3.534 \(\int \frac{x \left (1+a+a^2+x^2+a x^2+b x^2+2 a b x^2+b x^4+b^2 x^4\right )}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{1}{b \sqrt{a+b x^2}}-\frac{1}{3 b \left (a+b x^2\right )^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]

[Out]

-1/(3*b*(a + b*x^2)^(3/2)) - 1/(b*Sqrt[a + b*x^2]) - ArcTanh[Sqrt[a + b*x^2]/Sqr
t[a - b]]/Sqrt[a - b]

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Rubi [A]  time = 0.970305, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 58, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ -\frac{1}{b \sqrt{a+b x^2}}-\frac{1}{3 b \left (a+b x^2\right )^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]

Antiderivative was successfully verified.

[In]  Int[(x*(1 + a + a^2 + x^2 + a*x^2 + b*x^2 + 2*a*b*x^2 + b*x^4 + b^2*x^4))/((1 + x^2)*(a + b*x^2)^(5/2)),x]

[Out]

-1/(3*b*(a + b*x^2)^(3/2)) - 1/(b*Sqrt[a + b*x^2]) - ArcTanh[Sqrt[a + b*x^2]/Sqr
t[a - b]]/Sqrt[a - b]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b**2*x**4+b*x**4+2*a*b*x**2+a*x**2+b*x**2+a**2+x**2+a+1)/(x**2+1)/(b*x**2+a)**(5/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0893021, size = 63, normalized size = 0.93 \[ -\frac{3 a+3 b x^2+1}{3 b \left (a+b x^2\right )^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(1 + a + a^2 + x^2 + a*x^2 + b*x^2 + 2*a*b*x^2 + b*x^4 + b^2*x^4))/((1 + x^2)*(a + b*x^2)^(5/2)),x]

[Out]

-(1 + 3*a + 3*b*x^2)/(3*b*(a + b*x^2)^(3/2)) - ArcTanh[Sqrt[a + b*x^2]/Sqrt[a -
b]]/Sqrt[a - b]

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Maple [B]  time = 0.029, size = 314, normalized size = 4.6 \[ -{b{x}^{2} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{{x}^{2} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,a}{3} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{a}{b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{b}{3} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{3\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{{a}^{2}}{ \left ( a-b \right ) ^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-2\,{\frac{ab}{ \left ( a-b \right ) ^{2}\sqrt{b{x}^{2}+a}}}+{\frac{{b}^{2}}{ \left ( a-b \right ) ^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{{a}^{2}}{ \left ( a-b \right ) ^{2}}\arctan \left ({1\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{-a+b}}}} \right ){\frac{1}{\sqrt{-a+b}}}}-2\,{\frac{ab}{ \left ( a-b \right ) ^{2}\sqrt{-a+b}}\arctan \left ({\frac{\sqrt{b{x}^{2}+a}}{\sqrt{-a+b}}} \right ) }+{\frac{{b}^{2}}{ \left ( a-b \right ) ^{2}}\arctan \left ({1\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{-a+b}}}} \right ){\frac{1}{\sqrt{-a+b}}}}+{\frac{{a}^{2}}{3\,a-3\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,ab}{3\,a-3\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{{b}^{2}}{3\,a-3\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b^2*x^4+b*x^4+2*a*b*x^2+a*x^2+b*x^2+a^2+x^2+a+1)/(x^2+1)/(b*x^2+a)^(5/2),x)

[Out]

