Optimal. Leaf size=242 \[ -\frac {\sqrt {x^2+2} x (a-2 b)}{b^2 \sqrt {x^2+1}}-\frac {\sqrt {x^2+2} (3 a-7 b) F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {2} b^2 \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}}+\frac {\sqrt {2} \sqrt {x^2+2} (a-2 b) E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{b^2 \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}}+\frac {\sqrt {x^2+2} (a-2 b) (a-b) \Pi \left (1-\frac {b}{a};\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} a b^2 \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}}+\frac {\sqrt {x^2+1} \sqrt {x^2+2} x}{3 b} \]
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Rubi [A] time = 0.15, 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 = {543, 539, 528, 531, 418, 492, 411} \[ -\frac {x \sqrt {x^2+2} (a-2 b)}{b^2 \sqrt {x^2+1}}-\frac {\sqrt {2} \sqrt {x^2+2} (3 a-5 b) F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 b^2 \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}}+\frac {\sqrt {2} \sqrt {x^2+2} (a-2 b) E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{b^2 \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}}+\frac {2 \sqrt {x^2+1} (a-b)^2 \Pi \left (1-\frac {2 b}{a};\left .\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-1\right )}{a b^2 \sqrt {\frac {x^2+1}{x^2+2}} \sqrt {x^2+2}}+\frac {x \sqrt {x^2+1} \sqrt {x^2+2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 492
Rule 528
Rule 531
Rule 539
Rule 543
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] time = 0.51, size = 204, normalized size = 0.84 \[ \frac {3 i a^3 \Pi \left (\frac {2 b}{a};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-i a \left (3 a^2-9 a b+7 b^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-12 i a^2 b \Pi \left (\frac {2 b}{a};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-6 i b^3 \Pi \left (\frac {2 b}{a};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+a b^2 x \sqrt {x^2+1} \sqrt {x^2+2}+15 i a b^2 \Pi \left (\frac {2 b}{a};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+3 i a b (a-2 b) E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )}{3 a b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 22.28, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{2} + 2} {\left (x^{2} + 1\right )}^{\frac {3}{2}}}{b x^{2} + a}, x\right ) \]
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 \[ \int \frac {\sqrt {x^{2} + 2} {\left (x^{2} + 1\right )}^{\frac {3}{2}}}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 370, normalized size = 1.53 \[ -\frac {\sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, \left (-a \,b^{2} x^{5}-3 a \,b^{2} x^{3}+3 i \sqrt {x^{2}+2}\, \sqrt {x^{2}+1}\, a^{3} \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-3 i \sqrt {x^{2}+2}\, \sqrt {x^{2}+1}\, a^{3} \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 b}{a}, \sqrt {2}\right )-3 i \sqrt {x^{2}+2}\, \sqrt {x^{2}+1}\, a^{2} b \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-9 i \sqrt {x^{2}+2}\, \sqrt {x^{2}+1}\, a^{2} b \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+12 i \sqrt {x^{2}+2}\, \sqrt {x^{2}+1}\, a^{2} b \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 b}{a}, \sqrt {2}\right )-2 a \,b^{2} x +6 i \sqrt {x^{2}+2}\, \sqrt {x^{2}+1}\, a \,b^{2} \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+7 i \sqrt {x^{2}+2}\, \sqrt {x^{2}+1}\, a \,b^{2} \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-15 i \sqrt {x^{2}+2}\, \sqrt {x^{2}+1}\, a \,b^{2} \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 b}{a}, \sqrt {2}\right )+6 i \sqrt {x^{2}+2}\, \sqrt {x^{2}+1}\, b^{3} \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 b}{a}, \sqrt {2}\right )\right )}{3 \left (x^{4}+3 x^{2}+2\right ) a \,b^{3}} \]
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 \[ \int \frac {\sqrt {x^{2} + 2} {\left (x^{2} + 1\right )}^{\frac {3}{2}}}{b x^{2} + a}\,{d x} \]
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 \[ \int \frac {{\left (x^2+1\right )}^{3/2}\,\sqrt {x^2+2}}{b\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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