Optimal. Leaf size=241 \[ -\frac {2 (3 b-d) x \sqrt {2+b x^2}}{3 b d \sqrt {3+d x^2}}+\frac {x \sqrt {2+b x^2} \sqrt {3+d x^2}}{3 d}+\frac {2 \sqrt {2} (3 b-d) \sqrt {2+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{3 b d^{3/2} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}}-\frac {\sqrt {2} \sqrt {2+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{d^{3/2} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {489, 545, 429,
506, 422} \begin {gather*} -\frac {\sqrt {2} \sqrt {b x^2+2} F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{d^{3/2} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}}+\frac {2 \sqrt {2} (3 b-d) \sqrt {b x^2+2} E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{3 b d^{3/2} \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}}{3 d}-\frac {2 x (3 b-d) \sqrt {b x^2+2}}{3 b d \sqrt {d x^2+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 489
Rule 506
Rule 545
Rubi steps
\begin {align*} \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}}{3 d}-\frac {\int \frac {6+2 (3 b-d) x^2}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx}{3 d}\\ &=\frac {x \sqrt {2+b x^2} \sqrt {3+d x^2}}{3 d}-\frac {2 \int \frac {1}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx}{d}-\frac {(2 (3 b-d)) \int \frac {x^2}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx}{3 d}\\ &=-\frac {2 (3 b-d) x \sqrt {2+b x^2}}{3 b d \sqrt {3+d x^2}}+\frac {x \sqrt {2+b x^2} \sqrt {3+d x^2}}{3 d}-\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 )}{d^{3/2} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}}+\frac {(2 (3 b-d)) \int \frac {\sqrt {2+b x^2}}{\left (3+d x^2\right )^{3/2}} \, dx}{b d}\\ &=-\frac {2 (3 b-d) x \sqrt {2+b x^2}}{3 b d \sqrt {3+d x^2}}+\frac {x \sqrt {2+b x^2} \sqrt {3+d x^2}}{3 d}+\frac {2 \sqrt {2} (3 b-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 d^{3/2} \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 )}{d^{3/2} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.82, size = 127, normalized size = 0.53 \begin {gather*} \frac {\sqrt {b} d x \sqrt {2+b x^2} \sqrt {3+d x^2}+2 i \sqrt {3} (3 b-d) E\left (i \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2}}\right )|\frac {2 d}{3 b}\right )-2 i \sqrt {3} (3 b-2 d) F\left (i \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2}}\right )|\frac {2 d}{3 b}\right )}{3 \sqrt {b} d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 306, normalized size = 1.27
method | result | size |
risch | \(\frac {x \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}}{3 d}-\frac {2 \left (\frac {3 \sqrt {3 d \,x^{2}+9}\, \sqrt {2 b \,x^{2}+4}\, \EllipticF \left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )}{2 \sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}}-\frac {\left (3 b -d \right ) \sqrt {3 d \,x^{2}+9}\, \sqrt {2 b \,x^{2}+4}\, \left (\EllipticF \left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )\right )}{\sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}\, b}\right ) \sqrt {\left (b \,x^{2}+2\right ) \left (d \,x^{2}+3\right )}}{3 d \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}}\) | \(259\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+2\right ) \left (d \,x^{2}+3\right )}\, \left (\frac {x \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}}{3 d}-\frac {\sqrt {3 d \,x^{2}+9}\, \sqrt {2 b \,x^{2}+4}\, \EllipticF \left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )}{d \sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}}-\frac {\left (2-\frac {6 b +4 d}{3 d}\right ) \sqrt {3 d \,x^{2}+9}\, \sqrt {2 b \,x^{2}+4}\, \left (\EllipticF \left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )\right )}{\sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}\, b}\right )}{\sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}}\) | \(268\) |
default | \(\frac {\sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}\, \left (b^{2} d \,x^{5} \sqrt {-d}+3 b^{2} x^{3} \sqrt {-d}+2 b d \,x^{3} \sqrt {-d}+3 \sqrt {2}\, \EllipticF \left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right ) b \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}-2 \sqrt {2}\, \EllipticF \left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right ) d \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}-6 \sqrt {2}\, \EllipticE \left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right ) b \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}+2 \sqrt {2}\, \EllipticE \left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right ) d \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}+6 b x \sqrt {-d}\right )}{3 \left (b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6\right ) d \sqrt {-d}\, b}\) | \(306\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {x^{2} \sqrt {b x^{2} + 2}}{\sqrt {d x^{2} + 3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^2\,\sqrt {b\,x^2+2}}{\sqrt {d\,x^2+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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