Optimal. Leaf size=215 \[ \frac {2 b^2 \sqrt {x^2+1} \Pi \left (1-\frac {2 b}{a};\left .\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-1\right )}{a \sqrt {\frac {x^2+1}{x^2+2}} \sqrt {x^2+2} (a-b)^2}+\frac {x \sqrt {x^2+2}}{3 \left (x^2+1\right )^{3/2} (a-b)}-\frac {\sqrt {2} \sqrt {x^2+2} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}} (a-b)}+\frac {\sqrt {2} \sqrt {x^2+2} (a-2 b) E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}} (a-b)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {546, 539, 526, 525, 418, 411} \[ \frac {2 b^2 \sqrt {x^2+1} \Pi \left (1-\frac {2 b}{a};\left .\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-1\right )}{a \sqrt {\frac {x^2+1}{x^2+2}} \sqrt {x^2+2} (a-b)^2}+\frac {x \sqrt {x^2+2}}{3 \left (x^2+1\right )^{3/2} (a-b)}-\frac {\sqrt {2} \sqrt {x^2+2} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}} (a-b)}+\frac {\sqrt {2} \sqrt {x^2+2} (a-2 b) E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}} (a-b)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 411
Rule 418
Rule 525
Rule 526
Rule 539
Rule 546
Rubi steps
\begin {align*} \int \frac {\sqrt {2+x^2}}{\left (1+x^2\right )^{5/2} \left (a+b x^2\right )} \, dx &=-\frac {\int \frac {\sqrt {2+x^2} \left (-a+2 b+b x^2\right )}{\left (1+x^2\right )^{5/2}} \, dx}{(a-b)^2}+\frac {b^2 \int \frac {\sqrt {2+x^2}}{\sqrt {1+x^2} \left (a+b x^2\right )} \, dx}{(a-b)^2}\\ &=\frac {x \sqrt {2+x^2}}{3 (a-b) \left (1+x^2\right )^{3/2}}+\frac {2 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 (a-b)^2 \sqrt {\frac {1+x^2}{2+x^2}} \sqrt {2+x^2}}+\frac {\int \frac {2 (2 a-5 b)+(a-4 b) x^2}{\left (1+x^2\right )^{3/2} \sqrt {2+x^2}} \, dx}{3 (a-b)^2}\\ &=\frac {x \sqrt {2+x^2}}{3 (a-b) \left (1+x^2\right )^{3/2}}+\frac {2 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 (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}{(a-b)^2}-\frac {2 \int \frac {1}{\sqrt {1+x^2} \sqrt {2+x^2}} \, dx}{3 (a-b)}\\ &=\frac {x \sqrt {2+x^2}}{3 (a-b) \left (1+x^2\right )^{3/2}}+\frac {\sqrt {2} (a-2 b) \sqrt {2+x^2} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{(a-b)^2 \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}}-\frac {\sqrt {2} \sqrt {2+x^2} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 (a-b) \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}}+\frac {2 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 (a-b)^2 \sqrt {\frac {1+x^2}{2+x^2}} \sqrt {2+x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.50, size = 357, normalized size = 1.66 \[ \frac {8 a^2 \sqrt {x^2+1} \sqrt {x^2+2} x+6 a^2 \sqrt {x^2+1} \sqrt {x^2+2} x^3-6 i \sqrt {2} b^2 x^4 \Pi \left (\frac {b}{a};i \sinh ^{-1}(x)|\frac {1}{2}\right )-12 i \sqrt {2} b^2 x^2 \Pi \left (\frac {b}{a};i \sinh ^{-1}(x)|\frac {1}{2}\right )-6 i \sqrt {2} b^2 \Pi \left (\frac {b}{a};i \sinh ^{-1}(x)|\frac {1}{2}\right )+3 i \sqrt {2} a b x^4 \Pi \left (\frac {b}{a};i \sinh ^{-1}(x)|\frac {1}{2}\right )-14 a b \sqrt {x^2+1} \sqrt {x^2+2} x-i \sqrt {2} a \left (x^2+1\right )^2 (4 a-7 b) F\left (i \sinh ^{-1}(x)|\frac {1}{2}\right )+6 i \sqrt {2} a \left (x^2+1\right )^2 (a-2 b) E\left (i \sinh ^{-1}(x)|\frac {1}{2}\right )+6 i \sqrt {2} a b x^2 \Pi \left (\frac {b}{a};i \sinh ^{-1}(x)|\frac {1}{2}\right )-12 a b \sqrt {x^2+1} \sqrt {x^2+2} x^3+3 i \sqrt {2} a b \Pi \left (\frac {b}{a};i \sinh ^{-1}(x)|\frac {1}{2}\right )}{6 a \left (x^2+1\right )^2 (a-b)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 18.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{2} + 2} \sqrt {x^{2} + 1}}{b x^{8} + {\left (a + 3 \, b\right )} x^{6} + 3 \, {\left (a + b\right )} x^{4} + {\left (3 \, a + b\right )} x^{2} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x^{2} + 2}}{{\left (b x^{2} + a\right )} {\left (x^{2} + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 477, normalized size = 2.22 \[ \frac {3 a^{2} x^{5}-6 a b \,x^{5}+10 a^{2} x^{3}+3 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, a^{2} x^{2} \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, a^{2} x^{2} \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-19 a b \,x^{3}-6 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, a b \,x^{2} \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, a b \,x^{2} \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+3 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, a b \,x^{2} \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 b}{a}, \sqrt {2}\right )-6 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, b^{2} x^{2} \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 b}{a}, \sqrt {2}\right )+8 a^{2} x +3 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, a^{2} \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, a^{2} \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-14 a b x -6 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, a b \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, a b \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+3 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, a b \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 b}{a}, \sqrt {2}\right )-6 i \sqrt {x^{2}+1}\, \sqrt {x^{2}+2}\, b^{2} \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 b}{a}, \sqrt {2}\right )}{3 \sqrt {x^{2}+2}\, \left (a -b \right )^{2} \left (x^{2}+1\right )^{\frac {3}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x^{2} + 2}}{{\left (b x^{2} + a\right )} {\left (x^{2} + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {x^2+2}}{{\left (x^2+1\right )}^{5/2}\,\left (b\,x^2+a\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________