Optimal. Leaf size=242 \[ -\frac {(a-2 b) x \sqrt {2+x^2}}{b^2 \sqrt {1+x^2}}+\frac {x \sqrt {1+x^2} \sqrt {2+x^2}}{3 b}+\frac {\sqrt {2} (a-2 b) \sqrt {2+x^2} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{b^2 \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}}-\frac {(3 a-7 b) \sqrt {2+x^2} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {2} b^2 \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}}+\frac {(a-2 b) (a-b) \sqrt {2+x^2} \Pi \left (1-\frac {b}{a};\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} a b^2 \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}} \]
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Rubi [A]
time = 0.09, antiderivative size = 239, normalized size of antiderivative = 0.99, number of steps
used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {557, 553, 542,
545, 429, 506, 422} \begin {gather*} -\frac {\sqrt {2} \sqrt {x^2+2} (3 a-5 b) F\left (\text {ArcTan}(x)\left |\frac {1}{2}\right .\right )}{3 b^2 \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}}+\frac {\sqrt {2} \sqrt {x^2+2} (a-2 b) E\left (\text {ArcTan}(x)\left |\frac {1}{2}\right .\right )}{b^2 \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}}+\frac {2 \sqrt {x^2+1} (a-b)^2 \Pi \left (1-\frac {2 b}{a};\left .\text {ArcTan}\left (\frac {x}{\sqrt {2}}\right )\right |-1\right )}{a b^2 \sqrt {\frac {x^2+1}{x^2+2}} \sqrt {x^2+2}}-\frac {x \sqrt {x^2+2} (a-2 b)}{b^2 \sqrt {x^2+1}}+\frac {x \sqrt {x^2+1} \sqrt {x^2+2}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 506
Rule 542
Rule 545
Rule 553
Rule 557
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {2+x^2}}{a+b x^2} \, dx &=\frac {\int \frac {\sqrt {2+x^2} \left (-a+2 b+b x^2\right )}{\sqrt {1+x^2}} \, dx}{b^2}+\frac {(a-b)^2 \int \frac {\sqrt {2+x^2}}{\sqrt {1+x^2} \left (a+b x^2\right )} \, dx}{b^2}\\ &=\frac {x \sqrt {1+x^2} \sqrt {2+x^2}}{3 b}+\frac {2 (a-b)^2 \sqrt {1+x^2} \Pi \left (1-\frac {2 b}{a};\left .\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-1\right )}{a b^2 \sqrt {\frac {1+x^2}{2+x^2}} \sqrt {2+x^2}}+\frac {\int \frac {-2 (3 a-5 b)-3 (a-2 b) x^2}{\sqrt {1+x^2} \sqrt {2+x^2}} \, dx}{3 b^2}\\ &=\frac {x \sqrt {1+x^2} \sqrt {2+x^2}}{3 b}+\frac {2 (a-b)^2 \sqrt {1+x^2} \Pi \left (1-\frac {2 b}{a};\left .\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-1\right )}{a b^2 \sqrt {\frac {1+x^2}{2+x^2}} \sqrt {2+x^2}}-\frac {(2 (3 a-5 b)) \int \frac {1}{\sqrt {1+x^2} \sqrt {2+x^2}} \, dx}{3 b^2}-\frac {(a-2 b) \int \frac {x^2}{\sqrt {1+x^2} \sqrt {2+x^2}} \, dx}{b^2}\\ &=-\frac {(a-2 b) x \sqrt {2+x^2}}{b^2 \sqrt {1+x^2}}+\frac {x \sqrt {1+x^2} \sqrt {2+x^2}}{3 b}-\frac {\sqrt {2} (3 a-5 b) \sqrt {2+x^2} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 b^2 \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}}+\frac {2 (a-b)^2 \sqrt {1+x^2} \Pi \left (1-\frac {2 b}{a};\left .\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-1\right )}{a b^2 \sqrt {\frac {1+x^2}{2+x^2}} \sqrt {2+x^2}}+\frac {(a-2 b) \int \frac {\sqrt {2+x^2}}{\left (1+x^2\right )^{3/2}} \, dx}{b^2}\\ &=-\frac {(a-2 b) x \sqrt {2+x^2}}{b^2 \sqrt {1+x^2}}+\frac {x \sqrt {1+x^2} \sqrt {2+x^2}}{3 b}+\frac {\sqrt {2} (a-2 b) \sqrt {2+x^2} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{b^2 \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}}-\frac {\sqrt {2} (3 a-5 b) \sqrt {2+x^2} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 b^2 \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}}+\frac {2 (a-b)^2 \sqrt {1+x^2} \Pi \left (1-\frac {2 b}{a};\left .\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-1\right )}{a b^2 \sqrt {\frac {1+x^2}{2+x^2}} \sqrt {2+x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.22, size = 204, normalized size = 0.84 \begin {gather*} \frac {a b^2 x \sqrt {1+x^2} \sqrt {2+x^2}+3 i a (a-2 b) b E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-i a \left (3 a^2-9 a b+7 b^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+3 i a^3 \Pi \left (\frac {2 b}{a};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-12 i a^2 b \Pi \left (\frac {2 b}{a};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+15 i a b^2 \Pi \left (\frac {2 b}{a};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-6 i b^3 \Pi \left (\frac {2 b}{a};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )}{3 a b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.22, size = 370, normalized size = 1.53
