x^{2} + 6}, x\right ) \]
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 \[ \int \frac {b x^{2} + a}{\sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 105, normalized size = 0.55 \[ \frac {\sqrt {2}\, \left (a d \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )+2 b \EllipticE \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )-2 b \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )\right )}{2 \sqrt {-f}\, d} \]
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 \[ \int \frac {b x^{2} + a}{\sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}}\,{d x} \]
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 \[ \int \frac {b\,x^2+a}{\sqrt {d\,x^2+2}\,\sqrt {f\,x^2+3}} \,d x \]
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 \[ \int \frac {a + b x^{2}}{\sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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