Optimal. Leaf size=235 \[ \frac {2 \sqrt {2} \sqrt {b x^2+2} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right ),1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}}+\frac {1}{3} x \sqrt {b x^2+2} \sqrt {d x^2+3}+\frac {x (3 b+2 d) \sqrt {b x^2+2}}{3 b \sqrt {d x^2+3}}-\frac {\sqrt {2} (3 b+2 d) \sqrt {b x^2+2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{3 b \sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}} \]
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Rubi [A] time = 0.13, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {417, 531, 418, 492, 411} \[ \frac {1}{3} x \sqrt {b x^2+2} \sqrt {d x^2+3}+\frac {x (3 b+2 d) \sqrt {b x^2+2}}{3 b \sqrt {d x^2+3}}+\frac {2 \sqrt {2} \sqrt {b x^2+2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}}-\frac {\sqrt {2} (3 b+2 d) \sqrt {b x^2+2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{3 b \sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 417
Rule 418
Rule 492
Rule 531
Rubi steps
\begin {align*} \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx &=\frac {1}{3} x \sqrt {2+b x^2} \sqrt {3+d x^2}+\frac {2}{3} \int \frac {6+\frac {1}{2} (3 b+2 d) x^2}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx\\ &=\frac {1}{3} x \sqrt {2+b x^2} \sqrt {3+d x^2}+4 \int \frac {1}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx+\frac {1}{3} (3 b+2 d) \int \frac {x^2}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx\\ &=\frac {(3 b+2 d) x \sqrt {2+b x^2}}{3 b \sqrt {3+d x^2}}+\frac {1}{3} x \sqrt {2+b x^2} \sqrt {3+d x^2}+\frac {2 \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}}+\frac {(-3 b-2 d) \int \frac {\sqrt {2+b x^2}}{\left (3+d x^2\right )^{3/2}} \, dx}{b}\\ &=\frac {(3 b+2 d) x \sqrt {2+b x^2}}{3 b \sqrt {3+d x^2}}+\frac {1}{3} x \sqrt {2+b x^2} \sqrt {3+d x^2}-\frac {\sqrt {2} (3 b+2 d) \sqrt {2+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{3 b \sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}}+\frac {2 \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 [C] time = 0.11, size = 127, normalized size = 0.54 \[ \frac {i \sqrt {3} (3 b-2 d) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2}}\right ),\frac {2 d}{3 b}\right )+\sqrt {b} d x \sqrt {b x^2+2} \sqrt {d x^2+3}-i \sqrt {3} (3 b+2 d) E\left (i \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2}}\right )|\frac {2 d}{3 b}\right )}{3 \sqrt {b} d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}, x\right ) \]
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 \[ \int \sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 303, normalized size = 1.29 \[ \frac {\sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}\, \left (\sqrt {-d}\, b^{2} d \,x^{5}+3 \sqrt {-d}\, b^{2} x^{3}+2 \sqrt {-d}\, b d \,x^{3}+6 \sqrt {-d}\, b x +3 \sqrt {2}\, \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}\, b \EllipticE \left (\frac {\sqrt {3}\, \sqrt {-d}\, x}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right )+3 \sqrt {2}\, \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}\, b \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-d}\, x}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right )+2 \sqrt {2}\, \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}\, d \EllipticE \left (\frac {\sqrt {3}\, \sqrt {-d}\, x}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right )-2 \sqrt {2}\, \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}\, d \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-d}\, x}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right )\right )}{3 \left (b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6\right ) \sqrt {-d}\, b} \]
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 \[ \int \sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {b\,x^2+2}\,\sqrt {d\,x^2+3} \,d x \]
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 \[ \int \sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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