-x^2*b/(b*x^2+a)^(3/2)-x^2/(b*x^2+a)^(3/2)-4/3*a/(b*x^2+a)^(3/2)-a/b/(b*x^2+a)^(
3/2)+1/3*b/(b*x^2+a)^(3/2)-1/3/b/(b*x^2+a)^(3/2)+1/(a-b)^2/(b*x^2+a)^(1/2)*a^2-2
/(a-b)^2/(b*x^2+a)^(1/2)*a*b+1/(a-b)^2/(b*x^2+a)^(1/2)*b^2+1/(a-b)^2/(-a+b)^(1/2
)*arctan((b*x^2+a)^(1/2)/(-a+b)^(1/2))*a^2-2/(a-b)^2/(-a+b)^(1/2)*arctan((b*x^2+
a)^(1/2)/(-a+b)^(1/2))*a*b+1/(a-b)^2/(-a+b)^(1/2)*arctan((b*x^2+a)^(1/2)/(-a+b)^
(1/2))*b^2+1/3/(a-b)/(b*x^2+a)^(3/2)*a^2-2/3/(a-b)/(b*x^2+a)^(3/2)*a*b+1/3/(a-b)
/(b*x^2+a)^(3/2)*b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + b*x^4 + 2*a*b*x^2 + a*x^2 + b*x^2 + a^2 + x^2 + a + 1)*x/((b*x^2 + a)^(5/2)*(x^2 + 1)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292535, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (3 \, b x^{2} + 3 \, a + 1\right )} \sqrt{b x^{2} + a} \sqrt{a - b} - 3 \,{\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \log \left (-\frac{4 \,{\left ({\left (a b - b^{2}\right )} x^{2} + 2 \, a^{2} - 3 \, a b + b^{2}\right )} \sqrt{b x^{2} + a} -{\left (b^{2} x^{4} + 2 \,{\left (4 \, a b - 3 \, b^{2}\right )} x^{2} + 8 \, a^{2} - 8 \, a b + b^{2}\right )} \sqrt{a - b}}{x^{4} + 2 \, x^{2} + 1}\right )}{12 \,{\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sqrt{a - b}}, -\frac{2 \,{\left (3 \, b x^{2} + 3 \, a + 1\right )} \sqrt{b x^{2} + a} \sqrt{-a + b} - 3 \,{\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \arctan \left (-\frac{{\left (b x^{2} + 2 \, a - b\right )} \sqrt{-a + b}}{2 \, \sqrt{b x^{2} + a}{\left (a - b\right )}}\right )}{6 \,{\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sqrt{-a + b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + b*x^4 + 2*a*b*x^2 + a*x^2 + b*x^2 + a^2 + x^2 + a + 1)*x/((b*x^2 + a)^(5/2)*(x^2 + 1)),x, algorithm="fricas")

[Out]

[-1/12*(4*(3*b*x^2 + 3*a + 1)*sqrt(b*x^2 + a)*sqrt(a - b) - 3*(b^3*x^4 + 2*a*b^2
*x^2 + a^2*b)*log(-(4*((a*b - b^2)*x^2 + 2*a^2 - 3*a*b + b^2)*sqrt(b*x^2 + a) -
(b^2*x^4 + 2*(4*a*b - 3*b^2)*x^2 + 8*a^2 - 8*a*b + b^2)*sqrt(a - b))/(x^4 + 2*x^
2 + 1)))/((b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*sqrt(a - b)), -1/6*(2*(3*b*x^2 + 3*a +
1)*sqrt(b*x^2 + a)*sqrt(-a + b) - 3*(b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*arctan(-1/2
*(b*x^2 + 2*a - b)*sqrt(-a + b)/(sqrt(b*x^2 + a)*(a - b))))/((b^3*x^4 + 2*a*b^2*
x^2 + a^2*b)*sqrt(-a + b))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b**2*x**4+b*x**4+2*a*b*x**2+a*x**2+b*x**2+a**2+x**2+a+1)/(x**2+1)/(b*x**2+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.2764, size = 70, normalized size = 1.03 \[ \frac{\arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a + b}}\right )}{\sqrt{-a + b}} - \frac{3 \, b x^{2} + 3 \, a + 1}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + b*x^4 + 2*a*b*x^2 + a*x^2 + b*x^2 + a^2 + x^2 + a + 1)*x/((b*x^2 + a)^(5/2)*(x^2 + 1)),x, algorithm="giac")

[Out]

arctan(sqrt(b*x^2 + a)/sqrt(-a + b))/sqrt(-a + b) - 1/3*(3*b*x^2 + 3*a + 1)/((b*
x^2 + a)^(3/2)*b)