method | result | size |
risch | \(\frac {x \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}}{3 b}-\frac {\left (\frac {3 i \left (a -2 b \right ) \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )-\EllipticE \left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )\right )}{2 b \sqrt {x^{4}+3 x^{2}+2}}+\frac {i \left (3 a^{2}-12 a b +13 b^{2}\right ) \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )}{2 b^{2} \sqrt {x^{4}+3 x^{2}+2}}-\frac {3 i \left (a^{3}-4 a^{2} b +5 a \,b^{2}-2 b^{3}\right ) \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {i x \sqrt {2}}{2}, \frac {2 b}{a}, \sqrt {2}\right )}{b^{2} a \sqrt {x^{4}+3 x^{2}+2}}\right ) \sqrt {\left (x^{2}+1\right ) \left (x^{2}+2\right )}}{3 b \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}}\) | \(263\) |
default | \(-\frac {\sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, \left (-a \,b^{2} x^{5}+3 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, \EllipticF \left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right ) a^{3}-9 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, \EllipticF \left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right ) a^{2} b +7 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, \EllipticF \left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right ) a \,b^{2}-3 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, \EllipticE \left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right ) a^{2} b +6 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, \EllipticE \left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right ) a \,b^{2}-3 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, \EllipticPi \left (\frac {i x \sqrt {2}}{2}, \frac {2 b}{a}, \sqrt {2}\right ) a^{3}+12 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, \EllipticPi \left (\frac {i x \sqrt {2}}{2}, \frac {2 b}{a}, \sqrt {2}\right ) a^{2} b -15 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, \EllipticPi \left (\frac {i x \sqrt {2}}{2}, \frac {2 b}{a}, \sqrt {2}\right ) a \,b^{2}+6 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, \EllipticPi \left (\frac {i x \sqrt {2}}{2}, \frac {2 b}{a}, \sqrt {2}\right ) b^{3}-3 a \,b^{2} x^{3}-2 a \,b^{2} x \right )}{3 \left (x^{4}+3 x^{2}+2\right ) b^{3} a}\) | \(370\) |
elliptic | \(\frac {\sqrt {\left (x^{2}+1\right ) \left (x^{2}+2\right )}\, \left (\frac {x \sqrt {x^{4}+3 x^{2}+2}}{3 b}+\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, a \EllipticE \left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )}{2 \sqrt {x^{4}+3 x^{2}+2}\, b^{2}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right ) a^{2}}{2 \sqrt {x^{4}+3 x^{2}+2}\, b^{3}}-\frac {7 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )}{6 b \sqrt {x^{4}+3 x^{2}+2}}-\frac {4 i a \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {i x \sqrt {2}}{2}, \frac {2 b}{a}, \sqrt {2}\right )}{b^{2} \sqrt {x^{4}+3 x^{2}+2}}+\frac {3 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right ) a}{2 \sqrt {x^{4}+3 x^{2}+2}\, b^{2}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticE \left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )}{b \sqrt {x^{4}+3 x^{2}+2}}+\frac {5 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {i x \sqrt {2}}{2}, \frac {2 b}{a}, \sqrt {2}\right )}{b \sqrt {x^{4}+3 x^{2}+2}}-\frac {2 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {i x \sqrt {2}}{2}, \frac {2 b}{a}, \sqrt {2}\right )}{a \sqrt {x^{4}+3 x^{2}+2}}+\frac {i a^{2} \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {i x \sqrt {2}}{2}, \frac {2 b}{a}, \sqrt {2}\right )}{b^{3} \sqrt {x^{4}+3 x^{2}+2}}\right )}{\sqrt {x^{2}+1}\, \sqrt {x^{2}+2}}\) | \(513\) |
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} + 1\right )^{\frac {3}{2}} \sqrt {x^{2} + 2}}{a + b x^{2}}\, 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 {{\left (x^2+1\right )}^{3/2}\,\sqrt {x^2+2}}{b\,x^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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